Require terminaison.

Require Relations.

Require term.

Require List.

Require equational_theory.

Require rpo_extension.

Require equational_extension.

Require closure_extension.

Require term_extension.

Require dp.

Require Inclusion.

Require or_ext_generated.

Require ZArith.

Require ring_extention.

Require Zwf.

Require Inverse_Image.

Require matrix.

Require more_list_extention.

Import List.

Import ZArith.

Set Implicit Arguments.

Module algebra.
 Module F
  <:term.Signature.
  Inductive symb  :
   Set := 
     (* id_a__zeros *)
    | id_a__zeros : symb
     (* id_U51 *)
    | id_U51 : symb
     (* id_a__U84 *)
    | id_a__U84 : symb
     (* id_a__U32 *)
    | id_a__U32 : symb
     (* id_isNatList *)
    | id_isNatList : symb
     (* id_a__U13 *)
    | id_a__U13 : symb
     (* id_U85 *)
    | id_U85 : symb
     (* id_a__length *)
    | id_a__length : symb
     (* id_a__isNatIList *)
    | id_a__isNatIList : symb
     (* id_U33 *)
    | id_U33 : symb
     (* id_a__U11 *)
    | id_a__U11 : symb
     (* id_U81 *)
    | id_U81 : symb
     (* id_a__U92 *)
    | id_a__U92 : symb
     (* id_a__U43 *)
    | id_a__U43 : symb
     (* id_U23 *)
    | id_U23 : symb
     (* id_a__isNatKind *)
    | id_a__isNatKind : symb
     (* id_U93 *)
    | id_U93 : symb
     (* id_U11 *)
    | id_U11 : symb
     (* id_a__U71 *)
    | id_a__U71 : symb
     (* id_U44 *)
    | id_U44 : symb
     (* id_0 *)
    | id_0 : symb
     (* id_U61 *)
    | id_U61 : symb
     (* id_a__U86 *)
    | id_a__U86 : symb
     (* id_a__U41 *)
    | id_a__U41 : symb
     (* id_U22 *)
    | id_U22 : symb
     (* id_a__U21 *)
    | id_a__U21 : symb
     (* id_U91 *)
    | id_U91 : symb
     (* id_length *)
    | id_length : symb
     (* id_a__U52 *)
    | id_a__U52 : symb
     (* id_U42 *)
    | id_U42 : symb
     (* id_a__U12 *)
    | id_a__U12 : symb
     (* id_U83 *)
    | id_U83 : symb
     (* id_a__U94 *)
    | id_a__U94 : symb
     (* id_a__U45 *)
    | id_a__U45 : symb
     (* id_U31 *)
    | id_U31 : symb
     (* id_a__isNat *)
    | id_a__isNat : symb
     (* id_isNatIListKind *)
    | id_isNatIListKind : symb
     (* id_a__U82 *)
    | id_a__U82 : symb
     (* id_U46 *)
    | id_U46 : symb
     (* id_cons *)
    | id_cons : symb
     (* id_U52 *)
    | id_U52 : symb
     (* id_a__U85 *)
    | id_a__U85 : symb
     (* id_a__U33 *)
    | id_a__U33 : symb
     (* id_U21 *)
    | id_U21 : symb
     (* id_a__isNatList *)
    | id_a__isNatList : symb
     (* id_U86 *)
    | id_U86 : symb
     (* id_mark *)
    | id_mark : symb
     (* id_a__U51 *)
    | id_a__U51 : symb
     (* id_U41 *)
    | id_U41 : symb
     (* id_tt *)
    | id_tt : symb
     (* id_U82 *)
    | id_U82 : symb
     (* id_a__U93 *)
    | id_a__U93 : symb
     (* id_a__U44 *)
    | id_a__U44 : symb
     (* id_isNat *)
    | id_isNat : symb
     (* id_a__U23 *)
    | id_a__U23 : symb
     (* id_U94 *)
    | id_U94 : symb
     (* id_U12 *)
    | id_U12 : symb
     (* id_a__U81 *)
    | id_a__U81 : symb
     (* id_U45 *)
    | id_U45 : symb
     (* id_zeros *)
    | id_zeros : symb
     (* id_U71 *)
    | id_U71 : symb
     (* id_a__U91 *)
    | id_a__U91 : symb
     (* id_a__U42 *)
    | id_a__U42 : symb
     (* id_isNatKind *)
    | id_isNatKind : symb
     (* id_a__U22 *)
    | id_a__U22 : symb
     (* id_U92 *)
    | id_U92 : symb
     (* id_nil *)
    | id_nil : symb
     (* id_a__U61 *)
    | id_a__U61 : symb
     (* id_U43 *)
    | id_U43 : symb
     (* id_a__isNatIListKind *)
    | id_a__isNatIListKind : symb
     (* id_U84 *)
    | id_U84 : symb
     (* id_s *)
    | id_s : symb
     (* id_a__U46 *)
    | id_a__U46 : symb
     (* id_U32 *)
    | id_U32 : symb
     (* id_a__U31 *)
    | id_a__U31 : symb
     (* id_U13 *)
    | id_U13 : symb
     (* id_a__U83 *)
    | id_a__U83 : symb
     (* id_isNatIList *)
    | id_isNatIList : symb
  .
  
  
  Definition symb_eq_bool (f1 f2:symb) : bool := 
    match f1,f2 with
      | id_a__zeros,id_a__zeros => true
      | id_U51,id_U51 => true
      | id_a__U84,id_a__U84 => true
      | id_a__U32,id_a__U32 => true
      | id_isNatList,id_isNatList => true
      | id_a__U13,id_a__U13 => true
      | id_U85,id_U85 => true
      | id_a__length,id_a__length => true
      | id_a__isNatIList,id_a__isNatIList => true
      | id_U33,id_U33 => true
      | id_a__U11,id_a__U11 => true
      | id_U81,id_U81 => true
      | id_a__U92,id_a__U92 => true
      | id_a__U43,id_a__U43 => true
      | id_U23,id_U23 => true
      | id_a__isNatKind,id_a__isNatKind => true
      | id_U93,id_U93 => true
      | id_U11,id_U11 => true
      | id_a__U71,id_a__U71 => true
      | id_U44,id_U44 => true
      | id_0,id_0 => true
      | id_U61,id_U61 => true
      | id_a__U86,id_a__U86 => true
      | id_a__U41,id_a__U41 => true
      | id_U22,id_U22 => true
      | id_a__U21,id_a__U21 => true
      | id_U91,id_U91 => true
      | id_length,id_length => true
      | id_a__U52,id_a__U52 => true
      | id_U42,id_U42 => true
      | id_a__U12,id_a__U12 => true
      | id_U83,id_U83 => true
      | id_a__U94,id_a__U94 => true
      | id_a__U45,id_a__U45 => true
      | id_U31,id_U31 => true
      | id_a__isNat,id_a__isNat => true
      | id_isNatIListKind,id_isNatIListKind => true
      | id_a__U82,id_a__U82 => true
      | id_U46,id_U46 => true
      | id_cons,id_cons => true
      | id_U52,id_U52 => true
      | id_a__U85,id_a__U85 => true
      | id_a__U33,id_a__U33 => true
      | id_U21,id_U21 => true
      | id_a__isNatList,id_a__isNatList => true
      | id_U86,id_U86 => true
      | id_mark,id_mark => true
      | id_a__U51,id_a__U51 => true
      | id_U41,id_U41 => true
      | id_tt,id_tt => true
      | id_U82,id_U82 => true
      | id_a__U93,id_a__U93 => true
      | id_a__U44,id_a__U44 => true
      | id_isNat,id_isNat => true
      | id_a__U23,id_a__U23 => true
      | id_U94,id_U94 => true
      | id_U12,id_U12 => true
      | id_a__U81,id_a__U81 => true
      | id_U45,id_U45 => true
      | id_zeros,id_zeros => true
      | id_U71,id_U71 => true
      | id_a__U91,id_a__U91 => true
      | id_a__U42,id_a__U42 => true
      | id_isNatKind,id_isNatKind => true
      | id_a__U22,id_a__U22 => true
      | id_U92,id_U92 => true
      | id_nil,id_nil => true
      | id_a__U61,id_a__U61 => true
      | id_U43,id_U43 => true
      | id_a__isNatIListKind,id_a__isNatIListKind => true
      | id_U84,id_U84 => true
      | id_s,id_s => true
      | id_a__U46,id_a__U46 => true
      | id_U32,id_U32 => true
      | id_a__U31,id_a__U31 => true
      | id_U13,id_U13 => true
      | id_a__U83,id_a__U83 => true
      | id_isNatIList,id_isNatIList => true
      | _,_ => false
      end.
  
  
   (* Proof of decidability of equality over symb *)
  Definition symb_eq_bool_ok(f1 f2:symb) :
   match symb_eq_bool f1 f2 with
     | true => f1 = f2
     | false => f1 <> f2
     end.
  Proof.
    intros f1 f2.
    
    refine match f1 as u1,f2 as u2 return 
             match symb_eq_bool u1 u2 return 
               Prop with
               | true => u1 = u2
               | false => u1 <> u2
               end with
             | id_a__zeros,id_a__zeros => refl_equal _
             | id_U51,id_U51 => refl_equal _
             | id_a__U84,id_a__U84 => refl_equal _
             | id_a__U32,id_a__U32 => refl_equal _
             | id_isNatList,id_isNatList => refl_equal _
             | id_a__U13,id_a__U13 => refl_equal _
             | id_U85,id_U85 => refl_equal _
             | id_a__length,id_a__length => refl_equal _
             | id_a__isNatIList,id_a__isNatIList => refl_equal _
             | id_U33,id_U33 => refl_equal _
             | id_a__U11,id_a__U11 => refl_equal _
             | id_U81,id_U81 => refl_equal _
             | id_a__U92,id_a__U92 => refl_equal _
             | id_a__U43,id_a__U43 => refl_equal _
             | id_U23,id_U23 => refl_equal _
             | id_a__isNatKind,id_a__isNatKind => refl_equal _
             | id_U93,id_U93 => refl_equal _
             | id_U11,id_U11 => refl_equal _
             | id_a__U71,id_a__U71 => refl_equal _
             | id_U44,id_U44 => refl_equal _
             | id_0,id_0 => refl_equal _
             | id_U61,id_U61 => refl_equal _
             | id_a__U86,id_a__U86 => refl_equal _
             | id_a__U41,id_a__U41 => refl_equal _
             | id_U22,id_U22 => refl_equal _
             | id_a__U21,id_a__U21 => refl_equal _
             | id_U91,id_U91 => refl_equal _
             | id_length,id_length => refl_equal _
             | id_a__U52,id_a__U52 => refl_equal _
             | id_U42,id_U42 => refl_equal _
             | id_a__U12,id_a__U12 => refl_equal _
             | id_U83,id_U83 => refl_equal _
             | id_a__U94,id_a__U94 => refl_equal _
             | id_a__U45,id_a__U45 => refl_equal _
             | id_U31,id_U31 => refl_equal _
             | id_a__isNat,id_a__isNat => refl_equal _
             | id_isNatIListKind,id_isNatIListKind => refl_equal _
             | id_a__U82,id_a__U82 => refl_equal _
             | id_U46,id_U46 => refl_equal _
             | id_cons,id_cons => refl_equal _
             | id_U52,id_U52 => refl_equal _
             | id_a__U85,id_a__U85 => refl_equal _
             | id_a__U33,id_a__U33 => refl_equal _
             | id_U21,id_U21 => refl_equal _
             | id_a__isNatList,id_a__isNatList => refl_equal _
             | id_U86,id_U86 => refl_equal _
             | id_mark,id_mark => refl_equal _
             | id_a__U51,id_a__U51 => refl_equal _
             | id_U41,id_U41 => refl_equal _
             | id_tt,id_tt => refl_equal _
             | id_U82,id_U82 => refl_equal _
             | id_a__U93,id_a__U93 => refl_equal _
             | id_a__U44,id_a__U44 => refl_equal _
             | id_isNat,id_isNat => refl_equal _
             | id_a__U23,id_a__U23 => refl_equal _
             | id_U94,id_U94 => refl_equal _
             | id_U12,id_U12 => refl_equal _
             | id_a__U81,id_a__U81 => refl_equal _
             | id_U45,id_U45 => refl_equal _
             | id_zeros,id_zeros => refl_equal _
             | id_U71,id_U71 => refl_equal _
             | id_a__U91,id_a__U91 => refl_equal _
             | id_a__U42,id_a__U42 => refl_equal _
             | id_isNatKind,id_isNatKind => refl_equal _
             | id_a__U22,id_a__U22 => refl_equal _
             | id_U92,id_U92 => refl_equal _
             | id_nil,id_nil => refl_equal _
             | id_a__U61,id_a__U61 => refl_equal _
             | id_U43,id_U43 => refl_equal _
             | id_a__isNatIListKind,id_a__isNatIListKind => refl_equal _
             | id_U84,id_U84 => refl_equal _
             | id_s,id_s => refl_equal _
             | id_a__U46,id_a__U46 => refl_equal _
             | id_U32,id_U32 => refl_equal _
             | id_a__U31,id_a__U31 => refl_equal _
             | id_U13,id_U13 => refl_equal _
             | id_a__U83,id_a__U83 => refl_equal _
             | id_isNatIList,id_isNatIList => refl_equal _
             | _,_ => _
             end;intros abs;discriminate.
  Defined.
  
  
  Definition arity (f:symb) := 
    match f with
      | id_a__zeros => term.Free 0
      | id_U51 => term.Free 2
      | id_a__U84 => term.Free 3
      | id_a__U32 => term.Free 2
      | id_isNatList => term.Free 1
      | id_a__U13 => term.Free 1
      | id_U85 => term.Free 2
      | id_a__length => term.Free 1
      | id_a__isNatIList => term.Free 1
      | id_U33 => term.Free 1
      | id_a__U11 => term.Free 2
      | id_U81 => term.Free 3
      | id_a__U92 => term.Free 3
      | id_a__U43 => term.Free 3
      | id_U23 => term.Free 1
      | id_a__isNatKind => term.Free 1
      | id_U93 => term.Free 3
      | id_U11 => term.Free 2
      | id_a__U71 => term.Free 1
      | id_U44 => term.Free 3
      | id_0 => term.Free 0
      | id_U61 => term.Free 1
      | id_a__U86 => term.Free 1
      | id_a__U41 => term.Free 3
      | id_U22 => term.Free 2
      | id_a__U21 => term.Free 2
      | id_U91 => term.Free 3
      | id_length => term.Free 1
      | id_a__U52 => term.Free 1
      | id_U42 => term.Free 3
      | id_a__U12 => term.Free 2
      | id_U83 => term.Free 3
      | id_a__U94 => term.Free 2
      | id_a__U45 => term.Free 2
      | id_U31 => term.Free 2
      | id_a__isNat => term.Free 1
      | id_isNatIListKind => term.Free 1
      | id_a__U82 => term.Free 3
      | id_U46 => term.Free 1
      | id_cons => term.Free 2
      | id_U52 => term.Free 1
      | id_a__U85 => term.Free 2
      | id_a__U33 => term.Free 1
      | id_U21 => term.Free 2
      | id_a__isNatList => term.Free 1
      | id_U86 => term.Free 1
      | id_mark => term.Free 1
      | id_a__U51 => term.Free 2
      | id_U41 => term.Free 3
      | id_tt => term.Free 0
      | id_U82 => term.Free 3
      | id_a__U93 => term.Free 3
      | id_a__U44 => term.Free 3
      | id_isNat => term.Free 1
      | id_a__U23 => term.Free 1
      | id_U94 => term.Free 2
      | id_U12 => term.Free 2
      | id_a__U81 => term.Free 3
      | id_U45 => term.Free 2
      | id_zeros => term.Free 0
      | id_U71 => term.Free 1
      | id_a__U91 => term.Free 3
      | id_a__U42 => term.Free 3
      | id_isNatKind => term.Free 1
      | id_a__U22 => term.Free 2
      | id_U92 => term.Free 3
      | id_nil => term.Free 0
      | id_a__U61 => term.Free 1
      | id_U43 => term.Free 3
      | id_a__isNatIListKind => term.Free 1
      | id_U84 => term.Free 3
      | id_s => term.Free 1
      | id_a__U46 => term.Free 1
      | id_U32 => term.Free 2
      | id_a__U31 => term.Free 2
      | id_U13 => term.Free 1
      | id_a__U83 => term.Free 3
      | id_isNatIList => term.Free 1
      end.
  
  
  Definition symb_order (f1 f2:symb) : bool := 
    match f1,f2 with
      | id_a__zeros,id_a__zeros => true
      | id_a__zeros,id_U51 => false
      | id_a__zeros,id_a__U84 => false
      | id_a__zeros,id_a__U32 => false
      | id_a__zeros,id_isNatList => false
      | id_a__zeros,id_a__U13 => false
      | id_a__zeros,id_U85 => false
      | id_a__zeros,id_a__length => false
      | id_a__zeros,id_a__isNatIList => false
      | id_a__zeros,id_U33 => false
      | id_a__zeros,id_a__U11 => false
      | id_a__zeros,id_U81 => false
      | id_a__zeros,id_a__U92 => false
      | id_a__zeros,id_a__U43 => false
      | id_a__zeros,id_U23 => false
      | id_a__zeros,id_a__isNatKind => false
      | id_a__zeros,id_U93 => false
      | id_a__zeros,id_U11 => false
      | id_a__zeros,id_a__U71 => false
      | id_a__zeros,id_U44 => false
      | id_a__zeros,id_0 => false
      | id_a__zeros,id_U61 => false
      | id_a__zeros,id_a__U86 => false
      | id_a__zeros,id_a__U41 => false
      | id_a__zeros,id_U22 => false
      | id_a__zeros,id_a__U21 => false
      | id_a__zeros,id_U91 => false
      | id_a__zeros,id_length => false
      | id_a__zeros,id_a__U52 => false
      | id_a__zeros,id_U42 => false
      | id_a__zeros,id_a__U12 => false
      | id_a__zeros,id_U83 => false
      | id_a__zeros,id_a__U94 => false
      | id_a__zeros,id_a__U45 => false
      | id_a__zeros,id_U31 => false
      | id_a__zeros,id_a__isNat => false
      | id_a__zeros,id_isNatIListKind => false
      | id_a__zeros,id_a__U82 => false
      | id_a__zeros,id_U46 => false
      | id_a__zeros,id_cons => false
      | id_a__zeros,id_U52 => false
      | id_a__zeros,id_a__U85 => false
      | id_a__zeros,id_a__U33 => false
      | id_a__zeros,id_U21 => false
      | id_a__zeros,id_a__isNatList => false
      | id_a__zeros,id_U86 => false
      | id_a__zeros,id_mark => false
      | id_a__zeros,id_a__U51 => false
      | id_a__zeros,id_U41 => false
      | id_a__zeros,id_tt => false
      | id_a__zeros,id_U82 => false
      | id_a__zeros,id_a__U93 => false
      | id_a__zeros,id_a__U44 => false
      | id_a__zeros,id_isNat => false
      | id_a__zeros,id_a__U23 => false
      | id_a__zeros,id_U94 => false
      | id_a__zeros,id_U12 => false
      | id_a__zeros,id_a__U81 => false
      | id_a__zeros,id_U45 => false
      | id_a__zeros,id_zeros => false
      | id_a__zeros,id_U71 => false
      | id_a__zeros,id_a__U91 => false
      | id_a__zeros,id_a__U42 => false
      | id_a__zeros,id_isNatKind => false
      | id_a__zeros,id_a__U22 => false
      | id_a__zeros,id_U92 => false
      | id_a__zeros,id_nil => false
      | id_a__zeros,id_a__U61 => false
      | id_a__zeros,id_U43 => false
      | id_a__zeros,id_a__isNatIListKind => false
      | id_a__zeros,id_U84 => false
      | id_a__zeros,id_s => false
      | id_a__zeros,id_a__U46 => false
      | id_a__zeros,id_U32 => false
      | id_a__zeros,id_a__U31 => false
      | id_a__zeros,id_U13 => false
      | id_a__zeros,id_a__U83 => false
      | id_a__zeros,id_isNatIList => false
      | id_U51,id_a__zeros => true
      | id_U51,id_U51 => true
      | id_U51,id_a__U84 => false
      | id_U51,id_a__U32 => false
      | id_U51,id_isNatList => false
      | id_U51,id_a__U13 => false
      | id_U51,id_U85 => false
      | id_U51,id_a__length => false
      | id_U51,id_a__isNatIList => false
      | id_U51,id_U33 => false
      | id_U51,id_a__U11 => false
      | id_U51,id_U81 => false
      | id_U51,id_a__U92 => false
      | id_U51,id_a__U43 => false
      | id_U51,id_U23 => false
      | id_U51,id_a__isNatKind => false
      | id_U51,id_U93 => false
      | id_U51,id_U11 => false
      | id_U51,id_a__U71 => false
      | id_U51,id_U44 => false
      | id_U51,id_0 => false
      | id_U51,id_U61 => false
      | id_U51,id_a__U86 => false
      | id_U51,id_a__U41 => false
      | id_U51,id_U22 => false
      | id_U51,id_a__U21 => false
      | id_U51,id_U91 => false
      | id_U51,id_length => false
      | id_U51,id_a__U52 => false
      | id_U51,id_U42 => false
      | id_U51,id_a__U12 => false
      | id_U51,id_U83 => false
      | id_U51,id_a__U94 => false
      | id_U51,id_a__U45 => false
      | id_U51,id_U31 => false
      | id_U51,id_a__isNat => false
      | id_U51,id_isNatIListKind => false
      | id_U51,id_a__U82 => false
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      | id_a__isNatKind,id_U71 => false
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      | id_U93,id_U51 => true
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      | id_U93,id_a__U32 => true
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      | id_U93,id_a__isNatIList => true
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      | id_U93,id_a__U43 => true
      | id_U93,id_U23 => true
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      | id_U93,id_a__U45 => false
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      | id_U93,id_U71 => false
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      | id_a__U45,id_a__U93 => false
      | id_a__U45,id_a__U44 => false
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      | id_a__U45,id_U12 => false
      | id_a__U45,id_a__U81 => false
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      | id_a__U45,id_s => false
      | id_a__U45,id_a__U46 => false
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      | id_a__U45,id_isNatIList => false
      | id_U31,id_a__zeros => true
      | id_U31,id_U51 => true
      | id_U31,id_a__U84 => true
      | id_U31,id_a__U32 => true
      | id_U31,id_isNatList => true
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      | id_U31,id_a__length => true
      | id_U31,id_a__isNatIList => true
      | id_U31,id_U33 => true
      | id_U31,id_a__U11 => true
      | id_U31,id_U81 => true
      | id_U31,id_a__U92 => true
      | id_U31,id_a__U43 => true
      | id_U31,id_U23 => true
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      | id_U31,id_U11 => true
      | id_U31,id_a__U71 => true
      | id_U31,id_U44 => true
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      | id_U31,id_a__U94 => true
      | id_U31,id_a__U45 => true
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      | id_U31,id_U46 => false
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      | id_U31,id_a__U44 => false
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      | id_U31,id_a__U81 => false
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      | id_U31,id_U71 => false
      | id_U31,id_a__U91 => false
      | id_U31,id_a__U42 => false
      | id_U31,id_isNatKind => false
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      | id_U31,id_a__U61 => false
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      | id_U31,id_s => false
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      | id_U31,id_U32 => false
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      | id_U31,id_U13 => false
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      | id_a__isNat,id_a__zeros => true
      | id_a__isNat,id_U51 => true
      | id_a__isNat,id_a__U84 => true
      | id_a__isNat,id_a__U32 => true
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      | id_a__isNat,id_a__length => true
      | id_a__isNat,id_a__isNatIList => true
      | id_a__isNat,id_U33 => true
      | id_a__isNat,id_a__U11 => true
      | id_a__isNat,id_U81 => true
      | id_a__isNat,id_a__U92 => true
      | id_a__isNat,id_a__U43 => true
      | id_a__isNat,id_U23 => true
      | id_a__isNat,id_a__isNatKind => true
      | id_a__isNat,id_U93 => true
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      | id_a__isNat,id_a__U71 => true
      | id_a__isNat,id_U44 => true
      | id_a__isNat,id_0 => true
      | id_a__isNat,id_U61 => true
      | id_a__isNat,id_a__U86 => true
      | id_a__isNat,id_a__U41 => true
      | id_a__isNat,id_U22 => true
      | id_a__isNat,id_a__U21 => true
      | id_a__isNat,id_U91 => true
      | id_a__isNat,id_length => true
      | id_a__isNat,id_a__U52 => true
      | id_a__isNat,id_U42 => true
      | id_a__isNat,id_a__U12 => true
      | id_a__isNat,id_U83 => true
      | id_a__isNat,id_a__U94 => true
      | id_a__isNat,id_a__U45 => true
      | id_a__isNat,id_U31 => true
      | id_a__isNat,id_a__isNat => true
      | id_a__isNat,id_isNatIListKind => false
      | id_a__isNat,id_a__U82 => false
      | id_a__isNat,id_U46 => false
      | id_a__isNat,id_cons => false
      | id_a__isNat,id_U52 => false
      | id_a__isNat,id_a__U85 => false
      | id_a__isNat,id_a__U33 => false
      | id_a__isNat,id_U21 => false
      | id_a__isNat,id_a__isNatList => false
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      | id_a__isNat,id_mark => false
      | id_a__isNat,id_a__U51 => false
      | id_a__isNat,id_U41 => false
      | id_a__isNat,id_tt => false
      | id_a__isNat,id_U82 => false
      | id_a__isNat,id_a__U93 => false
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      | id_a__isNat,id_U71 => false
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      | id_isNatIListKind,id_isNatList => true
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      | id_isNatIListKind,id_U85 => true
      | id_isNatIListKind,id_a__length => true
      | id_isNatIListKind,id_a__isNatIList => true
      | id_isNatIListKind,id_U33 => true
      | id_isNatIListKind,id_a__U11 => true
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      | id_isNatIListKind,id_a__U92 => true
      | id_isNatIListKind,id_a__U43 => true
      | id_isNatIListKind,id_U23 => true
      | id_isNatIListKind,id_a__isNatKind => true
      | id_isNatIListKind,id_U93 => true
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      | id_isNatIListKind,id_a__U71 => true
      | id_isNatIListKind,id_U44 => true
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      | id_isNatIListKind,id_U91 => true
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      | id_isNatIListKind,id_a__U52 => true
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      | id_isNatIListKind,id_a__U12 => true
      | id_isNatIListKind,id_U83 => true
      | id_isNatIListKind,id_a__U94 => true
      | id_isNatIListKind,id_a__U45 => true
      | id_isNatIListKind,id_U31 => true
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      | id_isNatIListKind,id_isNatIListKind => true
      | id_isNatIListKind,id_a__U82 => false
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      | id_isNatIListKind,id_cons => false
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      | id_a__U82,id_U31 => true
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      | id_a__U82,id_a__U93 => false
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      | id_U46,id_a__zeros => true
      | id_U46,id_U51 => true
      | id_U46,id_a__U84 => true
      | id_U46,id_a__U32 => true
      | id_U46,id_isNatList => true
      | id_U46,id_a__U13 => true
      | id_U46,id_U85 => true
      | id_U46,id_a__length => true
      | id_U46,id_a__isNatIList => true
      | id_U46,id_U33 => true
      | id_U46,id_a__U11 => true
      | id_U46,id_U81 => true
      | id_U46,id_a__U92 => true
      | id_U46,id_a__U43 => true
      | id_U46,id_U23 => true
      | id_U46,id_a__isNatKind => true
      | id_U46,id_U93 => true
      | id_U46,id_U11 => true
      | id_U46,id_a__U71 => true
      | id_U46,id_U44 => true
      | id_U46,id_0 => true
      | id_U46,id_U61 => true
      | id_U46,id_a__U86 => true
      | id_U46,id_a__U41 => true
      | id_U46,id_U22 => true
      | id_U46,id_a__U21 => true
      | id_U46,id_U91 => true
      | id_U46,id_length => true
      | id_U46,id_a__U52 => true
      | id_U46,id_U42 => true
      | id_U46,id_a__U12 => true
      | id_U46,id_U83 => true
      | id_U46,id_a__U94 => true
      | id_U46,id_a__U45 => true
      | id_U46,id_U31 => true
      | id_U46,id_a__isNat => true
      | id_U46,id_isNatIListKind => true
      | id_U46,id_a__U82 => true
      | id_U46,id_U46 => true
      | id_U46,id_cons => false
      | id_U46,id_U52 => false
      | id_U46,id_a__U85 => false
      | id_U46,id_a__U33 => false
      | id_U46,id_U21 => false
      | id_U46,id_a__isNatList => false
      | id_U46,id_U86 => false
      | id_U46,id_mark => false
      | id_U46,id_a__U51 => false
      | id_U46,id_U41 => false
      | id_U46,id_tt => false
      | id_U46,id_U82 => false
      | id_U46,id_a__U93 => false
      | id_U46,id_a__U44 => false
      | id_U46,id_isNat => false
      | id_U46,id_a__U23 => false
      | id_U46,id_U94 => false
      | id_U46,id_U12 => false
      | id_U46,id_a__U81 => false
      | id_U46,id_U45 => false
      | id_U46,id_zeros => false
      | id_U46,id_U71 => false
      | id_U46,id_a__U91 => false
      | id_U46,id_a__U42 => false
      | id_U46,id_isNatKind => false
      | id_U46,id_a__U22 => false
      | id_U46,id_U92 => false
      | id_U46,id_nil => false
      | id_U46,id_a__U61 => false
      | id_U46,id_U43 => false
      | id_U46,id_a__isNatIListKind => false
      | id_U46,id_U84 => false
      | id_U46,id_s => false
      | id_U46,id_a__U46 => false
      | id_U46,id_U32 => false
      | id_U46,id_a__U31 => false
      | id_U46,id_U13 => false
      | id_U46,id_a__U83 => false
      | id_U46,id_isNatIList => false
      | id_cons,id_a__zeros => true
      | id_cons,id_U51 => true
      | id_cons,id_a__U84 => true
      | id_cons,id_a__U32 => true
      | id_cons,id_isNatList => true
      | id_cons,id_a__U13 => true
      | id_cons,id_U85 => true
      | id_cons,id_a__length => true
      | id_cons,id_a__isNatIList => true
      | id_cons,id_U33 => true
      | id_cons,id_a__U11 => true
      | id_cons,id_U81 => true
      | id_cons,id_a__U92 => true
      | id_cons,id_a__U43 => true
      | id_cons,id_U23 => true
      | id_cons,id_a__isNatKind => true
      | id_cons,id_U93 => true
      | id_cons,id_U11 => true
      | id_cons,id_a__U71 => true
      | id_cons,id_U44 => true
      | id_cons,id_0 => true
      | id_cons,id_U61 => true
      | id_cons,id_a__U86 => true
      | id_cons,id_a__U41 => true
      | id_cons,id_U22 => true
      | id_cons,id_a__U21 => true
      | id_cons,id_U91 => true
      | id_cons,id_length => true
      | id_cons,id_a__U52 => true
      | id_cons,id_U42 => true
      | id_cons,id_a__U12 => true
      | id_cons,id_U83 => true
      | id_cons,id_a__U94 => true
      | id_cons,id_a__U45 => true
      | id_cons,id_U31 => true
      | id_cons,id_a__isNat => true
      | id_cons,id_isNatIListKind => true
      | id_cons,id_a__U82 => true
      | id_cons,id_U46 => true
      | id_cons,id_cons => true
      | id_cons,id_U52 => false
      | id_cons,id_a__U85 => false
      | id_cons,id_a__U33 => false
      | id_cons,id_U21 => false
      | id_cons,id_a__isNatList => false
      | id_cons,id_U86 => false
      | id_cons,id_mark => false
      | id_cons,id_a__U51 => false
      | id_cons,id_U41 => false
      | id_cons,id_tt => false
      | id_cons,id_U82 => false
      | id_cons,id_a__U93 => false
      | id_cons,id_a__U44 => false
      | id_cons,id_isNat => false
      | id_cons,id_a__U23 => false
      | id_cons,id_U94 => false
      | id_cons,id_U12 => false
      | id_cons,id_a__U81 => false
      | id_cons,id_U45 => false
      | id_cons,id_zeros => false
      | id_cons,id_U71 => false
      | id_cons,id_a__U91 => false
      | id_cons,id_a__U42 => false
      | id_cons,id_isNatKind => false
      | id_cons,id_a__U22 => false
      | id_cons,id_U92 => false
      | id_cons,id_nil => false
      | id_cons,id_a__U61 => false
      | id_cons,id_U43 => false
      | id_cons,id_a__isNatIListKind => false
      | id_cons,id_U84 => false
      | id_cons,id_s => false
      | id_cons,id_a__U46 => false
      | id_cons,id_U32 => false
      | id_cons,id_a__U31 => false
      | id_cons,id_U13 => false
      | id_cons,id_a__U83 => false
      | id_cons,id_isNatIList => false
      | id_U52,id_a__zeros => true
      | id_U52,id_U51 => true
      | id_U52,id_a__U84 => true
      | id_U52,id_a__U32 => true
      | id_U52,id_isNatList => true
      | id_U52,id_a__U13 => true
      | id_U52,id_U85 => true
      | id_U52,id_a__length => true
      | id_U52,id_a__isNatIList => true
      | id_U52,id_U33 => true
      | id_U52,id_a__U11 => true
      | id_U52,id_U81 => true
      | id_U52,id_a__U92 => true
      | id_U52,id_a__U43 => true
      | id_U52,id_U23 => true
      | id_U52,id_a__isNatKind => true
      | id_U52,id_U93 => true
      | id_U52,id_U11 => true
      | id_U52,id_a__U71 => true
      | id_U52,id_U44 => true
      | id_U52,id_0 => true
      | id_U52,id_U61 => true
      | id_U52,id_a__U86 => true
      | id_U52,id_a__U41 => true
      | id_U52,id_U22 => true
      | id_U52,id_a__U21 => true
      | id_U52,id_U91 => true
      | id_U52,id_length => true
      | id_U52,id_a__U52 => true
      | id_U52,id_U42 => true
      | id_U52,id_a__U12 => true
      | id_U52,id_U83 => true
      | id_U52,id_a__U94 => true
      | id_U52,id_a__U45 => true
      | id_U52,id_U31 => true
      | id_U52,id_a__isNat => true
      | id_U52,id_isNatIListKind => true
      | id_U52,id_a__U82 => true
      | id_U52,id_U46 => true
      | id_U52,id_cons => true
      | id_U52,id_U52 => true
      | id_U52,id_a__U85 => false
      | id_U52,id_a__U33 => false
      | id_U52,id_U21 => false
      | id_U52,id_a__isNatList => false
      | id_U52,id_U86 => false
      | id_U52,id_mark => false
      | id_U52,id_a__U51 => false
      | id_U52,id_U41 => false
      | id_U52,id_tt => false
      | id_U52,id_U82 => false
      | id_U52,id_a__U93 => false
      | id_U52,id_a__U44 => false
      | id_U52,id_isNat => false
      | id_U52,id_a__U23 => false
      | id_U52,id_U94 => false
      | id_U52,id_U12 => false
      | id_U52,id_a__U81 => false
      | id_U52,id_U45 => false
      | id_U52,id_zeros => false
      | id_U52,id_U71 => false
      | id_U52,id_a__U91 => false
      | id_U52,id_a__U42 => false
      | id_U52,id_isNatKind => false
      | id_U52,id_a__U22 => false
      | id_U52,id_U92 => false
      | id_U52,id_nil => false
      | id_U52,id_a__U61 => false
      | id_U52,id_U43 => false
      | id_U52,id_a__isNatIListKind => false
      | id_U52,id_U84 => false
      | id_U52,id_s => false
      | id_U52,id_a__U46 => false
      | id_U52,id_U32 => false
      | id_U52,id_a__U31 => false
      | id_U52,id_U13 => false
      | id_U52,id_a__U83 => false
      | id_U52,id_isNatIList => false
      | id_a__U85,id_a__zeros => true
      | id_a__U85,id_U51 => true
      | id_a__U85,id_a__U84 => true
      | id_a__U85,id_a__U32 => true
      | id_a__U85,id_isNatList => true
      | id_a__U85,id_a__U13 => true
      | id_a__U85,id_U85 => true
      | id_a__U85,id_a__length => true
      | id_a__U85,id_a__isNatIList => true
      | id_a__U85,id_U33 => true
      | id_a__U85,id_a__U11 => true
      | id_a__U85,id_U81 => true
      | id_a__U85,id_a__U92 => true
      | id_a__U85,id_a__U43 => true
      | id_a__U85,id_U23 => true
      | id_a__U85,id_a__isNatKind => true
      | id_a__U85,id_U93 => true
      | id_a__U85,id_U11 => true
      | id_a__U85,id_a__U71 => true
      | id_a__U85,id_U44 => true
      | id_a__U85,id_0 => true
      | id_a__U85,id_U61 => true
      | id_a__U85,id_a__U86 => true
      | id_a__U85,id_a__U41 => true
      | id_a__U85,id_U22 => true
      | id_a__U85,id_a__U21 => true
      | id_a__U85,id_U91 => true
      | id_a__U85,id_length => true
      | id_a__U85,id_a__U52 => true
      | id_a__U85,id_U42 => true
      | id_a__U85,id_a__U12 => true
      | id_a__U85,id_U83 => true
      | id_a__U85,id_a__U94 => true
      | id_a__U85,id_a__U45 => true
      | id_a__U85,id_U31 => true
      | id_a__U85,id_a__isNat => true
      | id_a__U85,id_isNatIListKind => true
      | id_a__U85,id_a__U82 => true
      | id_a__U85,id_U46 => true
      | id_a__U85,id_cons => true
      | id_a__U85,id_U52 => true
      | id_a__U85,id_a__U85 => true
      | id_a__U85,id_a__U33 => false
      | id_a__U85,id_U21 => false
      | id_a__U85,id_a__isNatList => false
      | id_a__U85,id_U86 => false
      | id_a__U85,id_mark => false
      | id_a__U85,id_a__U51 => false
      | id_a__U85,id_U41 => false
      | id_a__U85,id_tt => false
      | id_a__U85,id_U82 => false
      | id_a__U85,id_a__U93 => false
      | id_a__U85,id_a__U44 => false
      | id_a__U85,id_isNat => false
      | id_a__U85,id_a__U23 => false
      | id_a__U85,id_U94 => false
      | id_a__U85,id_U12 => false
      | id_a__U85,id_a__U81 => false
      | id_a__U85,id_U45 => false
      | id_a__U85,id_zeros => false
      | id_a__U85,id_U71 => false
      | id_a__U85,id_a__U91 => false
      | id_a__U85,id_a__U42 => false
      | id_a__U85,id_isNatKind => false
      | id_a__U85,id_a__U22 => false
      | id_a__U85,id_U92 => false
      | id_a__U85,id_nil => false
      | id_a__U85,id_a__U61 => false
      | id_a__U85,id_U43 => false
      | id_a__U85,id_a__isNatIListKind => false
      | id_a__U85,id_U84 => false
      | id_a__U85,id_s => false
      | id_a__U85,id_a__U46 => false
      | id_a__U85,id_U32 => false
      | id_a__U85,id_a__U31 => false
      | id_a__U85,id_U13 => false
      | id_a__U85,id_a__U83 => false
      | id_a__U85,id_isNatIList => false
      | id_a__U33,id_a__zeros => true
      | id_a__U33,id_U51 => true
      | id_a__U33,id_a__U84 => true
      | id_a__U33,id_a__U32 => true
      | id_a__U33,id_isNatList => true
      | id_a__U33,id_a__U13 => true
      | id_a__U33,id_U85 => true
      | id_a__U33,id_a__length => true
      | id_a__U33,id_a__isNatIList => true
      | id_a__U33,id_U33 => true
      | id_a__U33,id_a__U11 => true
      | id_a__U33,id_U81 => true
      | id_a__U33,id_a__U92 => true
      | id_a__U33,id_a__U43 => true
      | id_a__U33,id_U23 => true
      | id_a__U33,id_a__isNatKind => true
      | id_a__U33,id_U93 => true
      | id_a__U33,id_U11 => true
      | id_a__U33,id_a__U71 => true
      | id_a__U33,id_U44 => true
      | id_a__U33,id_0 => true
      | id_a__U33,id_U61 => true
      | id_a__U33,id_a__U86 => true
      | id_a__U33,id_a__U41 => true
      | id_a__U33,id_U22 => true
      | id_a__U33,id_a__U21 => true
      | id_a__U33,id_U91 => true
      | id_a__U33,id_length => true
      | id_a__U33,id_a__U52 => true
      | id_a__U33,id_U42 => true
      | id_a__U33,id_a__U12 => true
      | id_a__U33,id_U83 => true
      | id_a__U33,id_a__U94 => true
      | id_a__U33,id_a__U45 => true
      | id_a__U33,id_U31 => true
      | id_a__U33,id_a__isNat => true
      | id_a__U33,id_isNatIListKind => true
      | id_a__U33,id_a__U82 => true
      | id_a__U33,id_U46 => true
      | id_a__U33,id_cons => true
      | id_a__U33,id_U52 => true
      | id_a__U33,id_a__U85 => true
      | id_a__U33,id_a__U33 => true
      | id_a__U33,id_U21 => false
      | id_a__U33,id_a__isNatList => false
      | id_a__U33,id_U86 => false
      | id_a__U33,id_mark => false
      | id_a__U33,id_a__U51 => false
      | id_a__U33,id_U41 => false
      | id_a__U33,id_tt => false
      | id_a__U33,id_U82 => false
      | id_a__U33,id_a__U93 => false
      | id_a__U33,id_a__U44 => false
      | id_a__U33,id_isNat => false
      | id_a__U33,id_a__U23 => false
      | id_a__U33,id_U94 => false
      | id_a__U33,id_U12 => false
      | id_a__U33,id_a__U81 => false
      | id_a__U33,id_U45 => false
      | id_a__U33,id_zeros => false
      | id_a__U33,id_U71 => false
      | id_a__U33,id_a__U91 => false
      | id_a__U33,id_a__U42 => false
      | id_a__U33,id_isNatKind => false
      | id_a__U33,id_a__U22 => false
      | id_a__U33,id_U92 => false
      | id_a__U33,id_nil => false
      | id_a__U33,id_a__U61 => false
      | id_a__U33,id_U43 => false
      | id_a__U33,id_a__isNatIListKind => false
      | id_a__U33,id_U84 => false
      | id_a__U33,id_s => false
      | id_a__U33,id_a__U46 => false
      | id_a__U33,id_U32 => false
      | id_a__U33,id_a__U31 => false
      | id_a__U33,id_U13 => false
      | id_a__U33,id_a__U83 => false
      | id_a__U33,id_isNatIList => false
      | id_U21,id_a__zeros => true
      | id_U21,id_U51 => true
      | id_U21,id_a__U84 => true
      | id_U21,id_a__U32 => true
      | id_U21,id_isNatList => true
      | id_U21,id_a__U13 => true
      | id_U21,id_U85 => true
      | id_U21,id_a__length => true
      | id_U21,id_a__isNatIList => true
      | id_U21,id_U33 => true
      | id_U21,id_a__U11 => true
      | id_U21,id_U81 => true
      | id_U21,id_a__U92 => true
      | id_U21,id_a__U43 => true
      | id_U21,id_U23 => true
      | id_U21,id_a__isNatKind => true
      | id_U21,id_U93 => true
      | id_U21,id_U11 => true
      | id_U21,id_a__U71 => true
      | id_U21,id_U44 => true
      | id_U21,id_0 => true
      | id_U21,id_U61 => true
      | id_U21,id_a__U86 => true
      | id_U21,id_a__U41 => true
      | id_U21,id_U22 => true
      | id_U21,id_a__U21 => true
      | id_U21,id_U91 => true
      | id_U21,id_length => true
      | id_U21,id_a__U52 => true
      | id_U21,id_U42 => true
      | id_U21,id_a__U12 => true
      | id_U21,id_U83 => true
      | id_U21,id_a__U94 => true
      | id_U21,id_a__U45 => true
      | id_U21,id_U31 => true
      | id_U21,id_a__isNat => true
      | id_U21,id_isNatIListKind => true
      | id_U21,id_a__U82 => true
      | id_U21,id_U46 => true
      | id_U21,id_cons => true
      | id_U21,id_U52 => true
      | id_U21,id_a__U85 => true
      | id_U21,id_a__U33 => true
      | id_U21,id_U21 => true
      | id_U21,id_a__isNatList => false
      | id_U21,id_U86 => false
      | id_U21,id_mark => false
      | id_U21,id_a__U51 => false
      | id_U21,id_U41 => false
      | id_U21,id_tt => false
      | id_U21,id_U82 => false
      | id_U21,id_a__U93 => false
      | id_U21,id_a__U44 => false
      | id_U21,id_isNat => false
      | id_U21,id_a__U23 => false
      | id_U21,id_U94 => false
      | id_U21,id_U12 => false
      | id_U21,id_a__U81 => false
      | id_U21,id_U45 => false
      | id_U21,id_zeros => false
      | id_U21,id_U71 => false
      | id_U21,id_a__U91 => false
      | id_U21,id_a__U42 => false
      | id_U21,id_isNatKind => false
      | id_U21,id_a__U22 => false
      | id_U21,id_U92 => false
      | id_U21,id_nil => false
      | id_U21,id_a__U61 => false
      | id_U21,id_U43 => false
      | id_U21,id_a__isNatIListKind => false
      | id_U21,id_U84 => false
      | id_U21,id_s => false
      | id_U21,id_a__U46 => false
      | id_U21,id_U32 => false
      | id_U21,id_a__U31 => false
      | id_U21,id_U13 => false
      | id_U21,id_a__U83 => false
      | id_U21,id_isNatIList => false
      | id_a__isNatList,id_a__zeros => true
      | id_a__isNatList,id_U51 => true
      | id_a__isNatList,id_a__U84 => true
      | id_a__isNatList,id_a__U32 => true
      | id_a__isNatList,id_isNatList => true
      | id_a__isNatList,id_a__U13 => true
      | id_a__isNatList,id_U85 => true
      | id_a__isNatList,id_a__length => true
      | id_a__isNatList,id_a__isNatIList => true
      | id_a__isNatList,id_U33 => true
      | id_a__isNatList,id_a__U11 => true
      | id_a__isNatList,id_U81 => true
      | id_a__isNatList,id_a__U92 => true
      | id_a__isNatList,id_a__U43 => true
      | id_a__isNatList,id_U23 => true
      | id_a__isNatList,id_a__isNatKind => true
      | id_a__isNatList,id_U93 => true
      | id_a__isNatList,id_U11 => true
      | id_a__isNatList,id_a__U71 => true
      | id_a__isNatList,id_U44 => true
      | id_a__isNatList,id_0 => true
      | id_a__isNatList,id_U61 => true
      | id_a__isNatList,id_a__U86 => true
      | id_a__isNatList,id_a__U41 => true
      | id_a__isNatList,id_U22 => true
      | id_a__isNatList,id_a__U21 => true
      | id_a__isNatList,id_U91 => true
      | id_a__isNatList,id_length => true
      | id_a__isNatList,id_a__U52 => true
      | id_a__isNatList,id_U42 => true
      | id_a__isNatList,id_a__U12 => true
      | id_a__isNatList,id_U83 => true
      | id_a__isNatList,id_a__U94 => true
      | id_a__isNatList,id_a__U45 => true
      | id_a__isNatList,id_U31 => true
      | id_a__isNatList,id_a__isNat => true
      | id_a__isNatList,id_isNatIListKind => true
      | id_a__isNatList,id_a__U82 => true
      | id_a__isNatList,id_U46 => true
      | id_a__isNatList,id_cons => true
      | id_a__isNatList,id_U52 => true
      | id_a__isNatList,id_a__U85 => true
      | id_a__isNatList,id_a__U33 => true
      | id_a__isNatList,id_U21 => true
      | id_a__isNatList,id_a__isNatList => true
      | id_a__isNatList,id_U86 => false
      | id_a__isNatList,id_mark => false
      | id_a__isNatList,id_a__U51 => false
      | id_a__isNatList,id_U41 => false
      | id_a__isNatList,id_tt => false
      | id_a__isNatList,id_U82 => false
      | id_a__isNatList,id_a__U93 => false
      | id_a__isNatList,id_a__U44 => false
      | id_a__isNatList,id_isNat => false
      | id_a__isNatList,id_a__U23 => false
      | id_a__isNatList,id_U94 => false
      | id_a__isNatList,id_U12 => false
      | id_a__isNatList,id_a__U81 => false
      | id_a__isNatList,id_U45 => false
      | id_a__isNatList,id_zeros => false
      | id_a__isNatList,id_U71 => false
      | id_a__isNatList,id_a__U91 => false
      | id_a__isNatList,id_a__U42 => false
      | id_a__isNatList,id_isNatKind => false
      | id_a__isNatList,id_a__U22 => false
      | id_a__isNatList,id_U92 => false
      | id_a__isNatList,id_nil => false
      | id_a__isNatList,id_a__U61 => false
      | id_a__isNatList,id_U43 => false
      | id_a__isNatList,id_a__isNatIListKind => false
      | id_a__isNatList,id_U84 => false
      | id_a__isNatList,id_s => false
      | id_a__isNatList,id_a__U46 => false
      | id_a__isNatList,id_U32 => false
      | id_a__isNatList,id_a__U31 => false
      | id_a__isNatList,id_U13 => false
      | id_a__isNatList,id_a__U83 => false
      | id_a__isNatList,id_isNatIList => false
      | id_U86,id_a__zeros => true
      | id_U86,id_U51 => true
      | id_U86,id_a__U84 => true
      | id_U86,id_a__U32 => true
      | id_U86,id_isNatList => true
      | id_U86,id_a__U13 => true
      | id_U86,id_U85 => true
      | id_U86,id_a__length => true
      | id_U86,id_a__isNatIList => true
      | id_U86,id_U33 => true
      | id_U86,id_a__U11 => true
      | id_U86,id_U81 => true
      | id_U86,id_a__U92 => true
      | id_U86,id_a__U43 => true
      | id_U86,id_U23 => true
      | id_U86,id_a__isNatKind => true
      | id_U86,id_U93 => true
      | id_U86,id_U11 => true
      | id_U86,id_a__U71 => true
      | id_U86,id_U44 => true
      | id_U86,id_0 => true
      | id_U86,id_U61 => true
      | id_U86,id_a__U86 => true
      | id_U86,id_a__U41 => true
      | id_U86,id_U22 => true
      | id_U86,id_a__U21 => true
      | id_U86,id_U91 => true
      | id_U86,id_length => true
      | id_U86,id_a__U52 => true
      | id_U86,id_U42 => true
      | id_U86,id_a__U12 => true
      | id_U86,id_U83 => true
      | id_U86,id_a__U94 => true
      | id_U86,id_a__U45 => true
      | id_U86,id_U31 => true
      | id_U86,id_a__isNat => true
      | id_U86,id_isNatIListKind => true
      | id_U86,id_a__U82 => true
      | id_U86,id_U46 => true
      | id_U86,id_cons => true
      | id_U86,id_U52 => true
      | id_U86,id_a__U85 => true
      | id_U86,id_a__U33 => true
      | id_U86,id_U21 => true
      | id_U86,id_a__isNatList => true
      | id_U86,id_U86 => true
      | id_U86,id_mark => false
      | id_U86,id_a__U51 => false
      | id_U86,id_U41 => false
      | id_U86,id_tt => false
      | id_U86,id_U82 => false
      | id_U86,id_a__U93 => false
      | id_U86,id_a__U44 => false
      | id_U86,id_isNat => false
      | id_U86,id_a__U23 => false
      | id_U86,id_U94 => false
      | id_U86,id_U12 => false
      | id_U86,id_a__U81 => false
      | id_U86,id_U45 => false
      | id_U86,id_zeros => false
      | id_U86,id_U71 => false
      | id_U86,id_a__U91 => false
      | id_U86,id_a__U42 => false
      | id_U86,id_isNatKind => false
      | id_U86,id_a__U22 => false
      | id_U86,id_U92 => false
      | id_U86,id_nil => false
      | id_U86,id_a__U61 => false
      | id_U86,id_U43 => false
      | id_U86,id_a__isNatIListKind => false
      | id_U86,id_U84 => false
      | id_U86,id_s => false
      | id_U86,id_a__U46 => false
      | id_U86,id_U32 => false
      | id_U86,id_a__U31 => false
      | id_U86,id_U13 => false
      | id_U86,id_a__U83 => false
      | id_U86,id_isNatIList => false
      | id_mark,id_a__zeros => true
      | id_mark,id_U51 => true
      | id_mark,id_a__U84 => true
      | id_mark,id_a__U32 => true
      | id_mark,id_isNatList => true
      | id_mark,id_a__U13 => true
      | id_mark,id_U85 => true
      | id_mark,id_a__length => true
      | id_mark,id_a__isNatIList => true
      | id_mark,id_U33 => true
      | id_mark,id_a__U11 => true
      | id_mark,id_U81 => true
      | id_mark,id_a__U92 => true
      | id_mark,id_a__U43 => true
      | id_mark,id_U23 => true
      | id_mark,id_a__isNatKind => true
      | id_mark,id_U93 => true
      | id_mark,id_U11 => true
      | id_mark,id_a__U71 => true
      | id_mark,id_U44 => true
      | id_mark,id_0 => true
      | id_mark,id_U61 => true
      | id_mark,id_a__U86 => true
      | id_mark,id_a__U41 => true
      | id_mark,id_U22 => true
      | id_mark,id_a__U21 => true
      | id_mark,id_U91 => true
      | id_mark,id_length => true
      | id_mark,id_a__U52 => true
      | id_mark,id_U42 => true
      | id_mark,id_a__U12 => true
      | id_mark,id_U83 => true
      | id_mark,id_a__U94 => true
      | id_mark,id_a__U45 => true
      | id_mark,id_U31 => true
      | id_mark,id_a__isNat => true
      | id_mark,id_isNatIListKind => true
      | id_mark,id_a__U82 => true
      | id_mark,id_U46 => true
      | id_mark,id_cons => true
      | id_mark,id_U52 => true
      | id_mark,id_a__U85 => true
      | id_mark,id_a__U33 => true
      | id_mark,id_U21 => true
      | id_mark,id_a__isNatList => true
      | id_mark,id_U86 => true
      | id_mark,id_mark => true
      | id_mark,id_a__U51 => false
      | id_mark,id_U41 => false
      | id_mark,id_tt => false
      | id_mark,id_U82 => false
      | id_mark,id_a__U93 => false
      | id_mark,id_a__U44 => false
      | id_mark,id_isNat => false
      | id_mark,id_a__U23 => false
      | id_mark,id_U94 => false
      | id_mark,id_U12 => false
      | id_mark,id_a__U81 => false
      | id_mark,id_U45 => false
      | id_mark,id_zeros => false
      | id_mark,id_U71 => false
      | id_mark,id_a__U91 => false
      | id_mark,id_a__U42 => false
      | id_mark,id_isNatKind => false
      | id_mark,id_a__U22 => false
      | id_mark,id_U92 => false
      | id_mark,id_nil => false
      | id_mark,id_a__U61 => false
      | id_mark,id_U43 => false
      | id_mark,id_a__isNatIListKind => false
      | id_mark,id_U84 => false
      | id_mark,id_s => false
      | id_mark,id_a__U46 => false
      | id_mark,id_U32 => false
      | id_mark,id_a__U31 => false
      | id_mark,id_U13 => false
      | id_mark,id_a__U83 => false
      | id_mark,id_isNatIList => false
      | id_a__U51,id_a__zeros => true
      | id_a__U51,id_U51 => true
      | id_a__U51,id_a__U84 => true
      | id_a__U51,id_a__U32 => true
      | id_a__U51,id_isNatList => true
      | id_a__U51,id_a__U13 => true
      | id_a__U51,id_U85 => true
      | id_a__U51,id_a__length => true
      | id_a__U51,id_a__isNatIList => true
      | id_a__U51,id_U33 => true
      | id_a__U51,id_a__U11 => true
      | id_a__U51,id_U81 => true
      | id_a__U51,id_a__U92 => true
      | id_a__U51,id_a__U43 => true
      | id_a__U51,id_U23 => true
      | id_a__U51,id_a__isNatKind => true
      | id_a__U51,id_U93 => true
      | id_a__U51,id_U11 => true
      | id_a__U51,id_a__U71 => true
      | id_a__U51,id_U44 => true
      | id_a__U51,id_0 => true
      | id_a__U51,id_U61 => true
      | id_a__U51,id_a__U86 => true
      | id_a__U51,id_a__U41 => true
      | id_a__U51,id_U22 => true
      | id_a__U51,id_a__U21 => true
      | id_a__U51,id_U91 => true
      | id_a__U51,id_length => true
      | id_a__U51,id_a__U52 => true
      | id_a__U51,id_U42 => true
      | id_a__U51,id_a__U12 => true
      | id_a__U51,id_U83 => true
      | id_a__U51,id_a__U94 => true
      | id_a__U51,id_a__U45 => true
      | id_a__U51,id_U31 => true
      | id_a__U51,id_a__isNat => true
      | id_a__U51,id_isNatIListKind => true
      | id_a__U51,id_a__U82 => true
      | id_a__U51,id_U46 => true
      | id_a__U51,id_cons => true
      | id_a__U51,id_U52 => true
      | id_a__U51,id_a__U85 => true
      | id_a__U51,id_a__U33 => true
      | id_a__U51,id_U21 => true
      | id_a__U51,id_a__isNatList => true
      | id_a__U51,id_U86 => true
      | id_a__U51,id_mark => true
      | id_a__U51,id_a__U51 => true
      | id_a__U51,id_U41 => false
      | id_a__U51,id_tt => false
      | id_a__U51,id_U82 => false
      | id_a__U51,id_a__U93 => false
      | id_a__U51,id_a__U44 => false
      | id_a__U51,id_isNat => false
      | id_a__U51,id_a__U23 => false
      | id_a__U51,id_U94 => false
      | id_a__U51,id_U12 => false
      | id_a__U51,id_a__U81 => false
      | id_a__U51,id_U45 => false
      | id_a__U51,id_zeros => false
      | id_a__U51,id_U71 => false
      | id_a__U51,id_a__U91 => false
      | id_a__U51,id_a__U42 => false
      | id_a__U51,id_isNatKind => false
      | id_a__U51,id_a__U22 => false
      | id_a__U51,id_U92 => false
      | id_a__U51,id_nil => false
      | id_a__U51,id_a__U61 => false
      | id_a__U51,id_U43 => false
      | id_a__U51,id_a__isNatIListKind => false
      | id_a__U51,id_U84 => false
      | id_a__U51,id_s => false
      | id_a__U51,id_a__U46 => false
      | id_a__U51,id_U32 => false
      | id_a__U51,id_a__U31 => false
      | id_a__U51,id_U13 => false
      | id_a__U51,id_a__U83 => false
      | id_a__U51,id_isNatIList => false
      | id_U41,id_a__zeros => true
      | id_U41,id_U51 => true
      | id_U41,id_a__U84 => true
      | id_U41,id_a__U32 => true
      | id_U41,id_isNatList => true
      | id_U41,id_a__U13 => true
      | id_U41,id_U85 => true
      | id_U41,id_a__length => true
      | id_U41,id_a__isNatIList => true
      | id_U41,id_U33 => true
      | id_U41,id_a__U11 => true
      | id_U41,id_U81 => true
      | id_U41,id_a__U92 => true
      | id_U41,id_a__U43 => true
      | id_U41,id_U23 => true
      | id_U41,id_a__isNatKind => true
      | id_U41,id_U93 => true
      | id_U41,id_U11 => true
      | id_U41,id_a__U71 => true
      | id_U41,id_U44 => true
      | id_U41,id_0 => true
      | id_U41,id_U61 => true
      | id_U41,id_a__U86 => true
      | id_U41,id_a__U41 => true
      | id_U41,id_U22 => true
      | id_U41,id_a__U21 => true
      | id_U41,id_U91 => true
      | id_U41,id_length => true
      | id_U41,id_a__U52 => true
      | id_U41,id_U42 => true
      | id_U41,id_a__U12 => true
      | id_U41,id_U83 => true
      | id_U41,id_a__U94 => true
      | id_U41,id_a__U45 => true
      | id_U41,id_U31 => true
      | id_U41,id_a__isNat => true
      | id_U41,id_isNatIListKind => true
      | id_U41,id_a__U82 => true
      | id_U41,id_U46 => true
      | id_U41,id_cons => true
      | id_U41,id_U52 => true
      | id_U41,id_a__U85 => true
      | id_U41,id_a__U33 => true
      | id_U41,id_U21 => true
      | id_U41,id_a__isNatList => true
      | id_U41,id_U86 => true
      | id_U41,id_mark => true
      | id_U41,id_a__U51 => true
      | id_U41,id_U41 => true
      | id_U41,id_tt => false
      | id_U41,id_U82 => false
      | id_U41,id_a__U93 => false
      | id_U41,id_a__U44 => false
      | id_U41,id_isNat => false
      | id_U41,id_a__U23 => false
      | id_U41,id_U94 => false
      | id_U41,id_U12 => false
      | id_U41,id_a__U81 => false
      | id_U41,id_U45 => false
      | id_U41,id_zeros => false
      | id_U41,id_U71 => false
      | id_U41,id_a__U91 => false
      | id_U41,id_a__U42 => false
      | id_U41,id_isNatKind => false
      | id_U41,id_a__U22 => false
      | id_U41,id_U92 => false
      | id_U41,id_nil => false
      | id_U41,id_a__U61 => false
      | id_U41,id_U43 => false
      | id_U41,id_a__isNatIListKind => false
      | id_U41,id_U84 => false
      | id_U41,id_s => false
      | id_U41,id_a__U46 => false
      | id_U41,id_U32 => false
      | id_U41,id_a__U31 => false
      | id_U41,id_U13 => false
      | id_U41,id_a__U83 => false
      | id_U41,id_isNatIList => false
      | id_tt,id_a__zeros => true
      | id_tt,id_U51 => true
      | id_tt,id_a__U84 => true
      | id_tt,id_a__U32 => true
      | id_tt,id_isNatList => true
      | id_tt,id_a__U13 => true
      | id_tt,id_U85 => true
      | id_tt,id_a__length => true
      | id_tt,id_a__isNatIList => true
      | id_tt,id_U33 => true
      | id_tt,id_a__U11 => true
      | id_tt,id_U81 => true
      | id_tt,id_a__U92 => true
      | id_tt,id_a__U43 => true
      | id_tt,id_U23 => true
      | id_tt,id_a__isNatKind => true
      | id_tt,id_U93 => true
      | id_tt,id_U11 => true
      | id_tt,id_a__U71 => true
      | id_tt,id_U44 => true
      | id_tt,id_0 => true
      | id_tt,id_U61 => true
      | id_tt,id_a__U86 => true
      | id_tt,id_a__U41 => true
      | id_tt,id_U22 => true
      | id_tt,id_a__U21 => true
      | id_tt,id_U91 => true
      | id_tt,id_length => true
      | id_tt,id_a__U52 => true
      | id_tt,id_U42 => true
      | id_tt,id_a__U12 => true
      | id_tt,id_U83 => true
      | id_tt,id_a__U94 => true
      | id_tt,id_a__U45 => true
      | id_tt,id_U31 => true
      | id_tt,id_a__isNat => true
      | id_tt,id_isNatIListKind => true
      | id_tt,id_a__U82 => true
      | id_tt,id_U46 => true
      | id_tt,id_cons => true
      | id_tt,id_U52 => true
      | id_tt,id_a__U85 => true
      | id_tt,id_a__U33 => true
      | id_tt,id_U21 => true
      | id_tt,id_a__isNatList => true
      | id_tt,id_U86 => true
      | id_tt,id_mark => true
      | id_tt,id_a__U51 => true
      | id_tt,id_U41 => true
      | id_tt,id_tt => true
      | id_tt,id_U82 => false
      | id_tt,id_a__U93 => false
      | id_tt,id_a__U44 => false
      | id_tt,id_isNat => false
      | id_tt,id_a__U23 => false
      | id_tt,id_U94 => false
      | id_tt,id_U12 => false
      | id_tt,id_a__U81 => false
      | id_tt,id_U45 => false
      | id_tt,id_zeros => false
      | id_tt,id_U71 => false
      | id_tt,id_a__U91 => false
      | id_tt,id_a__U42 => false
      | id_tt,id_isNatKind => false
      | id_tt,id_a__U22 => false
      | id_tt,id_U92 => false
      | id_tt,id_nil => false
      | id_tt,id_a__U61 => false
      | id_tt,id_U43 => false
      | id_tt,id_a__isNatIListKind => false
      | id_tt,id_U84 => false
      | id_tt,id_s => false
      | id_tt,id_a__U46 => false
      | id_tt,id_U32 => false
      | id_tt,id_a__U31 => false
      | id_tt,id_U13 => false
      | id_tt,id_a__U83 => false
      | id_tt,id_isNatIList => false
      | id_U82,id_a__zeros => true
      | id_U82,id_U51 => true
      | id_U82,id_a__U84 => true
      | id_U82,id_a__U32 => true
      | id_U82,id_isNatList => true
      | id_U82,id_a__U13 => true
      | id_U82,id_U85 => true
      | id_U82,id_a__length => true
      | id_U82,id_a__isNatIList => true
      | id_U82,id_U33 => true
      | id_U82,id_a__U11 => true
      | id_U82,id_U81 => true
      | id_U82,id_a__U92 => true
      | id_U82,id_a__U43 => true
      | id_U82,id_U23 => true
      | id_U82,id_a__isNatKind => true
      | id_U82,id_U93 => true
      | id_U82,id_U11 => true
      | id_U82,id_a__U71 => true
      | id_U82,id_U44 => true
      | id_U82,id_0 => true
      | id_U82,id_U61 => true
      | id_U82,id_a__U86 => true
      | id_U82,id_a__U41 => true
      | id_U82,id_U22 => true
      | id_U82,id_a__U21 => true
      | id_U82,id_U91 => true
      | id_U82,id_length => true
      | id_U82,id_a__U52 => true
      | id_U82,id_U42 => true
      | id_U82,id_a__U12 => true
      | id_U82,id_U83 => true
      | id_U82,id_a__U94 => true
      | id_U82,id_a__U45 => true
      | id_U82,id_U31 => true
      | id_U82,id_a__isNat => true
      | id_U82,id_isNatIListKind => true
      | id_U82,id_a__U82 => true
      | id_U82,id_U46 => true
      | id_U82,id_cons => true
      | id_U82,id_U52 => true
      | id_U82,id_a__U85 => true
      | id_U82,id_a__U33 => true
      | id_U82,id_U21 => true
      | id_U82,id_a__isNatList => true
      | id_U82,id_U86 => true
      | id_U82,id_mark => true
      | id_U82,id_a__U51 => true
      | id_U82,id_U41 => true
      | id_U82,id_tt => true
      | id_U82,id_U82 => true
      | id_U82,id_a__U93 => false
      | id_U82,id_a__U44 => false
      | id_U82,id_isNat => false
      | id_U82,id_a__U23 => false
      | id_U82,id_U94 => false
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      | id_a__U44,id_a__U92 => true
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      | id_a__U44,id_a__isNatKind => true
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      | id_a__U44,id_a__U71 => true
      | id_a__U44,id_U44 => true
      | id_a__U44,id_0 => true
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      | id_a__U44,id_a__U45 => true
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      | id_a__U44,id_a__isNat => true
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      | id_a__U44,id_a__U82 => true
      | id_a__U44,id_U46 => true
      | id_a__U44,id_cons => true
      | id_a__U44,id_U52 => true
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      | id_a__U44,id_a__U33 => true
      | id_a__U44,id_U21 => true
      | id_a__U44,id_a__isNatList => true
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      | id_a__U44,id_a__U93 => true
      | id_a__U44,id_a__U44 => true
      | id_a__U44,id_isNat => false
      | id_a__U44,id_a__U23 => false
      | id_a__U44,id_U94 => false
      | id_a__U44,id_U12 => false
      | id_a__U44,id_a__U81 => false
      | id_a__U44,id_U45 => false
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      | id_a__U44,id_U71 => false
      | id_a__U44,id_a__U91 => false
      | id_a__U44,id_a__U42 => false
      | id_a__U44,id_isNatKind => false
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      | id_a__U44,id_a__U61 => false
      | id_a__U44,id_U43 => false
      | id_a__U44,id_a__isNatIListKind => false
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      | id_a__U44,id_s => false
      | id_a__U44,id_a__U46 => false
      | id_a__U44,id_U32 => false
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      | id_a__U44,id_U13 => false
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      | id_isNat,id_a__zeros => true
      | id_isNat,id_U51 => true
      | id_isNat,id_a__U84 => true
      | id_isNat,id_a__U32 => true
      | id_isNat,id_isNatList => true
      | id_isNat,id_a__U13 => true
      | id_isNat,id_U85 => true
      | id_isNat,id_a__length => true
      | id_isNat,id_a__isNatIList => true
      | id_isNat,id_U33 => true
      | id_isNat,id_a__U11 => true
      | id_isNat,id_U81 => true
      | id_isNat,id_a__U92 => true
      | id_isNat,id_a__U43 => true
      | id_isNat,id_U23 => true
      | id_isNat,id_a__isNatKind => true
      | id_isNat,id_U93 => true
      | id_isNat,id_U11 => true
      | id_isNat,id_a__U71 => true
      | id_isNat,id_U44 => true
      | id_isNat,id_0 => true
      | id_isNat,id_U61 => true
      | id_isNat,id_a__U86 => true
      | id_isNat,id_a__U41 => true
      | id_isNat,id_U22 => true
      | id_isNat,id_a__U21 => true
      | id_isNat,id_U91 => true
      | id_isNat,id_length => true
      | id_isNat,id_a__U52 => true
      | id_isNat,id_U42 => true
      | id_isNat,id_a__U12 => true
      | id_isNat,id_U83 => true
      | id_isNat,id_a__U94 => true
      | id_isNat,id_a__U45 => true
      | id_isNat,id_U31 => true
      | id_isNat,id_a__isNat => true
      | id_isNat,id_isNatIListKind => true
      | id_isNat,id_a__U82 => true
      | id_isNat,id_U46 => true
      | id_isNat,id_cons => true
      | id_isNat,id_U52 => true
      | id_isNat,id_a__U85 => true
      | id_isNat,id_a__U33 => true
      | id_isNat,id_U21 => true
      | id_isNat,id_a__isNatList => true
      | id_isNat,id_U86 => true
      | id_isNat,id_mark => true
      | id_isNat,id_a__U51 => true
      | id_isNat,id_U41 => true
      | id_isNat,id_tt => true
      | id_isNat,id_U82 => true
      | id_isNat,id_a__U93 => true
      | id_isNat,id_a__U44 => true
      | id_isNat,id_isNat => true
      | id_isNat,id_a__U23 => false
      | id_isNat,id_U94 => false
      | id_isNat,id_U12 => false
      | id_isNat,id_a__U81 => false
      | id_isNat,id_U45 => false
      | id_isNat,id_zeros => false
      | id_isNat,id_U71 => false
      | id_isNat,id_a__U91 => false
      | id_isNat,id_a__U42 => false
      | id_isNat,id_isNatKind => false
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      | id_isNat,id_U92 => false
      | id_isNat,id_nil => false
      | id_isNat,id_a__U61 => false
      | id_isNat,id_U43 => false
      | id_isNat,id_a__isNatIListKind => false
      | id_isNat,id_U84 => false
      | id_isNat,id_s => false
      | id_isNat,id_a__U46 => false
      | id_isNat,id_U32 => false
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      | id_isNat,id_U13 => false
      | id_isNat,id_a__U83 => false
      | id_isNat,id_isNatIList => false
      | id_a__U23,id_a__zeros => true
      | id_a__U23,id_U51 => true
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      | id_a__U23,id_a__U32 => true
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      | id_a__U23,id_a__length => true
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      | id_a__U23,id_a__U92 => true
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      | id_a__U23,id_U23 => true
      | id_a__U23,id_a__isNatKind => true
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      | id_a__U23,id_U44 => true
      | id_a__U23,id_0 => true
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      | id_a__U23,id_a__U33 => true
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      | id_a__U23,id_U12 => false
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      | id_U94,id_a__U33 => true
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      | id_U12,id_a__isNatIListKind => false
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      | id_U12,id_a__U31 => false
      | id_U12,id_U13 => false
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      | id_a__U81,id_a__isNatIList => true
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      | id_a__U81,id_a__U11 => true
      | id_a__U81,id_U81 => true
      | id_a__U81,id_a__U92 => true
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      | id_a__U81,id_U23 => true
      | id_a__U81,id_a__isNatKind => true
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      | id_a__U81,id_U11 => true
      | id_a__U81,id_a__U71 => true
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      | id_a__U81,id_U22 => true
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      | id_a__U81,id_U91 => true
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      | id_a__U81,id_a__U52 => true
      | id_a__U81,id_U42 => true
      | id_a__U81,id_a__U12 => true
      | id_a__U81,id_U83 => true
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      | id_a__U81,id_a__U45 => true
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      | id_a__U81,id_a__isNat => true
      | id_a__U81,id_isNatIListKind => true
      | id_a__U81,id_a__U82 => true
      | id_a__U81,id_U46 => true
      | id_a__U81,id_cons => true
      | id_a__U81,id_U52 => true
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      | id_a__U81,id_a__U33 => true
      | id_a__U81,id_U21 => true
      | id_a__U81,id_a__isNatList => true
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      | id_a__U81,id_U82 => true
      | id_a__U81,id_a__U93 => true
      | id_a__U81,id_a__U44 => true
      | id_a__U81,id_isNat => true
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      | id_a__U81,id_U12 => true
      | id_a__U81,id_a__U81 => true
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      | id_a__U81,id_U71 => false
      | id_a__U81,id_a__U91 => false
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      | id_a__U81,id_isNatKind => false
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      | id_a__U81,id_a__U61 => false
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      | id_a__U81,id_a__isNatIListKind => false
      | id_a__U81,id_U84 => false
      | id_a__U81,id_s => false
      | id_a__U81,id_a__U46 => false
      | id_a__U81,id_U32 => false
      | id_a__U81,id_a__U31 => false
      | id_a__U81,id_U13 => false
      | id_a__U81,id_a__U83 => false
      | id_a__U81,id_isNatIList => false
      | id_U45,id_a__zeros => true
      | id_U45,id_U51 => true
      | id_U45,id_a__U84 => true
      | id_U45,id_a__U32 => true
      | id_U45,id_isNatList => true
      | id_U45,id_a__U13 => true
      | id_U45,id_U85 => true
      | id_U45,id_a__length => true
      | id_U45,id_a__isNatIList => true
      | id_U45,id_U33 => true
      | id_U45,id_a__U11 => true
      | id_U45,id_U81 => true
      | id_U45,id_a__U92 => true
      | id_U45,id_a__U43 => true
      | id_U45,id_U23 => true
      | id_U45,id_a__isNatKind => true
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      | id_U45,id_U11 => true
      | id_U45,id_a__U71 => true
      | id_U45,id_U44 => true
      | id_U45,id_0 => true
      | id_U45,id_U61 => true
      | id_U45,id_a__U86 => true
      | id_U45,id_a__U41 => true
      | id_U45,id_U22 => true
      | id_U45,id_a__U21 => true
      | id_U45,id_U91 => true
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      | id_U45,id_a__U52 => true
      | id_U45,id_U42 => true
      | id_U45,id_a__U12 => true
      | id_U45,id_U83 => true
      | id_U45,id_a__U94 => true
      | id_U45,id_a__U45 => true
      | id_U45,id_U31 => true
      | id_U45,id_a__isNat => true
      | id_U45,id_isNatIListKind => true
      | id_U45,id_a__U82 => true
      | id_U45,id_U46 => true
      | id_U45,id_cons => true
      | id_U45,id_U52 => true
      | id_U45,id_a__U85 => true
      | id_U45,id_a__U33 => true
      | id_U45,id_U21 => true
      | id_U45,id_a__isNatList => true
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      | id_U45,id_mark => true
      | id_U45,id_a__U51 => true
      | id_U45,id_U41 => true
      | id_U45,id_tt => true
      | id_U45,id_U82 => true
      | id_U45,id_a__U93 => true
      | id_U45,id_a__U44 => true
      | id_U45,id_isNat => true
      | id_U45,id_a__U23 => true
      | id_U45,id_U94 => true
      | id_U45,id_U12 => true
      | id_U45,id_a__U81 => true
      | id_U45,id_U45 => true
      | id_U45,id_zeros => false
      | id_U45,id_U71 => false
      | id_U45,id_a__U91 => false
      | id_U45,id_a__U42 => false
      | id_U45,id_isNatKind => false
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      | id_U45,id_U92 => false
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      | id_U45,id_a__U61 => false
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      | id_U45,id_U84 => false
      | id_U45,id_s => false
      | id_U45,id_a__U46 => false
      | id_U45,id_U32 => false
      | id_U45,id_a__U31 => false
      | id_U45,id_U13 => false
      | id_U45,id_a__U83 => false
      | id_U45,id_isNatIList => false
      | id_zeros,id_a__zeros => true
      | id_zeros,id_U51 => true
      | id_zeros,id_a__U84 => true
      | id_zeros,id_a__U32 => true
      | id_zeros,id_isNatList => true
      | id_zeros,id_a__U13 => true
      | id_zeros,id_U85 => true
      | id_zeros,id_a__length => true
      | id_zeros,id_a__isNatIList => true
      | id_zeros,id_U33 => true
      | id_zeros,id_a__U11 => true
      | id_zeros,id_U81 => true
      | id_zeros,id_a__U92 => true
      | id_zeros,id_a__U43 => true
      | id_zeros,id_U23 => true
      | id_zeros,id_a__isNatKind => true
      | id_zeros,id_U93 => true
      | id_zeros,id_U11 => true
      | id_zeros,id_a__U71 => true
      | id_zeros,id_U44 => true
      | id_zeros,id_0 => true
      | id_zeros,id_U61 => true
      | id_zeros,id_a__U86 => true
      | id_zeros,id_a__U41 => true
      | id_zeros,id_U22 => true
      | id_zeros,id_a__U21 => true
      | id_zeros,id_U91 => true
      | id_zeros,id_length => true
      | id_zeros,id_a__U52 => true
      | id_zeros,id_U42 => true
      | id_zeros,id_a__U12 => true
      | id_zeros,id_U83 => true
      | id_zeros,id_a__U94 => true
      | id_zeros,id_a__U45 => true
      | id_zeros,id_U31 => true
      | id_zeros,id_a__isNat => true
      | id_zeros,id_isNatIListKind => true
      | id_zeros,id_a__U82 => true
      | id_zeros,id_U46 => true
      | id_zeros,id_cons => true
      | id_zeros,id_U52 => true
      | id_zeros,id_a__U85 => true
      | id_zeros,id_a__U33 => true
      | id_zeros,id_U21 => true
      | id_zeros,id_a__isNatList => true
      | id_zeros,id_U86 => true
      | id_zeros,id_mark => true
      | id_zeros,id_a__U51 => true
      | id_zeros,id_U41 => true
      | id_zeros,id_tt => true
      | id_zeros,id_U82 => true
      | id_zeros,id_a__U93 => true
      | id_zeros,id_a__U44 => true
      | id_zeros,id_isNat => true
      | id_zeros,id_a__U23 => true
      | id_zeros,id_U94 => true
      | id_zeros,id_U12 => true
      | id_zeros,id_a__U81 => true
      | id_zeros,id_U45 => true
      | id_zeros,id_zeros => true
      | id_zeros,id_U71 => false
      | id_zeros,id_a__U91 => false
      | id_zeros,id_a__U42 => false
      | id_zeros,id_isNatKind => false
      | id_zeros,id_a__U22 => false
      | id_zeros,id_U92 => false
      | id_zeros,id_nil => false
      | id_zeros,id_a__U61 => false
      | id_zeros,id_U43 => false
      | id_zeros,id_a__isNatIListKind => false
      | id_zeros,id_U84 => false
      | id_zeros,id_s => false
      | id_zeros,id_a__U46 => false
      | id_zeros,id_U32 => false
      | id_zeros,id_a__U31 => false
      | id_zeros,id_U13 => false
      | id_zeros,id_a__U83 => false
      | id_zeros,id_isNatIList => false
      | id_U71,id_a__zeros => true
      | id_U71,id_U51 => true
      | id_U71,id_a__U84 => true
      | id_U71,id_a__U32 => true
      | id_U71,id_isNatList => true
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      | id_U71,id_a__isNat => true
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      | id_isNatKind,id_a__zeros => true
      | id_isNatKind,id_U51 => true
      | id_isNatKind,id_a__U84 => true
      | id_isNatKind,id_a__U32 => true
      | id_isNatKind,id_isNatList => true
      | id_isNatKind,id_a__U13 => true
      | id_isNatKind,id_U85 => true
      | id_isNatKind,id_a__length => true
      | id_isNatKind,id_a__isNatIList => true
      | id_isNatKind,id_U33 => true
      | id_isNatKind,id_a__U11 => true
      | id_isNatKind,id_U81 => true
      | id_isNatKind,id_a__U92 => true
      | id_isNatKind,id_a__U43 => true
      | id_isNatKind,id_U23 => true
      | id_isNatKind,id_a__isNatKind => true
      | id_isNatKind,id_U93 => true
      | id_isNatKind,id_U11 => true
      | id_isNatKind,id_a__U71 => true
      | id_isNatKind,id_U44 => true
      | id_isNatKind,id_0 => true
      | id_isNatKind,id_U61 => true
      | id_isNatKind,id_a__U86 => true
      | id_isNatKind,id_a__U41 => true
      | id_isNatKind,id_U22 => true
      | id_isNatKind,id_a__U21 => true
      | id_isNatKind,id_U91 => true
      | id_isNatKind,id_length => true
      | id_isNatKind,id_a__U52 => true
      | id_isNatKind,id_U42 => true
      | id_isNatKind,id_a__U12 => true
      | id_isNatKind,id_U83 => true
      | id_isNatKind,id_a__U94 => true
      | id_isNatKind,id_a__U45 => true
      | id_isNatKind,id_U31 => true
      | id_isNatKind,id_a__isNat => true
      | id_isNatKind,id_isNatIListKind => true
      | id_isNatKind,id_a__U82 => true
      | id_isNatKind,id_U46 => true
      | id_isNatKind,id_cons => true
      | id_isNatKind,id_U52 => true
      | id_isNatKind,id_a__U85 => true
      | id_isNatKind,id_a__U33 => true
      | id_isNatKind,id_U21 => true
      | id_isNatKind,id_a__isNatList => true
      | id_isNatKind,id_U86 => true
      | id_isNatKind,id_mark => true
      | id_isNatKind,id_a__U51 => true
      | id_isNatKind,id_U41 => true
      | id_isNatKind,id_tt => true
      | id_isNatKind,id_U82 => true
      | id_isNatKind,id_a__U93 => true
      | id_isNatKind,id_a__U44 => true
      | id_isNatKind,id_isNat => true
      | id_isNatKind,id_a__U23 => true
      | id_isNatKind,id_U94 => true
      | id_isNatKind,id_U12 => true
      | id_isNatKind,id_a__U81 => true
      | id_isNatKind,id_U45 => true
      | id_isNatKind,id_zeros => true
      | id_isNatKind,id_U71 => true
      | id_isNatKind,id_a__U91 => true
      | id_isNatKind,id_a__U42 => true
      | id_isNatKind,id_isNatKind => true
      | id_isNatKind,id_a__U22 => false
      | id_isNatKind,id_U92 => false
      | id_isNatKind,id_nil => false
      | id_isNatKind,id_a__U61 => false
      | id_isNatKind,id_U43 => false
      | id_isNatKind,id_a__isNatIListKind => false
      | id_isNatKind,id_U84 => false
      | id_isNatKind,id_s => false
      | id_isNatKind,id_a__U46 => false
      | id_isNatKind,id_U32 => false
      | id_isNatKind,id_a__U31 => false
      | id_isNatKind,id_U13 => false
      | id_isNatKind,id_a__U83 => false
      | id_isNatKind,id_isNatIList => false
      | id_a__U22,id_a__zeros => true
      | id_a__U22,id_U51 => true
      | id_a__U22,id_a__U84 => true
      | id_a__U22,id_a__U32 => true
      | id_a__U22,id_isNatList => true
      | id_a__U22,id_a__U13 => true
      | id_a__U22,id_U85 => true
      | id_a__U22,id_a__length => true
      | id_a__U22,id_a__isNatIList => true
      | id_a__U22,id_U33 => true
      | id_a__U22,id_a__U11 => true
      | id_a__U22,id_U81 => true
      | id_a__U22,id_a__U92 => true
      | id_a__U22,id_a__U43 => true
      | id_a__U22,id_U23 => true
      | id_a__U22,id_a__isNatKind => true
      | id_a__U22,id_U93 => true
      | id_a__U22,id_U11 => true
      | id_a__U22,id_a__U71 => true
      | id_a__U22,id_U44 => true
      | id_a__U22,id_0 => true
      | id_a__U22,id_U61 => true
      | id_a__U22,id_a__U86 => true
      | id_a__U22,id_a__U41 => true
      | id_a__U22,id_U22 => true
      | id_a__U22,id_a__U21 => true
      | id_a__U22,id_U91 => true
      | id_a__U22,id_length => true
      | id_a__U22,id_a__U52 => true
      | id_a__U22,id_U42 => true
      | id_a__U22,id_a__U12 => true
      | id_a__U22,id_U83 => true
      | id_a__U22,id_a__U94 => true
      | id_a__U22,id_a__U45 => true
      | id_a__U22,id_U31 => true
      | id_a__U22,id_a__isNat => true
      | id_a__U22,id_isNatIListKind => true
      | id_a__U22,id_a__U82 => true
      | id_a__U22,id_U46 => true
      | id_a__U22,id_cons => true
      | id_a__U22,id_U52 => true
      | id_a__U22,id_a__U85 => true
      | id_a__U22,id_a__U33 => true
      | id_a__U22,id_U21 => true
      | id_a__U22,id_a__isNatList => true
      | id_a__U22,id_U86 => true
      | id_a__U22,id_mark => true
      | id_a__U22,id_a__U51 => true
      | id_a__U22,id_U41 => true
      | id_a__U22,id_tt => true
      | id_a__U22,id_U82 => true
      | id_a__U22,id_a__U93 => true
      | id_a__U22,id_a__U44 => true
      | id_a__U22,id_isNat => true
      | id_a__U22,id_a__U23 => true
      | id_a__U22,id_U94 => true
      | id_a__U22,id_U12 => true
      | id_a__U22,id_a__U81 => true
      | id_a__U22,id_U45 => true
      | id_a__U22,id_zeros => true
      | id_a__U22,id_U71 => true
      | id_a__U22,id_a__U91 => true
      | id_a__U22,id_a__U42 => true
      | id_a__U22,id_isNatKind => true
      | id_a__U22,id_a__U22 => true
      | id_a__U22,id_U92 => false
      | id_a__U22,id_nil => false
      | id_a__U22,id_a__U61 => false
      | id_a__U22,id_U43 => false
      | id_a__U22,id_a__isNatIListKind => false
      | id_a__U22,id_U84 => false
      | id_a__U22,id_s => false
      | id_a__U22,id_a__U46 => false
      | id_a__U22,id_U32 => false
      | id_a__U22,id_a__U31 => false
      | id_a__U22,id_U13 => false
      | id_a__U22,id_a__U83 => false
      | id_a__U22,id_isNatIList => false
      | id_U92,id_a__zeros => true
      | id_U92,id_U51 => true
      | id_U92,id_a__U84 => true
      | id_U92,id_a__U32 => true
      | id_U92,id_isNatList => true
      | id_U92,id_a__U13 => true
      | id_U92,id_U85 => true
      | id_U92,id_a__length => true
      | id_U92,id_a__isNatIList => true
      | id_U92,id_U33 => true
      | id_U92,id_a__U11 => true
      | id_U92,id_U81 => true
      | id_U92,id_a__U92 => true
      | id_U92,id_a__U43 => true
      | id_U92,id_U23 => true
      | id_U92,id_a__isNatKind => true
      | id_U92,id_U93 => true
      | id_U92,id_U11 => true
      | id_U92,id_a__U71 => true
      | id_U92,id_U44 => true
      | id_U92,id_0 => true
      | id_U92,id_U61 => true
      | id_U92,id_a__U86 => true
      | id_U92,id_a__U41 => true
      | id_U92,id_U22 => true
      | id_U92,id_a__U21 => true
      | id_U92,id_U91 => true
      | id_U92,id_length => true
      | id_U92,id_a__U52 => true
      | id_U92,id_U42 => true
      | id_U92,id_a__U12 => true
      | id_U92,id_U83 => true
      | id_U92,id_a__U94 => true
      | id_U92,id_a__U45 => true
      | id_U92,id_U31 => true
      | id_U92,id_a__isNat => true
      | id_U92,id_isNatIListKind => true
      | id_U92,id_a__U82 => true
      | id_U92,id_U46 => true
      | id_U92,id_cons => true
      | id_U92,id_U52 => true
      | id_U92,id_a__U85 => true
      | id_U92,id_a__U33 => true
      | id_U92,id_U21 => true
      | id_U92,id_a__isNatList => true
      | id_U92,id_U86 => true
      | id_U92,id_mark => true
      | id_U92,id_a__U51 => true
      | id_U92,id_U41 => true
      | id_U92,id_tt => true
      | id_U92,id_U82 => true
      | id_U92,id_a__U93 => true
      | id_U92,id_a__U44 => true
      | id_U92,id_isNat => true
      | id_U92,id_a__U23 => true
      | id_U92,id_U94 => true
      | id_U92,id_U12 => true
      | id_U92,id_a__U81 => true
      | id_U92,id_U45 => true
      | id_U92,id_zeros => true
      | id_U92,id_U71 => true
      | id_U92,id_a__U91 => true
      | id_U92,id_a__U42 => true
      | id_U92,id_isNatKind => true
      | id_U92,id_a__U22 => true
      | id_U92,id_U92 => true
      | id_U92,id_nil => false
      | id_U92,id_a__U61 => false
      | id_U92,id_U43 => false
      | id_U92,id_a__isNatIListKind => false
      | id_U92,id_U84 => false
      | id_U92,id_s => false
      | id_U92,id_a__U46 => false
      | id_U92,id_U32 => false
      | id_U92,id_a__U31 => false
      | id_U92,id_U13 => false
      | id_U92,id_a__U83 => false
      | id_U92,id_isNatIList => false
      | id_nil,id_a__zeros => true
      | id_nil,id_U51 => true
      | id_nil,id_a__U84 => true
      | id_nil,id_a__U32 => true
      | id_nil,id_isNatList => true
      | id_nil,id_a__U13 => true
      | id_nil,id_U85 => true
      | id_nil,id_a__length => true
      | id_nil,id_a__isNatIList => true
      | id_nil,id_U33 => true
      | id_nil,id_a__U11 => true
      | id_nil,id_U81 => true
      | id_nil,id_a__U92 => true
      | id_nil,id_a__U43 => true
      | id_nil,id_U23 => true
      | id_nil,id_a__isNatKind => true
      | id_nil,id_U93 => true
      | id_nil,id_U11 => true
      | id_nil,id_a__U71 => true
      | id_nil,id_U44 => true
      | id_nil,id_0 => true
      | id_nil,id_U61 => true
      | id_nil,id_a__U86 => true
      | id_nil,id_a__U41 => true
      | id_nil,id_U22 => true
      | id_nil,id_a__U21 => true
      | id_nil,id_U91 => true
      | id_nil,id_length => true
      | id_nil,id_a__U52 => true
      | id_nil,id_U42 => true
      | id_nil,id_a__U12 => true
      | id_nil,id_U83 => true
      | id_nil,id_a__U94 => true
      | id_nil,id_a__U45 => true
      | id_nil,id_U31 => true
      | id_nil,id_a__isNat => true
      | id_nil,id_isNatIListKind => true
      | id_nil,id_a__U82 => true
      | id_nil,id_U46 => true
      | id_nil,id_cons => true
      | id_nil,id_U52 => true
      | id_nil,id_a__U85 => true
      | id_nil,id_a__U33 => true
      | id_nil,id_U21 => true
      | id_nil,id_a__isNatList => true
      | id_nil,id_U86 => true
      | id_nil,id_mark => true
      | id_nil,id_a__U51 => true
      | id_nil,id_U41 => true
      | id_nil,id_tt => true
      | id_nil,id_U82 => true
      | id_nil,id_a__U93 => true
      | id_nil,id_a__U44 => true
      | id_nil,id_isNat => true
      | id_nil,id_a__U23 => true
      | id_nil,id_U94 => true
      | id_nil,id_U12 => true
      | id_nil,id_a__U81 => true
      | id_nil,id_U45 => true
      | id_nil,id_zeros => true
      | id_nil,id_U71 => true
      | id_nil,id_a__U91 => true
      | id_nil,id_a__U42 => true
      | id_nil,id_isNatKind => true
      | id_nil,id_a__U22 => true
      | id_nil,id_U92 => true
      | id_nil,id_nil => true
      | id_nil,id_a__U61 => false
      | id_nil,id_U43 => false
      | id_nil,id_a__isNatIListKind => false
      | id_nil,id_U84 => false
      | id_nil,id_s => false
      | id_nil,id_a__U46 => false
      | id_nil,id_U32 => false
      | id_nil,id_a__U31 => false
      | id_nil,id_U13 => false
      | id_nil,id_a__U83 => false
      | id_nil,id_isNatIList => false
      | id_a__U61,id_a__zeros => true
      | id_a__U61,id_U51 => true
      | id_a__U61,id_a__U84 => true
      | id_a__U61,id_a__U32 => true
      | id_a__U61,id_isNatList => true
      | id_a__U61,id_a__U13 => true
      | id_a__U61,id_U85 => true
      | id_a__U61,id_a__length => true
      | id_a__U61,id_a__isNatIList => true
      | id_a__U61,id_U33 => true
      | id_a__U61,id_a__U11 => true
      | id_a__U61,id_U81 => true
      | id_a__U61,id_a__U92 => true
      | id_a__U61,id_a__U43 => true
      | id_a__U61,id_U23 => true
      | id_a__U61,id_a__isNatKind => true
      | id_a__U61,id_U93 => true
      | id_a__U61,id_U11 => true
      | id_a__U61,id_a__U71 => true
      | id_a__U61,id_U44 => true
      | id_a__U61,id_0 => true
      | id_a__U61,id_U61 => true
      | id_a__U61,id_a__U86 => true
      | id_a__U61,id_a__U41 => true
      | id_a__U61,id_U22 => true
      | id_a__U61,id_a__U21 => true
      | id_a__U61,id_U91 => true
      | id_a__U61,id_length => true
      | id_a__U61,id_a__U52 => true
      | id_a__U61,id_U42 => true
      | id_a__U61,id_a__U12 => true
      | id_a__U61,id_U83 => true
      | id_a__U61,id_a__U94 => true
      | id_a__U61,id_a__U45 => true
      | id_a__U61,id_U31 => true
      | id_a__U61,id_a__isNat => true
      | id_a__U61,id_isNatIListKind => true
      | id_a__U61,id_a__U82 => true
      | id_a__U61,id_U46 => true
      | id_a__U61,id_cons => true
      | id_a__U61,id_U52 => true
      | id_a__U61,id_a__U85 => true
      | id_a__U61,id_a__U33 => true
      | id_a__U61,id_U21 => true
      | id_a__U61,id_a__isNatList => true
      | id_a__U61,id_U86 => true
      | id_a__U61,id_mark => true
      | id_a__U61,id_a__U51 => true
      | id_a__U61,id_U41 => true
      | id_a__U61,id_tt => true
      | id_a__U61,id_U82 => true
      | id_a__U61,id_a__U93 => true
      | id_a__U61,id_a__U44 => true
      | id_a__U61,id_isNat => true
      | id_a__U61,id_a__U23 => true
      | id_a__U61,id_U94 => true
      | id_a__U61,id_U12 => true
      | id_a__U61,id_a__U81 => true
      | id_a__U61,id_U45 => true
      | id_a__U61,id_zeros => true
      | id_a__U61,id_U71 => true
      | id_a__U61,id_a__U91 => true
      | id_a__U61,id_a__U42 => true
      | id_a__U61,id_isNatKind => true
      | id_a__U61,id_a__U22 => true
      | id_a__U61,id_U92 => true
      | id_a__U61,id_nil => true
      | id_a__U61,id_a__U61 => true
      | id_a__U61,id_U43 => false
      | id_a__U61,id_a__isNatIListKind => false
      | id_a__U61,id_U84 => false
      | id_a__U61,id_s => false
      | id_a__U61,id_a__U46 => false
      | id_a__U61,id_U32 => false
      | id_a__U61,id_a__U31 => false
      | id_a__U61,id_U13 => false
      | id_a__U61,id_a__U83 => false
      | id_a__U61,id_isNatIList => false
      | id_U43,id_a__zeros => true
      | id_U43,id_U51 => true
      | id_U43,id_a__U84 => true
      | id_U43,id_a__U32 => true
      | id_U43,id_isNatList => true
      | id_U43,id_a__U13 => true
      | id_U43,id_U85 => true
      | id_U43,id_a__length => true
      | id_U43,id_a__isNatIList => true
      | id_U43,id_U33 => true
      | id_U43,id_a__U11 => true
      | id_U43,id_U81 => true
      | id_U43,id_a__U92 => true
      | id_U43,id_a__U43 => true
      | id_U43,id_U23 => true
      | id_U43,id_a__isNatKind => true
      | id_U43,id_U93 => true
      | id_U43,id_U11 => true
      | id_U43,id_a__U71 => true
      | id_U43,id_U44 => true
      | id_U43,id_0 => true
      | id_U43,id_U61 => true
      | id_U43,id_a__U86 => true
      | id_U43,id_a__U41 => true
      | id_U43,id_U22 => true
      | id_U43,id_a__U21 => true
      | id_U43,id_U91 => true
      | id_U43,id_length => true
      | id_U43,id_a__U52 => true
      | id_U43,id_U42 => true
      | id_U43,id_a__U12 => true
      | id_U43,id_U83 => true
      | id_U43,id_a__U94 => true
      | id_U43,id_a__U45 => true
      | id_U43,id_U31 => true
      | id_U43,id_a__isNat => true
      | id_U43,id_isNatIListKind => true
      | id_U43,id_a__U82 => true
      | id_U43,id_U46 => true
      | id_U43,id_cons => true
      | id_U43,id_U52 => true
      | id_U43,id_a__U85 => true
      | id_U43,id_a__U33 => true
      | id_U43,id_U21 => true
      | id_U43,id_a__isNatList => true
      | id_U43,id_U86 => true
      | id_U43,id_mark => true
      | id_U43,id_a__U51 => true
      | id_U43,id_U41 => true
      | id_U43,id_tt => true
      | id_U43,id_U82 => true
      | id_U43,id_a__U93 => true
      | id_U43,id_a__U44 => true
      | id_U43,id_isNat => true
      | id_U43,id_a__U23 => true
      | id_U43,id_U94 => true
      | id_U43,id_U12 => true
      | id_U43,id_a__U81 => true
      | id_U43,id_U45 => true
      | id_U43,id_zeros => true
      | id_U43,id_U71 => true
      | id_U43,id_a__U91 => true
      | id_U43,id_a__U42 => true
      | id_U43,id_isNatKind => true
      | id_U43,id_a__U22 => true
      | id_U43,id_U92 => true
      | id_U43,id_nil => true
      | id_U43,id_a__U61 => true
      | id_U43,id_U43 => true
      | id_U43,id_a__isNatIListKind => false
      | id_U43,id_U84 => false
      | id_U43,id_s => false
      | id_U43,id_a__U46 => false
      | id_U43,id_U32 => false
      | id_U43,id_a__U31 => false
      | id_U43,id_U13 => false
      | id_U43,id_a__U83 => false
      | id_U43,id_isNatIList => false
      | id_a__isNatIListKind,id_a__zeros => true
      | id_a__isNatIListKind,id_U51 => true
      | id_a__isNatIListKind,id_a__U84 => true
      | id_a__isNatIListKind,id_a__U32 => true
      | id_a__isNatIListKind,id_isNatList => true
      | id_a__isNatIListKind,id_a__U13 => true
      | id_a__isNatIListKind,id_U85 => true
      | id_a__isNatIListKind,id_a__length => true
      | id_a__isNatIListKind,id_a__isNatIList => true
      | id_a__isNatIListKind,id_U33 => true
      | id_a__isNatIListKind,id_a__U11 => true
      | id_a__isNatIListKind,id_U81 => true
      | id_a__isNatIListKind,id_a__U92 => true
      | id_a__isNatIListKind,id_a__U43 => true
      | id_a__isNatIListKind,id_U23 => true
      | id_a__isNatIListKind,id_a__isNatKind => true
      | id_a__isNatIListKind,id_U93 => true
      | id_a__isNatIListKind,id_U11 => true
      | id_a__isNatIListKind,id_a__U71 => true
      | id_a__isNatIListKind,id_U44 => true
      | id_a__isNatIListKind,id_0 => true
      | id_a__isNatIListKind,id_U61 => true
      | id_a__isNatIListKind,id_a__U86 => true
      | id_a__isNatIListKind,id_a__U41 => true
      | id_a__isNatIListKind,id_U22 => true
      | id_a__isNatIListKind,id_a__U21 => true
      | id_a__isNatIListKind,id_U91 => true
      | id_a__isNatIListKind,id_length => true
      | id_a__isNatIListKind,id_a__U52 => true
      | id_a__isNatIListKind,id_U42 => true
      | id_a__isNatIListKind,id_a__U12 => true
      | id_a__isNatIListKind,id_U83 => true
      | id_a__isNatIListKind,id_a__U94 => true
      | id_a__isNatIListKind,id_a__U45 => true
      | id_a__isNatIListKind,id_U31 => true
      | id_a__isNatIListKind,id_a__isNat => true
      | id_a__isNatIListKind,id_isNatIListKind => true
      | id_a__isNatIListKind,id_a__U82 => true
      | id_a__isNatIListKind,id_U46 => true
      | id_a__isNatIListKind,id_cons => true
      | id_a__isNatIListKind,id_U52 => true
      | id_a__isNatIListKind,id_a__U85 => true
      | id_a__isNatIListKind,id_a__U33 => true
      | id_a__isNatIListKind,id_U21 => true
      | id_a__isNatIListKind,id_a__isNatList => true
      | id_a__isNatIListKind,id_U86 => true
      | id_a__isNatIListKind,id_mark => true
      | id_a__isNatIListKind,id_a__U51 => true
      | id_a__isNatIListKind,id_U41 => true
      | id_a__isNatIListKind,id_tt => true
      | id_a__isNatIListKind,id_U82 => true
      | id_a__isNatIListKind,id_a__U93 => true
      | id_a__isNatIListKind,id_a__U44 => true
      | id_a__isNatIListKind,id_isNat => true
      | id_a__isNatIListKind,id_a__U23 => true
      | id_a__isNatIListKind,id_U94 => true
      | id_a__isNatIListKind,id_U12 => true
      | id_a__isNatIListKind,id_a__U81 => true
      | id_a__isNatIListKind,id_U45 => true
      | id_a__isNatIListKind,id_zeros => true
      | id_a__isNatIListKind,id_U71 => true
      | id_a__isNatIListKind,id_a__U91 => true
      | id_a__isNatIListKind,id_a__U42 => true
      | id_a__isNatIListKind,id_isNatKind => true
      | id_a__isNatIListKind,id_a__U22 => true
      | id_a__isNatIListKind,id_U92 => true
      | id_a__isNatIListKind,id_nil => true
      | id_a__isNatIListKind,id_a__U61 => true
      | id_a__isNatIListKind,id_U43 => true
      | id_a__isNatIListKind,id_a__isNatIListKind => true
      | id_a__isNatIListKind,id_U84 => false
      | id_a__isNatIListKind,id_s => false
      | id_a__isNatIListKind,id_a__U46 => false
      | id_a__isNatIListKind,id_U32 => false
      | id_a__isNatIListKind,id_a__U31 => false
      | id_a__isNatIListKind,id_U13 => false
      | id_a__isNatIListKind,id_a__U83 => false
      | id_a__isNatIListKind,id_isNatIList => false
      | id_U84,id_a__zeros => true
      | id_U84,id_U51 => true
      | id_U84,id_a__U84 => true
      | id_U84,id_a__U32 => true
      | id_U84,id_isNatList => true
      | id_U84,id_a__U13 => true
      | id_U84,id_U85 => true
      | id_U84,id_a__length => true
      | id_U84,id_a__isNatIList => true
      | id_U84,id_U33 => true
      | id_U84,id_a__U11 => true
      | id_U84,id_U81 => true
      | id_U84,id_a__U92 => true
      | id_U84,id_a__U43 => true
      | id_U84,id_U23 => true
      | id_U84,id_a__isNatKind => true
      | id_U84,id_U93 => true
      | id_U84,id_U11 => true
      | id_U84,id_a__U71 => true
      | id_U84,id_U44 => true
      | id_U84,id_0 => true
      | id_U84,id_U61 => true
      | id_U84,id_a__U86 => true
      | id_U84,id_a__U41 => true
      | id_U84,id_U22 => true
      | id_U84,id_a__U21 => true
      | id_U84,id_U91 => true
      | id_U84,id_length => true
      | id_U84,id_a__U52 => true
      | id_U84,id_U42 => true
      | id_U84,id_a__U12 => true
      | id_U84,id_U83 => true
      | id_U84,id_a__U94 => true
      | id_U84,id_a__U45 => true
      | id_U84,id_U31 => true
      | id_U84,id_a__isNat => true
      | id_U84,id_isNatIListKind => true
      | id_U84,id_a__U82 => true
      | id_U84,id_U46 => true
      | id_U84,id_cons => true
      | id_U84,id_U52 => true
      | id_U84,id_a__U85 => true
      | id_U84,id_a__U33 => true
      | id_U84,id_U21 => true
      | id_U84,id_a__isNatList => true
      | id_U84,id_U86 => true
      | id_U84,id_mark => true
      | id_U84,id_a__U51 => true
      | id_U84,id_U41 => true
      | id_U84,id_tt => true
      | id_U84,id_U82 => true
      | id_U84,id_a__U93 => true
      | id_U84,id_a__U44 => true
      | id_U84,id_isNat => true
      | id_U84,id_a__U23 => true
      | id_U84,id_U94 => true
      | id_U84,id_U12 => true
      | id_U84,id_a__U81 => true
      | id_U84,id_U45 => true
      | id_U84,id_zeros => true
      | id_U84,id_U71 => true
      | id_U84,id_a__U91 => true
      | id_U84,id_a__U42 => true
      | id_U84,id_isNatKind => true
      | id_U84,id_a__U22 => true
      | id_U84,id_U92 => true
      | id_U84,id_nil => true
      | id_U84,id_a__U61 => true
      | id_U84,id_U43 => true
      | id_U84,id_a__isNatIListKind => true
      | id_U84,id_U84 => true
      | id_U84,id_s => false
      | id_U84,id_a__U46 => false
      | id_U84,id_U32 => false
      | id_U84,id_a__U31 => false
      | id_U84,id_U13 => false
      | id_U84,id_a__U83 => false
      | id_U84,id_isNatIList => false
      | id_s,id_a__zeros => true
      | id_s,id_U51 => true
      | id_s,id_a__U84 => true
      | id_s,id_a__U32 => true
      | id_s,id_isNatList => true
      | id_s,id_a__U13 => true
      | id_s,id_U85 => true
      | id_s,id_a__length => true
      | id_s,id_a__isNatIList => true
      | id_s,id_U33 => true
      | id_s,id_a__U11 => true
      | id_s,id_U81 => true
      | id_s,id_a__U92 => true
      | id_s,id_a__U43 => true
      | id_s,id_U23 => true
      | id_s,id_a__isNatKind => true
      | id_s,id_U93 => true
      | id_s,id_U11 => true
      | id_s,id_a__U71 => true
      | id_s,id_U44 => true
      | id_s,id_0 => true
      | id_s,id_U61 => true
      | id_s,id_a__U86 => true
      | id_s,id_a__U41 => true
      | id_s,id_U22 => true
      | id_s,id_a__U21 => true
      | id_s,id_U91 => true
      | id_s,id_length => true
      | id_s,id_a__U52 => true
      | id_s,id_U42 => true
      | id_s,id_a__U12 => true
      | id_s,id_U83 => true
      | id_s,id_a__U94 => true
      | id_s,id_a__U45 => true
      | id_s,id_U31 => true
      | id_s,id_a__isNat => true
      | id_s,id_isNatIListKind => true
      | id_s,id_a__U82 => true
      | id_s,id_U46 => true
      | id_s,id_cons => true
      | id_s,id_U52 => true
      | id_s,id_a__U85 => true
      | id_s,id_a__U33 => true
      | id_s,id_U21 => true
      | id_s,id_a__isNatList => true
      | id_s,id_U86 => true
      | id_s,id_mark => true
      | id_s,id_a__U51 => true
      | id_s,id_U41 => true
      | id_s,id_tt => true
      | id_s,id_U82 => true
      | id_s,id_a__U93 => true
      | id_s,id_a__U44 => true
      | id_s,id_isNat => true
      | id_s,id_a__U23 => true
      | id_s,id_U94 => true
      | id_s,id_U12 => true
      | id_s,id_a__U81 => true
      | id_s,id_U45 => true
      | id_s,id_zeros => true
      | id_s,id_U71 => true
      | id_s,id_a__U91 => true
      | id_s,id_a__U42 => true
      | id_s,id_isNatKind => true
      | id_s,id_a__U22 => true
      | id_s,id_U92 => true
      | id_s,id_nil => true
      | id_s,id_a__U61 => true
      | id_s,id_U43 => true
      | id_s,id_a__isNatIListKind => true
      | id_s,id_U84 => true
      | id_s,id_s => true
      | id_s,id_a__U46 => false
      | id_s,id_U32 => false
      | id_s,id_a__U31 => false
      | id_s,id_U13 => false
      | id_s,id_a__U83 => false
      | id_s,id_isNatIList => false
      | id_a__U46,id_a__zeros => true
      | id_a__U46,id_U51 => true
      | id_a__U46,id_a__U84 => true
      | id_a__U46,id_a__U32 => true
      | id_a__U46,id_isNatList => true
      | id_a__U46,id_a__U13 => true
      | id_a__U46,id_U85 => true
      | id_a__U46,id_a__length => true
      | id_a__U46,id_a__isNatIList => true
      | id_a__U46,id_U33 => true
      | id_a__U46,id_a__U11 => true
      | id_a__U46,id_U81 => true
      | id_a__U46,id_a__U92 => true
      | id_a__U46,id_a__U43 => true
      | id_a__U46,id_U23 => true
      | id_a__U46,id_a__isNatKind => true
      | id_a__U46,id_U93 => true
      | id_a__U46,id_U11 => true
      | id_a__U46,id_a__U71 => true
      | id_a__U46,id_U44 => true
      | id_a__U46,id_0 => true
      | id_a__U46,id_U61 => true
      | id_a__U46,id_a__U86 => true
      | id_a__U46,id_a__U41 => true
      | id_a__U46,id_U22 => true
      | id_a__U46,id_a__U21 => true
      | id_a__U46,id_U91 => true
      | id_a__U46,id_length => true
      | id_a__U46,id_a__U52 => true
      | id_a__U46,id_U42 => true
      | id_a__U46,id_a__U12 => true
      | id_a__U46,id_U83 => true
      | id_a__U46,id_a__U94 => true
      | id_a__U46,id_a__U45 => true
      | id_a__U46,id_U31 => true
      | id_a__U46,id_a__isNat => true
      | id_a__U46,id_isNatIListKind => true
      | id_a__U46,id_a__U82 => true
      | id_a__U46,id_U46 => true
      | id_a__U46,id_cons => true
      | id_a__U46,id_U52 => true
      | id_a__U46,id_a__U85 => true
      | id_a__U46,id_a__U33 => true
      | id_a__U46,id_U21 => true
      | id_a__U46,id_a__isNatList => true
      | id_a__U46,id_U86 => true
      | id_a__U46,id_mark => true
      | id_a__U46,id_a__U51 => true
      | id_a__U46,id_U41 => true
      | id_a__U46,id_tt => true
      | id_a__U46,id_U82 => true
      | id_a__U46,id_a__U93 => true
      | id_a__U46,id_a__U44 => true
      | id_a__U46,id_isNat => true
      | id_a__U46,id_a__U23 => true
      | id_a__U46,id_U94 => true
      | id_a__U46,id_U12 => true
      | id_a__U46,id_a__U81 => true
      | id_a__U46,id_U45 => true
      | id_a__U46,id_zeros => true
      | id_a__U46,id_U71 => true
      | id_a__U46,id_a__U91 => true
      | id_a__U46,id_a__U42 => true
      | id_a__U46,id_isNatKind => true
      | id_a__U46,id_a__U22 => true
      | id_a__U46,id_U92 => true
      | id_a__U46,id_nil => true
      | id_a__U46,id_a__U61 => true
      | id_a__U46,id_U43 => true
      | id_a__U46,id_a__isNatIListKind => true
      | id_a__U46,id_U84 => true
      | id_a__U46,id_s => true
      | id_a__U46,id_a__U46 => true
      | id_a__U46,id_U32 => false
      | id_a__U46,id_a__U31 => false
      | id_a__U46,id_U13 => false
      | id_a__U46,id_a__U83 => false
      | id_a__U46,id_isNatIList => false
      | id_U32,id_a__zeros => true
      | id_U32,id_U51 => true
      | id_U32,id_a__U84 => true
      | id_U32,id_a__U32 => true
      | id_U32,id_isNatList => true
      | id_U32,id_a__U13 => true
      | id_U32,id_U85 => true
      | id_U32,id_a__length => true
      | id_U32,id_a__isNatIList => true
      | id_U32,id_U33 => true
      | id_U32,id_a__U11 => true
      | id_U32,id_U81 => true
      | id_U32,id_a__U92 => true
      | id_U32,id_a__U43 => true
      | id_U32,id_U23 => true
      | id_U32,id_a__isNatKind => true
      | id_U32,id_U93 => true
      | id_U32,id_U11 => true
      | id_U32,id_a__U71 => true
      | id_U32,id_U44 => true
      | id_U32,id_0 => true
      | id_U32,id_U61 => true
      | id_U32,id_a__U86 => true
      | id_U32,id_a__U41 => true
      | id_U32,id_U22 => true
      | id_U32,id_a__U21 => true
      | id_U32,id_U91 => true
      | id_U32,id_length => true
      | id_U32,id_a__U52 => true
      | id_U32,id_U42 => true
      | id_U32,id_a__U12 => true
      | id_U32,id_U83 => true
      | id_U32,id_a__U94 => true
      | id_U32,id_a__U45 => true
      | id_U32,id_U31 => true
      | id_U32,id_a__isNat => true
      | id_U32,id_isNatIListKind => true
      | id_U32,id_a__U82 => true
      | id_U32,id_U46 => true
      | id_U32,id_cons => true
      | id_U32,id_U52 => true
      | id_U32,id_a__U85 => true
      | id_U32,id_a__U33 => true
      | id_U32,id_U21 => true
      | id_U32,id_a__isNatList => true
      | id_U32,id_U86 => true
      | id_U32,id_mark => true
      | id_U32,id_a__U51 => true
      | id_U32,id_U41 => true
      | id_U32,id_tt => true
      | id_U32,id_U82 => true
      | id_U32,id_a__U93 => true
      | id_U32,id_a__U44 => true
      | id_U32,id_isNat => true
      | id_U32,id_a__U23 => true
      | id_U32,id_U94 => true
      | id_U32,id_U12 => true
      | id_U32,id_a__U81 => true
      | id_U32,id_U45 => true
      | id_U32,id_zeros => true
      | id_U32,id_U71 => true
      | id_U32,id_a__U91 => true
      | id_U32,id_a__U42 => true
      | id_U32,id_isNatKind => true
      | id_U32,id_a__U22 => true
      | id_U32,id_U92 => true
      | id_U32,id_nil => true
      | id_U32,id_a__U61 => true
      | id_U32,id_U43 => true
      | id_U32,id_a__isNatIListKind => true
      | id_U32,id_U84 => true
      | id_U32,id_s => true
      | id_U32,id_a__U46 => true
      | id_U32,id_U32 => true
      | id_U32,id_a__U31 => false
      | id_U32,id_U13 => false
      | id_U32,id_a__U83 => false
      | id_U32,id_isNatIList => false
      | id_a__U31,id_a__zeros => true
      | id_a__U31,id_U51 => true
      | id_a__U31,id_a__U84 => true
      | id_a__U31,id_a__U32 => true
      | id_a__U31,id_isNatList => true
      | id_a__U31,id_a__U13 => true
      | id_a__U31,id_U85 => true
      | id_a__U31,id_a__length => true
      | id_a__U31,id_a__isNatIList => true
      | id_a__U31,id_U33 => true
      | id_a__U31,id_a__U11 => true
      | id_a__U31,id_U81 => true
      | id_a__U31,id_a__U92 => true
      | id_a__U31,id_a__U43 => true
      | id_a__U31,id_U23 => true
      | id_a__U31,id_a__isNatKind => true
      | id_a__U31,id_U93 => true
      | id_a__U31,id_U11 => true
      | id_a__U31,id_a__U71 => true
      | id_a__U31,id_U44 => true
      | id_a__U31,id_0 => true
      | id_a__U31,id_U61 => true
      | id_a__U31,id_a__U86 => true
      | id_a__U31,id_a__U41 => true
      | id_a__U31,id_U22 => true
      | id_a__U31,id_a__U21 => true
      | id_a__U31,id_U91 => true
      | id_a__U31,id_length => true
      | id_a__U31,id_a__U52 => true
      | id_a__U31,id_U42 => true
      | id_a__U31,id_a__U12 => true
      | id_a__U31,id_U83 => true
      | id_a__U31,id_a__U94 => true
      | id_a__U31,id_a__U45 => true
      | id_a__U31,id_U31 => true
      | id_a__U31,id_a__isNat => true
      | id_a__U31,id_isNatIListKind => true
      | id_a__U31,id_a__U82 => true
      | id_a__U31,id_U46 => true
      | id_a__U31,id_cons => true
      | id_a__U31,id_U52 => true
      | id_a__U31,id_a__U85 => true
      | id_a__U31,id_a__U33 => true
      | id_a__U31,id_U21 => true
      | id_a__U31,id_a__isNatList => true
      | id_a__U31,id_U86 => true
      | id_a__U31,id_mark => true
      | id_a__U31,id_a__U51 => true
      | id_a__U31,id_U41 => true
      | id_a__U31,id_tt => true
      | id_a__U31,id_U82 => true
      | id_a__U31,id_a__U93 => true
      | id_a__U31,id_a__U44 => true
      | id_a__U31,id_isNat => true
      | id_a__U31,id_a__U23 => true
      | id_a__U31,id_U94 => true
      | id_a__U31,id_U12 => true
      | id_a__U31,id_a__U81 => true
      | id_a__U31,id_U45 => true
      | id_a__U31,id_zeros => true
      | id_a__U31,id_U71 => true
      | id_a__U31,id_a__U91 => true
      | id_a__U31,id_a__U42 => true
      | id_a__U31,id_isNatKind => true
      | id_a__U31,id_a__U22 => true
      | id_a__U31,id_U92 => true
      | id_a__U31,id_nil => true
      | id_a__U31,id_a__U61 => true
      | id_a__U31,id_U43 => true
      | id_a__U31,id_a__isNatIListKind => true
      | id_a__U31,id_U84 => true
      | id_a__U31,id_s => true
      | id_a__U31,id_a__U46 => true
      | id_a__U31,id_U32 => true
      | id_a__U31,id_a__U31 => true
      | id_a__U31,id_U13 => false
      | id_a__U31,id_a__U83 => false
      | id_a__U31,id_isNatIList => false
      | id_U13,id_a__zeros => true
      | id_U13,id_U51 => true
      | id_U13,id_a__U84 => true
      | id_U13,id_a__U32 => true
      | id_U13,id_isNatList => true
      | id_U13,id_a__U13 => true
      | id_U13,id_U85 => true
      | id_U13,id_a__length => true
      | id_U13,id_a__isNatIList => true
      | id_U13,id_U33 => true
      | id_U13,id_a__U11 => true
      | id_U13,id_U81 => true
      | id_U13,id_a__U92 => true
      | id_U13,id_a__U43 => true
      | id_U13,id_U23 => true
      | id_U13,id_a__isNatKind => true
      | id_U13,id_U93 => true
      | id_U13,id_U11 => true
      | id_U13,id_a__U71 => true
      | id_U13,id_U44 => true
      | id_U13,id_0 => true
      | id_U13,id_U61 => true
      | id_U13,id_a__U86 => true
      | id_U13,id_a__U41 => true
      | id_U13,id_U22 => true
      | id_U13,id_a__U21 => true
      | id_U13,id_U91 => true
      | id_U13,id_length => true
      | id_U13,id_a__U52 => true
      | id_U13,id_U42 => true
      | id_U13,id_a__U12 => true
      | id_U13,id_U83 => true
      | id_U13,id_a__U94 => true
      | id_U13,id_a__U45 => true
      | id_U13,id_U31 => true
      | id_U13,id_a__isNat => true
      | id_U13,id_isNatIListKind => true
      | id_U13,id_a__U82 => true
      | id_U13,id_U46 => true
      | id_U13,id_cons => true
      | id_U13,id_U52 => true
      | id_U13,id_a__U85 => true
      | id_U13,id_a__U33 => true
      | id_U13,id_U21 => true
      | id_U13,id_a__isNatList => true
      | id_U13,id_U86 => true
      | id_U13,id_mark => true
      | id_U13,id_a__U51 => true
      | id_U13,id_U41 => true
      | id_U13,id_tt => true
      | id_U13,id_U82 => true
      | id_U13,id_a__U93 => true
      | id_U13,id_a__U44 => true
      | id_U13,id_isNat => true
      | id_U13,id_a__U23 => true
      | id_U13,id_U94 => true
      | id_U13,id_U12 => true
      | id_U13,id_a__U81 => true
      | id_U13,id_U45 => true
      | id_U13,id_zeros => true
      | id_U13,id_U71 => true
      | id_U13,id_a__U91 => true
      | id_U13,id_a__U42 => true
      | id_U13,id_isNatKind => true
      | id_U13,id_a__U22 => true
      | id_U13,id_U92 => true
      | id_U13,id_nil => true
      | id_U13,id_a__U61 => true
      | id_U13,id_U43 => true
      | id_U13,id_a__isNatIListKind => true
      | id_U13,id_U84 => true
      | id_U13,id_s => true
      | id_U13,id_a__U46 => true
      | id_U13,id_U32 => true
      | id_U13,id_a__U31 => true
      | id_U13,id_U13 => true
      | id_U13,id_a__U83 => false
      | id_U13,id_isNatIList => false
      | id_a__U83,id_a__zeros => true
      | id_a__U83,id_U51 => true
      | id_a__U83,id_a__U84 => true
      | id_a__U83,id_a__U32 => true
      | id_a__U83,id_isNatList => true
      | id_a__U83,id_a__U13 => true
      | id_a__U83,id_U85 => true
      | id_a__U83,id_a__length => true
      | id_a__U83,id_a__isNatIList => true
      | id_a__U83,id_U33 => true
      | id_a__U83,id_a__U11 => true
      | id_a__U83,id_U81 => true
      | id_a__U83,id_a__U92 => true
      | id_a__U83,id_a__U43 => true
      | id_a__U83,id_U23 => true
      | id_a__U83,id_a__isNatKind => true
      | id_a__U83,id_U93 => true
      | id_a__U83,id_U11 => true
      | id_a__U83,id_a__U71 => true
      | id_a__U83,id_U44 => true
      | id_a__U83,id_0 => true
      | id_a__U83,id_U61 => true
      | id_a__U83,id_a__U86 => true
      | id_a__U83,id_a__U41 => true
      | id_a__U83,id_U22 => true
      | id_a__U83,id_a__U21 => true
      | id_a__U83,id_U91 => true
      | id_a__U83,id_length => true
      | id_a__U83,id_a__U52 => true
      | id_a__U83,id_U42 => true
      | id_a__U83,id_a__U12 => true
      | id_a__U83,id_U83 => true
      | id_a__U83,id_a__U94 => true
      | id_a__U83,id_a__U45 => true
      | id_a__U83,id_U31 => true
      | id_a__U83,id_a__isNat => true
      | id_a__U83,id_isNatIListKind => true
      | id_a__U83,id_a__U82 => true
      | id_a__U83,id_U46 => true
      | id_a__U83,id_cons => true
      | id_a__U83,id_U52 => true
      | id_a__U83,id_a__U85 => true
      | id_a__U83,id_a__U33 => true
      | id_a__U83,id_U21 => true
      | id_a__U83,id_a__isNatList => true
      | id_a__U83,id_U86 => true
      | id_a__U83,id_mark => true
      | id_a__U83,id_a__U51 => true
      | id_a__U83,id_U41 => true
      | id_a__U83,id_tt => true
      | id_a__U83,id_U82 => true
      | id_a__U83,id_a__U93 => true
      | id_a__U83,id_a__U44 => true
      | id_a__U83,id_isNat => true
      | id_a__U83,id_a__U23 => true
      | id_a__U83,id_U94 => true
      | id_a__U83,id_U12 => true
      | id_a__U83,id_a__U81 => true
      | id_a__U83,id_U45 => true
      | id_a__U83,id_zeros => true
      | id_a__U83,id_U71 => true
      | id_a__U83,id_a__U91 => true
      | id_a__U83,id_a__U42 => true
      | id_a__U83,id_isNatKind => true
      | id_a__U83,id_a__U22 => true
      | id_a__U83,id_U92 => true
      | id_a__U83,id_nil => true
      | id_a__U83,id_a__U61 => true
      | id_a__U83,id_U43 => true
      | id_a__U83,id_a__isNatIListKind => true
      | id_a__U83,id_U84 => true
      | id_a__U83,id_s => true
      | id_a__U83,id_a__U46 => true
      | id_a__U83,id_U32 => true
      | id_a__U83,id_a__U31 => true
      | id_a__U83,id_U13 => true
      | id_a__U83,id_a__U83 => true
      | id_a__U83,id_isNatIList => false
      | id_isNatIList,id_a__zeros => true
      | id_isNatIList,id_U51 => true
      | id_isNatIList,id_a__U84 => true
      | id_isNatIList,id_a__U32 => true
      | id_isNatIList,id_isNatList => true
      | id_isNatIList,id_a__U13 => true
      | id_isNatIList,id_U85 => true
      | id_isNatIList,id_a__length => true
      | id_isNatIList,id_a__isNatIList => true
      | id_isNatIList,id_U33 => true
      | id_isNatIList,id_a__U11 => true
      | id_isNatIList,id_U81 => true
      | id_isNatIList,id_a__U92 => true
      | id_isNatIList,id_a__U43 => true
      | id_isNatIList,id_U23 => true
      | id_isNatIList,id_a__isNatKind => true
      | id_isNatIList,id_U93 => true
      | id_isNatIList,id_U11 => true
      | id_isNatIList,id_a__U71 => true
      | id_isNatIList,id_U44 => true
      | id_isNatIList,id_0 => true
      | id_isNatIList,id_U61 => true
      | id_isNatIList,id_a__U86 => true
      | id_isNatIList,id_a__U41 => true
      | id_isNatIList,id_U22 => true
      | id_isNatIList,id_a__U21 => true
      | id_isNatIList,id_U91 => true
      | id_isNatIList,id_length => true
      | id_isNatIList,id_a__U52 => true
      | id_isNatIList,id_U42 => true
      | id_isNatIList,id_a__U12 => true
      | id_isNatIList,id_U83 => true
      | id_isNatIList,id_a__U94 => true
      | id_isNatIList,id_a__U45 => true
      | id_isNatIList,id_U31 => true
      | id_isNatIList,id_a__isNat => true
      | id_isNatIList,id_isNatIListKind => true
      | id_isNatIList,id_a__U82 => true
      | id_isNatIList,id_U46 => true
      | id_isNatIList,id_cons => true
      | id_isNatIList,id_U52 => true
      | id_isNatIList,id_a__U85 => true
      | id_isNatIList,id_a__U33 => true
      | id_isNatIList,id_U21 => true
      | id_isNatIList,id_a__isNatList => true
      | id_isNatIList,id_U86 => true
      | id_isNatIList,id_mark => true
      | id_isNatIList,id_a__U51 => true
      | id_isNatIList,id_U41 => true
      | id_isNatIList,id_tt => true
      | id_isNatIList,id_U82 => true
      | id_isNatIList,id_a__U93 => true
      | id_isNatIList,id_a__U44 => true
      | id_isNatIList,id_isNat => true
      | id_isNatIList,id_a__U23 => true
      | id_isNatIList,id_U94 => true
      | id_isNatIList,id_U12 => true
      | id_isNatIList,id_a__U81 => true
      | id_isNatIList,id_U45 => true
      | id_isNatIList,id_zeros => true
      | id_isNatIList,id_U71 => true
      | id_isNatIList,id_a__U91 => true
      | id_isNatIList,id_a__U42 => true
      | id_isNatIList,id_isNatKind => true
      | id_isNatIList,id_a__U22 => true
      | id_isNatIList,id_U92 => true
      | id_isNatIList,id_nil => true
      | id_isNatIList,id_a__U61 => true
      | id_isNatIList,id_U43 => true
      | id_isNatIList,id_a__isNatIListKind => true
      | id_isNatIList,id_U84 => true
      | id_isNatIList,id_s => true
      | id_isNatIList,id_a__U46 => true
      | id_isNatIList,id_U32 => true
      | id_isNatIList,id_a__U31 => true
      | id_isNatIList,id_U13 => true
      | id_isNatIList,id_a__U83 => true
      | id_isNatIList,id_isNatIList => true
      end.
  
  
  Module Symb.
   Definition A  := symb.
   
   Definition eq_A  := @eq A.
   
   
   Definition eq_proof : equivalence A eq_A.
   Proof.
     constructor.
     red ;reflexivity .
     red ;intros ;transitivity y ;assumption.
     red ;intros ;symmetry ;assumption.
   Defined.
   
   
   Add Relation A eq_A 
  reflexivity proved by (@equiv_refl _ _ eq_proof)
    symmetry proved by (@equiv_sym _ _ eq_proof)
      transitivity proved by (@equiv_trans _ _ eq_proof) as EQA
.
   
   Definition eq_bool  := symb_eq_bool.
   
   Definition eq_bool_ok  := symb_eq_bool_ok.
  End Symb.
  
  Export Symb.
 End F.
 
 Module Alg := term.Make'(F)(term_extension.IntVars).
 
 Module Alg_ext := term_extension.Make(Alg).
 
 Module EQT := equational_theory.Make(Alg).
 
 Module EQT_ext := equational_extension.Make(EQT).
End algebra.

Module R_xml_0_deep_rew.
 Inductive R_xml_0_rules  :
  algebra.Alg.term ->algebra.Alg.term ->Prop := 
    (* a__zeros -> cons(0,zeros) *)
   | R_xml_0_rule_0 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term 
                   algebra.F.id_0 nil)::(algebra.Alg.Term algebra.F.id_zeros 
                   nil)::nil)) (algebra.Alg.Term algebra.F.id_a__zeros nil)
   
    (* a__U11(tt,V1_) -> a__U12(a__isNatIListKind(V1_),V1_) *)
   | R_xml_0_rule_1 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U12 ((algebra.Alg.Term 
                   algebra.F.id_a__isNatIListKind 
                   ((algebra.Alg.Var 1)::nil))::(algebra.Alg.Var 1)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U11 ((algebra.Alg.Term 
      algebra.F.id_tt nil)::(algebra.Alg.Var 1)::nil))
    (* a__U12(tt,V1_) -> a__U13(a__isNatList(V1_)) *)
   | R_xml_0_rule_2 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U13 ((algebra.Alg.Term 
                   algebra.F.id_a__isNatList 
                   ((algebra.Alg.Var 1)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U12 ((algebra.Alg.Term 
      algebra.F.id_tt nil)::(algebra.Alg.Var 1)::nil))
    (* a__U13(tt) -> tt *)
   | R_xml_0_rule_3 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_tt nil) 
     (algebra.Alg.Term algebra.F.id_a__U13 ((algebra.Alg.Term 
      algebra.F.id_tt nil)::nil))
    (* a__U21(tt,V1_) -> a__U22(a__isNatKind(V1_),V1_) *)
   | R_xml_0_rule_4 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U22 ((algebra.Alg.Term 
                   algebra.F.id_a__isNatKind ((algebra.Alg.Var 1)::nil))::
                   (algebra.Alg.Var 1)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U21 ((algebra.Alg.Term 
      algebra.F.id_tt nil)::(algebra.Alg.Var 1)::nil))
    (* a__U22(tt,V1_) -> a__U23(a__isNat(V1_)) *)
   | R_xml_0_rule_5 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U23 ((algebra.Alg.Term 
                   algebra.F.id_a__isNat ((algebra.Alg.Var 1)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U22 ((algebra.Alg.Term 
      algebra.F.id_tt nil)::(algebra.Alg.Var 1)::nil))
    (* a__U23(tt) -> tt *)
   | R_xml_0_rule_6 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_tt nil) 
     (algebra.Alg.Term algebra.F.id_a__U23 ((algebra.Alg.Term 
      algebra.F.id_tt nil)::nil))
    (* a__U31(tt,V_) -> a__U32(a__isNatIListKind(V_),V_) *)
   | R_xml_0_rule_7 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U32 ((algebra.Alg.Term 
                   algebra.F.id_a__isNatIListKind 
                   ((algebra.Alg.Var 2)::nil))::(algebra.Alg.Var 2)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U31 ((algebra.Alg.Term 
      algebra.F.id_tt nil)::(algebra.Alg.Var 2)::nil))
    (* a__U32(tt,V_) -> a__U33(a__isNatList(V_)) *)
   | R_xml_0_rule_8 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U33 ((algebra.Alg.Term 
                   algebra.F.id_a__isNatList 
                   ((algebra.Alg.Var 2)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U32 ((algebra.Alg.Term 
      algebra.F.id_tt nil)::(algebra.Alg.Var 2)::nil))
    (* a__U33(tt) -> tt *)
   | R_xml_0_rule_9 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_tt nil) 
     (algebra.Alg.Term algebra.F.id_a__U33 ((algebra.Alg.Term 
      algebra.F.id_tt nil)::nil))
   
    (* a__U41(tt,V1_,V2_) -> a__U42(a__isNatKind(V1_),V1_,V2_) *)
   | R_xml_0_rule_10 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U42 ((algebra.Alg.Term 
                   algebra.F.id_a__isNatKind ((algebra.Alg.Var 1)::nil))::
                   (algebra.Alg.Var 1)::(algebra.Alg.Var 3)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U41 ((algebra.Alg.Term 
      algebra.F.id_tt nil)::(algebra.Alg.Var 1)::(algebra.Alg.Var 3)::nil))
   
    (* a__U42(tt,V1_,V2_) -> a__U43(a__isNatIListKind(V2_),V1_,V2_) *)
   | R_xml_0_rule_11 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U43 ((algebra.Alg.Term 
                   algebra.F.id_a__isNatIListKind 
                   ((algebra.Alg.Var 3)::nil))::(algebra.Alg.Var 1)::
                   (algebra.Alg.Var 3)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U42 ((algebra.Alg.Term 
      algebra.F.id_tt nil)::(algebra.Alg.Var 1)::(algebra.Alg.Var 3)::nil))
   
    (* a__U43(tt,V1_,V2_) -> a__U44(a__isNatIListKind(V2_),V1_,V2_) *)
   | R_xml_0_rule_12 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U44 ((algebra.Alg.Term 
                   algebra.F.id_a__isNatIListKind 
                   ((algebra.Alg.Var 3)::nil))::(algebra.Alg.Var 1)::
                   (algebra.Alg.Var 3)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U43 ((algebra.Alg.Term 
      algebra.F.id_tt nil)::(algebra.Alg.Var 1)::(algebra.Alg.Var 3)::nil))
    (* a__U44(tt,V1_,V2_) -> a__U45(a__isNat(V1_),V2_) *)
   | R_xml_0_rule_13 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U45 ((algebra.Alg.Term 
                   algebra.F.id_a__isNat ((algebra.Alg.Var 1)::nil))::
                   (algebra.Alg.Var 3)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U44 ((algebra.Alg.Term 
      algebra.F.id_tt nil)::(algebra.Alg.Var 1)::(algebra.Alg.Var 3)::nil))
    (* a__U45(tt,V2_) -> a__U46(a__isNatIList(V2_)) *)
   | R_xml_0_rule_14 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U46 ((algebra.Alg.Term 
                   algebra.F.id_a__isNatIList 
                   ((algebra.Alg.Var 3)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U45 ((algebra.Alg.Term 
      algebra.F.id_tt nil)::(algebra.Alg.Var 3)::nil))
    (* a__U46(tt) -> tt *)
   | R_xml_0_rule_15 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_tt nil) 
     (algebra.Alg.Term algebra.F.id_a__U46 ((algebra.Alg.Term 
      algebra.F.id_tt nil)::nil))
    (* a__U51(tt,V2_) -> a__U52(a__isNatIListKind(V2_)) *)
   | R_xml_0_rule_16 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U52 ((algebra.Alg.Term 
                   algebra.F.id_a__isNatIListKind 
                   ((algebra.Alg.Var 3)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U51 ((algebra.Alg.Term 
      algebra.F.id_tt nil)::(algebra.Alg.Var 3)::nil))
    (* a__U52(tt) -> tt *)
   | R_xml_0_rule_17 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_tt nil) 
     (algebra.Alg.Term algebra.F.id_a__U52 ((algebra.Alg.Term 
      algebra.F.id_tt nil)::nil))
    (* a__U61(tt) -> tt *)
   | R_xml_0_rule_18 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_tt nil) 
     (algebra.Alg.Term algebra.F.id_a__U61 ((algebra.Alg.Term 
      algebra.F.id_tt nil)::nil))
    (* a__U71(tt) -> tt *)
   | R_xml_0_rule_19 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_tt nil) 
     (algebra.Alg.Term algebra.F.id_a__U71 ((algebra.Alg.Term 
      algebra.F.id_tt nil)::nil))
   
    (* a__U81(tt,V1_,V2_) -> a__U82(a__isNatKind(V1_),V1_,V2_) *)
   | R_xml_0_rule_20 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U82 ((algebra.Alg.Term 
                   algebra.F.id_a__isNatKind ((algebra.Alg.Var 1)::nil))::
                   (algebra.Alg.Var 1)::(algebra.Alg.Var 3)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U81 ((algebra.Alg.Term 
      algebra.F.id_tt nil)::(algebra.Alg.Var 1)::(algebra.Alg.Var 3)::nil))
   
    (* a__U82(tt,V1_,V2_) -> a__U83(a__isNatIListKind(V2_),V1_,V2_) *)
   | R_xml_0_rule_21 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U83 ((algebra.Alg.Term 
                   algebra.F.id_a__isNatIListKind 
                   ((algebra.Alg.Var 3)::nil))::(algebra.Alg.Var 1)::
                   (algebra.Alg.Var 3)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U82 ((algebra.Alg.Term 
      algebra.F.id_tt nil)::(algebra.Alg.Var 1)::(algebra.Alg.Var 3)::nil))
   
    (* a__U83(tt,V1_,V2_) -> a__U84(a__isNatIListKind(V2_),V1_,V2_) *)
   | R_xml_0_rule_22 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U84 ((algebra.Alg.Term 
                   algebra.F.id_a__isNatIListKind 
                   ((algebra.Alg.Var 3)::nil))::(algebra.Alg.Var 1)::
                   (algebra.Alg.Var 3)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U83 ((algebra.Alg.Term 
      algebra.F.id_tt nil)::(algebra.Alg.Var 1)::(algebra.Alg.Var 3)::nil))
    (* a__U84(tt,V1_,V2_) -> a__U85(a__isNat(V1_),V2_) *)
   | R_xml_0_rule_23 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U85 ((algebra.Alg.Term 
                   algebra.F.id_a__isNat ((algebra.Alg.Var 1)::nil))::
                   (algebra.Alg.Var 3)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U84 ((algebra.Alg.Term 
      algebra.F.id_tt nil)::(algebra.Alg.Var 1)::(algebra.Alg.Var 3)::nil))
    (* a__U85(tt,V2_) -> a__U86(a__isNatList(V2_)) *)
   | R_xml_0_rule_24 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U86 ((algebra.Alg.Term 
                   algebra.F.id_a__isNatList 
                   ((algebra.Alg.Var 3)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U85 ((algebra.Alg.Term 
      algebra.F.id_tt nil)::(algebra.Alg.Var 3)::nil))
    (* a__U86(tt) -> tt *)
   | R_xml_0_rule_25 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_tt nil) 
     (algebra.Alg.Term algebra.F.id_a__U86 ((algebra.Alg.Term 
      algebra.F.id_tt nil)::nil))
   
    (* a__U91(tt,L_,N_) -> a__U92(a__isNatIListKind(L_),L_,N_) *)
   | R_xml_0_rule_26 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U92 ((algebra.Alg.Term 
                   algebra.F.id_a__isNatIListKind 
                   ((algebra.Alg.Var 4)::nil))::(algebra.Alg.Var 4)::
                   (algebra.Alg.Var 5)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U91 ((algebra.Alg.Term 
      algebra.F.id_tt nil)::(algebra.Alg.Var 4)::(algebra.Alg.Var 5)::nil))
    (* a__U92(tt,L_,N_) -> a__U93(a__isNat(N_),L_,N_) *)
   | R_xml_0_rule_27 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U93 ((algebra.Alg.Term 
                   algebra.F.id_a__isNat ((algebra.Alg.Var 5)::nil))::
                   (algebra.Alg.Var 4)::(algebra.Alg.Var 5)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U92 ((algebra.Alg.Term 
      algebra.F.id_tt nil)::(algebra.Alg.Var 4)::(algebra.Alg.Var 5)::nil))
    (* a__U93(tt,L_,N_) -> a__U94(a__isNatKind(N_),L_) *)
   | R_xml_0_rule_28 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U94 ((algebra.Alg.Term 
                   algebra.F.id_a__isNatKind ((algebra.Alg.Var 5)::nil))::
                   (algebra.Alg.Var 4)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U93 ((algebra.Alg.Term 
      algebra.F.id_tt nil)::(algebra.Alg.Var 4)::(algebra.Alg.Var 5)::nil))
    (* a__U94(tt,L_) -> s(a__length(mark(L_))) *)
   | R_xml_0_rule_29 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_s ((algebra.Alg.Term 
                   algebra.F.id_a__length ((algebra.Alg.Term 
                   algebra.F.id_mark 
                   ((algebra.Alg.Var 4)::nil))::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U94 ((algebra.Alg.Term 
      algebra.F.id_tt nil)::(algebra.Alg.Var 4)::nil))
    (* a__isNat(0) -> tt *)
   | R_xml_0_rule_30 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_tt nil) 
     (algebra.Alg.Term algebra.F.id_a__isNat ((algebra.Alg.Term 
      algebra.F.id_0 nil)::nil))
   
    (* a__isNat(length(V1_)) -> a__U11(a__isNatIListKind(V1_),V1_) *)
   | R_xml_0_rule_31 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U11 ((algebra.Alg.Term 
                   algebra.F.id_a__isNatIListKind 
                   ((algebra.Alg.Var 1)::nil))::(algebra.Alg.Var 1)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__isNat ((algebra.Alg.Term 
      algebra.F.id_length ((algebra.Alg.Var 1)::nil))::nil))
   
    (* a__isNat(s(V1_)) -> a__U21(a__isNatKind(V1_),V1_) *)
   | R_xml_0_rule_32 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U21 ((algebra.Alg.Term 
                   algebra.F.id_a__isNatKind ((algebra.Alg.Var 1)::nil))::
                   (algebra.Alg.Var 1)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__isNat ((algebra.Alg.Term 
      algebra.F.id_s ((algebra.Alg.Var 1)::nil))::nil))
   
    (* a__isNatIList(V_) -> a__U31(a__isNatIListKind(V_),V_) *)
   | R_xml_0_rule_33 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U31 ((algebra.Alg.Term 
                   algebra.F.id_a__isNatIListKind 
                   ((algebra.Alg.Var 2)::nil))::(algebra.Alg.Var 2)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__isNatIList ((algebra.Alg.Var 2)::nil))
    (* a__isNatIList(zeros) -> tt *)
   | R_xml_0_rule_34 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_tt nil) 
     (algebra.Alg.Term algebra.F.id_a__isNatIList ((algebra.Alg.Term 
      algebra.F.id_zeros nil)::nil))
   
    (* a__isNatIList(cons(V1_,V2_)) -> a__U41(a__isNatKind(V1_),V1_,V2_) *)
   | R_xml_0_rule_35 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U41 ((algebra.Alg.Term 
                   algebra.F.id_a__isNatKind ((algebra.Alg.Var 1)::nil))::
                   (algebra.Alg.Var 1)::(algebra.Alg.Var 3)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__isNatIList ((algebra.Alg.Term 
      algebra.F.id_cons ((algebra.Alg.Var 1)::
      (algebra.Alg.Var 3)::nil))::nil))
    (* a__isNatIListKind(nil) -> tt *)
   | R_xml_0_rule_36 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_tt nil) 
     (algebra.Alg.Term algebra.F.id_a__isNatIListKind ((algebra.Alg.Term 
      algebra.F.id_nil nil)::nil))
    (* a__isNatIListKind(zeros) -> tt *)
   | R_xml_0_rule_37 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_tt nil) 
     (algebra.Alg.Term algebra.F.id_a__isNatIListKind ((algebra.Alg.Term 
      algebra.F.id_zeros nil)::nil))
   
    (* a__isNatIListKind(cons(V1_,V2_)) -> a__U51(a__isNatKind(V1_),V2_) *)
   | R_xml_0_rule_38 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U51 ((algebra.Alg.Term 
                   algebra.F.id_a__isNatKind ((algebra.Alg.Var 1)::nil))::
                   (algebra.Alg.Var 3)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__isNatIListKind ((algebra.Alg.Term 
      algebra.F.id_cons ((algebra.Alg.Var 1)::
      (algebra.Alg.Var 3)::nil))::nil))
    (* a__isNatKind(0) -> tt *)
   | R_xml_0_rule_39 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_tt nil) 
     (algebra.Alg.Term algebra.F.id_a__isNatKind ((algebra.Alg.Term 
      algebra.F.id_0 nil)::nil))
   
    (* a__isNatKind(length(V1_)) -> a__U61(a__isNatIListKind(V1_)) *)
   | R_xml_0_rule_40 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U61 ((algebra.Alg.Term 
                   algebra.F.id_a__isNatIListKind 
                   ((algebra.Alg.Var 1)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_a__isNatKind ((algebra.Alg.Term 
      algebra.F.id_length ((algebra.Alg.Var 1)::nil))::nil))
   
    (* a__isNatKind(s(V1_)) -> a__U71(a__isNatKind(V1_)) *)
   | R_xml_0_rule_41 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U71 ((algebra.Alg.Term 
                   algebra.F.id_a__isNatKind 
                   ((algebra.Alg.Var 1)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_a__isNatKind ((algebra.Alg.Term 
      algebra.F.id_s ((algebra.Alg.Var 1)::nil))::nil))
    (* a__isNatList(nil) -> tt *)
   | R_xml_0_rule_42 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_tt nil) 
     (algebra.Alg.Term algebra.F.id_a__isNatList ((algebra.Alg.Term 
      algebra.F.id_nil nil)::nil))
   
    (* a__isNatList(cons(V1_,V2_)) -> a__U81(a__isNatKind(V1_),V1_,V2_) *)
   | R_xml_0_rule_43 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U81 ((algebra.Alg.Term 
                   algebra.F.id_a__isNatKind ((algebra.Alg.Var 1)::nil))::
                   (algebra.Alg.Var 1)::(algebra.Alg.Var 3)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__isNatList ((algebra.Alg.Term 
      algebra.F.id_cons ((algebra.Alg.Var 1)::
      (algebra.Alg.Var 3)::nil))::nil))
    (* a__length(nil) -> 0 *)
   | R_xml_0_rule_44 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_0 nil) 
     (algebra.Alg.Term algebra.F.id_a__length ((algebra.Alg.Term 
      algebra.F.id_nil nil)::nil))
   
    (* a__length(cons(N_,L_)) -> a__U91(a__isNatList(L_),L_,N_) *)
   | R_xml_0_rule_45 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U91 ((algebra.Alg.Term 
                   algebra.F.id_a__isNatList ((algebra.Alg.Var 4)::nil))::
                   (algebra.Alg.Var 4)::(algebra.Alg.Var 5)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__length ((algebra.Alg.Term 
      algebra.F.id_cons ((algebra.Alg.Var 5)::
      (algebra.Alg.Var 4)::nil))::nil))
    (* mark(zeros) -> a__zeros *)
   | R_xml_0_rule_46 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__zeros nil) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term 
      algebra.F.id_zeros nil)::nil))
    (* mark(U11(X1_,X2_)) -> a__U11(mark(X1_),X2_) *)
   | R_xml_0_rule_47 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U11 ((algebra.Alg.Term 
                   algebra.F.id_mark ((algebra.Alg.Var 6)::nil))::
                   (algebra.Alg.Var 7)::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U11 
      ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::nil))::nil))
    (* mark(U12(X1_,X2_)) -> a__U12(mark(X1_),X2_) *)
   | R_xml_0_rule_48 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U12 ((algebra.Alg.Term 
                   algebra.F.id_mark ((algebra.Alg.Var 6)::nil))::
                   (algebra.Alg.Var 7)::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U12 
      ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::nil))::nil))
   
    (* mark(isNatIListKind(X_)) -> a__isNatIListKind(X_) *)
   | R_xml_0_rule_49 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
                   ((algebra.Alg.Var 8)::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term 
      algebra.F.id_isNatIListKind ((algebra.Alg.Var 8)::nil))::nil))
    (* mark(U13(X_)) -> a__U13(mark(X_)) *)
   | R_xml_0_rule_50 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U13 ((algebra.Alg.Term 
                   algebra.F.id_mark ((algebra.Alg.Var 8)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U13 
      ((algebra.Alg.Var 8)::nil))::nil))
    (* mark(isNatList(X_)) -> a__isNatList(X_) *)
   | R_xml_0_rule_51 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__isNatList 
                   ((algebra.Alg.Var 8)::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term 
      algebra.F.id_isNatList ((algebra.Alg.Var 8)::nil))::nil))
    (* mark(U21(X1_,X2_)) -> a__U21(mark(X1_),X2_) *)
   | R_xml_0_rule_52 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U21 ((algebra.Alg.Term 
                   algebra.F.id_mark ((algebra.Alg.Var 6)::nil))::
                   (algebra.Alg.Var 7)::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U21 
      ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::nil))::nil))
    (* mark(U22(X1_,X2_)) -> a__U22(mark(X1_),X2_) *)
   | R_xml_0_rule_53 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U22 ((algebra.Alg.Term 
                   algebra.F.id_mark ((algebra.Alg.Var 6)::nil))::
                   (algebra.Alg.Var 7)::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U22 
      ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::nil))::nil))
    (* mark(isNatKind(X_)) -> a__isNatKind(X_) *)
   | R_xml_0_rule_54 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__isNatKind 
                   ((algebra.Alg.Var 8)::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term 
      algebra.F.id_isNatKind ((algebra.Alg.Var 8)::nil))::nil))
    (* mark(U23(X_)) -> a__U23(mark(X_)) *)
   | R_xml_0_rule_55 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U23 ((algebra.Alg.Term 
                   algebra.F.id_mark ((algebra.Alg.Var 8)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U23 
      ((algebra.Alg.Var 8)::nil))::nil))
    (* mark(isNat(X_)) -> a__isNat(X_) *)
   | R_xml_0_rule_56 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__isNat 
                   ((algebra.Alg.Var 8)::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term 
      algebra.F.id_isNat ((algebra.Alg.Var 8)::nil))::nil))
    (* mark(U31(X1_,X2_)) -> a__U31(mark(X1_),X2_) *)
   | R_xml_0_rule_57 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U31 ((algebra.Alg.Term 
                   algebra.F.id_mark ((algebra.Alg.Var 6)::nil))::
                   (algebra.Alg.Var 7)::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U31 
      ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::nil))::nil))
    (* mark(U32(X1_,X2_)) -> a__U32(mark(X1_),X2_) *)
   | R_xml_0_rule_58 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U32 ((algebra.Alg.Term 
                   algebra.F.id_mark ((algebra.Alg.Var 6)::nil))::
                   (algebra.Alg.Var 7)::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U32 
      ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::nil))::nil))
    (* mark(U33(X_)) -> a__U33(mark(X_)) *)
   | R_xml_0_rule_59 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U33 ((algebra.Alg.Term 
                   algebra.F.id_mark ((algebra.Alg.Var 8)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U33 
      ((algebra.Alg.Var 8)::nil))::nil))
   
    (* mark(U41(X1_,X2_,X3_)) -> a__U41(mark(X1_),X2_,X3_) *)
   | R_xml_0_rule_60 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U41 ((algebra.Alg.Term 
                   algebra.F.id_mark ((algebra.Alg.Var 6)::nil))::
                   (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U41 
      ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::
      (algebra.Alg.Var 9)::nil))::nil))
   
    (* mark(U42(X1_,X2_,X3_)) -> a__U42(mark(X1_),X2_,X3_) *)
   | R_xml_0_rule_61 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U42 ((algebra.Alg.Term 
                   algebra.F.id_mark ((algebra.Alg.Var 6)::nil))::
                   (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U42 
      ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::
      (algebra.Alg.Var 9)::nil))::nil))
   
    (* mark(U43(X1_,X2_,X3_)) -> a__U43(mark(X1_),X2_,X3_) *)
   | R_xml_0_rule_62 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U43 ((algebra.Alg.Term 
                   algebra.F.id_mark ((algebra.Alg.Var 6)::nil))::
                   (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U43 
      ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::
      (algebra.Alg.Var 9)::nil))::nil))
   
    (* mark(U44(X1_,X2_,X3_)) -> a__U44(mark(X1_),X2_,X3_) *)
   | R_xml_0_rule_63 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U44 ((algebra.Alg.Term 
                   algebra.F.id_mark ((algebra.Alg.Var 6)::nil))::
                   (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U44 
      ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::
      (algebra.Alg.Var 9)::nil))::nil))
    (* mark(U45(X1_,X2_)) -> a__U45(mark(X1_),X2_) *)
   | R_xml_0_rule_64 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U45 ((algebra.Alg.Term 
                   algebra.F.id_mark ((algebra.Alg.Var 6)::nil))::
                   (algebra.Alg.Var 7)::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U45 
      ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::nil))::nil))
    (* mark(U46(X_)) -> a__U46(mark(X_)) *)
   | R_xml_0_rule_65 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U46 ((algebra.Alg.Term 
                   algebra.F.id_mark ((algebra.Alg.Var 8)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U46 
      ((algebra.Alg.Var 8)::nil))::nil))
    (* mark(isNatIList(X_)) -> a__isNatIList(X_) *)
   | R_xml_0_rule_66 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__isNatIList 
                   ((algebra.Alg.Var 8)::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term 
      algebra.F.id_isNatIList ((algebra.Alg.Var 8)::nil))::nil))
    (* mark(U51(X1_,X2_)) -> a__U51(mark(X1_),X2_) *)
   | R_xml_0_rule_67 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U51 ((algebra.Alg.Term 
                   algebra.F.id_mark ((algebra.Alg.Var 6)::nil))::
                   (algebra.Alg.Var 7)::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U51 
      ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::nil))::nil))
    (* mark(U52(X_)) -> a__U52(mark(X_)) *)
   | R_xml_0_rule_68 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U52 ((algebra.Alg.Term 
                   algebra.F.id_mark ((algebra.Alg.Var 8)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U52 
      ((algebra.Alg.Var 8)::nil))::nil))
    (* mark(U61(X_)) -> a__U61(mark(X_)) *)
   | R_xml_0_rule_69 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U61 ((algebra.Alg.Term 
                   algebra.F.id_mark ((algebra.Alg.Var 8)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U61 
      ((algebra.Alg.Var 8)::nil))::nil))
    (* mark(U71(X_)) -> a__U71(mark(X_)) *)
   | R_xml_0_rule_70 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U71 ((algebra.Alg.Term 
                   algebra.F.id_mark ((algebra.Alg.Var 8)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U71 
      ((algebra.Alg.Var 8)::nil))::nil))
   
    (* mark(U81(X1_,X2_,X3_)) -> a__U81(mark(X1_),X2_,X3_) *)
   | R_xml_0_rule_71 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U81 ((algebra.Alg.Term 
                   algebra.F.id_mark ((algebra.Alg.Var 6)::nil))::
                   (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U81 
      ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::
      (algebra.Alg.Var 9)::nil))::nil))
   
    (* mark(U82(X1_,X2_,X3_)) -> a__U82(mark(X1_),X2_,X3_) *)
   | R_xml_0_rule_72 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U82 ((algebra.Alg.Term 
                   algebra.F.id_mark ((algebra.Alg.Var 6)::nil))::
                   (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U82 
      ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::
      (algebra.Alg.Var 9)::nil))::nil))
   
    (* mark(U83(X1_,X2_,X3_)) -> a__U83(mark(X1_),X2_,X3_) *)
   | R_xml_0_rule_73 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U83 ((algebra.Alg.Term 
                   algebra.F.id_mark ((algebra.Alg.Var 6)::nil))::
                   (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U83 
      ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::
      (algebra.Alg.Var 9)::nil))::nil))
   
    (* mark(U84(X1_,X2_,X3_)) -> a__U84(mark(X1_),X2_,X3_) *)
   | R_xml_0_rule_74 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U84 ((algebra.Alg.Term 
                   algebra.F.id_mark ((algebra.Alg.Var 6)::nil))::
                   (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U84 
      ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::
      (algebra.Alg.Var 9)::nil))::nil))
    (* mark(U85(X1_,X2_)) -> a__U85(mark(X1_),X2_) *)
   | R_xml_0_rule_75 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U85 ((algebra.Alg.Term 
                   algebra.F.id_mark ((algebra.Alg.Var 6)::nil))::
                   (algebra.Alg.Var 7)::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U85 
      ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::nil))::nil))
    (* mark(U86(X_)) -> a__U86(mark(X_)) *)
   | R_xml_0_rule_76 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U86 ((algebra.Alg.Term 
                   algebra.F.id_mark ((algebra.Alg.Var 8)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U86 
      ((algebra.Alg.Var 8)::nil))::nil))
   
    (* mark(U91(X1_,X2_,X3_)) -> a__U91(mark(X1_),X2_,X3_) *)
   | R_xml_0_rule_77 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U91 ((algebra.Alg.Term 
                   algebra.F.id_mark ((algebra.Alg.Var 6)::nil))::
                   (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U91 
      ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::
      (algebra.Alg.Var 9)::nil))::nil))
   
    (* mark(U92(X1_,X2_,X3_)) -> a__U92(mark(X1_),X2_,X3_) *)
   | R_xml_0_rule_78 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U92 ((algebra.Alg.Term 
                   algebra.F.id_mark ((algebra.Alg.Var 6)::nil))::
                   (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U92 
      ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::
      (algebra.Alg.Var 9)::nil))::nil))
   
    (* mark(U93(X1_,X2_,X3_)) -> a__U93(mark(X1_),X2_,X3_) *)
   | R_xml_0_rule_79 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U93 ((algebra.Alg.Term 
                   algebra.F.id_mark ((algebra.Alg.Var 6)::nil))::
                   (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U93 
      ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::
      (algebra.Alg.Var 9)::nil))::nil))
    (* mark(U94(X1_,X2_)) -> a__U94(mark(X1_),X2_) *)
   | R_xml_0_rule_80 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U94 ((algebra.Alg.Term 
                   algebra.F.id_mark ((algebra.Alg.Var 6)::nil))::
                   (algebra.Alg.Var 7)::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U94 
      ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::nil))::nil))
    (* mark(length(X_)) -> a__length(mark(X_)) *)
   | R_xml_0_rule_81 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__length 
                   ((algebra.Alg.Term algebra.F.id_mark 
                   ((algebra.Alg.Var 8)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term 
      algebra.F.id_length ((algebra.Alg.Var 8)::nil))::nil))
    (* mark(cons(X1_,X2_)) -> cons(mark(X1_),X2_) *)
   | R_xml_0_rule_82 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term 
                   algebra.F.id_mark ((algebra.Alg.Var 6)::nil))::
                   (algebra.Alg.Var 7)::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term 
      algebra.F.id_cons ((algebra.Alg.Var 6)::
      (algebra.Alg.Var 7)::nil))::nil))
    (* mark(0) -> 0 *)
   | R_xml_0_rule_83 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_0 nil) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_0 
      nil)::nil))
    (* mark(tt) -> tt *)
   | R_xml_0_rule_84 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_tt nil) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_tt 
      nil)::nil))
    (* mark(s(X_)) -> s(mark(X_)) *)
   | R_xml_0_rule_85 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_s ((algebra.Alg.Term 
                   algebra.F.id_mark ((algebra.Alg.Var 8)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_s 
      ((algebra.Alg.Var 8)::nil))::nil))
    (* mark(nil) -> nil *)
   | R_xml_0_rule_86 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_nil nil) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_nil 
      nil)::nil))
    (* a__zeros -> zeros *)
   | R_xml_0_rule_87 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_zeros nil) 
     (algebra.Alg.Term algebra.F.id_a__zeros nil)
    (* a__U11(X1_,X2_) -> U11(X1_,X2_) *)
   | R_xml_0_rule_88 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_U11 ((algebra.Alg.Var 6)::
                   (algebra.Alg.Var 7)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U11 ((algebra.Alg.Var 6)::
      (algebra.Alg.Var 7)::nil))
    (* a__U12(X1_,X2_) -> U12(X1_,X2_) *)
   | R_xml_0_rule_89 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_U12 ((algebra.Alg.Var 6)::
                   (algebra.Alg.Var 7)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U12 ((algebra.Alg.Var 6)::
      (algebra.Alg.Var 7)::nil))
    (* a__isNatIListKind(X_) -> isNatIListKind(X_) *)
   | R_xml_0_rule_90 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_isNatIListKind 
                   ((algebra.Alg.Var 8)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
      ((algebra.Alg.Var 8)::nil))
    (* a__U13(X_) -> U13(X_) *)
   | R_xml_0_rule_91 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_U13 
                   ((algebra.Alg.Var 8)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U13 ((algebra.Alg.Var 8)::nil))
    (* a__isNatList(X_) -> isNatList(X_) *)
   | R_xml_0_rule_92 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_isNatList 
                   ((algebra.Alg.Var 8)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__isNatList ((algebra.Alg.Var 8)::nil))
    (* a__U21(X1_,X2_) -> U21(X1_,X2_) *)
   | R_xml_0_rule_93 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_U21 ((algebra.Alg.Var 6)::
                   (algebra.Alg.Var 7)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U21 ((algebra.Alg.Var 6)::
      (algebra.Alg.Var 7)::nil))
    (* a__U22(X1_,X2_) -> U22(X1_,X2_) *)
   | R_xml_0_rule_94 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_U22 ((algebra.Alg.Var 6)::
                   (algebra.Alg.Var 7)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U22 ((algebra.Alg.Var 6)::
      (algebra.Alg.Var 7)::nil))
    (* a__isNatKind(X_) -> isNatKind(X_) *)
   | R_xml_0_rule_95 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_isNatKind 
                   ((algebra.Alg.Var 8)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__isNatKind ((algebra.Alg.Var 8)::nil))
    (* a__U23(X_) -> U23(X_) *)
   | R_xml_0_rule_96 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_U23 
                   ((algebra.Alg.Var 8)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U23 ((algebra.Alg.Var 8)::nil))
    (* a__isNat(X_) -> isNat(X_) *)
   | R_xml_0_rule_97 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_isNat 
                   ((algebra.Alg.Var 8)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__isNat ((algebra.Alg.Var 8)::nil))
    (* a__U31(X1_,X2_) -> U31(X1_,X2_) *)
   | R_xml_0_rule_98 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_U31 ((algebra.Alg.Var 6)::
                   (algebra.Alg.Var 7)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U31 ((algebra.Alg.Var 6)::
      (algebra.Alg.Var 7)::nil))
    (* a__U32(X1_,X2_) -> U32(X1_,X2_) *)
   | R_xml_0_rule_99 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_U32 ((algebra.Alg.Var 6)::
                   (algebra.Alg.Var 7)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U32 ((algebra.Alg.Var 6)::
      (algebra.Alg.Var 7)::nil))
    (* a__U33(X_) -> U33(X_) *)
   | R_xml_0_rule_100 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_U33 
                   ((algebra.Alg.Var 8)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U33 ((algebra.Alg.Var 8)::nil))
    (* a__U41(X1_,X2_,X3_) -> U41(X1_,X2_,X3_) *)
   | R_xml_0_rule_101 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_U41 ((algebra.Alg.Var 6)::
                   (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U41 ((algebra.Alg.Var 6)::
      (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil))
    (* a__U42(X1_,X2_,X3_) -> U42(X1_,X2_,X3_) *)
   | R_xml_0_rule_102 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_U42 ((algebra.Alg.Var 6)::
                   (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U42 ((algebra.Alg.Var 6)::
      (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil))
    (* a__U43(X1_,X2_,X3_) -> U43(X1_,X2_,X3_) *)
   | R_xml_0_rule_103 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_U43 ((algebra.Alg.Var 6)::
                   (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U43 ((algebra.Alg.Var 6)::
      (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil))
    (* a__U44(X1_,X2_,X3_) -> U44(X1_,X2_,X3_) *)
   | R_xml_0_rule_104 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_U44 ((algebra.Alg.Var 6)::
                   (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U44 ((algebra.Alg.Var 6)::
      (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil))
    (* a__U45(X1_,X2_) -> U45(X1_,X2_) *)
   | R_xml_0_rule_105 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_U45 ((algebra.Alg.Var 6)::
                   (algebra.Alg.Var 7)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U45 ((algebra.Alg.Var 6)::
      (algebra.Alg.Var 7)::nil))
    (* a__U46(X_) -> U46(X_) *)
   | R_xml_0_rule_106 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_U46 
                   ((algebra.Alg.Var 8)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U46 ((algebra.Alg.Var 8)::nil))
    (* a__isNatIList(X_) -> isNatIList(X_) *)
   | R_xml_0_rule_107 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_isNatIList 
                   ((algebra.Alg.Var 8)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__isNatIList ((algebra.Alg.Var 8)::nil))
    (* a__U51(X1_,X2_) -> U51(X1_,X2_) *)
   | R_xml_0_rule_108 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_U51 ((algebra.Alg.Var 6)::
                   (algebra.Alg.Var 7)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U51 ((algebra.Alg.Var 6)::
      (algebra.Alg.Var 7)::nil))
    (* a__U52(X_) -> U52(X_) *)
   | R_xml_0_rule_109 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_U52 
                   ((algebra.Alg.Var 8)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U52 ((algebra.Alg.Var 8)::nil))
    (* a__U61(X_) -> U61(X_) *)
   | R_xml_0_rule_110 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_U61 
                   ((algebra.Alg.Var 8)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U61 ((algebra.Alg.Var 8)::nil))
    (* a__U71(X_) -> U71(X_) *)
   | R_xml_0_rule_111 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_U71 
                   ((algebra.Alg.Var 8)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U71 ((algebra.Alg.Var 8)::nil))
    (* a__U81(X1_,X2_,X3_) -> U81(X1_,X2_,X3_) *)
   | R_xml_0_rule_112 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_U81 ((algebra.Alg.Var 6)::
                   (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U81 ((algebra.Alg.Var 6)::
      (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil))
    (* a__U82(X1_,X2_,X3_) -> U82(X1_,X2_,X3_) *)
   | R_xml_0_rule_113 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_U82 ((algebra.Alg.Var 6)::
                   (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U82 ((algebra.Alg.Var 6)::
      (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil))
    (* a__U83(X1_,X2_,X3_) -> U83(X1_,X2_,X3_) *)
   | R_xml_0_rule_114 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_U83 ((algebra.Alg.Var 6)::
                   (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U83 ((algebra.Alg.Var 6)::
      (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil))
    (* a__U84(X1_,X2_,X3_) -> U84(X1_,X2_,X3_) *)
   | R_xml_0_rule_115 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_U84 ((algebra.Alg.Var 6)::
                   (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U84 ((algebra.Alg.Var 6)::
      (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil))
    (* a__U85(X1_,X2_) -> U85(X1_,X2_) *)
   | R_xml_0_rule_116 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_U85 ((algebra.Alg.Var 6)::
                   (algebra.Alg.Var 7)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U85 ((algebra.Alg.Var 6)::
      (algebra.Alg.Var 7)::nil))
    (* a__U86(X_) -> U86(X_) *)
   | R_xml_0_rule_117 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_U86 
                   ((algebra.Alg.Var 8)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U86 ((algebra.Alg.Var 8)::nil))
    (* a__U91(X1_,X2_,X3_) -> U91(X1_,X2_,X3_) *)
   | R_xml_0_rule_118 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_U91 ((algebra.Alg.Var 6)::
                   (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U91 ((algebra.Alg.Var 6)::
      (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil))
    (* a__U92(X1_,X2_,X3_) -> U92(X1_,X2_,X3_) *)
   | R_xml_0_rule_119 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_U92 ((algebra.Alg.Var 6)::
                   (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U92 ((algebra.Alg.Var 6)::
      (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil))
    (* a__U93(X1_,X2_,X3_) -> U93(X1_,X2_,X3_) *)
   | R_xml_0_rule_120 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_U93 ((algebra.Alg.Var 6)::
                   (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U93 ((algebra.Alg.Var 6)::
      (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil))
    (* a__U94(X1_,X2_) -> U94(X1_,X2_) *)
   | R_xml_0_rule_121 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_U94 ((algebra.Alg.Var 6)::
                   (algebra.Alg.Var 7)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U94 ((algebra.Alg.Var 6)::
      (algebra.Alg.Var 7)::nil))
    (* a__length(X_) -> length(X_) *)
   | R_xml_0_rule_122 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_length 
                   ((algebra.Alg.Var 8)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__length ((algebra.Alg.Var 8)::nil))
 .
 
 
 Definition R_xml_0_rule_as_list_0  := 
   ((algebra.Alg.Term algebra.F.id_a__zeros nil),
    (algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_0 
     nil)::(algebra.Alg.Term algebra.F.id_zeros nil)::nil)))::nil.
 
 
 Definition R_xml_0_rule_as_list_1  := 
   ((algebra.Alg.Term algebra.F.id_a__U11 ((algebra.Alg.Term algebra.F.id_tt 
     nil)::(algebra.Alg.Var 1)::nil)),
    (algebra.Alg.Term algebra.F.id_a__U12 ((algebra.Alg.Term 
     algebra.F.id_a__isNatIListKind ((algebra.Alg.Var 1)::nil))::
     (algebra.Alg.Var 1)::nil)))::R_xml_0_rule_as_list_0.
 
 
 Definition R_xml_0_rule_as_list_2  := 
   ((algebra.Alg.Term algebra.F.id_a__U12 ((algebra.Alg.Term algebra.F.id_tt 
     nil)::(algebra.Alg.Var 1)::nil)),
    (algebra.Alg.Term algebra.F.id_a__U13 ((algebra.Alg.Term 
     algebra.F.id_a__isNatList ((algebra.Alg.Var 1)::nil))::nil)))::
    R_xml_0_rule_as_list_1.
 
 
 Definition R_xml_0_rule_as_list_3  := 
   ((algebra.Alg.Term algebra.F.id_a__U13 ((algebra.Alg.Term algebra.F.id_tt 
     nil)::nil)),(algebra.Alg.Term algebra.F.id_tt nil))::
    R_xml_0_rule_as_list_2.
 
 
 Definition R_xml_0_rule_as_list_4  := 
   ((algebra.Alg.Term algebra.F.id_a__U21 ((algebra.Alg.Term algebra.F.id_tt 
     nil)::(algebra.Alg.Var 1)::nil)),
    (algebra.Alg.Term algebra.F.id_a__U22 ((algebra.Alg.Term 
     algebra.F.id_a__isNatKind ((algebra.Alg.Var 1)::nil))::
     (algebra.Alg.Var 1)::nil)))::R_xml_0_rule_as_list_3.
 
 
 Definition R_xml_0_rule_as_list_5  := 
   ((algebra.Alg.Term algebra.F.id_a__U22 ((algebra.Alg.Term algebra.F.id_tt 
     nil)::(algebra.Alg.Var 1)::nil)),
    (algebra.Alg.Term algebra.F.id_a__U23 ((algebra.Alg.Term 
     algebra.F.id_a__isNat ((algebra.Alg.Var 1)::nil))::nil)))::
    R_xml_0_rule_as_list_4.
 
 
 Definition R_xml_0_rule_as_list_6  := 
   ((algebra.Alg.Term algebra.F.id_a__U23 ((algebra.Alg.Term algebra.F.id_tt 
     nil)::nil)),(algebra.Alg.Term algebra.F.id_tt nil))::
    R_xml_0_rule_as_list_5.
 
 
 Definition R_xml_0_rule_as_list_7  := 
   ((algebra.Alg.Term algebra.F.id_a__U31 ((algebra.Alg.Term algebra.F.id_tt 
     nil)::(algebra.Alg.Var 2)::nil)),
    (algebra.Alg.Term algebra.F.id_a__U32 ((algebra.Alg.Term 
     algebra.F.id_a__isNatIListKind ((algebra.Alg.Var 2)::nil))::
     (algebra.Alg.Var 2)::nil)))::R_xml_0_rule_as_list_6.
 
 
 Definition R_xml_0_rule_as_list_8  := 
   ((algebra.Alg.Term algebra.F.id_a__U32 ((algebra.Alg.Term algebra.F.id_tt 
     nil)::(algebra.Alg.Var 2)::nil)),
    (algebra.Alg.Term algebra.F.id_a__U33 ((algebra.Alg.Term 
     algebra.F.id_a__isNatList ((algebra.Alg.Var 2)::nil))::nil)))::
    R_xml_0_rule_as_list_7.
 
 
 Definition R_xml_0_rule_as_list_9  := 
   ((algebra.Alg.Term algebra.F.id_a__U33 ((algebra.Alg.Term algebra.F.id_tt 
     nil)::nil)),(algebra.Alg.Term algebra.F.id_tt nil))::
    R_xml_0_rule_as_list_8.
 
 
 Definition R_xml_0_rule_as_list_10  := 
   ((algebra.Alg.Term algebra.F.id_a__U41 ((algebra.Alg.Term algebra.F.id_tt 
     nil)::(algebra.Alg.Var 1)::(algebra.Alg.Var 3)::nil)),
    (algebra.Alg.Term algebra.F.id_a__U42 ((algebra.Alg.Term 
     algebra.F.id_a__isNatKind ((algebra.Alg.Var 1)::nil))::
     (algebra.Alg.Var 1)::(algebra.Alg.Var 3)::nil)))::R_xml_0_rule_as_list_9
   .
 
 
 Definition R_xml_0_rule_as_list_11  := 
   ((algebra.Alg.Term algebra.F.id_a__U42 ((algebra.Alg.Term algebra.F.id_tt 
     nil)::(algebra.Alg.Var 1)::(algebra.Alg.Var 3)::nil)),
    (algebra.Alg.Term algebra.F.id_a__U43 ((algebra.Alg.Term 
     algebra.F.id_a__isNatIListKind ((algebra.Alg.Var 3)::nil))::
     (algebra.Alg.Var 1)::(algebra.Alg.Var 3)::nil)))::
    R_xml_0_rule_as_list_10.
 
 
 Definition R_xml_0_rule_as_list_12  := 
   ((algebra.Alg.Term algebra.F.id_a__U43 ((algebra.Alg.Term algebra.F.id_tt 
     nil)::(algebra.Alg.Var 1)::(algebra.Alg.Var 3)::nil)),
    (algebra.Alg.Term algebra.F.id_a__U44 ((algebra.Alg.Term 
     algebra.F.id_a__isNatIListKind ((algebra.Alg.Var 3)::nil))::
     (algebra.Alg.Var 1)::(algebra.Alg.Var 3)::nil)))::
    R_xml_0_rule_as_list_11.
 
 
 Definition R_xml_0_rule_as_list_13  := 
   ((algebra.Alg.Term algebra.F.id_a__U44 ((algebra.Alg.Term algebra.F.id_tt 
     nil)::(algebra.Alg.Var 1)::(algebra.Alg.Var 3)::nil)),
    (algebra.Alg.Term algebra.F.id_a__U45 ((algebra.Alg.Term 
     algebra.F.id_a__isNat ((algebra.Alg.Var 1)::nil))::
     (algebra.Alg.Var 3)::nil)))::R_xml_0_rule_as_list_12.
 
 
 Definition R_xml_0_rule_as_list_14  := 
   ((algebra.Alg.Term algebra.F.id_a__U45 ((algebra.Alg.Term algebra.F.id_tt 
     nil)::(algebra.Alg.Var 3)::nil)),
    (algebra.Alg.Term algebra.F.id_a__U46 ((algebra.Alg.Term 
     algebra.F.id_a__isNatIList ((algebra.Alg.Var 3)::nil))::nil)))::
    R_xml_0_rule_as_list_13.
 
 
 Definition R_xml_0_rule_as_list_15  := 
   ((algebra.Alg.Term algebra.F.id_a__U46 ((algebra.Alg.Term algebra.F.id_tt 
     nil)::nil)),(algebra.Alg.Term algebra.F.id_tt nil))::
    R_xml_0_rule_as_list_14.
 
 
 Definition R_xml_0_rule_as_list_16  := 
   ((algebra.Alg.Term algebra.F.id_a__U51 ((algebra.Alg.Term algebra.F.id_tt 
     nil)::(algebra.Alg.Var 3)::nil)),
    (algebra.Alg.Term algebra.F.id_a__U52 ((algebra.Alg.Term 
     algebra.F.id_a__isNatIListKind ((algebra.Alg.Var 3)::nil))::nil)))::
    R_xml_0_rule_as_list_15.
 
 
 Definition R_xml_0_rule_as_list_17  := 
   ((algebra.Alg.Term algebra.F.id_a__U52 ((algebra.Alg.Term algebra.F.id_tt 
     nil)::nil)),(algebra.Alg.Term algebra.F.id_tt nil))::
    R_xml_0_rule_as_list_16.
 
 
 Definition R_xml_0_rule_as_list_18  := 
   ((algebra.Alg.Term algebra.F.id_a__U61 ((algebra.Alg.Term algebra.F.id_tt 
     nil)::nil)),(algebra.Alg.Term algebra.F.id_tt nil))::
    R_xml_0_rule_as_list_17.
 
 
 Definition R_xml_0_rule_as_list_19  := 
   ((algebra.Alg.Term algebra.F.id_a__U71 ((algebra.Alg.Term algebra.F.id_tt 
     nil)::nil)),(algebra.Alg.Term algebra.F.id_tt nil))::
    R_xml_0_rule_as_list_18.
 
 
 Definition R_xml_0_rule_as_list_20  := 
   ((algebra.Alg.Term algebra.F.id_a__U81 ((algebra.Alg.Term algebra.F.id_tt 
     nil)::(algebra.Alg.Var 1)::(algebra.Alg.Var 3)::nil)),
    (algebra.Alg.Term algebra.F.id_a__U82 ((algebra.Alg.Term 
     algebra.F.id_a__isNatKind ((algebra.Alg.Var 1)::nil))::
     (algebra.Alg.Var 1)::(algebra.Alg.Var 3)::nil)))::
    R_xml_0_rule_as_list_19.
 
 
 Definition R_xml_0_rule_as_list_21  := 
   ((algebra.Alg.Term algebra.F.id_a__U82 ((algebra.Alg.Term algebra.F.id_tt 
     nil)::(algebra.Alg.Var 1)::(algebra.Alg.Var 3)::nil)),
    (algebra.Alg.Term algebra.F.id_a__U83 ((algebra.Alg.Term 
     algebra.F.id_a__isNatIListKind ((algebra.Alg.Var 3)::nil))::
     (algebra.Alg.Var 1)::(algebra.Alg.Var 3)::nil)))::
    R_xml_0_rule_as_list_20.
 
 
 Definition R_xml_0_rule_as_list_22  := 
   ((algebra.Alg.Term algebra.F.id_a__U83 ((algebra.Alg.Term algebra.F.id_tt 
     nil)::(algebra.Alg.Var 1)::(algebra.Alg.Var 3)::nil)),
    (algebra.Alg.Term algebra.F.id_a__U84 ((algebra.Alg.Term 
     algebra.F.id_a__isNatIListKind ((algebra.Alg.Var 3)::nil))::
     (algebra.Alg.Var 1)::(algebra.Alg.Var 3)::nil)))::
    R_xml_0_rule_as_list_21.
 
 
 Definition R_xml_0_rule_as_list_23  := 
   ((algebra.Alg.Term algebra.F.id_a__U84 ((algebra.Alg.Term algebra.F.id_tt 
     nil)::(algebra.Alg.Var 1)::(algebra.Alg.Var 3)::nil)),
    (algebra.Alg.Term algebra.F.id_a__U85 ((algebra.Alg.Term 
     algebra.F.id_a__isNat ((algebra.Alg.Var 1)::nil))::
     (algebra.Alg.Var 3)::nil)))::R_xml_0_rule_as_list_22.
 
 
 Definition R_xml_0_rule_as_list_24  := 
   ((algebra.Alg.Term algebra.F.id_a__U85 ((algebra.Alg.Term algebra.F.id_tt 
     nil)::(algebra.Alg.Var 3)::nil)),
    (algebra.Alg.Term algebra.F.id_a__U86 ((algebra.Alg.Term 
     algebra.F.id_a__isNatList ((algebra.Alg.Var 3)::nil))::nil)))::
    R_xml_0_rule_as_list_23.
 
 
 Definition R_xml_0_rule_as_list_25  := 
   ((algebra.Alg.Term algebra.F.id_a__U86 ((algebra.Alg.Term algebra.F.id_tt 
     nil)::nil)),(algebra.Alg.Term algebra.F.id_tt nil))::
    R_xml_0_rule_as_list_24.
 
 
 Definition R_xml_0_rule_as_list_26  := 
   ((algebra.Alg.Term algebra.F.id_a__U91 ((algebra.Alg.Term algebra.F.id_tt 
     nil)::(algebra.Alg.Var 4)::(algebra.Alg.Var 5)::nil)),
    (algebra.Alg.Term algebra.F.id_a__U92 ((algebra.Alg.Term 
     algebra.F.id_a__isNatIListKind ((algebra.Alg.Var 4)::nil))::
     (algebra.Alg.Var 4)::(algebra.Alg.Var 5)::nil)))::
    R_xml_0_rule_as_list_25.
 
 
 Definition R_xml_0_rule_as_list_27  := 
   ((algebra.Alg.Term algebra.F.id_a__U92 ((algebra.Alg.Term algebra.F.id_tt 
     nil)::(algebra.Alg.Var 4)::(algebra.Alg.Var 5)::nil)),
    (algebra.Alg.Term algebra.F.id_a__U93 ((algebra.Alg.Term 
     algebra.F.id_a__isNat ((algebra.Alg.Var 5)::nil))::(algebra.Alg.Var 4)::
     (algebra.Alg.Var 5)::nil)))::R_xml_0_rule_as_list_26.
 
 
 Definition R_xml_0_rule_as_list_28  := 
   ((algebra.Alg.Term algebra.F.id_a__U93 ((algebra.Alg.Term algebra.F.id_tt 
     nil)::(algebra.Alg.Var 4)::(algebra.Alg.Var 5)::nil)),
    (algebra.Alg.Term algebra.F.id_a__U94 ((algebra.Alg.Term 
     algebra.F.id_a__isNatKind ((algebra.Alg.Var 5)::nil))::
     (algebra.Alg.Var 4)::nil)))::R_xml_0_rule_as_list_27.
 
 
 Definition R_xml_0_rule_as_list_29  := 
   ((algebra.Alg.Term algebra.F.id_a__U94 ((algebra.Alg.Term algebra.F.id_tt 
     nil)::(algebra.Alg.Var 4)::nil)),
    (algebra.Alg.Term algebra.F.id_s ((algebra.Alg.Term 
     algebra.F.id_a__length ((algebra.Alg.Term algebra.F.id_mark 
     ((algebra.Alg.Var 4)::nil))::nil))::nil)))::R_xml_0_rule_as_list_28.
 
 
 Definition R_xml_0_rule_as_list_30  := 
   ((algebra.Alg.Term algebra.F.id_a__isNat ((algebra.Alg.Term 
     algebra.F.id_0 nil)::nil)),(algebra.Alg.Term algebra.F.id_tt nil))::
    R_xml_0_rule_as_list_29.
 
 
 Definition R_xml_0_rule_as_list_31  := 
   ((algebra.Alg.Term algebra.F.id_a__isNat ((algebra.Alg.Term 
     algebra.F.id_length ((algebra.Alg.Var 1)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__U11 ((algebra.Alg.Term 
     algebra.F.id_a__isNatIListKind ((algebra.Alg.Var 1)::nil))::
     (algebra.Alg.Var 1)::nil)))::R_xml_0_rule_as_list_30.
 
 
 Definition R_xml_0_rule_as_list_32  := 
   ((algebra.Alg.Term algebra.F.id_a__isNat ((algebra.Alg.Term 
     algebra.F.id_s ((algebra.Alg.Var 1)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__U21 ((algebra.Alg.Term 
     algebra.F.id_a__isNatKind ((algebra.Alg.Var 1)::nil))::
     (algebra.Alg.Var 1)::nil)))::R_xml_0_rule_as_list_31.
 
 
 Definition R_xml_0_rule_as_list_33  := 
   ((algebra.Alg.Term algebra.F.id_a__isNatIList ((algebra.Alg.Var 2)::nil)),
    (algebra.Alg.Term algebra.F.id_a__U31 ((algebra.Alg.Term 
     algebra.F.id_a__isNatIListKind ((algebra.Alg.Var 2)::nil))::
     (algebra.Alg.Var 2)::nil)))::R_xml_0_rule_as_list_32.
 
 
 Definition R_xml_0_rule_as_list_34  := 
   ((algebra.Alg.Term algebra.F.id_a__isNatIList ((algebra.Alg.Term 
     algebra.F.id_zeros nil)::nil)),(algebra.Alg.Term algebra.F.id_tt nil))::
    R_xml_0_rule_as_list_33.
 
 
 Definition R_xml_0_rule_as_list_35  := 
   ((algebra.Alg.Term algebra.F.id_a__isNatIList ((algebra.Alg.Term 
     algebra.F.id_cons ((algebra.Alg.Var 1)::
     (algebra.Alg.Var 3)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__U41 ((algebra.Alg.Term 
     algebra.F.id_a__isNatKind ((algebra.Alg.Var 1)::nil))::
     (algebra.Alg.Var 1)::(algebra.Alg.Var 3)::nil)))::
    R_xml_0_rule_as_list_34.
 
 
 Definition R_xml_0_rule_as_list_36  := 
   ((algebra.Alg.Term algebra.F.id_a__isNatIListKind ((algebra.Alg.Term 
     algebra.F.id_nil nil)::nil)),(algebra.Alg.Term algebra.F.id_tt nil))::
    R_xml_0_rule_as_list_35.
 
 
 Definition R_xml_0_rule_as_list_37  := 
   ((algebra.Alg.Term algebra.F.id_a__isNatIListKind ((algebra.Alg.Term 
     algebra.F.id_zeros nil)::nil)),(algebra.Alg.Term algebra.F.id_tt nil))::
    R_xml_0_rule_as_list_36.
 
 
 Definition R_xml_0_rule_as_list_38  := 
   ((algebra.Alg.Term algebra.F.id_a__isNatIListKind ((algebra.Alg.Term 
     algebra.F.id_cons ((algebra.Alg.Var 1)::
     (algebra.Alg.Var 3)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__U51 ((algebra.Alg.Term 
     algebra.F.id_a__isNatKind ((algebra.Alg.Var 1)::nil))::
     (algebra.Alg.Var 3)::nil)))::R_xml_0_rule_as_list_37.
 
 
 Definition R_xml_0_rule_as_list_39  := 
   ((algebra.Alg.Term algebra.F.id_a__isNatKind ((algebra.Alg.Term 
     algebra.F.id_0 nil)::nil)),(algebra.Alg.Term algebra.F.id_tt nil))::
    R_xml_0_rule_as_list_38.
 
 
 Definition R_xml_0_rule_as_list_40  := 
   ((algebra.Alg.Term algebra.F.id_a__isNatKind ((algebra.Alg.Term 
     algebra.F.id_length ((algebra.Alg.Var 1)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__U61 ((algebra.Alg.Term 
     algebra.F.id_a__isNatIListKind ((algebra.Alg.Var 1)::nil))::nil)))::
    R_xml_0_rule_as_list_39.
 
 
 Definition R_xml_0_rule_as_list_41  := 
   ((algebra.Alg.Term algebra.F.id_a__isNatKind ((algebra.Alg.Term 
     algebra.F.id_s ((algebra.Alg.Var 1)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__U71 ((algebra.Alg.Term 
     algebra.F.id_a__isNatKind ((algebra.Alg.Var 1)::nil))::nil)))::
    R_xml_0_rule_as_list_40.
 
 
 Definition R_xml_0_rule_as_list_42  := 
   ((algebra.Alg.Term algebra.F.id_a__isNatList ((algebra.Alg.Term 
     algebra.F.id_nil nil)::nil)),(algebra.Alg.Term algebra.F.id_tt nil))::
    R_xml_0_rule_as_list_41.
 
 
 Definition R_xml_0_rule_as_list_43  := 
   ((algebra.Alg.Term algebra.F.id_a__isNatList ((algebra.Alg.Term 
     algebra.F.id_cons ((algebra.Alg.Var 1)::
     (algebra.Alg.Var 3)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__U81 ((algebra.Alg.Term 
     algebra.F.id_a__isNatKind ((algebra.Alg.Var 1)::nil))::
     (algebra.Alg.Var 1)::(algebra.Alg.Var 3)::nil)))::
    R_xml_0_rule_as_list_42.
 
 
 Definition R_xml_0_rule_as_list_44  := 
   ((algebra.Alg.Term algebra.F.id_a__length ((algebra.Alg.Term 
     algebra.F.id_nil nil)::nil)),(algebra.Alg.Term algebra.F.id_0 nil))::
    R_xml_0_rule_as_list_43.
 
 
 Definition R_xml_0_rule_as_list_45  := 
   ((algebra.Alg.Term algebra.F.id_a__length ((algebra.Alg.Term 
     algebra.F.id_cons ((algebra.Alg.Var 5)::
     (algebra.Alg.Var 4)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__U91 ((algebra.Alg.Term 
     algebra.F.id_a__isNatList ((algebra.Alg.Var 4)::nil))::
     (algebra.Alg.Var 4)::(algebra.Alg.Var 5)::nil)))::
    R_xml_0_rule_as_list_44.
 
 
 Definition R_xml_0_rule_as_list_46  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term 
     algebra.F.id_zeros nil)::nil)),
    (algebra.Alg.Term algebra.F.id_a__zeros nil))::R_xml_0_rule_as_list_45.
 
 
 Definition R_xml_0_rule_as_list_47  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U11 
     ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__U11 ((algebra.Alg.Term 
     algebra.F.id_mark ((algebra.Alg.Var 6)::nil))::
     (algebra.Alg.Var 7)::nil)))::R_xml_0_rule_as_list_46.
 
 
 Definition R_xml_0_rule_as_list_48  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U12 
     ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__U12 ((algebra.Alg.Term 
     algebra.F.id_mark ((algebra.Alg.Var 6)::nil))::
     (algebra.Alg.Var 7)::nil)))::R_xml_0_rule_as_list_47.
 
 
 Definition R_xml_0_rule_as_list_49  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term 
     algebra.F.id_isNatIListKind ((algebra.Alg.Var 8)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
     ((algebra.Alg.Var 8)::nil)))::R_xml_0_rule_as_list_48.
 
 
 Definition R_xml_0_rule_as_list_50  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U13 
     ((algebra.Alg.Var 8)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__U13 ((algebra.Alg.Term 
     algebra.F.id_mark ((algebra.Alg.Var 8)::nil))::nil)))::
    R_xml_0_rule_as_list_49.
 
 
 Definition R_xml_0_rule_as_list_51  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term 
     algebra.F.id_isNatList ((algebra.Alg.Var 8)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__isNatList ((algebra.Alg.Var 8)::nil)))::
    R_xml_0_rule_as_list_50.
 
 
 Definition R_xml_0_rule_as_list_52  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U21 
     ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__U21 ((algebra.Alg.Term 
     algebra.F.id_mark ((algebra.Alg.Var 6)::nil))::
     (algebra.Alg.Var 7)::nil)))::R_xml_0_rule_as_list_51.
 
 
 Definition R_xml_0_rule_as_list_53  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U22 
     ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__U22 ((algebra.Alg.Term 
     algebra.F.id_mark ((algebra.Alg.Var 6)::nil))::
     (algebra.Alg.Var 7)::nil)))::R_xml_0_rule_as_list_52.
 
 
 Definition R_xml_0_rule_as_list_54  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term 
     algebra.F.id_isNatKind ((algebra.Alg.Var 8)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__isNatKind ((algebra.Alg.Var 8)::nil)))::
    R_xml_0_rule_as_list_53.
 
 
 Definition R_xml_0_rule_as_list_55  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U23 
     ((algebra.Alg.Var 8)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__U23 ((algebra.Alg.Term 
     algebra.F.id_mark ((algebra.Alg.Var 8)::nil))::nil)))::
    R_xml_0_rule_as_list_54.
 
 
 Definition R_xml_0_rule_as_list_56  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term 
     algebra.F.id_isNat ((algebra.Alg.Var 8)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__isNat ((algebra.Alg.Var 8)::nil)))::
    R_xml_0_rule_as_list_55.
 
 
 Definition R_xml_0_rule_as_list_57  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U31 
     ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__U31 ((algebra.Alg.Term 
     algebra.F.id_mark ((algebra.Alg.Var 6)::nil))::
     (algebra.Alg.Var 7)::nil)))::R_xml_0_rule_as_list_56.
 
 
 Definition R_xml_0_rule_as_list_58  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U32 
     ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__U32 ((algebra.Alg.Term 
     algebra.F.id_mark ((algebra.Alg.Var 6)::nil))::
     (algebra.Alg.Var 7)::nil)))::R_xml_0_rule_as_list_57.
 
 
 Definition R_xml_0_rule_as_list_59  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U33 
     ((algebra.Alg.Var 8)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__U33 ((algebra.Alg.Term 
     algebra.F.id_mark ((algebra.Alg.Var 8)::nil))::nil)))::
    R_xml_0_rule_as_list_58.
 
 
 Definition R_xml_0_rule_as_list_60  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U41 
     ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::
     (algebra.Alg.Var 9)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__U41 ((algebra.Alg.Term 
     algebra.F.id_mark ((algebra.Alg.Var 6)::nil))::(algebra.Alg.Var 7)::
     (algebra.Alg.Var 9)::nil)))::R_xml_0_rule_as_list_59.
 
 
 Definition R_xml_0_rule_as_list_61  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U42 
     ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::
     (algebra.Alg.Var 9)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__U42 ((algebra.Alg.Term 
     algebra.F.id_mark ((algebra.Alg.Var 6)::nil))::(algebra.Alg.Var 7)::
     (algebra.Alg.Var 9)::nil)))::R_xml_0_rule_as_list_60.
 
 
 Definition R_xml_0_rule_as_list_62  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U43 
     ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::
     (algebra.Alg.Var 9)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__U43 ((algebra.Alg.Term 
     algebra.F.id_mark ((algebra.Alg.Var 6)::nil))::(algebra.Alg.Var 7)::
     (algebra.Alg.Var 9)::nil)))::R_xml_0_rule_as_list_61.
 
 
 Definition R_xml_0_rule_as_list_63  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U44 
     ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::
     (algebra.Alg.Var 9)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__U44 ((algebra.Alg.Term 
     algebra.F.id_mark ((algebra.Alg.Var 6)::nil))::(algebra.Alg.Var 7)::
     (algebra.Alg.Var 9)::nil)))::R_xml_0_rule_as_list_62.
 
 
 Definition R_xml_0_rule_as_list_64  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U45 
     ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__U45 ((algebra.Alg.Term 
     algebra.F.id_mark ((algebra.Alg.Var 6)::nil))::
     (algebra.Alg.Var 7)::nil)))::R_xml_0_rule_as_list_63.
 
 
 Definition R_xml_0_rule_as_list_65  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U46 
     ((algebra.Alg.Var 8)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__U46 ((algebra.Alg.Term 
     algebra.F.id_mark ((algebra.Alg.Var 8)::nil))::nil)))::
    R_xml_0_rule_as_list_64.
 
 
 Definition R_xml_0_rule_as_list_66  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term 
     algebra.F.id_isNatIList ((algebra.Alg.Var 8)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__isNatIList ((algebra.Alg.Var 8)::nil)))::
    R_xml_0_rule_as_list_65.
 
 
 Definition R_xml_0_rule_as_list_67  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U51 
     ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__U51 ((algebra.Alg.Term 
     algebra.F.id_mark ((algebra.Alg.Var 6)::nil))::
     (algebra.Alg.Var 7)::nil)))::R_xml_0_rule_as_list_66.
 
 
 Definition R_xml_0_rule_as_list_68  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U52 
     ((algebra.Alg.Var 8)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__U52 ((algebra.Alg.Term 
     algebra.F.id_mark ((algebra.Alg.Var 8)::nil))::nil)))::
    R_xml_0_rule_as_list_67.
 
 
 Definition R_xml_0_rule_as_list_69  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U61 
     ((algebra.Alg.Var 8)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__U61 ((algebra.Alg.Term 
     algebra.F.id_mark ((algebra.Alg.Var 8)::nil))::nil)))::
    R_xml_0_rule_as_list_68.
 
 
 Definition R_xml_0_rule_as_list_70  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U71 
     ((algebra.Alg.Var 8)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__U71 ((algebra.Alg.Term 
     algebra.F.id_mark ((algebra.Alg.Var 8)::nil))::nil)))::
    R_xml_0_rule_as_list_69.
 
 
 Definition R_xml_0_rule_as_list_71  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U81 
     ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::
     (algebra.Alg.Var 9)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__U81 ((algebra.Alg.Term 
     algebra.F.id_mark ((algebra.Alg.Var 6)::nil))::(algebra.Alg.Var 7)::
     (algebra.Alg.Var 9)::nil)))::R_xml_0_rule_as_list_70.
 
 
 Definition R_xml_0_rule_as_list_72  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U82 
     ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::
     (algebra.Alg.Var 9)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__U82 ((algebra.Alg.Term 
     algebra.F.id_mark ((algebra.Alg.Var 6)::nil))::(algebra.Alg.Var 7)::
     (algebra.Alg.Var 9)::nil)))::R_xml_0_rule_as_list_71.
 
 
 Definition R_xml_0_rule_as_list_73  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U83 
     ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::
     (algebra.Alg.Var 9)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__U83 ((algebra.Alg.Term 
     algebra.F.id_mark ((algebra.Alg.Var 6)::nil))::(algebra.Alg.Var 7)::
     (algebra.Alg.Var 9)::nil)))::R_xml_0_rule_as_list_72.
 
 
 Definition R_xml_0_rule_as_list_74  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U84 
     ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::
     (algebra.Alg.Var 9)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__U84 ((algebra.Alg.Term 
     algebra.F.id_mark ((algebra.Alg.Var 6)::nil))::(algebra.Alg.Var 7)::
     (algebra.Alg.Var 9)::nil)))::R_xml_0_rule_as_list_73.
 
 
 Definition R_xml_0_rule_as_list_75  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U85 
     ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__U85 ((algebra.Alg.Term 
     algebra.F.id_mark ((algebra.Alg.Var 6)::nil))::
     (algebra.Alg.Var 7)::nil)))::R_xml_0_rule_as_list_74.
 
 
 Definition R_xml_0_rule_as_list_76  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U86 
     ((algebra.Alg.Var 8)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__U86 ((algebra.Alg.Term 
     algebra.F.id_mark ((algebra.Alg.Var 8)::nil))::nil)))::
    R_xml_0_rule_as_list_75.
 
 
 Definition R_xml_0_rule_as_list_77  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U91 
     ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::
     (algebra.Alg.Var 9)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__U91 ((algebra.Alg.Term 
     algebra.F.id_mark ((algebra.Alg.Var 6)::nil))::(algebra.Alg.Var 7)::
     (algebra.Alg.Var 9)::nil)))::R_xml_0_rule_as_list_76.
 
 
 Definition R_xml_0_rule_as_list_78  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U92 
     ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::
     (algebra.Alg.Var 9)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__U92 ((algebra.Alg.Term 
     algebra.F.id_mark ((algebra.Alg.Var 6)::nil))::(algebra.Alg.Var 7)::
     (algebra.Alg.Var 9)::nil)))::R_xml_0_rule_as_list_77.
 
 
 Definition R_xml_0_rule_as_list_79  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U93 
     ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::
     (algebra.Alg.Var 9)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__U93 ((algebra.Alg.Term 
     algebra.F.id_mark ((algebra.Alg.Var 6)::nil))::(algebra.Alg.Var 7)::
     (algebra.Alg.Var 9)::nil)))::R_xml_0_rule_as_list_78.
 
 
 Definition R_xml_0_rule_as_list_80  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U94 
     ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__U94 ((algebra.Alg.Term 
     algebra.F.id_mark ((algebra.Alg.Var 6)::nil))::
     (algebra.Alg.Var 7)::nil)))::R_xml_0_rule_as_list_79.
 
 
 Definition R_xml_0_rule_as_list_81  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term 
     algebra.F.id_length ((algebra.Alg.Var 8)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__length ((algebra.Alg.Term 
     algebra.F.id_mark ((algebra.Alg.Var 8)::nil))::nil)))::
    R_xml_0_rule_as_list_80.
 
 
 Definition R_xml_0_rule_as_list_82  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_cons 
     ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_mark 
     ((algebra.Alg.Var 6)::nil))::(algebra.Alg.Var 7)::nil)))::
    R_xml_0_rule_as_list_81.
 
 
 Definition R_xml_0_rule_as_list_83  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_0 
     nil)::nil)),(algebra.Alg.Term algebra.F.id_0 nil))::
    R_xml_0_rule_as_list_82.
 
 
 Definition R_xml_0_rule_as_list_84  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_tt 
     nil)::nil)),(algebra.Alg.Term algebra.F.id_tt nil))::
    R_xml_0_rule_as_list_83.
 
 
 Definition R_xml_0_rule_as_list_85  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_s 
     ((algebra.Alg.Var 8)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_s ((algebra.Alg.Term algebra.F.id_mark 
     ((algebra.Alg.Var 8)::nil))::nil)))::R_xml_0_rule_as_list_84.
 
 
 Definition R_xml_0_rule_as_list_86  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_nil 
     nil)::nil)),(algebra.Alg.Term algebra.F.id_nil nil))::
    R_xml_0_rule_as_list_85.
 
 
 Definition R_xml_0_rule_as_list_87  := 
   ((algebra.Alg.Term algebra.F.id_a__zeros nil),
    (algebra.Alg.Term algebra.F.id_zeros nil))::R_xml_0_rule_as_list_86.
 
 
 Definition R_xml_0_rule_as_list_88  := 
   ((algebra.Alg.Term algebra.F.id_a__U11 ((algebra.Alg.Var 6)::
     (algebra.Alg.Var 7)::nil)),
    (algebra.Alg.Term algebra.F.id_U11 ((algebra.Alg.Var 6)::
     (algebra.Alg.Var 7)::nil)))::R_xml_0_rule_as_list_87.
 
 
 Definition R_xml_0_rule_as_list_89  := 
   ((algebra.Alg.Term algebra.F.id_a__U12 ((algebra.Alg.Var 6)::
     (algebra.Alg.Var 7)::nil)),
    (algebra.Alg.Term algebra.F.id_U12 ((algebra.Alg.Var 6)::
     (algebra.Alg.Var 7)::nil)))::R_xml_0_rule_as_list_88.
 
 
 Definition R_xml_0_rule_as_list_90  := 
   ((algebra.Alg.Term algebra.F.id_a__isNatIListKind 
     ((algebra.Alg.Var 8)::nil)),
    (algebra.Alg.Term algebra.F.id_isNatIListKind ((algebra.Alg.Var 8)::nil)))::
    R_xml_0_rule_as_list_89.
 
 
 Definition R_xml_0_rule_as_list_91  := 
   ((algebra.Alg.Term algebra.F.id_a__U13 ((algebra.Alg.Var 8)::nil)),
    (algebra.Alg.Term algebra.F.id_U13 ((algebra.Alg.Var 8)::nil)))::
    R_xml_0_rule_as_list_90.
 
 
 Definition R_xml_0_rule_as_list_92  := 
   ((algebra.Alg.Term algebra.F.id_a__isNatList ((algebra.Alg.Var 8)::nil)),
    (algebra.Alg.Term algebra.F.id_isNatList ((algebra.Alg.Var 8)::nil)))::
    R_xml_0_rule_as_list_91.
 
 
 Definition R_xml_0_rule_as_list_93  := 
   ((algebra.Alg.Term algebra.F.id_a__U21 ((algebra.Alg.Var 6)::
     (algebra.Alg.Var 7)::nil)),
    (algebra.Alg.Term algebra.F.id_U21 ((algebra.Alg.Var 6)::
     (algebra.Alg.Var 7)::nil)))::R_xml_0_rule_as_list_92.
 
 
 Definition R_xml_0_rule_as_list_94  := 
   ((algebra.Alg.Term algebra.F.id_a__U22 ((algebra.Alg.Var 6)::
     (algebra.Alg.Var 7)::nil)),
    (algebra.Alg.Term algebra.F.id_U22 ((algebra.Alg.Var 6)::
     (algebra.Alg.Var 7)::nil)))::R_xml_0_rule_as_list_93.
 
 
 Definition R_xml_0_rule_as_list_95  := 
   ((algebra.Alg.Term algebra.F.id_a__isNatKind ((algebra.Alg.Var 8)::nil)),
    (algebra.Alg.Term algebra.F.id_isNatKind ((algebra.Alg.Var 8)::nil)))::
    R_xml_0_rule_as_list_94.
 
 
 Definition R_xml_0_rule_as_list_96  := 
   ((algebra.Alg.Term algebra.F.id_a__U23 ((algebra.Alg.Var 8)::nil)),
    (algebra.Alg.Term algebra.F.id_U23 ((algebra.Alg.Var 8)::nil)))::
    R_xml_0_rule_as_list_95.
 
 
 Definition R_xml_0_rule_as_list_97  := 
   ((algebra.Alg.Term algebra.F.id_a__isNat ((algebra.Alg.Var 8)::nil)),
    (algebra.Alg.Term algebra.F.id_isNat ((algebra.Alg.Var 8)::nil)))::
    R_xml_0_rule_as_list_96.
 
 
 Definition R_xml_0_rule_as_list_98  := 
   ((algebra.Alg.Term algebra.F.id_a__U31 ((algebra.Alg.Var 6)::
     (algebra.Alg.Var 7)::nil)),
    (algebra.Alg.Term algebra.F.id_U31 ((algebra.Alg.Var 6)::
     (algebra.Alg.Var 7)::nil)))::R_xml_0_rule_as_list_97.
 
 
 Definition R_xml_0_rule_as_list_99  := 
   ((algebra.Alg.Term algebra.F.id_a__U32 ((algebra.Alg.Var 6)::
     (algebra.Alg.Var 7)::nil)),
    (algebra.Alg.Term algebra.F.id_U32 ((algebra.Alg.Var 6)::
     (algebra.Alg.Var 7)::nil)))::R_xml_0_rule_as_list_98.
 
 
 Definition R_xml_0_rule_as_list_100  := 
   ((algebra.Alg.Term algebra.F.id_a__U33 ((algebra.Alg.Var 8)::nil)),
    (algebra.Alg.Term algebra.F.id_U33 ((algebra.Alg.Var 8)::nil)))::
    R_xml_0_rule_as_list_99.
 
 
 Definition R_xml_0_rule_as_list_101  := 
   ((algebra.Alg.Term algebra.F.id_a__U41 ((algebra.Alg.Var 6)::
     (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil)),
    (algebra.Alg.Term algebra.F.id_U41 ((algebra.Alg.Var 6)::
     (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil)))::
    R_xml_0_rule_as_list_100.
 
 
 Definition R_xml_0_rule_as_list_102  := 
   ((algebra.Alg.Term algebra.F.id_a__U42 ((algebra.Alg.Var 6)::
     (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil)),
    (algebra.Alg.Term algebra.F.id_U42 ((algebra.Alg.Var 6)::
     (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil)))::
    R_xml_0_rule_as_list_101.
 
 
 Definition R_xml_0_rule_as_list_103  := 
   ((algebra.Alg.Term algebra.F.id_a__U43 ((algebra.Alg.Var 6)::
     (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil)),
    (algebra.Alg.Term algebra.F.id_U43 ((algebra.Alg.Var 6)::
     (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil)))::
    R_xml_0_rule_as_list_102.
 
 
 Definition R_xml_0_rule_as_list_104  := 
   ((algebra.Alg.Term algebra.F.id_a__U44 ((algebra.Alg.Var 6)::
     (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil)),
    (algebra.Alg.Term algebra.F.id_U44 ((algebra.Alg.Var 6)::
     (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil)))::
    R_xml_0_rule_as_list_103.
 
 
 Definition R_xml_0_rule_as_list_105  := 
   ((algebra.Alg.Term algebra.F.id_a__U45 ((algebra.Alg.Var 6)::
     (algebra.Alg.Var 7)::nil)),
    (algebra.Alg.Term algebra.F.id_U45 ((algebra.Alg.Var 6)::
     (algebra.Alg.Var 7)::nil)))::R_xml_0_rule_as_list_104.
 
 
 Definition R_xml_0_rule_as_list_106  := 
   ((algebra.Alg.Term algebra.F.id_a__U46 ((algebra.Alg.Var 8)::nil)),
    (algebra.Alg.Term algebra.F.id_U46 ((algebra.Alg.Var 8)::nil)))::
    R_xml_0_rule_as_list_105.
 
 
 Definition R_xml_0_rule_as_list_107  := 
   ((algebra.Alg.Term algebra.F.id_a__isNatIList ((algebra.Alg.Var 8)::nil)),
    (algebra.Alg.Term algebra.F.id_isNatIList ((algebra.Alg.Var 8)::nil)))::
    R_xml_0_rule_as_list_106.
 
 
 Definition R_xml_0_rule_as_list_108  := 
   ((algebra.Alg.Term algebra.F.id_a__U51 ((algebra.Alg.Var 6)::
     (algebra.Alg.Var 7)::nil)),
    (algebra.Alg.Term algebra.F.id_U51 ((algebra.Alg.Var 6)::
     (algebra.Alg.Var 7)::nil)))::R_xml_0_rule_as_list_107.
 
 
 Definition R_xml_0_rule_as_list_109  := 
   ((algebra.Alg.Term algebra.F.id_a__U52 ((algebra.Alg.Var 8)::nil)),
    (algebra.Alg.Term algebra.F.id_U52 ((algebra.Alg.Var 8)::nil)))::
    R_xml_0_rule_as_list_108.
 
 
 Definition R_xml_0_rule_as_list_110  := 
   ((algebra.Alg.Term algebra.F.id_a__U61 ((algebra.Alg.Var 8)::nil)),
    (algebra.Alg.Term algebra.F.id_U61 ((algebra.Alg.Var 8)::nil)))::
    R_xml_0_rule_as_list_109.
 
 
 Definition R_xml_0_rule_as_list_111  := 
   ((algebra.Alg.Term algebra.F.id_a__U71 ((algebra.Alg.Var 8)::nil)),
    (algebra.Alg.Term algebra.F.id_U71 ((algebra.Alg.Var 8)::nil)))::
    R_xml_0_rule_as_list_110.
 
 
 Definition R_xml_0_rule_as_list_112  := 
   ((algebra.Alg.Term algebra.F.id_a__U81 ((algebra.Alg.Var 6)::
     (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil)),
    (algebra.Alg.Term algebra.F.id_U81 ((algebra.Alg.Var 6)::
     (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil)))::
    R_xml_0_rule_as_list_111.
 
 
 Definition R_xml_0_rule_as_list_113  := 
   ((algebra.Alg.Term algebra.F.id_a__U82 ((algebra.Alg.Var 6)::
     (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil)),
    (algebra.Alg.Term algebra.F.id_U82 ((algebra.Alg.Var 6)::
     (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil)))::
    R_xml_0_rule_as_list_112.
 
 
 Definition R_xml_0_rule_as_list_114  := 
   ((algebra.Alg.Term algebra.F.id_a__U83 ((algebra.Alg.Var 6)::
     (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil)),
    (algebra.Alg.Term algebra.F.id_U83 ((algebra.Alg.Var 6)::
     (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil)))::
    R_xml_0_rule_as_list_113.
 
 
 Definition R_xml_0_rule_as_list_115  := 
   ((algebra.Alg.Term algebra.F.id_a__U84 ((algebra.Alg.Var 6)::
     (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil)),
    (algebra.Alg.Term algebra.F.id_U84 ((algebra.Alg.Var 6)::
     (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil)))::
    R_xml_0_rule_as_list_114.
 
 
 Definition R_xml_0_rule_as_list_116  := 
   ((algebra.Alg.Term algebra.F.id_a__U85 ((algebra.Alg.Var 6)::
     (algebra.Alg.Var 7)::nil)),
    (algebra.Alg.Term algebra.F.id_U85 ((algebra.Alg.Var 6)::
     (algebra.Alg.Var 7)::nil)))::R_xml_0_rule_as_list_115.
 
 
 Definition R_xml_0_rule_as_list_117  := 
   ((algebra.Alg.Term algebra.F.id_a__U86 ((algebra.Alg.Var 8)::nil)),
    (algebra.Alg.Term algebra.F.id_U86 ((algebra.Alg.Var 8)::nil)))::
    R_xml_0_rule_as_list_116.
 
 
 Definition R_xml_0_rule_as_list_118  := 
   ((algebra.Alg.Term algebra.F.id_a__U91 ((algebra.Alg.Var 6)::
     (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil)),
    (algebra.Alg.Term algebra.F.id_U91 ((algebra.Alg.Var 6)::
     (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil)))::
    R_xml_0_rule_as_list_117.
 
 
 Definition R_xml_0_rule_as_list_119  := 
   ((algebra.Alg.Term algebra.F.id_a__U92 ((algebra.Alg.Var 6)::
     (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil)),
    (algebra.Alg.Term algebra.F.id_U92 ((algebra.Alg.Var 6)::
     (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil)))::
    R_xml_0_rule_as_list_118.
 
 
 Definition R_xml_0_rule_as_list_120  := 
   ((algebra.Alg.Term algebra.F.id_a__U93 ((algebra.Alg.Var 6)::
     (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil)),
    (algebra.Alg.Term algebra.F.id_U93 ((algebra.Alg.Var 6)::
     (algebra.Alg.Var 7)::(algebra.Alg.Var 9)::nil)))::
    R_xml_0_rule_as_list_119.
 
 
 Definition R_xml_0_rule_as_list_121  := 
   ((algebra.Alg.Term algebra.F.id_a__U94 ((algebra.Alg.Var 6)::
     (algebra.Alg.Var 7)::nil)),
    (algebra.Alg.Term algebra.F.id_U94 ((algebra.Alg.Var 6)::
     (algebra.Alg.Var 7)::nil)))::R_xml_0_rule_as_list_120.
 
 
 Definition R_xml_0_rule_as_list_122  := 
   ((algebra.Alg.Term algebra.F.id_a__length ((algebra.Alg.Var 8)::nil)),
    (algebra.Alg.Term algebra.F.id_length ((algebra.Alg.Var 8)::nil)))::
    R_xml_0_rule_as_list_121.
 
 Definition R_xml_0_rule_as_list  := R_xml_0_rule_as_list_122.
 
 
 Lemma R_xml_0_rules_included :
  forall l r, R_xml_0_rules r l <-> In (l,r) R_xml_0_rule_as_list.
 Proof.
   intros l r.
   constructor.
   intros H.
   
   case H;clear H;
    (apply (more_list.mem_impl_in (@eq (algebra.Alg.term*algebra.Alg.term)));
     [tauto|idtac]);
    match goal with
      |  |- _ _ _ ?t ?l =>
       let u := fresh "u" in 
        (generalize (more_list.mem_bool_ok _ _ 
                      algebra.Alg_ext.eq_term_term_bool_ok t l);
          set (u:=more_list.mem_bool algebra.Alg_ext.eq_term_term_bool t l) in *;
          vm_compute in u|-;unfold u in *;clear u;intros H;refine H)
      end
    .
   intros H.
   vm_compute in H|-.
   rewrite  <- or_ext_generated.or25_equiv in H|-.
   case H;clear H;intros H.
   injection H;intros ;subst;constructor 123.
   injection H;intros ;subst;constructor 122.
   injection H;intros ;subst;constructor 121.
   injection H;intros ;subst;constructor 120.
   injection H;intros ;subst;constructor 119.
   injection H;intros ;subst;constructor 118.
   injection H;intros ;subst;constructor 117.
   injection H;intros ;subst;constructor 116.
   injection H;intros ;subst;constructor 115.
   injection H;intros ;subst;constructor 114.
   injection H;intros ;subst;constructor 113.
   injection H;intros ;subst;constructor 112.
   injection H;intros ;subst;constructor 111.
   injection H;intros ;subst;constructor 110.
   injection H;intros ;subst;constructor 109.
   injection H;intros ;subst;constructor 108.
   injection H;intros ;subst;constructor 107.
   injection H;intros ;subst;constructor 106.
   injection H;intros ;subst;constructor 105.
   injection H;intros ;subst;constructor 104.
   injection H;intros ;subst;constructor 103.
   injection H;intros ;subst;constructor 102.
   injection H;intros ;subst;constructor 101.
   injection H;intros ;subst;constructor 100.
   rewrite  <- or_ext_generated.or25_equiv in H|-.
   case H;clear H;intros H.
   injection H;intros ;subst;constructor 99.
   injection H;intros ;subst;constructor 98.
   injection H;intros ;subst;constructor 97.
   injection H;intros ;subst;constructor 96.
   injection H;intros ;subst;constructor 95.
   injection H;intros ;subst;constructor 94.
   injection H;intros ;subst;constructor 93.
   injection H;intros ;subst;constructor 92.
   injection H;intros ;subst;constructor 91.
   injection H;intros ;subst;constructor 90.
   injection H;intros ;subst;constructor 89.
   injection H;intros ;subst;constructor 88.
   injection H;intros ;subst;constructor 87.
   injection H;intros ;subst;constructor 86.
   injection H;intros ;subst;constructor 85.
   injection H;intros ;subst;constructor 84.
   injection H;intros ;subst;constructor 83.
   injection H;intros ;subst;constructor 82.
   injection H;intros ;subst;constructor 81.
   injection H;intros ;subst;constructor 80.
   injection H;intros ;subst;constructor 79.
   injection H;intros ;subst;constructor 78.
   injection H;intros ;subst;constructor 77.
   injection H;intros ;subst;constructor 76.
   rewrite  <- or_ext_generated.or25_equiv in H|-.
   case H;clear H;intros H.
   injection H;intros ;subst;constructor 75.
   injection H;intros ;subst;constructor 74.
   injection H;intros ;subst;constructor 73.
   injection H;intros ;subst;constructor 72.
   injection H;intros ;subst;constructor 71.
   injection H;intros ;subst;constructor 70.
   injection H;intros ;subst;constructor 69.
   injection H;intros ;subst;constructor 68.
   injection H;intros ;subst;constructor 67.
   injection H;intros ;subst;constructor 66.
   injection H;intros ;subst;constructor 65.
   injection H;intros ;subst;constructor 64.
   injection H;intros ;subst;constructor 63.
   injection H;intros ;subst;constructor 62.
   injection H;intros ;subst;constructor 61.
   injection H;intros ;subst;constructor 60.
   injection H;intros ;subst;constructor 59.
   injection H;intros ;subst;constructor 58.
   injection H;intros ;subst;constructor 57.
   injection H;intros ;subst;constructor 56.
   injection H;intros ;subst;constructor 55.
   injection H;intros ;subst;constructor 54.
   injection H;intros ;subst;constructor 53.
   injection H;intros ;subst;constructor 52.
   rewrite  <- or_ext_generated.or25_equiv in H|-.
   case H;clear H;intros H.
   injection H;intros ;subst;constructor 51.
   injection H;intros ;subst;constructor 50.
   injection H;intros ;subst;constructor 49.
   injection H;intros ;subst;constructor 48.
   injection H;intros ;subst;constructor 47.
   injection H;intros ;subst;constructor 46.
   injection H;intros ;subst;constructor 45.
   injection H;intros ;subst;constructor 44.
   injection H;intros ;subst;constructor 43.
   injection H;intros ;subst;constructor 42.
   injection H;intros ;subst;constructor 41.
   injection H;intros ;subst;constructor 40.
   injection H;intros ;subst;constructor 39.
   injection H;intros ;subst;constructor 38.
   injection H;intros ;subst;constructor 37.
   injection H;intros ;subst;constructor 36.
   injection H;intros ;subst;constructor 35.
   injection H;intros ;subst;constructor 34.
   injection H;intros ;subst;constructor 33.
   injection H;intros ;subst;constructor 32.
   injection H;intros ;subst;constructor 31.
   injection H;intros ;subst;constructor 30.
   injection H;intros ;subst;constructor 29.
   injection H;intros ;subst;constructor 28.
   rewrite  <- or_ext_generated.or25_equiv in H|-.
   case H;clear H;intros H.
   injection H;intros ;subst;constructor 27.
   injection H;intros ;subst;constructor 26.
   injection H;intros ;subst;constructor 25.
   injection H;intros ;subst;constructor 24.
   injection H;intros ;subst;constructor 23.
   injection H;intros ;subst;constructor 22.
   injection H;intros ;subst;constructor 21.
   injection H;intros ;subst;constructor 20.
   injection H;intros ;subst;constructor 19.
   injection H;intros ;subst;constructor 18.
   injection H;intros ;subst;constructor 17.
   injection H;intros ;subst;constructor 16.
   injection H;intros ;subst;constructor 15.
   injection H;intros ;subst;constructor 14.
   injection H;intros ;subst;constructor 13.
   injection H;intros ;subst;constructor 12.
   injection H;intros ;subst;constructor 11.
   injection H;intros ;subst;constructor 10.
   injection H;intros ;subst;constructor 9.
   injection H;intros ;subst;constructor 8.
   injection H;intros ;subst;constructor 7.
   injection H;intros ;subst;constructor 6.
   injection H;intros ;subst;constructor 5.
   injection H;intros ;subst;constructor 4.
   rewrite  <- or_ext_generated.or4_equiv in H|-.
   case H;clear H;intros H.
   injection H;intros ;subst;constructor 3.
   injection H;intros ;subst;constructor 2.
   injection H;intros ;subst;constructor 1.
   elim H.
 Qed.
 
 
 Lemma R_xml_0_non_var : forall x t, ~R_xml_0_rules t (algebra.EQT.T.Var x).
 Proof.
   intros x t H.
   inversion H.
 Qed.
 
 
 Lemma R_xml_0_reg :
  forall s t, 
   (R_xml_0_rules s t) ->
    forall x, In x (algebra.Alg.var_list s) ->In x (algebra.Alg.var_list t).
 Proof.
   intros s t H.
   
   inversion H;intros x Hx;
    (apply (more_list.mem_impl_in (@eq algebra.Alg.variable));[tauto|idtac]);
    apply (more_list.in_impl_mem (@eq algebra.Alg.variable)) in Hx;
    vm_compute in Hx|-*;tauto.
 Qed.
 
 
 Inductive and_41 (x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 x21 x22 x23 
 x24 x25 x26 x27 x28 x29 x30 x31 x32 x33 x34 x35 x36 x37 x38 x39 x40 
 x41 x42 x43 x44 x45 x46 x47 x48 x49 x50 x51:Prop) :
  Prop := 
   | conj_41 :
    x11->x12->x13->x14->x15->x16->x17->x18->x19->x20->x21->x22->x23->
     x24->x25->x26->x27->x28->x29->x30->x31->x32->x33->x34->x35->x36->
     x37->x38->x39->x40->x41->x42->x43->x44->x45->x46->x47->x48->x49->
     x50->x51->
     and_41 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 x21 x22 x23 x24 x25 
      x26 x27 x28 x29 x30 x31 x32 x33 x34 x35 x36 x37 x38 x39 x40 x41 
      x42 x43 x44 x45 x46 x47 x48 x49 x50 x51
 .
 
 
 Lemma are_constuctors_of_R_xml_0 :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    and_41 (forall x12 x14, 
            t = (algebra.Alg.Term algebra.F.id_U51 (x12::x14::nil)) ->
             exists x11,
               exists x13,
                 t' = (algebra.Alg.Term algebra.F.id_U51 (x11::x13::nil))/\ 
                 (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
                   x11 x12)/\ 
                 (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
                   x13 x14)) 
     (forall x12, 
      t = (algebra.Alg.Term algebra.F.id_isNatList (x12::nil)) ->
       exists x11,
         t' = (algebra.Alg.Term algebra.F.id_isNatList (x11::nil))/\ 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
           x11 x12)) 
     (forall x12 x14, 
      t = (algebra.Alg.Term algebra.F.id_U85 (x12::x14::nil)) ->
       exists x11,
         exists x13,
           t' = (algebra.Alg.Term algebra.F.id_U85 (x11::x13::nil))/\ 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
             x11 x12)/\ 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
             x13 x14)) 
     (forall x12, 
      t = (algebra.Alg.Term algebra.F.id_U33 (x12::nil)) ->
       exists x11,
         t' = (algebra.Alg.Term algebra.F.id_U33 (x11::nil))/\ 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
           x11 x12)) 
     (forall x12 x14 x16, 
      t = (algebra.Alg.Term algebra.F.id_U81 (x12::x14::x16::nil)) ->
       exists x11,
         exists x13,
           exists x15,
             t' = (algebra.Alg.Term algebra.F.id_U81 (x11::x13::x15::nil))/\ 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
               x11 x12)/\ 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
               x13 x14)/\ 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
               x15 x16)) 
     (forall x12, 
      t = (algebra.Alg.Term algebra.F.id_U23 (x12::nil)) ->
       exists x11,
         t' = (algebra.Alg.Term algebra.F.id_U23 (x11::nil))/\ 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
           x11 x12)) 
     (forall x12 x14 x16, 
      t = (algebra.Alg.Term algebra.F.id_U93 (x12::x14::x16::nil)) ->
       exists x11,
         exists x13,
           exists x15,
             t' = (algebra.Alg.Term algebra.F.id_U93 (x11::x13::x15::nil))/\ 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
               x11 x12)/\ 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
               x13 x14)/\ 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
               x15 x16)) 
     (forall x12 x14, 
      t = (algebra.Alg.Term algebra.F.id_U11 (x12::x14::nil)) ->
       exists x11,
         exists x13,
           t' = (algebra.Alg.Term algebra.F.id_U11 (x11::x13::nil))/\ 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
             x11 x12)/\ 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
             x13 x14)) 
     (forall x12 x14 x16, 
      t = (algebra.Alg.Term algebra.F.id_U44 (x12::x14::x16::nil)) ->
       exists x11,
         exists x13,
           exists x15,
             t' = (algebra.Alg.Term algebra.F.id_U44 (x11::x13::x15::nil))/\ 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
               x11 x12)/\ 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
               x13 x14)/\ 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
               x15 x16)) 
     (t = (algebra.Alg.Term algebra.F.id_0 nil) ->
      t' = (algebra.Alg.Term algebra.F.id_0 nil)) 
     (forall x12, 
      t = (algebra.Alg.Term algebra.F.id_U61 (x12::nil)) ->
       exists x11,
         t' = (algebra.Alg.Term algebra.F.id_U61 (x11::nil))/\ 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
           x11 x12)) 
     (forall x12 x14, 
      t = (algebra.Alg.Term algebra.F.id_U22 (x12::x14::nil)) ->
       exists x11,
         exists x13,
           t' = (algebra.Alg.Term algebra.F.id_U22 (x11::x13::nil))/\ 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
             x11 x12)/\ 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
             x13 x14)) 
     (forall x12 x14 x16, 
      t = (algebra.Alg.Term algebra.F.id_U91 (x12::x14::x16::nil)) ->
       exists x11,
         exists x13,
           exists x15,
             t' = (algebra.Alg.Term algebra.F.id_U91 (x11::x13::x15::nil))/\ 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
               x11 x12)/\ 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
               x13 x14)/\ 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
               x15 x16)) 
     (forall x12, 
      t = (algebra.Alg.Term algebra.F.id_length (x12::nil)) ->
       exists x11,
         t' = (algebra.Alg.Term algebra.F.id_length (x11::nil))/\ 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
           x11 x12)) 
     (forall x12 x14 x16, 
      t = (algebra.Alg.Term algebra.F.id_U42 (x12::x14::x16::nil)) ->
       exists x11,
         exists x13,
           exists x15,
             t' = (algebra.Alg.Term algebra.F.id_U42 (x11::x13::x15::nil))/\ 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
               x11 x12)/\ 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
               x13 x14)/\ 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
               x15 x16)) 
     (forall x12 x14 x16, 
      t = (algebra.Alg.Term algebra.F.id_U83 (x12::x14::x16::nil)) ->
       exists x11,
         exists x13,
           exists x15,
             t' = (algebra.Alg.Term algebra.F.id_U83 (x11::x13::x15::nil))/\ 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
               x11 x12)/\ 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
               x13 x14)/\ 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
               x15 x16)) 
     (forall x12 x14, 
      t = (algebra.Alg.Term algebra.F.id_U31 (x12::x14::nil)) ->
       exists x11,
         exists x13,
           t' = (algebra.Alg.Term algebra.F.id_U31 (x11::x13::nil))/\ 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
             x11 x12)/\ 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
             x13 x14)) 
     (forall x12, 
      t = (algebra.Alg.Term algebra.F.id_isNatIListKind (x12::nil)) ->
       exists x11,
         t' = (algebra.Alg.Term algebra.F.id_isNatIListKind (x11::nil))/\ 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
           x11 x12)) 
     (forall x12, 
      t = (algebra.Alg.Term algebra.F.id_U46 (x12::nil)) ->
       exists x11,
         t' = (algebra.Alg.Term algebra.F.id_U46 (x11::nil))/\ 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
           x11 x12)) 
     (forall x12 x14, 
      t = (algebra.Alg.Term algebra.F.id_cons (x12::x14::nil)) ->
       exists x11,
         exists x13,
           t' = (algebra.Alg.Term algebra.F.id_cons (x11::x13::nil))/\ 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
             x11 x12)/\ 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
             x13 x14)) 
     (forall x12, 
      t = (algebra.Alg.Term algebra.F.id_U52 (x12::nil)) ->
       exists x11,
         t' = (algebra.Alg.Term algebra.F.id_U52 (x11::nil))/\ 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
           x11 x12)) 
     (forall x12 x14, 
      t = (algebra.Alg.Term algebra.F.id_U21 (x12::x14::nil)) ->
       exists x11,
         exists x13,
           t' = (algebra.Alg.Term algebra.F.id_U21 (x11::x13::nil))/\ 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
             x11 x12)/\ 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
             x13 x14)) 
     (forall x12, 
      t = (algebra.Alg.Term algebra.F.id_U86 (x12::nil)) ->
       exists x11,
         t' = (algebra.Alg.Term algebra.F.id_U86 (x11::nil))/\ 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
           x11 x12)) 
     (forall x12 x14 x16, 
      t = (algebra.Alg.Term algebra.F.id_U41 (x12::x14::x16::nil)) ->
       exists x11,
         exists x13,
           exists x15,
             t' = (algebra.Alg.Term algebra.F.id_U41 (x11::x13::x15::nil))/\ 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
               x11 x12)/\ 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
               x13 x14)/\ 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
               x15 x16)) 
     (t = (algebra.Alg.Term algebra.F.id_tt nil) ->
      t' = (algebra.Alg.Term algebra.F.id_tt nil)) 
     (forall x12 x14 x16, 
      t = (algebra.Alg.Term algebra.F.id_U82 (x12::x14::x16::nil)) ->
       exists x11,
         exists x13,
           exists x15,
             t' = (algebra.Alg.Term algebra.F.id_U82 (x11::x13::x15::nil))/\ 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
               x11 x12)/\ 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
               x13 x14)/\ 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
               x15 x16)) 
     (forall x12, 
      t = (algebra.Alg.Term algebra.F.id_isNat (x12::nil)) ->
       exists x11,
         t' = (algebra.Alg.Term algebra.F.id_isNat (x11::nil))/\ 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
           x11 x12)) 
     (forall x12 x14, 
      t = (algebra.Alg.Term algebra.F.id_U94 (x12::x14::nil)) ->
       exists x11,
         exists x13,
           t' = (algebra.Alg.Term algebra.F.id_U94 (x11::x13::nil))/\ 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
             x11 x12)/\ 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
             x13 x14)) 
     (forall x12 x14, 
      t = (algebra.Alg.Term algebra.F.id_U12 (x12::x14::nil)) ->
       exists x11,
         exists x13,
           t' = (algebra.Alg.Term algebra.F.id_U12 (x11::x13::nil))/\ 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
             x11 x12)/\ 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
             x13 x14)) 
     (forall x12 x14, 
      t = (algebra.Alg.Term algebra.F.id_U45 (x12::x14::nil)) ->
       exists x11,
         exists x13,
           t' = (algebra.Alg.Term algebra.F.id_U45 (x11::x13::nil))/\ 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
             x11 x12)/\ 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
             x13 x14)) 
     (t = (algebra.Alg.Term algebra.F.id_zeros nil) ->
      t' = (algebra.Alg.Term algebra.F.id_zeros nil)) 
     (forall x12, 
      t = (algebra.Alg.Term algebra.F.id_U71 (x12::nil)) ->
       exists x11,
         t' = (algebra.Alg.Term algebra.F.id_U71 (x11::nil))/\ 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
           x11 x12)) 
     (forall x12, 
      t = (algebra.Alg.Term algebra.F.id_isNatKind (x12::nil)) ->
       exists x11,
         t' = (algebra.Alg.Term algebra.F.id_isNatKind (x11::nil))/\ 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
           x11 x12)) 
     (forall x12 x14 x16, 
      t = (algebra.Alg.Term algebra.F.id_U92 (x12::x14::x16::nil)) ->
       exists x11,
         exists x13,
           exists x15,
             t' = (algebra.Alg.Term algebra.F.id_U92 (x11::x13::x15::nil))/\ 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
               x11 x12)/\ 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
               x13 x14)/\ 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
               x15 x16)) 
     (t = (algebra.Alg.Term algebra.F.id_nil nil) ->
      t' = (algebra.Alg.Term algebra.F.id_nil nil)) 
     (forall x12 x14 x16, 
      t = (algebra.Alg.Term algebra.F.id_U43 (x12::x14::x16::nil)) ->
       exists x11,
         exists x13,
           exists x15,
             t' = (algebra.Alg.Term algebra.F.id_U43 (x11::x13::x15::nil))/\ 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
               x11 x12)/\ 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
               x13 x14)/\ 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
               x15 x16)) 
     (forall x12 x14 x16, 
      t = (algebra.Alg.Term algebra.F.id_U84 (x12::x14::x16::nil)) ->
       exists x11,
         exists x13,
           exists x15,
             t' = (algebra.Alg.Term algebra.F.id_U84 (x11::x13::x15::nil))/\ 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
               x11 x12)/\ 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
               x13 x14)/\ 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
               x15 x16)) 
     (forall x12, 
      t = (algebra.Alg.Term algebra.F.id_s (x12::nil)) ->
       exists x11,
         t' = (algebra.Alg.Term algebra.F.id_s (x11::nil))/\ 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
           x11 x12)) 
     (forall x12 x14, 
      t = (algebra.Alg.Term algebra.F.id_U32 (x12::x14::nil)) ->
       exists x11,
         exists x13,
           t' = (algebra.Alg.Term algebra.F.id_U32 (x11::x13::nil))/\ 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
             x11 x12)/\ 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
             x13 x14)) 
     (forall x12, 
      t = (algebra.Alg.Term algebra.F.id_U13 (x12::nil)) ->
       exists x11,
         t' = (algebra.Alg.Term algebra.F.id_U13 (x11::nil))/\ 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
           x11 x12)) 
     (forall x12, 
      t = (algebra.Alg.Term algebra.F.id_isNatIList (x12::nil)) ->
       exists x11,
         t' = (algebra.Alg.Term algebra.F.id_isNatIList (x11::nil))/\ 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
           x11 x12)).
 Proof.
   intros t t' H.
   
   induction H as [|y IH z z_to_y] using 
   closure_extension.refl_trans_clos_ind2.
   constructor 1.
   intros x12 x14 H;exists x12;exists x14;intuition;constructor 1.
   intros x12 H;exists x12;intuition;constructor 1.
   intros x12 x14 H;exists x12;exists x14;intuition;constructor 1.
   intros x12 H;exists x12;intuition;constructor 1.
   
   intros x12 x14 x16 H;exists x12;exists x14;exists x16;intuition;
    constructor 1.
   intros x12 H;exists x12;intuition;constructor 1.
   
   intros x12 x14 x16 H;exists x12;exists x14;exists x16;intuition;
    constructor 1.
   intros x12 x14 H;exists x12;exists x14;intuition;constructor 1.
   
   intros x12 x14 x16 H;exists x12;exists x14;exists x16;intuition;
    constructor 1.
   intros H;intuition;constructor 1.
   intros x12 H;exists x12;intuition;constructor 1.
   intros x12 x14 H;exists x12;exists x14;intuition;constructor 1.
   
   intros x12 x14 x16 H;exists x12;exists x14;exists x16;intuition;
    constructor 1.
   intros x12 H;exists x12;intuition;constructor 1.
   
   intros x12 x14 x16 H;exists x12;exists x14;exists x16;intuition;
    constructor 1.
   
   intros x12 x14 x16 H;exists x12;exists x14;exists x16;intuition;
    constructor 1.
   intros x12 x14 H;exists x12;exists x14;intuition;constructor 1.
   intros x12 H;exists x12;intuition;constructor 1.
   intros x12 H;exists x12;intuition;constructor 1.
   intros x12 x14 H;exists x12;exists x14;intuition;constructor 1.
   intros x12 H;exists x12;intuition;constructor 1.
   intros x12 x14 H;exists x12;exists x14;intuition;constructor 1.
   intros x12 H;exists x12;intuition;constructor 1.
   
   intros x12 x14 x16 H;exists x12;exists x14;exists x16;intuition;
    constructor 1.
   intros H;intuition;constructor 1.
   
   intros x12 x14 x16 H;exists x12;exists x14;exists x16;intuition;
    constructor 1.
   intros x12 H;exists x12;intuition;constructor 1.
   intros x12 x14 H;exists x12;exists x14;intuition;constructor 1.
   intros x12 x14 H;exists x12;exists x14;intuition;constructor 1.
   intros x12 x14 H;exists x12;exists x14;intuition;constructor 1.
   intros H;intuition;constructor 1.
   intros x12 H;exists x12;intuition;constructor 1.
   intros x12 H;exists x12;intuition;constructor 1.
   
   intros x12 x14 x16 H;exists x12;exists x14;exists x16;intuition;
    constructor 1.
   intros H;intuition;constructor 1.
   
   intros x12 x14 x16 H;exists x12;exists x14;exists x16;intuition;
    constructor 1.
   
   intros x12 x14 x16 H;exists x12;exists x14;exists x16;intuition;
    constructor 1.
   intros x12 H;exists x12;intuition;constructor 1.
   intros x12 x14 H;exists x12;exists x14;intuition;constructor 1.
   intros x12 H;exists x12;intuition;constructor 1.
   intros x12 H;exists x12;intuition;constructor 1.
   inversion z_to_y as [t1 t2 H H0 H1|f l1 l2 H0 H H2];clear z_to_y;subst.
   
   inversion H as [t1 t2 sigma H2 H1 H0];clear H IH;subst;inversion H2;
    clear ;constructor;try (intros until 0 );clear ;intros abs;
    discriminate abs.
   
   destruct IH as 
   [H_id_U51 H_id_isNatList H_id_U85 H_id_U33 H_id_U81 H_id_U23 H_id_U93 
    H_id_U11 H_id_U44 H_id_0 H_id_U61 H_id_U22 H_id_U91 H_id_length H_id_U42 
    H_id_U83 H_id_U31 H_id_isNatIListKind H_id_U46 H_id_cons H_id_U52 
    H_id_U21 H_id_U86 H_id_U41 H_id_tt H_id_U82 H_id_isNat H_id_U94 H_id_U12 
    H_id_U45 H_id_zeros H_id_U71 H_id_isNatKind H_id_U92 H_id_nil H_id_U43 
    H_id_U84 H_id_s H_id_U32 H_id_U13 H_id_isNatIList].
   constructor.
   
   clear H_id_isNatList H_id_U85 H_id_U33 H_id_U81 H_id_U23 H_id_U93 
   H_id_U11 H_id_U44 H_id_0 H_id_U61 H_id_U22 H_id_U91 H_id_length H_id_U42 
   H_id_U83 H_id_U31 H_id_isNatIListKind H_id_U46 H_id_cons H_id_U52 
   H_id_U21 H_id_U86 H_id_U41 H_id_tt H_id_U82 H_id_isNat H_id_U94 H_id_U12 
   H_id_U45 H_id_zeros H_id_U71 H_id_isNatKind H_id_U92 H_id_nil H_id_U43 
   H_id_U84 H_id_s H_id_U32 H_id_U13 H_id_isNatIList;intros x12 x14 H;
    injection H;clear H;intros ;subst;
    repeat (
    match goal with
      | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ =>
       inversion H;clear H;subst
      end
    ).
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x12 |- _ =>
      destruct (H_id_U51 y x14 (refl_equal _)) as [x11 [x13]];intros ;
       intuition;exists x11;exists x13;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x14 |- _ =>
      destruct (H_id_U51 x12 y (refl_equal _)) as [x11 [x13]];intros ;
       intuition;exists x11;exists x13;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   clear H_id_U51 H_id_U85 H_id_U33 H_id_U81 H_id_U23 H_id_U93 H_id_U11 
   H_id_U44 H_id_0 H_id_U61 H_id_U22 H_id_U91 H_id_length H_id_U42 H_id_U83 
   H_id_U31 H_id_isNatIListKind H_id_U46 H_id_cons H_id_U52 H_id_U21 
   H_id_U86 H_id_U41 H_id_tt H_id_U82 H_id_isNat H_id_U94 H_id_U12 H_id_U45 
   H_id_zeros H_id_U71 H_id_isNatKind H_id_U92 H_id_nil H_id_U43 H_id_U84 
   H_id_s H_id_U32 H_id_U13 H_id_isNatIList;intros x12 H;injection H;
    clear H;intros ;subst;
    repeat (
    match goal with
      | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ =>
       inversion H;clear H;subst
      end
    ).
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x12 |- _ =>
      destruct (H_id_isNatList y (refl_equal _)) as [x11];intros ;intuition;
       exists x11;intuition;eapply closure_extension.refl_trans_clos_R;
       eassumption
     end
   .
   
   clear H_id_U51 H_id_isNatList H_id_U33 H_id_U81 H_id_U23 H_id_U93 
   H_id_U11 H_id_U44 H_id_0 H_id_U61 H_id_U22 H_id_U91 H_id_length H_id_U42 
   H_id_U83 H_id_U31 H_id_isNatIListKind H_id_U46 H_id_cons H_id_U52 
   H_id_U21 H_id_U86 H_id_U41 H_id_tt H_id_U82 H_id_isNat H_id_U94 H_id_U12 
   H_id_U45 H_id_zeros H_id_U71 H_id_isNatKind H_id_U92 H_id_nil H_id_U43 
   H_id_U84 H_id_s H_id_U32 H_id_U13 H_id_isNatIList;intros x12 x14 H;
    injection H;clear H;intros ;subst;
    repeat (
    match goal with
      | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ =>
       inversion H;clear H;subst
      end
    ).
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x12 |- _ =>
      destruct (H_id_U85 y x14 (refl_equal _)) as [x11 [x13]];intros ;
       intuition;exists x11;exists x13;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x14 |- _ =>
      destruct (H_id_U85 x12 y (refl_equal _)) as [x11 [x13]];intros ;
       intuition;exists x11;exists x13;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   clear H_id_U51 H_id_isNatList H_id_U85 H_id_U81 H_id_U23 H_id_U93 
   H_id_U11 H_id_U44 H_id_0 H_id_U61 H_id_U22 H_id_U91 H_id_length H_id_U42 
   H_id_U83 H_id_U31 H_id_isNatIListKind H_id_U46 H_id_cons H_id_U52 
   H_id_U21 H_id_U86 H_id_U41 H_id_tt H_id_U82 H_id_isNat H_id_U94 H_id_U12 
   H_id_U45 H_id_zeros H_id_U71 H_id_isNatKind H_id_U92 H_id_nil H_id_U43 
   H_id_U84 H_id_s H_id_U32 H_id_U13 H_id_isNatIList;intros x12 H;
    injection H;clear H;intros ;subst;
    repeat (
    match goal with
      | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ =>
       inversion H;clear H;subst
      end
    ).
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x12 |- _ =>
      destruct (H_id_U33 y (refl_equal _)) as [x11];intros ;intuition;
       exists x11;intuition;eapply closure_extension.refl_trans_clos_R;
       eassumption
     end
   .
   
   clear H_id_U51 H_id_isNatList H_id_U85 H_id_U33 H_id_U23 H_id_U93 
   H_id_U11 H_id_U44 H_id_0 H_id_U61 H_id_U22 H_id_U91 H_id_length H_id_U42 
   H_id_U83 H_id_U31 H_id_isNatIListKind H_id_U46 H_id_cons H_id_U52 
   H_id_U21 H_id_U86 H_id_U41 H_id_tt H_id_U82 H_id_isNat H_id_U94 H_id_U12 
   H_id_U45 H_id_zeros H_id_U71 H_id_isNatKind H_id_U92 H_id_nil H_id_U43 
   H_id_U84 H_id_s H_id_U32 H_id_U13 H_id_isNatIList;intros x12 x14 x16 H;
    injection H;clear H;intros ;subst;
    repeat (
    match goal with
      | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ =>
       inversion H;clear H;subst
      end
    ).
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x12 |- _ =>
      destruct (H_id_U81 y x14 x16 (refl_equal _)) as [x11 [x13 [x15]]];
       intros ;intuition;exists x11;exists x13;exists x15;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x14 |- _ =>
      destruct (H_id_U81 x12 y x16 (refl_equal _)) as [x11 [x13 [x15]]];
       intros ;intuition;exists x11;exists x13;exists x15;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x16 |- _ =>
      destruct (H_id_U81 x12 x14 y (refl_equal _)) as [x11 [x13 [x15]]];
       intros ;intuition;exists x11;exists x13;exists x15;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   clear H_id_U51 H_id_isNatList H_id_U85 H_id_U33 H_id_U81 H_id_U93 
   H_id_U11 H_id_U44 H_id_0 H_id_U61 H_id_U22 H_id_U91 H_id_length H_id_U42 
   H_id_U83 H_id_U31 H_id_isNatIListKind H_id_U46 H_id_cons H_id_U52 
   H_id_U21 H_id_U86 H_id_U41 H_id_tt H_id_U82 H_id_isNat H_id_U94 H_id_U12 
   H_id_U45 H_id_zeros H_id_U71 H_id_isNatKind H_id_U92 H_id_nil H_id_U43 
   H_id_U84 H_id_s H_id_U32 H_id_U13 H_id_isNatIList;intros x12 H;
    injection H;clear H;intros ;subst;
    repeat (
    match goal with
      | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ =>
       inversion H;clear H;subst
      end
    ).
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x12 |- _ =>
      destruct (H_id_U23 y (refl_equal _)) as [x11];intros ;intuition;
       exists x11;intuition;eapply closure_extension.refl_trans_clos_R;
       eassumption
     end
   .
   
   clear H_id_U51 H_id_isNatList H_id_U85 H_id_U33 H_id_U81 H_id_U23 
   H_id_U11 H_id_U44 H_id_0 H_id_U61 H_id_U22 H_id_U91 H_id_length H_id_U42 
   H_id_U83 H_id_U31 H_id_isNatIListKind H_id_U46 H_id_cons H_id_U52 
   H_id_U21 H_id_U86 H_id_U41 H_id_tt H_id_U82 H_id_isNat H_id_U94 H_id_U12 
   H_id_U45 H_id_zeros H_id_U71 H_id_isNatKind H_id_U92 H_id_nil H_id_U43 
   H_id_U84 H_id_s H_id_U32 H_id_U13 H_id_isNatIList;intros x12 x14 x16 H;
    injection H;clear H;intros ;subst;
    repeat (
    match goal with
      | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ =>
       inversion H;clear H;subst
      end
    ).
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x12 |- _ =>
      destruct (H_id_U93 y x14 x16 (refl_equal _)) as [x11 [x13 [x15]]];
       intros ;intuition;exists x11;exists x13;exists x15;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x14 |- _ =>
      destruct (H_id_U93 x12 y x16 (refl_equal _)) as [x11 [x13 [x15]]];
       intros ;intuition;exists x11;exists x13;exists x15;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x16 |- _ =>
      destruct (H_id_U93 x12 x14 y (refl_equal _)) as [x11 [x13 [x15]]];
       intros ;intuition;exists x11;exists x13;exists x15;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   clear H_id_U51 H_id_isNatList H_id_U85 H_id_U33 H_id_U81 H_id_U23 
   H_id_U93 H_id_U44 H_id_0 H_id_U61 H_id_U22 H_id_U91 H_id_length H_id_U42 
   H_id_U83 H_id_U31 H_id_isNatIListKind H_id_U46 H_id_cons H_id_U52 
   H_id_U21 H_id_U86 H_id_U41 H_id_tt H_id_U82 H_id_isNat H_id_U94 H_id_U12 
   H_id_U45 H_id_zeros H_id_U71 H_id_isNatKind H_id_U92 H_id_nil H_id_U43 
   H_id_U84 H_id_s H_id_U32 H_id_U13 H_id_isNatIList;intros x12 x14 H;
    injection H;clear H;intros ;subst;
    repeat (
    match goal with
      | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ =>
       inversion H;clear H;subst
      end
    ).
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x12 |- _ =>
      destruct (H_id_U11 y x14 (refl_equal _)) as [x11 [x13]];intros ;
       intuition;exists x11;exists x13;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x14 |- _ =>
      destruct (H_id_U11 x12 y (refl_equal _)) as [x11 [x13]];intros ;
       intuition;exists x11;exists x13;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   clear H_id_U51 H_id_isNatList H_id_U85 H_id_U33 H_id_U81 H_id_U23 
   H_id_U93 H_id_U11 H_id_0 H_id_U61 H_id_U22 H_id_U91 H_id_length H_id_U42 
   H_id_U83 H_id_U31 H_id_isNatIListKind H_id_U46 H_id_cons H_id_U52 
   H_id_U21 H_id_U86 H_id_U41 H_id_tt H_id_U82 H_id_isNat H_id_U94 H_id_U12 
   H_id_U45 H_id_zeros H_id_U71 H_id_isNatKind H_id_U92 H_id_nil H_id_U43 
   H_id_U84 H_id_s H_id_U32 H_id_U13 H_id_isNatIList;intros x12 x14 x16 H;
    injection H;clear H;intros ;subst;
    repeat (
    match goal with
      | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ =>
       inversion H;clear H;subst
      end
    ).
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x12 |- _ =>
      destruct (H_id_U44 y x14 x16 (refl_equal _)) as [x11 [x13 [x15]]];
       intros ;intuition;exists x11;exists x13;exists x15;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x14 |- _ =>
      destruct (H_id_U44 x12 y x16 (refl_equal _)) as [x11 [x13 [x15]]];
       intros ;intuition;exists x11;exists x13;exists x15;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x16 |- _ =>
      destruct (H_id_U44 x12 x14 y (refl_equal _)) as [x11 [x13 [x15]]];
       intros ;intuition;exists x11;exists x13;exists x15;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   clear H_id_U51 H_id_isNatList H_id_U85 H_id_U33 H_id_U81 H_id_U23 
   H_id_U93 H_id_U11 H_id_U44 H_id_U61 H_id_U22 H_id_U91 H_id_length 
   H_id_U42 H_id_U83 H_id_U31 H_id_isNatIListKind H_id_U46 H_id_cons 
   H_id_U52 H_id_U21 H_id_U86 H_id_U41 H_id_tt H_id_U82 H_id_isNat H_id_U94 
   H_id_U12 H_id_U45 H_id_zeros H_id_U71 H_id_isNatKind H_id_U92 H_id_nil 
   H_id_U43 H_id_U84 H_id_s H_id_U32 H_id_U13 H_id_isNatIList;intros H;
    injection H;clear H;intros ;subst;
    repeat (
    match goal with
      | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ =>
       inversion H;clear H;subst
      end
    ).
   
   clear H_id_U51 H_id_isNatList H_id_U85 H_id_U33 H_id_U81 H_id_U23 
   H_id_U93 H_id_U11 H_id_U44 H_id_0 H_id_U22 H_id_U91 H_id_length H_id_U42 
   H_id_U83 H_id_U31 H_id_isNatIListKind H_id_U46 H_id_cons H_id_U52 
   H_id_U21 H_id_U86 H_id_U41 H_id_tt H_id_U82 H_id_isNat H_id_U94 H_id_U12 
   H_id_U45 H_id_zeros H_id_U71 H_id_isNatKind H_id_U92 H_id_nil H_id_U43 
   H_id_U84 H_id_s H_id_U32 H_id_U13 H_id_isNatIList;intros x12 H;
    injection H;clear H;intros ;subst;
    repeat (
    match goal with
      | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ =>
       inversion H;clear H;subst
      end
    ).
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x12 |- _ =>
      destruct (H_id_U61 y (refl_equal _)) as [x11];intros ;intuition;
       exists x11;intuition;eapply closure_extension.refl_trans_clos_R;
       eassumption
     end
   .
   
   clear H_id_U51 H_id_isNatList H_id_U85 H_id_U33 H_id_U81 H_id_U23 
   H_id_U93 H_id_U11 H_id_U44 H_id_0 H_id_U61 H_id_U91 H_id_length H_id_U42 
   H_id_U83 H_id_U31 H_id_isNatIListKind H_id_U46 H_id_cons H_id_U52 
   H_id_U21 H_id_U86 H_id_U41 H_id_tt H_id_U82 H_id_isNat H_id_U94 H_id_U12 
   H_id_U45 H_id_zeros H_id_U71 H_id_isNatKind H_id_U92 H_id_nil H_id_U43 
   H_id_U84 H_id_s H_id_U32 H_id_U13 H_id_isNatIList;intros x12 x14 H;
    injection H;clear H;intros ;subst;
    repeat (
    match goal with
      | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ =>
       inversion H;clear H;subst
      end
    ).
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x12 |- _ =>
      destruct (H_id_U22 y x14 (refl_equal _)) as [x11 [x13]];intros ;
       intuition;exists x11;exists x13;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x14 |- _ =>
      destruct (H_id_U22 x12 y (refl_equal _)) as [x11 [x13]];intros ;
       intuition;exists x11;exists x13;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   clear H_id_U51 H_id_isNatList H_id_U85 H_id_U33 H_id_U81 H_id_U23 
   H_id_U93 H_id_U11 H_id_U44 H_id_0 H_id_U61 H_id_U22 H_id_length H_id_U42 
   H_id_U83 H_id_U31 H_id_isNatIListKind H_id_U46 H_id_cons H_id_U52 
   H_id_U21 H_id_U86 H_id_U41 H_id_tt H_id_U82 H_id_isNat H_id_U94 H_id_U12 
   H_id_U45 H_id_zeros H_id_U71 H_id_isNatKind H_id_U92 H_id_nil H_id_U43 
   H_id_U84 H_id_s H_id_U32 H_id_U13 H_id_isNatIList;intros x12 x14 x16 H;
    injection H;clear H;intros ;subst;
    repeat (
    match goal with
      | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ =>
       inversion H;clear H;subst
      end
    ).
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x12 |- _ =>
      destruct (H_id_U91 y x14 x16 (refl_equal _)) as [x11 [x13 [x15]]];
       intros ;intuition;exists x11;exists x13;exists x15;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x14 |- _ =>
      destruct (H_id_U91 x12 y x16 (refl_equal _)) as [x11 [x13 [x15]]];
       intros ;intuition;exists x11;exists x13;exists x15;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x16 |- _ =>
      destruct (H_id_U91 x12 x14 y (refl_equal _)) as [x11 [x13 [x15]]];
       intros ;intuition;exists x11;exists x13;exists x15;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   clear H_id_U51 H_id_isNatList H_id_U85 H_id_U33 H_id_U81 H_id_U23 
   H_id_U93 H_id_U11 H_id_U44 H_id_0 H_id_U61 H_id_U22 H_id_U91 H_id_U42 
   H_id_U83 H_id_U31 H_id_isNatIListKind H_id_U46 H_id_cons H_id_U52 
   H_id_U21 H_id_U86 H_id_U41 H_id_tt H_id_U82 H_id_isNat H_id_U94 H_id_U12 
   H_id_U45 H_id_zeros H_id_U71 H_id_isNatKind H_id_U92 H_id_nil H_id_U43 
   H_id_U84 H_id_s H_id_U32 H_id_U13 H_id_isNatIList;intros x12 H;
    injection H;clear H;intros ;subst;
    repeat (
    match goal with
      | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ =>
       inversion H;clear H;subst
      end
    ).
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x12 |- _ =>
      destruct (H_id_length y (refl_equal _)) as [x11];intros ;intuition;
       exists x11;intuition;eapply closure_extension.refl_trans_clos_R;
       eassumption
     end
   .
   
   clear H_id_U51 H_id_isNatList H_id_U85 H_id_U33 H_id_U81 H_id_U23 
   H_id_U93 H_id_U11 H_id_U44 H_id_0 H_id_U61 H_id_U22 H_id_U91 H_id_length 
   H_id_U83 H_id_U31 H_id_isNatIListKind H_id_U46 H_id_cons H_id_U52 
   H_id_U21 H_id_U86 H_id_U41 H_id_tt H_id_U82 H_id_isNat H_id_U94 H_id_U12 
   H_id_U45 H_id_zeros H_id_U71 H_id_isNatKind H_id_U92 H_id_nil H_id_U43 
   H_id_U84 H_id_s H_id_U32 H_id_U13 H_id_isNatIList;intros x12 x14 x16 H;
    injection H;clear H;intros ;subst;
    repeat (
    match goal with
      | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ =>
       inversion H;clear H;subst
      end
    ).
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x12 |- _ =>
      destruct (H_id_U42 y x14 x16 (refl_equal _)) as [x11 [x13 [x15]]];
       intros ;intuition;exists x11;exists x13;exists x15;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x14 |- _ =>
      destruct (H_id_U42 x12 y x16 (refl_equal _)) as [x11 [x13 [x15]]];
       intros ;intuition;exists x11;exists x13;exists x15;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x16 |- _ =>
      destruct (H_id_U42 x12 x14 y (refl_equal _)) as [x11 [x13 [x15]]];
       intros ;intuition;exists x11;exists x13;exists x15;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   clear H_id_U51 H_id_isNatList H_id_U85 H_id_U33 H_id_U81 H_id_U23 
   H_id_U93 H_id_U11 H_id_U44 H_id_0 H_id_U61 H_id_U22 H_id_U91 H_id_length 
   H_id_U42 H_id_U31 H_id_isNatIListKind H_id_U46 H_id_cons H_id_U52 
   H_id_U21 H_id_U86 H_id_U41 H_id_tt H_id_U82 H_id_isNat H_id_U94 H_id_U12 
   H_id_U45 H_id_zeros H_id_U71 H_id_isNatKind H_id_U92 H_id_nil H_id_U43 
   H_id_U84 H_id_s H_id_U32 H_id_U13 H_id_isNatIList;intros x12 x14 x16 H;
    injection H;clear H;intros ;subst;
    repeat (
    match goal with
      | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ =>
       inversion H;clear H;subst
      end
    ).
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x12 |- _ =>
      destruct (H_id_U83 y x14 x16 (refl_equal _)) as [x11 [x13 [x15]]];
       intros ;intuition;exists x11;exists x13;exists x15;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x14 |- _ =>
      destruct (H_id_U83 x12 y x16 (refl_equal _)) as [x11 [x13 [x15]]];
       intros ;intuition;exists x11;exists x13;exists x15;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x16 |- _ =>
      destruct (H_id_U83 x12 x14 y (refl_equal _)) as [x11 [x13 [x15]]];
       intros ;intuition;exists x11;exists x13;exists x15;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   clear H_id_U51 H_id_isNatList H_id_U85 H_id_U33 H_id_U81 H_id_U23 
   H_id_U93 H_id_U11 H_id_U44 H_id_0 H_id_U61 H_id_U22 H_id_U91 H_id_length 
   H_id_U42 H_id_U83 H_id_isNatIListKind H_id_U46 H_id_cons H_id_U52 
   H_id_U21 H_id_U86 H_id_U41 H_id_tt H_id_U82 H_id_isNat H_id_U94 H_id_U12 
   H_id_U45 H_id_zeros H_id_U71 H_id_isNatKind H_id_U92 H_id_nil H_id_U43 
   H_id_U84 H_id_s H_id_U32 H_id_U13 H_id_isNatIList;intros x12 x14 H;
    injection H;clear H;intros ;subst;
    repeat (
    match goal with
      | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ =>
       inversion H;clear H;subst
      end
    ).
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x12 |- _ =>
      destruct (H_id_U31 y x14 (refl_equal _)) as [x11 [x13]];intros ;
       intuition;exists x11;exists x13;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x14 |- _ =>
      destruct (H_id_U31 x12 y (refl_equal _)) as [x11 [x13]];intros ;
       intuition;exists x11;exists x13;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   clear H_id_U51 H_id_isNatList H_id_U85 H_id_U33 H_id_U81 H_id_U23 
   H_id_U93 H_id_U11 H_id_U44 H_id_0 H_id_U61 H_id_U22 H_id_U91 H_id_length 
   H_id_U42 H_id_U83 H_id_U31 H_id_U46 H_id_cons H_id_U52 H_id_U21 H_id_U86 
   H_id_U41 H_id_tt H_id_U82 H_id_isNat H_id_U94 H_id_U12 H_id_U45 
   H_id_zeros H_id_U71 H_id_isNatKind H_id_U92 H_id_nil H_id_U43 H_id_U84 
   H_id_s H_id_U32 H_id_U13 H_id_isNatIList;intros x12 H;injection H;
    clear H;intros ;subst;
    repeat (
    match goal with
      | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ =>
       inversion H;clear H;subst
      end
    ).
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x12 |- _ =>
      destruct (H_id_isNatIListKind y (refl_equal _)) as [x11];intros ;
       intuition;exists x11;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   clear H_id_U51 H_id_isNatList H_id_U85 H_id_U33 H_id_U81 H_id_U23 
   H_id_U93 H_id_U11 H_id_U44 H_id_0 H_id_U61 H_id_U22 H_id_U91 H_id_length 
   H_id_U42 H_id_U83 H_id_U31 H_id_isNatIListKind H_id_cons H_id_U52 
   H_id_U21 H_id_U86 H_id_U41 H_id_tt H_id_U82 H_id_isNat H_id_U94 H_id_U12 
   H_id_U45 H_id_zeros H_id_U71 H_id_isNatKind H_id_U92 H_id_nil H_id_U43 
   H_id_U84 H_id_s H_id_U32 H_id_U13 H_id_isNatIList;intros x12 H;
    injection H;clear H;intros ;subst;
    repeat (
    match goal with
      | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ =>
       inversion H;clear H;subst
      end
    ).
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x12 |- _ =>
      destruct (H_id_U46 y (refl_equal _)) as [x11];intros ;intuition;
       exists x11;intuition;eapply closure_extension.refl_trans_clos_R;
       eassumption
     end
   .
   
   clear H_id_U51 H_id_isNatList H_id_U85 H_id_U33 H_id_U81 H_id_U23 
   H_id_U93 H_id_U11 H_id_U44 H_id_0 H_id_U61 H_id_U22 H_id_U91 H_id_length 
   H_id_U42 H_id_U83 H_id_U31 H_id_isNatIListKind H_id_U46 H_id_U52 H_id_U21 
   H_id_U86 H_id_U41 H_id_tt H_id_U82 H_id_isNat H_id_U94 H_id_U12 H_id_U45 
   H_id_zeros H_id_U71 H_id_isNatKind H_id_U92 H_id_nil H_id_U43 H_id_U84 
   H_id_s H_id_U32 H_id_U13 H_id_isNatIList;intros x12 x14 H;injection H;
    clear H;intros ;subst;
    repeat (
    match goal with
      | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ =>
       inversion H;clear H;subst
      end
    ).
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x12 |- _ =>
      destruct (H_id_cons y x14 (refl_equal _)) as [x11 [x13]];intros ;
       intuition;exists x11;exists x13;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x14 |- _ =>
      destruct (H_id_cons x12 y (refl_equal _)) as [x11 [x13]];intros ;
       intuition;exists x11;exists x13;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   clear H_id_U51 H_id_isNatList H_id_U85 H_id_U33 H_id_U81 H_id_U23 
   H_id_U93 H_id_U11 H_id_U44 H_id_0 H_id_U61 H_id_U22 H_id_U91 H_id_length 
   H_id_U42 H_id_U83 H_id_U31 H_id_isNatIListKind H_id_U46 H_id_cons 
   H_id_U21 H_id_U86 H_id_U41 H_id_tt H_id_U82 H_id_isNat H_id_U94 H_id_U12 
   H_id_U45 H_id_zeros H_id_U71 H_id_isNatKind H_id_U92 H_id_nil H_id_U43 
   H_id_U84 H_id_s H_id_U32 H_id_U13 H_id_isNatIList;intros x12 H;
    injection H;clear H;intros ;subst;
    repeat (
    match goal with
      | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ =>
       inversion H;clear H;subst
      end
    ).
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x12 |- _ =>
      destruct (H_id_U52 y (refl_equal _)) as [x11];intros ;intuition;
       exists x11;intuition;eapply closure_extension.refl_trans_clos_R;
       eassumption
     end
   .
   
   clear H_id_U51 H_id_isNatList H_id_U85 H_id_U33 H_id_U81 H_id_U23 
   H_id_U93 H_id_U11 H_id_U44 H_id_0 H_id_U61 H_id_U22 H_id_U91 H_id_length 
   H_id_U42 H_id_U83 H_id_U31 H_id_isNatIListKind H_id_U46 H_id_cons 
   H_id_U52 H_id_U86 H_id_U41 H_id_tt H_id_U82 H_id_isNat H_id_U94 H_id_U12 
   H_id_U45 H_id_zeros H_id_U71 H_id_isNatKind H_id_U92 H_id_nil H_id_U43 
   H_id_U84 H_id_s H_id_U32 H_id_U13 H_id_isNatIList;intros x12 x14 H;
    injection H;clear H;intros ;subst;
    repeat (
    match goal with
      | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ =>
       inversion H;clear H;subst
      end
    ).
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x12 |- _ =>
      destruct (H_id_U21 y x14 (refl_equal _)) as [x11 [x13]];intros ;
       intuition;exists x11;exists x13;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x14 |- _ =>
      destruct (H_id_U21 x12 y (refl_equal _)) as [x11 [x13]];intros ;
       intuition;exists x11;exists x13;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   clear H_id_U51 H_id_isNatList H_id_U85 H_id_U33 H_id_U81 H_id_U23 
   H_id_U93 H_id_U11 H_id_U44 H_id_0 H_id_U61 H_id_U22 H_id_U91 H_id_length 
   H_id_U42 H_id_U83 H_id_U31 H_id_isNatIListKind H_id_U46 H_id_cons 
   H_id_U52 H_id_U21 H_id_U41 H_id_tt H_id_U82 H_id_isNat H_id_U94 H_id_U12 
   H_id_U45 H_id_zeros H_id_U71 H_id_isNatKind H_id_U92 H_id_nil H_id_U43 
   H_id_U84 H_id_s H_id_U32 H_id_U13 H_id_isNatIList;intros x12 H;
    injection H;clear H;intros ;subst;
    repeat (
    match goal with
      | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ =>
       inversion H;clear H;subst
      end
    ).
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x12 |- _ =>
      destruct (H_id_U86 y (refl_equal _)) as [x11];intros ;intuition;
       exists x11;intuition;eapply closure_extension.refl_trans_clos_R;
       eassumption
     end
   .
   
   clear H_id_U51 H_id_isNatList H_id_U85 H_id_U33 H_id_U81 H_id_U23 
   H_id_U93 H_id_U11 H_id_U44 H_id_0 H_id_U61 H_id_U22 H_id_U91 H_id_length 
   H_id_U42 H_id_U83 H_id_U31 H_id_isNatIListKind H_id_U46 H_id_cons 
   H_id_U52 H_id_U21 H_id_U86 H_id_tt H_id_U82 H_id_isNat H_id_U94 H_id_U12 
   H_id_U45 H_id_zeros H_id_U71 H_id_isNatKind H_id_U92 H_id_nil H_id_U43 
   H_id_U84 H_id_s H_id_U32 H_id_U13 H_id_isNatIList;intros x12 x14 x16 H;
    injection H;clear H;intros ;subst;
    repeat (
    match goal with
      | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ =>
       inversion H;clear H;subst
      end
    ).
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x12 |- _ =>
      destruct (H_id_U41 y x14 x16 (refl_equal _)) as [x11 [x13 [x15]]];
       intros ;intuition;exists x11;exists x13;exists x15;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x14 |- _ =>
      destruct (H_id_U41 x12 y x16 (refl_equal _)) as [x11 [x13 [x15]]];
       intros ;intuition;exists x11;exists x13;exists x15;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x16 |- _ =>
      destruct (H_id_U41 x12 x14 y (refl_equal _)) as [x11 [x13 [x15]]];
       intros ;intuition;exists x11;exists x13;exists x15;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   clear H_id_U51 H_id_isNatList H_id_U85 H_id_U33 H_id_U81 H_id_U23 
   H_id_U93 H_id_U11 H_id_U44 H_id_0 H_id_U61 H_id_U22 H_id_U91 H_id_length 
   H_id_U42 H_id_U83 H_id_U31 H_id_isNatIListKind H_id_U46 H_id_cons 
   H_id_U52 H_id_U21 H_id_U86 H_id_U41 H_id_U82 H_id_isNat H_id_U94 H_id_U12 
   H_id_U45 H_id_zeros H_id_U71 H_id_isNatKind H_id_U92 H_id_nil H_id_U43 
   H_id_U84 H_id_s H_id_U32 H_id_U13 H_id_isNatIList;intros H;injection H;
    clear H;intros ;subst;
    repeat (
    match goal with
      | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ =>
       inversion H;clear H;subst
      end
    ).
   
   clear H_id_U51 H_id_isNatList H_id_U85 H_id_U33 H_id_U81 H_id_U23 
   H_id_U93 H_id_U11 H_id_U44 H_id_0 H_id_U61 H_id_U22 H_id_U91 H_id_length 
   H_id_U42 H_id_U83 H_id_U31 H_id_isNatIListKind H_id_U46 H_id_cons 
   H_id_U52 H_id_U21 H_id_U86 H_id_U41 H_id_tt H_id_isNat H_id_U94 H_id_U12 
   H_id_U45 H_id_zeros H_id_U71 H_id_isNatKind H_id_U92 H_id_nil H_id_U43 
   H_id_U84 H_id_s H_id_U32 H_id_U13 H_id_isNatIList;intros x12 x14 x16 H;
    injection H;clear H;intros ;subst;
    repeat (
    match goal with
      | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ =>
       inversion H;clear H;subst
      end
    ).
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x12 |- _ =>
      destruct (H_id_U82 y x14 x16 (refl_equal _)) as [x11 [x13 [x15]]];
       intros ;intuition;exists x11;exists x13;exists x15;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x14 |- _ =>
      destruct (H_id_U82 x12 y x16 (refl_equal _)) as [x11 [x13 [x15]]];
       intros ;intuition;exists x11;exists x13;exists x15;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x16 |- _ =>
      destruct (H_id_U82 x12 x14 y (refl_equal _)) as [x11 [x13 [x15]]];
       intros ;intuition;exists x11;exists x13;exists x15;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   clear H_id_U51 H_id_isNatList H_id_U85 H_id_U33 H_id_U81 H_id_U23 
   H_id_U93 H_id_U11 H_id_U44 H_id_0 H_id_U61 H_id_U22 H_id_U91 H_id_length 
   H_id_U42 H_id_U83 H_id_U31 H_id_isNatIListKind H_id_U46 H_id_cons 
   H_id_U52 H_id_U21 H_id_U86 H_id_U41 H_id_tt H_id_U82 H_id_U94 H_id_U12 
   H_id_U45 H_id_zeros H_id_U71 H_id_isNatKind H_id_U92 H_id_nil H_id_U43 
   H_id_U84 H_id_s H_id_U32 H_id_U13 H_id_isNatIList;intros x12 H;
    injection H;clear H;intros ;subst;
    repeat (
    match goal with
      | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ =>
       inversion H;clear H;subst
      end
    ).
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x12 |- _ =>
      destruct (H_id_isNat y (refl_equal _)) as [x11];intros ;intuition;
       exists x11;intuition;eapply closure_extension.refl_trans_clos_R;
       eassumption
     end
   .
   
   clear H_id_U51 H_id_isNatList H_id_U85 H_id_U33 H_id_U81 H_id_U23 
   H_id_U93 H_id_U11 H_id_U44 H_id_0 H_id_U61 H_id_U22 H_id_U91 H_id_length 
   H_id_U42 H_id_U83 H_id_U31 H_id_isNatIListKind H_id_U46 H_id_cons 
   H_id_U52 H_id_U21 H_id_U86 H_id_U41 H_id_tt H_id_U82 H_id_isNat H_id_U12 
   H_id_U45 H_id_zeros H_id_U71 H_id_isNatKind H_id_U92 H_id_nil H_id_U43 
   H_id_U84 H_id_s H_id_U32 H_id_U13 H_id_isNatIList;intros x12 x14 H;
    injection H;clear H;intros ;subst;
    repeat (
    match goal with
      | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ =>
       inversion H;clear H;subst
      end
    ).
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x12 |- _ =>
      destruct (H_id_U94 y x14 (refl_equal _)) as [x11 [x13]];intros ;
       intuition;exists x11;exists x13;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x14 |- _ =>
      destruct (H_id_U94 x12 y (refl_equal _)) as [x11 [x13]];intros ;
       intuition;exists x11;exists x13;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   clear H_id_U51 H_id_isNatList H_id_U85 H_id_U33 H_id_U81 H_id_U23 
   H_id_U93 H_id_U11 H_id_U44 H_id_0 H_id_U61 H_id_U22 H_id_U91 H_id_length 
   H_id_U42 H_id_U83 H_id_U31 H_id_isNatIListKind H_id_U46 H_id_cons 
   H_id_U52 H_id_U21 H_id_U86 H_id_U41 H_id_tt H_id_U82 H_id_isNat H_id_U94 
   H_id_U45 H_id_zeros H_id_U71 H_id_isNatKind H_id_U92 H_id_nil H_id_U43 
   H_id_U84 H_id_s H_id_U32 H_id_U13 H_id_isNatIList;intros x12 x14 H;
    injection H;clear H;intros ;subst;
    repeat (
    match goal with
      | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ =>
       inversion H;clear H;subst
      end
    ).
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x12 |- _ =>
      destruct (H_id_U12 y x14 (refl_equal _)) as [x11 [x13]];intros ;
       intuition;exists x11;exists x13;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x14 |- _ =>
      destruct (H_id_U12 x12 y (refl_equal _)) as [x11 [x13]];intros ;
       intuition;exists x11;exists x13;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   clear H_id_U51 H_id_isNatList H_id_U85 H_id_U33 H_id_U81 H_id_U23 
   H_id_U93 H_id_U11 H_id_U44 H_id_0 H_id_U61 H_id_U22 H_id_U91 H_id_length 
   H_id_U42 H_id_U83 H_id_U31 H_id_isNatIListKind H_id_U46 H_id_cons 
   H_id_U52 H_id_U21 H_id_U86 H_id_U41 H_id_tt H_id_U82 H_id_isNat H_id_U94 
   H_id_U12 H_id_zeros H_id_U71 H_id_isNatKind H_id_U92 H_id_nil H_id_U43 
   H_id_U84 H_id_s H_id_U32 H_id_U13 H_id_isNatIList;intros x12 x14 H;
    injection H;clear H;intros ;subst;
    repeat (
    match goal with
      | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ =>
       inversion H;clear H;subst
      end
    ).
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x12 |- _ =>
      destruct (H_id_U45 y x14 (refl_equal _)) as [x11 [x13]];intros ;
       intuition;exists x11;exists x13;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x14 |- _ =>
      destruct (H_id_U45 x12 y (refl_equal _)) as [x11 [x13]];intros ;
       intuition;exists x11;exists x13;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   clear H_id_U51 H_id_isNatList H_id_U85 H_id_U33 H_id_U81 H_id_U23 
   H_id_U93 H_id_U11 H_id_U44 H_id_0 H_id_U61 H_id_U22 H_id_U91 H_id_length 
   H_id_U42 H_id_U83 H_id_U31 H_id_isNatIListKind H_id_U46 H_id_cons 
   H_id_U52 H_id_U21 H_id_U86 H_id_U41 H_id_tt H_id_U82 H_id_isNat H_id_U94 
   H_id_U12 H_id_U45 H_id_U71 H_id_isNatKind H_id_U92 H_id_nil H_id_U43 
   H_id_U84 H_id_s H_id_U32 H_id_U13 H_id_isNatIList;intros H;injection H;
    clear H;intros ;subst;
    repeat (
    match goal with
      | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ =>
       inversion H;clear H;subst
      end
    ).
   
   clear H_id_U51 H_id_isNatList H_id_U85 H_id_U33 H_id_U81 H_id_U23 
   H_id_U93 H_id_U11 H_id_U44 H_id_0 H_id_U61 H_id_U22 H_id_U91 H_id_length 
   H_id_U42 H_id_U83 H_id_U31 H_id_isNatIListKind H_id_U46 H_id_cons 
   H_id_U52 H_id_U21 H_id_U86 H_id_U41 H_id_tt H_id_U82 H_id_isNat H_id_U94 
   H_id_U12 H_id_U45 H_id_zeros H_id_isNatKind H_id_U92 H_id_nil H_id_U43 
   H_id_U84 H_id_s H_id_U32 H_id_U13 H_id_isNatIList;intros x12 H;
    injection H;clear H;intros ;subst;
    repeat (
    match goal with
      | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ =>
       inversion H;clear H;subst
      end
    ).
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x12 |- _ =>
      destruct (H_id_U71 y (refl_equal _)) as [x11];intros ;intuition;
       exists x11;intuition;eapply closure_extension.refl_trans_clos_R;
       eassumption
     end
   .
   
   clear H_id_U51 H_id_isNatList H_id_U85 H_id_U33 H_id_U81 H_id_U23 
   H_id_U93 H_id_U11 H_id_U44 H_id_0 H_id_U61 H_id_U22 H_id_U91 H_id_length 
   H_id_U42 H_id_U83 H_id_U31 H_id_isNatIListKind H_id_U46 H_id_cons 
   H_id_U52 H_id_U21 H_id_U86 H_id_U41 H_id_tt H_id_U82 H_id_isNat H_id_U94 
   H_id_U12 H_id_U45 H_id_zeros H_id_U71 H_id_U92 H_id_nil H_id_U43 H_id_U84 
   H_id_s H_id_U32 H_id_U13 H_id_isNatIList;intros x12 H;injection H;
    clear H;intros ;subst;
    repeat (
    match goal with
      | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ =>
       inversion H;clear H;subst
      end
    ).
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x12 |- _ =>
      destruct (H_id_isNatKind y (refl_equal _)) as [x11];intros ;intuition;
       exists x11;intuition;eapply closure_extension.refl_trans_clos_R;
       eassumption
     end
   .
   
   clear H_id_U51 H_id_isNatList H_id_U85 H_id_U33 H_id_U81 H_id_U23 
   H_id_U93 H_id_U11 H_id_U44 H_id_0 H_id_U61 H_id_U22 H_id_U91 H_id_length 
   H_id_U42 H_id_U83 H_id_U31 H_id_isNatIListKind H_id_U46 H_id_cons 
   H_id_U52 H_id_U21 H_id_U86 H_id_U41 H_id_tt H_id_U82 H_id_isNat H_id_U94 
   H_id_U12 H_id_U45 H_id_zeros H_id_U71 H_id_isNatKind H_id_nil H_id_U43 
   H_id_U84 H_id_s H_id_U32 H_id_U13 H_id_isNatIList;intros x12 x14 x16 H;
    injection H;clear H;intros ;subst;
    repeat (
    match goal with
      | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ =>
       inversion H;clear H;subst
      end
    ).
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x12 |- _ =>
      destruct (H_id_U92 y x14 x16 (refl_equal _)) as [x11 [x13 [x15]]];
       intros ;intuition;exists x11;exists x13;exists x15;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x14 |- _ =>
      destruct (H_id_U92 x12 y x16 (refl_equal _)) as [x11 [x13 [x15]]];
       intros ;intuition;exists x11;exists x13;exists x15;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x16 |- _ =>
      destruct (H_id_U92 x12 x14 y (refl_equal _)) as [x11 [x13 [x15]]];
       intros ;intuition;exists x11;exists x13;exists x15;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   clear H_id_U51 H_id_isNatList H_id_U85 H_id_U33 H_id_U81 H_id_U23 
   H_id_U93 H_id_U11 H_id_U44 H_id_0 H_id_U61 H_id_U22 H_id_U91 H_id_length 
   H_id_U42 H_id_U83 H_id_U31 H_id_isNatIListKind H_id_U46 H_id_cons 
   H_id_U52 H_id_U21 H_id_U86 H_id_U41 H_id_tt H_id_U82 H_id_isNat H_id_U94 
   H_id_U12 H_id_U45 H_id_zeros H_id_U71 H_id_isNatKind H_id_U92 H_id_U43 
   H_id_U84 H_id_s H_id_U32 H_id_U13 H_id_isNatIList;intros H;injection H;
    clear H;intros ;subst;
    repeat (
    match goal with
      | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ =>
       inversion H;clear H;subst
      end
    ).
   
   clear H_id_U51 H_id_isNatList H_id_U85 H_id_U33 H_id_U81 H_id_U23 
   H_id_U93 H_id_U11 H_id_U44 H_id_0 H_id_U61 H_id_U22 H_id_U91 H_id_length 
   H_id_U42 H_id_U83 H_id_U31 H_id_isNatIListKind H_id_U46 H_id_cons 
   H_id_U52 H_id_U21 H_id_U86 H_id_U41 H_id_tt H_id_U82 H_id_isNat H_id_U94 
   H_id_U12 H_id_U45 H_id_zeros H_id_U71 H_id_isNatKind H_id_U92 H_id_nil 
   H_id_U84 H_id_s H_id_U32 H_id_U13 H_id_isNatIList;intros x12 x14 x16 H;
    injection H;clear H;intros ;subst;
    repeat (
    match goal with
      | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ =>
       inversion H;clear H;subst
      end
    ).
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x12 |- _ =>
      destruct (H_id_U43 y x14 x16 (refl_equal _)) as [x11 [x13 [x15]]];
       intros ;intuition;exists x11;exists x13;exists x15;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x14 |- _ =>
      destruct (H_id_U43 x12 y x16 (refl_equal _)) as [x11 [x13 [x15]]];
       intros ;intuition;exists x11;exists x13;exists x15;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x16 |- _ =>
      destruct (H_id_U43 x12 x14 y (refl_equal _)) as [x11 [x13 [x15]]];
       intros ;intuition;exists x11;exists x13;exists x15;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   clear H_id_U51 H_id_isNatList H_id_U85 H_id_U33 H_id_U81 H_id_U23 
   H_id_U93 H_id_U11 H_id_U44 H_id_0 H_id_U61 H_id_U22 H_id_U91 H_id_length 
   H_id_U42 H_id_U83 H_id_U31 H_id_isNatIListKind H_id_U46 H_id_cons 
   H_id_U52 H_id_U21 H_id_U86 H_id_U41 H_id_tt H_id_U82 H_id_isNat H_id_U94 
   H_id_U12 H_id_U45 H_id_zeros H_id_U71 H_id_isNatKind H_id_U92 H_id_nil 
   H_id_U43 H_id_s H_id_U32 H_id_U13 H_id_isNatIList;intros x12 x14 x16 H;
    injection H;clear H;intros ;subst;
    repeat (
    match goal with
      | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ =>
       inversion H;clear H;subst
      end
    ).
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x12 |- _ =>
      destruct (H_id_U84 y x14 x16 (refl_equal _)) as [x11 [x13 [x15]]];
       intros ;intuition;exists x11;exists x13;exists x15;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x14 |- _ =>
      destruct (H_id_U84 x12 y x16 (refl_equal _)) as [x11 [x13 [x15]]];
       intros ;intuition;exists x11;exists x13;exists x15;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x16 |- _ =>
      destruct (H_id_U84 x12 x14 y (refl_equal _)) as [x11 [x13 [x15]]];
       intros ;intuition;exists x11;exists x13;exists x15;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   clear H_id_U51 H_id_isNatList H_id_U85 H_id_U33 H_id_U81 H_id_U23 
   H_id_U93 H_id_U11 H_id_U44 H_id_0 H_id_U61 H_id_U22 H_id_U91 H_id_length 
   H_id_U42 H_id_U83 H_id_U31 H_id_isNatIListKind H_id_U46 H_id_cons 
   H_id_U52 H_id_U21 H_id_U86 H_id_U41 H_id_tt H_id_U82 H_id_isNat H_id_U94 
   H_id_U12 H_id_U45 H_id_zeros H_id_U71 H_id_isNatKind H_id_U92 H_id_nil 
   H_id_U43 H_id_U84 H_id_U32 H_id_U13 H_id_isNatIList;intros x12 H;
    injection H;clear H;intros ;subst;
    repeat (
    match goal with
      | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ =>
       inversion H;clear H;subst
      end
    ).
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x12 |- _ =>
      destruct (H_id_s y (refl_equal _)) as [x11];intros ;intuition;
       exists x11;intuition;eapply closure_extension.refl_trans_clos_R;
       eassumption
     end
   .
   
   clear H_id_U51 H_id_isNatList H_id_U85 H_id_U33 H_id_U81 H_id_U23 
   H_id_U93 H_id_U11 H_id_U44 H_id_0 H_id_U61 H_id_U22 H_id_U91 H_id_length 
   H_id_U42 H_id_U83 H_id_U31 H_id_isNatIListKind H_id_U46 H_id_cons 
   H_id_U52 H_id_U21 H_id_U86 H_id_U41 H_id_tt H_id_U82 H_id_isNat H_id_U94 
   H_id_U12 H_id_U45 H_id_zeros H_id_U71 H_id_isNatKind H_id_U92 H_id_nil 
   H_id_U43 H_id_U84 H_id_s H_id_U13 H_id_isNatIList;intros x12 x14 H;
    injection H;clear H;intros ;subst;
    repeat (
    match goal with
      | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ =>
       inversion H;clear H;subst
      end
    ).
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x12 |- _ =>
      destruct (H_id_U32 y x14 (refl_equal _)) as [x11 [x13]];intros ;
       intuition;exists x11;exists x13;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x14 |- _ =>
      destruct (H_id_U32 x12 y (refl_equal _)) as [x11 [x13]];intros ;
       intuition;exists x11;exists x13;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   clear H_id_U51 H_id_isNatList H_id_U85 H_id_U33 H_id_U81 H_id_U23 
   H_id_U93 H_id_U11 H_id_U44 H_id_0 H_id_U61 H_id_U22 H_id_U91 H_id_length 
   H_id_U42 H_id_U83 H_id_U31 H_id_isNatIListKind H_id_U46 H_id_cons 
   H_id_U52 H_id_U21 H_id_U86 H_id_U41 H_id_tt H_id_U82 H_id_isNat H_id_U94 
   H_id_U12 H_id_U45 H_id_zeros H_id_U71 H_id_isNatKind H_id_U92 H_id_nil 
   H_id_U43 H_id_U84 H_id_s H_id_U32 H_id_isNatIList;intros x12 H;
    injection H;clear H;intros ;subst;
    repeat (
    match goal with
      | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ =>
       inversion H;clear H;subst
      end
    ).
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x12 |- _ =>
      destruct (H_id_U13 y (refl_equal _)) as [x11];intros ;intuition;
       exists x11;intuition;eapply closure_extension.refl_trans_clos_R;
       eassumption
     end
   .
   
   clear H_id_U51 H_id_isNatList H_id_U85 H_id_U33 H_id_U81 H_id_U23 
   H_id_U93 H_id_U11 H_id_U44 H_id_0 H_id_U61 H_id_U22 H_id_U91 H_id_length 
   H_id_U42 H_id_U83 H_id_U31 H_id_isNatIListKind H_id_U46 H_id_cons 
   H_id_U52 H_id_U21 H_id_U86 H_id_U41 H_id_tt H_id_U82 H_id_isNat H_id_U94 
   H_id_U12 H_id_U45 H_id_zeros H_id_U71 H_id_isNatKind H_id_U92 H_id_nil 
   H_id_U43 H_id_U84 H_id_s H_id_U32 H_id_U13;intros x12 H;injection H;
    clear H;intros ;subst;
    repeat (
    match goal with
      | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ =>
       inversion H;clear H;subst
      end
    ).
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x12 |- _ =>
      destruct (H_id_isNatIList y (refl_equal _)) as [x11];intros ;intuition;
       exists x11;intuition;eapply closure_extension.refl_trans_clos_R;
       eassumption
     end
   .
 Qed.
 
 
 Lemma id_U51_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x12 x14, 
     t = (algebra.Alg.Term algebra.F.id_U51 (x12::x14::nil)) ->
      exists x11,
        exists x13,
          t' = (algebra.Alg.Term algebra.F.id_U51 (x11::x13::nil))/\ 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
            x11 x12)/\ 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
            x13 x14).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_isNatList_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x12, 
     t = (algebra.Alg.Term algebra.F.id_isNatList (x12::nil)) ->
      exists x11,
        t' = (algebra.Alg.Term algebra.F.id_isNatList (x11::nil))/\ 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x11 x12).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_U85_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x12 x14, 
     t = (algebra.Alg.Term algebra.F.id_U85 (x12::x14::nil)) ->
      exists x11,
        exists x13,
          t' = (algebra.Alg.Term algebra.F.id_U85 (x11::x13::nil))/\ 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
            x11 x12)/\ 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
            x13 x14).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_U33_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x12, 
     t = (algebra.Alg.Term algebra.F.id_U33 (x12::nil)) ->
      exists x11,
        t' = (algebra.Alg.Term algebra.F.id_U33 (x11::nil))/\ 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x11 x12).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_U81_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x12 x14 x16, 
     t = (algebra.Alg.Term algebra.F.id_U81 (x12::x14::x16::nil)) ->
      exists x11,
        exists x13,
          exists x15,
            t' = (algebra.Alg.Term algebra.F.id_U81 (x11::x13::x15::nil))/\ 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
              x11 x12)/\ 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
              x13 x14)/\ 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
              x15 x16).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_U23_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x12, 
     t = (algebra.Alg.Term algebra.F.id_U23 (x12::nil)) ->
      exists x11,
        t' = (algebra.Alg.Term algebra.F.id_U23 (x11::nil))/\ 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x11 x12).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_U93_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x12 x14 x16, 
     t = (algebra.Alg.Term algebra.F.id_U93 (x12::x14::x16::nil)) ->
      exists x11,
        exists x13,
          exists x15,
            t' = (algebra.Alg.Term algebra.F.id_U93 (x11::x13::x15::nil))/\ 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
              x11 x12)/\ 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
              x13 x14)/\ 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
              x15 x16).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_U11_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x12 x14, 
     t = (algebra.Alg.Term algebra.F.id_U11 (x12::x14::nil)) ->
      exists x11,
        exists x13,
          t' = (algebra.Alg.Term algebra.F.id_U11 (x11::x13::nil))/\ 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
            x11 x12)/\ 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
            x13 x14).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_U44_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x12 x14 x16, 
     t = (algebra.Alg.Term algebra.F.id_U44 (x12::x14::x16::nil)) ->
      exists x11,
        exists x13,
          exists x15,
            t' = (algebra.Alg.Term algebra.F.id_U44 (x11::x13::x15::nil))/\ 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
              x11 x12)/\ 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
              x13 x14)/\ 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
              x15 x16).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_0_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    t = (algebra.Alg.Term algebra.F.id_0 nil) ->
     t' = (algebra.Alg.Term algebra.F.id_0 nil).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_U61_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x12, 
     t = (algebra.Alg.Term algebra.F.id_U61 (x12::nil)) ->
      exists x11,
        t' = (algebra.Alg.Term algebra.F.id_U61 (x11::nil))/\ 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x11 x12).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_U22_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x12 x14, 
     t = (algebra.Alg.Term algebra.F.id_U22 (x12::x14::nil)) ->
      exists x11,
        exists x13,
          t' = (algebra.Alg.Term algebra.F.id_U22 (x11::x13::nil))/\ 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
            x11 x12)/\ 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
            x13 x14).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_U91_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x12 x14 x16, 
     t = (algebra.Alg.Term algebra.F.id_U91 (x12::x14::x16::nil)) ->
      exists x11,
        exists x13,
          exists x15,
            t' = (algebra.Alg.Term algebra.F.id_U91 (x11::x13::x15::nil))/\ 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
              x11 x12)/\ 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
              x13 x14)/\ 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
              x15 x16).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_length_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x12, 
     t = (algebra.Alg.Term algebra.F.id_length (x12::nil)) ->
      exists x11,
        t' = (algebra.Alg.Term algebra.F.id_length (x11::nil))/\ 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x11 x12).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_U42_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x12 x14 x16, 
     t = (algebra.Alg.Term algebra.F.id_U42 (x12::x14::x16::nil)) ->
      exists x11,
        exists x13,
          exists x15,
            t' = (algebra.Alg.Term algebra.F.id_U42 (x11::x13::x15::nil))/\ 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
              x11 x12)/\ 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
              x13 x14)/\ 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
              x15 x16).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_U83_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x12 x14 x16, 
     t = (algebra.Alg.Term algebra.F.id_U83 (x12::x14::x16::nil)) ->
      exists x11,
        exists x13,
          exists x15,
            t' = (algebra.Alg.Term algebra.F.id_U83 (x11::x13::x15::nil))/\ 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
              x11 x12)/\ 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
              x13 x14)/\ 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
              x15 x16).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_U31_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x12 x14, 
     t = (algebra.Alg.Term algebra.F.id_U31 (x12::x14::nil)) ->
      exists x11,
        exists x13,
          t' = (algebra.Alg.Term algebra.F.id_U31 (x11::x13::nil))/\ 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
            x11 x12)/\ 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
            x13 x14).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_isNatIListKind_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x12, 
     t = (algebra.Alg.Term algebra.F.id_isNatIListKind (x12::nil)) ->
      exists x11,
        t' = (algebra.Alg.Term algebra.F.id_isNatIListKind (x11::nil))/\ 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x11 x12).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_U46_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x12, 
     t = (algebra.Alg.Term algebra.F.id_U46 (x12::nil)) ->
      exists x11,
        t' = (algebra.Alg.Term algebra.F.id_U46 (x11::nil))/\ 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x11 x12).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_cons_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x12 x14, 
     t = (algebra.Alg.Term algebra.F.id_cons (x12::x14::nil)) ->
      exists x11,
        exists x13,
          t' = (algebra.Alg.Term algebra.F.id_cons (x11::x13::nil))/\ 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
            x11 x12)/\ 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
            x13 x14).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_U52_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x12, 
     t = (algebra.Alg.Term algebra.F.id_U52 (x12::nil)) ->
      exists x11,
        t' = (algebra.Alg.Term algebra.F.id_U52 (x11::nil))/\ 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x11 x12).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_U21_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x12 x14, 
     t = (algebra.Alg.Term algebra.F.id_U21 (x12::x14::nil)) ->
      exists x11,
        exists x13,
          t' = (algebra.Alg.Term algebra.F.id_U21 (x11::x13::nil))/\ 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
            x11 x12)/\ 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
            x13 x14).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_U86_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x12, 
     t = (algebra.Alg.Term algebra.F.id_U86 (x12::nil)) ->
      exists x11,
        t' = (algebra.Alg.Term algebra.F.id_U86 (x11::nil))/\ 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x11 x12).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_U41_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x12 x14 x16, 
     t = (algebra.Alg.Term algebra.F.id_U41 (x12::x14::x16::nil)) ->
      exists x11,
        exists x13,
          exists x15,
            t' = (algebra.Alg.Term algebra.F.id_U41 (x11::x13::x15::nil))/\ 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
              x11 x12)/\ 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
              x13 x14)/\ 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
              x15 x16).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_tt_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    t = (algebra.Alg.Term algebra.F.id_tt nil) ->
     t' = (algebra.Alg.Term algebra.F.id_tt nil).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_U82_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x12 x14 x16, 
     t = (algebra.Alg.Term algebra.F.id_U82 (x12::x14::x16::nil)) ->
      exists x11,
        exists x13,
          exists x15,
            t' = (algebra.Alg.Term algebra.F.id_U82 (x11::x13::x15::nil))/\ 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
              x11 x12)/\ 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
              x13 x14)/\ 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
              x15 x16).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_isNat_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x12, 
     t = (algebra.Alg.Term algebra.F.id_isNat (x12::nil)) ->
      exists x11,
        t' = (algebra.Alg.Term algebra.F.id_isNat (x11::nil))/\ 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x11 x12).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_U94_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x12 x14, 
     t = (algebra.Alg.Term algebra.F.id_U94 (x12::x14::nil)) ->
      exists x11,
        exists x13,
          t' = (algebra.Alg.Term algebra.F.id_U94 (x11::x13::nil))/\ 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
            x11 x12)/\ 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
            x13 x14).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_U12_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x12 x14, 
     t = (algebra.Alg.Term algebra.F.id_U12 (x12::x14::nil)) ->
      exists x11,
        exists x13,
          t' = (algebra.Alg.Term algebra.F.id_U12 (x11::x13::nil))/\ 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
            x11 x12)/\ 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
            x13 x14).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_U45_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x12 x14, 
     t = (algebra.Alg.Term algebra.F.id_U45 (x12::x14::nil)) ->
      exists x11,
        exists x13,
          t' = (algebra.Alg.Term algebra.F.id_U45 (x11::x13::nil))/\ 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
            x11 x12)/\ 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
            x13 x14).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_zeros_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    t = (algebra.Alg.Term algebra.F.id_zeros nil) ->
     t' = (algebra.Alg.Term algebra.F.id_zeros nil).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_U71_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x12, 
     t = (algebra.Alg.Term algebra.F.id_U71 (x12::nil)) ->
      exists x11,
        t' = (algebra.Alg.Term algebra.F.id_U71 (x11::nil))/\ 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x11 x12).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_isNatKind_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x12, 
     t = (algebra.Alg.Term algebra.F.id_isNatKind (x12::nil)) ->
      exists x11,
        t' = (algebra.Alg.Term algebra.F.id_isNatKind (x11::nil))/\ 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x11 x12).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_U92_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x12 x14 x16, 
     t = (algebra.Alg.Term algebra.F.id_U92 (x12::x14::x16::nil)) ->
      exists x11,
        exists x13,
          exists x15,
            t' = (algebra.Alg.Term algebra.F.id_U92 (x11::x13::x15::nil))/\ 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
              x11 x12)/\ 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
              x13 x14)/\ 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
              x15 x16).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_nil_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    t = (algebra.Alg.Term algebra.F.id_nil nil) ->
     t' = (algebra.Alg.Term algebra.F.id_nil nil).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_U43_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x12 x14 x16, 
     t = (algebra.Alg.Term algebra.F.id_U43 (x12::x14::x16::nil)) ->
      exists x11,
        exists x13,
          exists x15,
            t' = (algebra.Alg.Term algebra.F.id_U43 (x11::x13::x15::nil))/\ 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
              x11 x12)/\ 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
              x13 x14)/\ 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
              x15 x16).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_U84_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x12 x14 x16, 
     t = (algebra.Alg.Term algebra.F.id_U84 (x12::x14::x16::nil)) ->
      exists x11,
        exists x13,
          exists x15,
            t' = (algebra.Alg.Term algebra.F.id_U84 (x11::x13::x15::nil))/\ 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
              x11 x12)/\ 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
              x13 x14)/\ 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
              x15 x16).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_s_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x12, 
     t = (algebra.Alg.Term algebra.F.id_s (x12::nil)) ->
      exists x11,
        t' = (algebra.Alg.Term algebra.F.id_s (x11::nil))/\ 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x11 x12).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_U32_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x12 x14, 
     t = (algebra.Alg.Term algebra.F.id_U32 (x12::x14::nil)) ->
      exists x11,
        exists x13,
          t' = (algebra.Alg.Term algebra.F.id_U32 (x11::x13::nil))/\ 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
            x11 x12)/\ 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
            x13 x14).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_U13_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x12, 
     t = (algebra.Alg.Term algebra.F.id_U13 (x12::nil)) ->
      exists x11,
        t' = (algebra.Alg.Term algebra.F.id_U13 (x11::nil))/\ 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x11 x12).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_isNatIList_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x12, 
     t = (algebra.Alg.Term algebra.F.id_isNatIList (x12::nil)) ->
      exists x11,
        t' = (algebra.Alg.Term algebra.F.id_isNatIList (x11::nil))/\ 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x11 x12).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Ltac impossible_star_reduction_R_xml_0  :=
  match goal with
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U51 (?x12::?x11::nil)) |- _ =>
     let x12 := fresh "x" in 
      (let x11 := fresh "x" in 
        (let Heq := fresh "Heq" in 
          (let Hred1 := fresh "Hred" in 
            (let Hred2 := fresh "Hred" in 
              (destruct (id_U51_is_R_xml_0_constructor H (refl_equal _)) as 
               [x12 [x11 [Heq [Hred2 Hred1]]]];
                (discriminate Heq)||
                (injection Heq;intros ;subst;clear Heq;clear H;
                  impossible_star_reduction_R_xml_0 ))))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_isNatList (?x11::nil)) |- _ =>
     let x11 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_isNatList_is_R_xml_0_constructor H (refl_equal _))
            as [x11 [Heq Hred1]];
            (discriminate Heq)||
            (injection Heq;intros ;subst;clear Heq;clear H;
              impossible_star_reduction_R_xml_0 ))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U85 (?x12::?x11::nil)) |- _ =>
     let x12 := fresh "x" in 
      (let x11 := fresh "x" in 
        (let Heq := fresh "Heq" in 
          (let Hred1 := fresh "Hred" in 
            (let Hred2 := fresh "Hred" in 
              (destruct (id_U85_is_R_xml_0_constructor H (refl_equal _)) as 
               [x12 [x11 [Heq [Hred2 Hred1]]]];
                (discriminate Heq)||
                (injection Heq;intros ;subst;clear Heq;clear H;
                  impossible_star_reduction_R_xml_0 ))))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U33 (?x11::nil)) |- _ =>
     let x11 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_U33_is_R_xml_0_constructor H (refl_equal _)) as 
           [x11 [Heq Hred1]];
            (discriminate Heq)||
            (injection Heq;intros ;subst;clear Heq;clear H;
              impossible_star_reduction_R_xml_0 ))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U81 (?x13::?x12::?x11::nil)) |- 
    _ =>
     let x13 := fresh "x" in 
      (let x12 := fresh "x" in 
        (let x11 := fresh "x" in 
          (let Heq := fresh "Heq" in 
            (let Hred1 := fresh "Hred" in 
              (let Hred2 := fresh "Hred" in 
                (let Hred3 := fresh "Hred" in 
                  (destruct (id_U81_is_R_xml_0_constructor H (refl_equal _))
                    as [x13 [x12 [x11 [Heq [Hred3 [Hred2 Hred1]]]]]];
                    (discriminate Heq)||
                    (injection Heq;intros ;subst;clear Heq;clear H;
                      impossible_star_reduction_R_xml_0 ))))))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U23 (?x11::nil)) |- _ =>
     let x11 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_U23_is_R_xml_0_constructor H (refl_equal _)) as 
           [x11 [Heq Hred1]];
            (discriminate Heq)||
            (injection Heq;intros ;subst;clear Heq;clear H;
              impossible_star_reduction_R_xml_0 ))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U93 (?x13::?x12::?x11::nil)) |- 
    _ =>
     let x13 := fresh "x" in 
      (let x12 := fresh "x" in 
        (let x11 := fresh "x" in 
          (let Heq := fresh "Heq" in 
            (let Hred1 := fresh "Hred" in 
              (let Hred2 := fresh "Hred" in 
                (let Hred3 := fresh "Hred" in 
                  (destruct (id_U93_is_R_xml_0_constructor H (refl_equal _))
                    as [x13 [x12 [x11 [Heq [Hred3 [Hred2 Hred1]]]]]];
                    (discriminate Heq)||
                    (injection Heq;intros ;subst;clear Heq;clear H;
                      impossible_star_reduction_R_xml_0 ))))))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U11 (?x12::?x11::nil)) |- _ =>
     let x12 := fresh "x" in 
      (let x11 := fresh "x" in 
        (let Heq := fresh "Heq" in 
          (let Hred1 := fresh "Hred" in 
            (let Hred2 := fresh "Hred" in 
              (destruct (id_U11_is_R_xml_0_constructor H (refl_equal _)) as 
               [x12 [x11 [Heq [Hred2 Hred1]]]];
                (discriminate Heq)||
                (injection Heq;intros ;subst;clear Heq;clear H;
                  impossible_star_reduction_R_xml_0 ))))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U44 (?x13::?x12::?x11::nil)) |- 
    _ =>
     let x13 := fresh "x" in 
      (let x12 := fresh "x" in 
        (let x11 := fresh "x" in 
          (let Heq := fresh "Heq" in 
            (let Hred1 := fresh "Hred" in 
              (let Hred2 := fresh "Hred" in 
                (let Hred3 := fresh "Hred" in 
                  (destruct (id_U44_is_R_xml_0_constructor H (refl_equal _))
                    as [x13 [x12 [x11 [Heq [Hred3 [Hred2 Hred1]]]]]];
                    (discriminate Heq)||
                    (injection Heq;intros ;subst;clear Heq;clear H;
                      impossible_star_reduction_R_xml_0 ))))))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_0 nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_0_is_R_xml_0_constructor H (refl_equal _)) in *;
        (discriminate Heq)||
        (clearbody Heq;try (subst);try (clear Heq);clear H;
          impossible_star_reduction_R_xml_0 ))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U61 (?x11::nil)) |- _ =>
     let x11 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_U61_is_R_xml_0_constructor H (refl_equal _)) as 
           [x11 [Heq Hred1]];
            (discriminate Heq)||
            (injection Heq;intros ;subst;clear Heq;clear H;
              impossible_star_reduction_R_xml_0 ))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U22 (?x12::?x11::nil)) |- _ =>
     let x12 := fresh "x" in 
      (let x11 := fresh "x" in 
        (let Heq := fresh "Heq" in 
          (let Hred1 := fresh "Hred" in 
            (let Hred2 := fresh "Hred" in 
              (destruct (id_U22_is_R_xml_0_constructor H (refl_equal _)) as 
               [x12 [x11 [Heq [Hred2 Hred1]]]];
                (discriminate Heq)||
                (injection Heq;intros ;subst;clear Heq;clear H;
                  impossible_star_reduction_R_xml_0 ))))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U91 (?x13::?x12::?x11::nil)) |- 
    _ =>
     let x13 := fresh "x" in 
      (let x12 := fresh "x" in 
        (let x11 := fresh "x" in 
          (let Heq := fresh "Heq" in 
            (let Hred1 := fresh "Hred" in 
              (let Hred2 := fresh "Hred" in 
                (let Hred3 := fresh "Hred" in 
                  (destruct (id_U91_is_R_xml_0_constructor H (refl_equal _))
                    as [x13 [x12 [x11 [Heq [Hred3 [Hred2 Hred1]]]]]];
                    (discriminate Heq)||
                    (injection Heq;intros ;subst;clear Heq;clear H;
                      impossible_star_reduction_R_xml_0 ))))))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_length (?x11::nil)) |- _ =>
     let x11 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_length_is_R_xml_0_constructor H (refl_equal _)) as 
           [x11 [Heq Hred1]];
            (discriminate Heq)||
            (injection Heq;intros ;subst;clear Heq;clear H;
              impossible_star_reduction_R_xml_0 ))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U42 (?x13::?x12::?x11::nil)) |- 
    _ =>
     let x13 := fresh "x" in 
      (let x12 := fresh "x" in 
        (let x11 := fresh "x" in 
          (let Heq := fresh "Heq" in 
            (let Hred1 := fresh "Hred" in 
              (let Hred2 := fresh "Hred" in 
                (let Hred3 := fresh "Hred" in 
                  (destruct (id_U42_is_R_xml_0_constructor H (refl_equal _))
                    as [x13 [x12 [x11 [Heq [Hred3 [Hred2 Hred1]]]]]];
                    (discriminate Heq)||
                    (injection Heq;intros ;subst;clear Heq;clear H;
                      impossible_star_reduction_R_xml_0 ))))))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U83 (?x13::?x12::?x11::nil)) |- 
    _ =>
     let x13 := fresh "x" in 
      (let x12 := fresh "x" in 
        (let x11 := fresh "x" in 
          (let Heq := fresh "Heq" in 
            (let Hred1 := fresh "Hred" in 
              (let Hred2 := fresh "Hred" in 
                (let Hred3 := fresh "Hred" in 
                  (destruct (id_U83_is_R_xml_0_constructor H (refl_equal _))
                    as [x13 [x12 [x11 [Heq [Hred3 [Hred2 Hred1]]]]]];
                    (discriminate Heq)||
                    (injection Heq;intros ;subst;clear Heq;clear H;
                      impossible_star_reduction_R_xml_0 ))))))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U31 (?x12::?x11::nil)) |- _ =>
     let x12 := fresh "x" in 
      (let x11 := fresh "x" in 
        (let Heq := fresh "Heq" in 
          (let Hred1 := fresh "Hred" in 
            (let Hred2 := fresh "Hred" in 
              (destruct (id_U31_is_R_xml_0_constructor H (refl_equal _)) as 
               [x12 [x11 [Heq [Hred2 Hred1]]]];
                (discriminate Heq)||
                (injection Heq;intros ;subst;clear Heq;clear H;
                  impossible_star_reduction_R_xml_0 ))))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_isNatIListKind (?x11::nil)) |- 
    _ =>
     let x11 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_isNatIListKind_is_R_xml_0_constructor H 
                      (refl_equal _))
            as [x11 [Heq Hred1]];
            (discriminate Heq)||
            (injection Heq;intros ;subst;clear Heq;clear H;
              impossible_star_reduction_R_xml_0 ))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U46 (?x11::nil)) |- _ =>
     let x11 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_U46_is_R_xml_0_constructor H (refl_equal _)) as 
           [x11 [Heq Hred1]];
            (discriminate Heq)||
            (injection Heq;intros ;subst;clear Heq;clear H;
              impossible_star_reduction_R_xml_0 ))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_cons (?x12::?x11::nil)) |- 
    _ =>
     let x12 := fresh "x" in 
      (let x11 := fresh "x" in 
        (let Heq := fresh "Heq" in 
          (let Hred1 := fresh "Hred" in 
            (let Hred2 := fresh "Hred" in 
              (destruct (id_cons_is_R_xml_0_constructor H (refl_equal _)) as 
               [x12 [x11 [Heq [Hred2 Hred1]]]];
                (discriminate Heq)||
                (injection Heq;intros ;subst;clear Heq;clear H;
                  impossible_star_reduction_R_xml_0 ))))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U52 (?x11::nil)) |- _ =>
     let x11 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_U52_is_R_xml_0_constructor H (refl_equal _)) as 
           [x11 [Heq Hred1]];
            (discriminate Heq)||
            (injection Heq;intros ;subst;clear Heq;clear H;
              impossible_star_reduction_R_xml_0 ))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U21 (?x12::?x11::nil)) |- _ =>
     let x12 := fresh "x" in 
      (let x11 := fresh "x" in 
        (let Heq := fresh "Heq" in 
          (let Hred1 := fresh "Hred" in 
            (let Hred2 := fresh "Hred" in 
              (destruct (id_U21_is_R_xml_0_constructor H (refl_equal _)) as 
               [x12 [x11 [Heq [Hred2 Hred1]]]];
                (discriminate Heq)||
                (injection Heq;intros ;subst;clear Heq;clear H;
                  impossible_star_reduction_R_xml_0 ))))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U86 (?x11::nil)) |- _ =>
     let x11 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_U86_is_R_xml_0_constructor H (refl_equal _)) as 
           [x11 [Heq Hred1]];
            (discriminate Heq)||
            (injection Heq;intros ;subst;clear Heq;clear H;
              impossible_star_reduction_R_xml_0 ))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U41 (?x13::?x12::?x11::nil)) |- 
    _ =>
     let x13 := fresh "x" in 
      (let x12 := fresh "x" in 
        (let x11 := fresh "x" in 
          (let Heq := fresh "Heq" in 
            (let Hred1 := fresh "Hred" in 
              (let Hred2 := fresh "Hred" in 
                (let Hred3 := fresh "Hred" in 
                  (destruct (id_U41_is_R_xml_0_constructor H (refl_equal _))
                    as [x13 [x12 [x11 [Heq [Hred3 [Hred2 Hred1]]]]]];
                    (discriminate Heq)||
                    (injection Heq;intros ;subst;clear Heq;clear H;
                      impossible_star_reduction_R_xml_0 ))))))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_tt nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_tt_is_R_xml_0_constructor H (refl_equal _)) in *;
        (discriminate Heq)||
        (clearbody Heq;try (subst);try (clear Heq);clear H;
          impossible_star_reduction_R_xml_0 ))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U82 (?x13::?x12::?x11::nil)) |- 
    _ =>
     let x13 := fresh "x" in 
      (let x12 := fresh "x" in 
        (let x11 := fresh "x" in 
          (let Heq := fresh "Heq" in 
            (let Hred1 := fresh "Hred" in 
              (let Hred2 := fresh "Hred" in 
                (let Hred3 := fresh "Hred" in 
                  (destruct (id_U82_is_R_xml_0_constructor H (refl_equal _))
                    as [x13 [x12 [x11 [Heq [Hred3 [Hred2 Hred1]]]]]];
                    (discriminate Heq)||
                    (injection Heq;intros ;subst;clear Heq;clear H;
                      impossible_star_reduction_R_xml_0 ))))))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_isNat (?x11::nil)) |- _ =>
     let x11 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_isNat_is_R_xml_0_constructor H (refl_equal _)) as 
           [x11 [Heq Hred1]];
            (discriminate Heq)||
            (injection Heq;intros ;subst;clear Heq;clear H;
              impossible_star_reduction_R_xml_0 ))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U94 (?x12::?x11::nil)) |- _ =>
     let x12 := fresh "x" in 
      (let x11 := fresh "x" in 
        (let Heq := fresh "Heq" in 
          (let Hred1 := fresh "Hred" in 
            (let Hred2 := fresh "Hred" in 
              (destruct (id_U94_is_R_xml_0_constructor H (refl_equal _)) as 
               [x12 [x11 [Heq [Hred2 Hred1]]]];
                (discriminate Heq)||
                (injection Heq;intros ;subst;clear Heq;clear H;
                  impossible_star_reduction_R_xml_0 ))))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U12 (?x12::?x11::nil)) |- _ =>
     let x12 := fresh "x" in 
      (let x11 := fresh "x" in 
        (let Heq := fresh "Heq" in 
          (let Hred1 := fresh "Hred" in 
            (let Hred2 := fresh "Hred" in 
              (destruct (id_U12_is_R_xml_0_constructor H (refl_equal _)) as 
               [x12 [x11 [Heq [Hred2 Hred1]]]];
                (discriminate Heq)||
                (injection Heq;intros ;subst;clear Heq;clear H;
                  impossible_star_reduction_R_xml_0 ))))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U45 (?x12::?x11::nil)) |- _ =>
     let x12 := fresh "x" in 
      (let x11 := fresh "x" in 
        (let Heq := fresh "Heq" in 
          (let Hred1 := fresh "Hred" in 
            (let Hred2 := fresh "Hred" in 
              (destruct (id_U45_is_R_xml_0_constructor H (refl_equal _)) as 
               [x12 [x11 [Heq [Hred2 Hred1]]]];
                (discriminate Heq)||
                (injection Heq;intros ;subst;clear Heq;clear H;
                  impossible_star_reduction_R_xml_0 ))))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_zeros nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_zeros_is_R_xml_0_constructor H (refl_equal _)) in *;
        (discriminate Heq)||
        (clearbody Heq;try (subst);try (clear Heq);clear H;
          impossible_star_reduction_R_xml_0 ))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U71 (?x11::nil)) |- _ =>
     let x11 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_U71_is_R_xml_0_constructor H (refl_equal _)) as 
           [x11 [Heq Hred1]];
            (discriminate Heq)||
            (injection Heq;intros ;subst;clear Heq;clear H;
              impossible_star_reduction_R_xml_0 ))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_isNatKind (?x11::nil)) |- _ =>
     let x11 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_isNatKind_is_R_xml_0_constructor H (refl_equal _))
            as [x11 [Heq Hred1]];
            (discriminate Heq)||
            (injection Heq;intros ;subst;clear Heq;clear H;
              impossible_star_reduction_R_xml_0 ))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U92 (?x13::?x12::?x11::nil)) |- 
    _ =>
     let x13 := fresh "x" in 
      (let x12 := fresh "x" in 
        (let x11 := fresh "x" in 
          (let Heq := fresh "Heq" in 
            (let Hred1 := fresh "Hred" in 
              (let Hred2 := fresh "Hred" in 
                (let Hred3 := fresh "Hred" in 
                  (destruct (id_U92_is_R_xml_0_constructor H (refl_equal _))
                    as [x13 [x12 [x11 [Heq [Hred3 [Hred2 Hred1]]]]]];
                    (discriminate Heq)||
                    (injection Heq;intros ;subst;clear Heq;clear H;
                      impossible_star_reduction_R_xml_0 ))))))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_nil nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_nil_is_R_xml_0_constructor H (refl_equal _)) in *;
        (discriminate Heq)||
        (clearbody Heq;try (subst);try (clear Heq);clear H;
          impossible_star_reduction_R_xml_0 ))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U43 (?x13::?x12::?x11::nil)) |- 
    _ =>
     let x13 := fresh "x" in 
      (let x12 := fresh "x" in 
        (let x11 := fresh "x" in 
          (let Heq := fresh "Heq" in 
            (let Hred1 := fresh "Hred" in 
              (let Hred2 := fresh "Hred" in 
                (let Hred3 := fresh "Hred" in 
                  (destruct (id_U43_is_R_xml_0_constructor H (refl_equal _))
                    as [x13 [x12 [x11 [Heq [Hred3 [Hred2 Hred1]]]]]];
                    (discriminate Heq)||
                    (injection Heq;intros ;subst;clear Heq;clear H;
                      impossible_star_reduction_R_xml_0 ))))))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U84 (?x13::?x12::?x11::nil)) |- 
    _ =>
     let x13 := fresh "x" in 
      (let x12 := fresh "x" in 
        (let x11 := fresh "x" in 
          (let Heq := fresh "Heq" in 
            (let Hred1 := fresh "Hred" in 
              (let Hred2 := fresh "Hred" in 
                (let Hred3 := fresh "Hred" in 
                  (destruct (id_U84_is_R_xml_0_constructor H (refl_equal _))
                    as [x13 [x12 [x11 [Heq [Hred3 [Hred2 Hred1]]]]]];
                    (discriminate Heq)||
                    (injection Heq;intros ;subst;clear Heq;clear H;
                      impossible_star_reduction_R_xml_0 ))))))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_s (?x11::nil)) |- _ =>
     let x11 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_s_is_R_xml_0_constructor H (refl_equal _)) as 
           [x11 [Heq Hred1]];
            (discriminate Heq)||
            (injection Heq;intros ;subst;clear Heq;clear H;
              impossible_star_reduction_R_xml_0 ))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U32 (?x12::?x11::nil)) |- _ =>
     let x12 := fresh "x" in 
      (let x11 := fresh "x" in 
        (let Heq := fresh "Heq" in 
          (let Hred1 := fresh "Hred" in 
            (let Hred2 := fresh "Hred" in 
              (destruct (id_U32_is_R_xml_0_constructor H (refl_equal _)) as 
               [x12 [x11 [Heq [Hred2 Hred1]]]];
                (discriminate Heq)||
                (injection Heq;intros ;subst;clear Heq;clear H;
                  impossible_star_reduction_R_xml_0 ))))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U13 (?x11::nil)) |- _ =>
     let x11 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_U13_is_R_xml_0_constructor H (refl_equal _)) as 
           [x11 [Heq Hred1]];
            (discriminate Heq)||
            (injection Heq;intros ;subst;clear Heq;clear H;
              impossible_star_reduction_R_xml_0 ))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_isNatIList (?x11::nil)) |- 
    _ =>
     let x11 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_isNatIList_is_R_xml_0_constructor H (refl_equal _))
            as [x11 [Heq Hred1]];
            (discriminate Heq)||
            (injection Heq;intros ;subst;clear Heq;clear H;
              impossible_star_reduction_R_xml_0 ))))
    end
  .
 
 
 Ltac simplify_star_reduction_R_xml_0  :=
  match goal with
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U51 (?x12::?x11::nil)) |- _ =>
     let x12 := fresh "x" in 
      (let x11 := fresh "x" in 
        (let Heq := fresh "Heq" in 
          (let Hred1 := fresh "Hred" in 
            (let Hred2 := fresh "Hred" in 
              (destruct (id_U51_is_R_xml_0_constructor H (refl_equal _)) as 
               [x12 [x11 [Heq [Hred2 Hred1]]]];
                (discriminate Heq)||
                (injection Heq;intros ;subst;clear Heq;clear H;
                  try (simplify_star_reduction_R_xml_0 )))))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_isNatList (?x11::nil)) |- _ =>
     let x11 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_isNatList_is_R_xml_0_constructor H (refl_equal _))
            as [x11 [Heq Hred1]];
            (discriminate Heq)||
            (injection Heq;intros ;subst;clear Heq;clear H;
              try (simplify_star_reduction_R_xml_0 )))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U85 (?x12::?x11::nil)) |- _ =>
     let x12 := fresh "x" in 
      (let x11 := fresh "x" in 
        (let Heq := fresh "Heq" in 
          (let Hred1 := fresh "Hred" in 
            (let Hred2 := fresh "Hred" in 
              (destruct (id_U85_is_R_xml_0_constructor H (refl_equal _)) as 
               [x12 [x11 [Heq [Hred2 Hred1]]]];
                (discriminate Heq)||
                (injection Heq;intros ;subst;clear Heq;clear H;
                  try (simplify_star_reduction_R_xml_0 )))))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U33 (?x11::nil)) |- _ =>
     let x11 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_U33_is_R_xml_0_constructor H (refl_equal _)) as 
           [x11 [Heq Hred1]];
            (discriminate Heq)||
            (injection Heq;intros ;subst;clear Heq;clear H;
              try (simplify_star_reduction_R_xml_0 )))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U81 (?x13::?x12::?x11::nil)) |- 
    _ =>
     let x13 := fresh "x" in 
      (let x12 := fresh "x" in 
        (let x11 := fresh "x" in 
          (let Heq := fresh "Heq" in 
            (let Hred1 := fresh "Hred" in 
              (let Hred2 := fresh "Hred" in 
                (let Hred3 := fresh "Hred" in 
                  (destruct (id_U81_is_R_xml_0_constructor H (refl_equal _))
                    as [x13 [x12 [x11 [Heq [Hred3 [Hred2 Hred1]]]]]];
                    (discriminate Heq)||
                    (injection Heq;intros ;subst;clear Heq;clear H;
                      try (simplify_star_reduction_R_xml_0 )))))))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U23 (?x11::nil)) |- _ =>
     let x11 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_U23_is_R_xml_0_constructor H (refl_equal _)) as 
           [x11 [Heq Hred1]];
            (discriminate Heq)||
            (injection Heq;intros ;subst;clear Heq;clear H;
              try (simplify_star_reduction_R_xml_0 )))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U93 (?x13::?x12::?x11::nil)) |- 
    _ =>
     let x13 := fresh "x" in 
      (let x12 := fresh "x" in 
        (let x11 := fresh "x" in 
          (let Heq := fresh "Heq" in 
            (let Hred1 := fresh "Hred" in 
              (let Hred2 := fresh "Hred" in 
                (let Hred3 := fresh "Hred" in 
                  (destruct (id_U93_is_R_xml_0_constructor H (refl_equal _))
                    as [x13 [x12 [x11 [Heq [Hred3 [Hred2 Hred1]]]]]];
                    (discriminate Heq)||
                    (injection Heq;intros ;subst;clear Heq;clear H;
                      try (simplify_star_reduction_R_xml_0 )))))))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U11 (?x12::?x11::nil)) |- _ =>
     let x12 := fresh "x" in 
      (let x11 := fresh "x" in 
        (let Heq := fresh "Heq" in 
          (let Hred1 := fresh "Hred" in 
            (let Hred2 := fresh "Hred" in 
              (destruct (id_U11_is_R_xml_0_constructor H (refl_equal _)) as 
               [x12 [x11 [Heq [Hred2 Hred1]]]];
                (discriminate Heq)||
                (injection Heq;intros ;subst;clear Heq;clear H;
                  try (simplify_star_reduction_R_xml_0 )))))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U44 (?x13::?x12::?x11::nil)) |- 
    _ =>
     let x13 := fresh "x" in 
      (let x12 := fresh "x" in 
        (let x11 := fresh "x" in 
          (let Heq := fresh "Heq" in 
            (let Hred1 := fresh "Hred" in 
              (let Hred2 := fresh "Hred" in 
                (let Hred3 := fresh "Hred" in 
                  (destruct (id_U44_is_R_xml_0_constructor H (refl_equal _))
                    as [x13 [x12 [x11 [Heq [Hred3 [Hred2 Hred1]]]]]];
                    (discriminate Heq)||
                    (injection Heq;intros ;subst;clear Heq;clear H;
                      try (simplify_star_reduction_R_xml_0 )))))))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_0 nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_0_is_R_xml_0_constructor H (refl_equal _)) in *;
        (discriminate Heq)||
        (clearbody Heq;try (subst);try (clear Heq);clear H;
          try (simplify_star_reduction_R_xml_0 )))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U61 (?x11::nil)) |- _ =>
     let x11 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_U61_is_R_xml_0_constructor H (refl_equal _)) as 
           [x11 [Heq Hred1]];
            (discriminate Heq)||
            (injection Heq;intros ;subst;clear Heq;clear H;
              try (simplify_star_reduction_R_xml_0 )))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U22 (?x12::?x11::nil)) |- _ =>
     let x12 := fresh "x" in 
      (let x11 := fresh "x" in 
        (let Heq := fresh "Heq" in 
          (let Hred1 := fresh "Hred" in 
            (let Hred2 := fresh "Hred" in 
              (destruct (id_U22_is_R_xml_0_constructor H (refl_equal _)) as 
               [x12 [x11 [Heq [Hred2 Hred1]]]];
                (discriminate Heq)||
                (injection Heq;intros ;subst;clear Heq;clear H;
                  try (simplify_star_reduction_R_xml_0 )))))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U91 (?x13::?x12::?x11::nil)) |- 
    _ =>
     let x13 := fresh "x" in 
      (let x12 := fresh "x" in 
        (let x11 := fresh "x" in 
          (let Heq := fresh "Heq" in 
            (let Hred1 := fresh "Hred" in 
              (let Hred2 := fresh "Hred" in 
                (let Hred3 := fresh "Hred" in 
                  (destruct (id_U91_is_R_xml_0_constructor H (refl_equal _))
                    as [x13 [x12 [x11 [Heq [Hred3 [Hred2 Hred1]]]]]];
                    (discriminate Heq)||
                    (injection Heq;intros ;subst;clear Heq;clear H;
                      try (simplify_star_reduction_R_xml_0 )))))))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_length (?x11::nil)) |- _ =>
     let x11 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_length_is_R_xml_0_constructor H (refl_equal _)) as 
           [x11 [Heq Hred1]];
            (discriminate Heq)||
            (injection Heq;intros ;subst;clear Heq;clear H;
              try (simplify_star_reduction_R_xml_0 )))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U42 (?x13::?x12::?x11::nil)) |- 
    _ =>
     let x13 := fresh "x" in 
      (let x12 := fresh "x" in 
        (let x11 := fresh "x" in 
          (let Heq := fresh "Heq" in 
            (let Hred1 := fresh "Hred" in 
              (let Hred2 := fresh "Hred" in 
                (let Hred3 := fresh "Hred" in 
                  (destruct (id_U42_is_R_xml_0_constructor H (refl_equal _))
                    as [x13 [x12 [x11 [Heq [Hred3 [Hred2 Hred1]]]]]];
                    (discriminate Heq)||
                    (injection Heq;intros ;subst;clear Heq;clear H;
                      try (simplify_star_reduction_R_xml_0 )))))))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U83 (?x13::?x12::?x11::nil)) |- 
    _ =>
     let x13 := fresh "x" in 
      (let x12 := fresh "x" in 
        (let x11 := fresh "x" in 
          (let Heq := fresh "Heq" in 
            (let Hred1 := fresh "Hred" in 
              (let Hred2 := fresh "Hred" in 
                (let Hred3 := fresh "Hred" in 
                  (destruct (id_U83_is_R_xml_0_constructor H (refl_equal _))
                    as [x13 [x12 [x11 [Heq [Hred3 [Hred2 Hred1]]]]]];
                    (discriminate Heq)||
                    (injection Heq;intros ;subst;clear Heq;clear H;
                      try (simplify_star_reduction_R_xml_0 )))))))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U31 (?x12::?x11::nil)) |- _ =>
     let x12 := fresh "x" in 
      (let x11 := fresh "x" in 
        (let Heq := fresh "Heq" in 
          (let Hred1 := fresh "Hred" in 
            (let Hred2 := fresh "Hred" in 
              (destruct (id_U31_is_R_xml_0_constructor H (refl_equal _)) as 
               [x12 [x11 [Heq [Hred2 Hred1]]]];
                (discriminate Heq)||
                (injection Heq;intros ;subst;clear Heq;clear H;
                  try (simplify_star_reduction_R_xml_0 )))))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_isNatIListKind (?x11::nil)) |- 
    _ =>
     let x11 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_isNatIListKind_is_R_xml_0_constructor H 
                      (refl_equal _))
            as [x11 [Heq Hred1]];
            (discriminate Heq)||
            (injection Heq;intros ;subst;clear Heq;clear H;
              try (simplify_star_reduction_R_xml_0 )))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U46 (?x11::nil)) |- _ =>
     let x11 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_U46_is_R_xml_0_constructor H (refl_equal _)) as 
           [x11 [Heq Hred1]];
            (discriminate Heq)||
            (injection Heq;intros ;subst;clear Heq;clear H;
              try (simplify_star_reduction_R_xml_0 )))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_cons (?x12::?x11::nil)) |- 
    _ =>
     let x12 := fresh "x" in 
      (let x11 := fresh "x" in 
        (let Heq := fresh "Heq" in 
          (let Hred1 := fresh "Hred" in 
            (let Hred2 := fresh "Hred" in 
              (destruct (id_cons_is_R_xml_0_constructor H (refl_equal _)) as 
               [x12 [x11 [Heq [Hred2 Hred1]]]];
                (discriminate Heq)||
                (injection Heq;intros ;subst;clear Heq;clear H;
                  try (simplify_star_reduction_R_xml_0 )))))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U52 (?x11::nil)) |- _ =>
     let x11 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_U52_is_R_xml_0_constructor H (refl_equal _)) as 
           [x11 [Heq Hred1]];
            (discriminate Heq)||
            (injection Heq;intros ;subst;clear Heq;clear H;
              try (simplify_star_reduction_R_xml_0 )))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U21 (?x12::?x11::nil)) |- _ =>
     let x12 := fresh "x" in 
      (let x11 := fresh "x" in 
        (let Heq := fresh "Heq" in 
          (let Hred1 := fresh "Hred" in 
            (let Hred2 := fresh "Hred" in 
              (destruct (id_U21_is_R_xml_0_constructor H (refl_equal _)) as 
               [x12 [x11 [Heq [Hred2 Hred1]]]];
                (discriminate Heq)||
                (injection Heq;intros ;subst;clear Heq;clear H;
                  try (simplify_star_reduction_R_xml_0 )))))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U86 (?x11::nil)) |- _ =>
     let x11 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_U86_is_R_xml_0_constructor H (refl_equal _)) as 
           [x11 [Heq Hred1]];
            (discriminate Heq)||
            (injection Heq;intros ;subst;clear Heq;clear H;
              try (simplify_star_reduction_R_xml_0 )))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U41 (?x13::?x12::?x11::nil)) |- 
    _ =>
     let x13 := fresh "x" in 
      (let x12 := fresh "x" in 
        (let x11 := fresh "x" in 
          (let Heq := fresh "Heq" in 
            (let Hred1 := fresh "Hred" in 
              (let Hred2 := fresh "Hred" in 
                (let Hred3 := fresh "Hred" in 
                  (destruct (id_U41_is_R_xml_0_constructor H (refl_equal _))
                    as [x13 [x12 [x11 [Heq [Hred3 [Hred2 Hred1]]]]]];
                    (discriminate Heq)||
                    (injection Heq;intros ;subst;clear Heq;clear H;
                      try (simplify_star_reduction_R_xml_0 )))))))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_tt nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_tt_is_R_xml_0_constructor H (refl_equal _)) in *;
        (discriminate Heq)||
        (clearbody Heq;try (subst);try (clear Heq);clear H;
          try (simplify_star_reduction_R_xml_0 )))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U82 (?x13::?x12::?x11::nil)) |- 
    _ =>
     let x13 := fresh "x" in 
      (let x12 := fresh "x" in 
        (let x11 := fresh "x" in 
          (let Heq := fresh "Heq" in 
            (let Hred1 := fresh "Hred" in 
              (let Hred2 := fresh "Hred" in 
                (let Hred3 := fresh "Hred" in 
                  (destruct (id_U82_is_R_xml_0_constructor H (refl_equal _))
                    as [x13 [x12 [x11 [Heq [Hred3 [Hred2 Hred1]]]]]];
                    (discriminate Heq)||
                    (injection Heq;intros ;subst;clear Heq;clear H;
                      try (simplify_star_reduction_R_xml_0 )))))))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_isNat (?x11::nil)) |- _ =>
     let x11 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_isNat_is_R_xml_0_constructor H (refl_equal _)) as 
           [x11 [Heq Hred1]];
            (discriminate Heq)||
            (injection Heq;intros ;subst;clear Heq;clear H;
              try (simplify_star_reduction_R_xml_0 )))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U94 (?x12::?x11::nil)) |- _ =>
     let x12 := fresh "x" in 
      (let x11 := fresh "x" in 
        (let Heq := fresh "Heq" in 
          (let Hred1 := fresh "Hred" in 
            (let Hred2 := fresh "Hred" in 
              (destruct (id_U94_is_R_xml_0_constructor H (refl_equal _)) as 
               [x12 [x11 [Heq [Hred2 Hred1]]]];
                (discriminate Heq)||
                (injection Heq;intros ;subst;clear Heq;clear H;
                  try (simplify_star_reduction_R_xml_0 )))))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U12 (?x12::?x11::nil)) |- _ =>
     let x12 := fresh "x" in 
      (let x11 := fresh "x" in 
        (let Heq := fresh "Heq" in 
          (let Hred1 := fresh "Hred" in 
            (let Hred2 := fresh "Hred" in 
              (destruct (id_U12_is_R_xml_0_constructor H (refl_equal _)) as 
               [x12 [x11 [Heq [Hred2 Hred1]]]];
                (discriminate Heq)||
                (injection Heq;intros ;subst;clear Heq;clear H;
                  try (simplify_star_reduction_R_xml_0 )))))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U45 (?x12::?x11::nil)) |- _ =>
     let x12 := fresh "x" in 
      (let x11 := fresh "x" in 
        (let Heq := fresh "Heq" in 
          (let Hred1 := fresh "Hred" in 
            (let Hred2 := fresh "Hred" in 
              (destruct (id_U45_is_R_xml_0_constructor H (refl_equal _)) as 
               [x12 [x11 [Heq [Hred2 Hred1]]]];
                (discriminate Heq)||
                (injection Heq;intros ;subst;clear Heq;clear H;
                  try (simplify_star_reduction_R_xml_0 )))))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_zeros nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_zeros_is_R_xml_0_constructor H (refl_equal _)) in *;
        (discriminate Heq)||
        (clearbody Heq;try (subst);try (clear Heq);clear H;
          try (simplify_star_reduction_R_xml_0 )))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U71 (?x11::nil)) |- _ =>
     let x11 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_U71_is_R_xml_0_constructor H (refl_equal _)) as 
           [x11 [Heq Hred1]];
            (discriminate Heq)||
            (injection Heq;intros ;subst;clear Heq;clear H;
              try (simplify_star_reduction_R_xml_0 )))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_isNatKind (?x11::nil)) |- _ =>
     let x11 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_isNatKind_is_R_xml_0_constructor H (refl_equal _))
            as [x11 [Heq Hred1]];
            (discriminate Heq)||
            (injection Heq;intros ;subst;clear Heq;clear H;
              try (simplify_star_reduction_R_xml_0 )))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U92 (?x13::?x12::?x11::nil)) |- 
    _ =>
     let x13 := fresh "x" in 
      (let x12 := fresh "x" in 
        (let x11 := fresh "x" in 
          (let Heq := fresh "Heq" in 
            (let Hred1 := fresh "Hred" in 
              (let Hred2 := fresh "Hred" in 
                (let Hred3 := fresh "Hred" in 
                  (destruct (id_U92_is_R_xml_0_constructor H (refl_equal _))
                    as [x13 [x12 [x11 [Heq [Hred3 [Hred2 Hred1]]]]]];
                    (discriminate Heq)||
                    (injection Heq;intros ;subst;clear Heq;clear H;
                      try (simplify_star_reduction_R_xml_0 )))))))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_nil nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_nil_is_R_xml_0_constructor H (refl_equal _)) in *;
        (discriminate Heq)||
        (clearbody Heq;try (subst);try (clear Heq);clear H;
          try (simplify_star_reduction_R_xml_0 )))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U43 (?x13::?x12::?x11::nil)) |- 
    _ =>
     let x13 := fresh "x" in 
      (let x12 := fresh "x" in 
        (let x11 := fresh "x" in 
          (let Heq := fresh "Heq" in 
            (let Hred1 := fresh "Hred" in 
              (let Hred2 := fresh "Hred" in 
                (let Hred3 := fresh "Hred" in 
                  (destruct (id_U43_is_R_xml_0_constructor H (refl_equal _))
                    as [x13 [x12 [x11 [Heq [Hred3 [Hred2 Hred1]]]]]];
                    (discriminate Heq)||
                    (injection Heq;intros ;subst;clear Heq;clear H;
                      try (simplify_star_reduction_R_xml_0 )))))))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U84 (?x13::?x12::?x11::nil)) |- 
    _ =>
     let x13 := fresh "x" in 
      (let x12 := fresh "x" in 
        (let x11 := fresh "x" in 
          (let Heq := fresh "Heq" in 
            (let Hred1 := fresh "Hred" in 
              (let Hred2 := fresh "Hred" in 
                (let Hred3 := fresh "Hred" in 
                  (destruct (id_U84_is_R_xml_0_constructor H (refl_equal _))
                    as [x13 [x12 [x11 [Heq [Hred3 [Hred2 Hred1]]]]]];
                    (discriminate Heq)||
                    (injection Heq;intros ;subst;clear Heq;clear H;
                      try (simplify_star_reduction_R_xml_0 )))))))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_s (?x11::nil)) |- _ =>
     let x11 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_s_is_R_xml_0_constructor H (refl_equal _)) as 
           [x11 [Heq Hred1]];
            (discriminate Heq)||
            (injection Heq;intros ;subst;clear Heq;clear H;
              try (simplify_star_reduction_R_xml_0 )))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U32 (?x12::?x11::nil)) |- _ =>
     let x12 := fresh "x" in 
      (let x11 := fresh "x" in 
        (let Heq := fresh "Heq" in 
          (let Hred1 := fresh "Hred" in 
            (let Hred2 := fresh "Hred" in 
              (destruct (id_U32_is_R_xml_0_constructor H (refl_equal _)) as 
               [x12 [x11 [Heq [Hred2 Hred1]]]];
                (discriminate Heq)||
                (injection Heq;intros ;subst;clear Heq;clear H;
                  try (simplify_star_reduction_R_xml_0 )))))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U13 (?x11::nil)) |- _ =>
     let x11 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_U13_is_R_xml_0_constructor H (refl_equal _)) as 
           [x11 [Heq Hred1]];
            (discriminate Heq)||
            (injection Heq;intros ;subst;clear Heq;clear H;
              try (simplify_star_reduction_R_xml_0 )))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_isNatIList (?x11::nil)) |- 
    _ =>
     let x11 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_isNatIList_is_R_xml_0_constructor H (refl_equal _))
            as [x11 [Heq Hred1]];
            (discriminate Heq)||
            (injection Heq;intros ;subst;clear Heq;clear H;
              try (simplify_star_reduction_R_xml_0 )))))
    end
  .
End R_xml_0_deep_rew.

Module InterpGen := interp.Interp(algebra.EQT).

Module ddp := dp.MakeDP(algebra.EQT).

Module SymbType. Definition A := algebra.Alg.F.Symb.A. End SymbType.

Module Symb_more_list := more_list_extention.Make(SymbType)(algebra.Alg.F.Symb).

Module SymbSet := list_set.Make(algebra.F.Symb).

Module Interp.
 Section S.
   Require Import interp.
   
   Hypothesis A : Type.
   
   Hypothesis Ale Alt Aeq : A -> A -> Prop.
   
   Hypothesis Aop : interp.ordering_pair Aeq Alt Ale.
   
   Hypothesis A0 : A.
   
   Notation Local "a <= b" := (Ale a b).
   
   Hypothesis P_id_a__zeros : A.
   
   Hypothesis P_id_U51 : A ->A ->A.
   
   Hypothesis P_id_a__U84 : A ->A ->A ->A.
   
   Hypothesis P_id_a__U32 : A ->A ->A.
   
   Hypothesis P_id_isNatList : A ->A.
   
   Hypothesis P_id_a__U13 : A ->A.
   
   Hypothesis P_id_U85 : A ->A ->A.
   
   Hypothesis P_id_a__length : A ->A.
   
   Hypothesis P_id_a__isNatIList : A ->A.
   
   Hypothesis P_id_U33 : A ->A.
   
   Hypothesis P_id_a__U11 : A ->A ->A.
   
   Hypothesis P_id_U81 : A ->A ->A ->A.
   
   Hypothesis P_id_a__U92 : A ->A ->A ->A.
   
   Hypothesis P_id_a__U43 : A ->A ->A ->A.
   
   Hypothesis P_id_U23 : A ->A.
   
   Hypothesis P_id_a__isNatKind : A ->A.
   
   Hypothesis P_id_U93 : A ->A ->A ->A.
   
   Hypothesis P_id_U11 : A ->A ->A.
   
   Hypothesis P_id_a__U71 : A ->A.
   
   Hypothesis P_id_U44 : A ->A ->A ->A.
   
   Hypothesis P_id_0 : A.
   
   Hypothesis P_id_U61 : A ->A.
   
   Hypothesis P_id_a__U86 : A ->A.
   
   Hypothesis P_id_a__U41 : A ->A ->A ->A.
   
   Hypothesis P_id_U22 : A ->A ->A.
   
   Hypothesis P_id_a__U21 : A ->A ->A.
   
   Hypothesis P_id_U91 : A ->A ->A ->A.
   
   Hypothesis P_id_length : A ->A.
   
   Hypothesis P_id_a__U52 : A ->A.
   
   Hypothesis P_id_U42 : A ->A ->A ->A.
   
   Hypothesis P_id_a__U12 : A ->A ->A.
   
   Hypothesis P_id_U83 : A ->A ->A ->A.
   
   Hypothesis P_id_a__U94 : A ->A ->A.
   
   Hypothesis P_id_a__U45 : A ->A ->A.
   
   Hypothesis P_id_U31 : A ->A ->A.
   
   Hypothesis P_id_a__isNat : A ->A.
   
   Hypothesis P_id_isNatIListKind : A ->A.
   
   Hypothesis P_id_a__U82 : A ->A ->A ->A.
   
   Hypothesis P_id_U46 : A ->A.
   
   Hypothesis P_id_cons : A ->A ->A.
   
   Hypothesis P_id_U52 : A ->A.
   
   Hypothesis P_id_a__U85 : A ->A ->A.
   
   Hypothesis P_id_a__U33 : A ->A.
   
   Hypothesis P_id_U21 : A ->A ->A.
   
   Hypothesis P_id_a__isNatList : A ->A.
   
   Hypothesis P_id_U86 : A ->A.
   
   Hypothesis P_id_mark : A ->A.
   
   Hypothesis P_id_a__U51 : A ->A ->A.
   
   Hypothesis P_id_U41 : A ->A ->A ->A.
   
   Hypothesis P_id_tt : A.
   
   Hypothesis P_id_U82 : A ->A ->A ->A.
   
   Hypothesis P_id_a__U93 : A ->A ->A ->A.
   
   Hypothesis P_id_a__U44 : A ->A ->A ->A.
   
   Hypothesis P_id_isNat : A ->A.
   
   Hypothesis P_id_a__U23 : A ->A.
   
   Hypothesis P_id_U94 : A ->A ->A.
   
   Hypothesis P_id_U12 : A ->A ->A.
   
   Hypothesis P_id_a__U81 : A ->A ->A ->A.
   
   Hypothesis P_id_U45 : A ->A ->A.
   
   Hypothesis P_id_zeros : A.
   
   Hypothesis P_id_U71 : A ->A.
   
   Hypothesis P_id_a__U91 : A ->A ->A ->A.
   
   Hypothesis P_id_a__U42 : A ->A ->A ->A.
   
   Hypothesis P_id_isNatKind : A ->A.
   
   Hypothesis P_id_a__U22 : A ->A ->A.
   
   Hypothesis P_id_U92 : A ->A ->A ->A.
   
   Hypothesis P_id_nil : A.
   
   Hypothesis P_id_a__U61 : A ->A.
   
   Hypothesis P_id_U43 : A ->A ->A ->A.
   
   Hypothesis P_id_a__isNatIListKind : A ->A.
   
   Hypothesis P_id_U84 : A ->A ->A ->A.
   
   Hypothesis P_id_s : A ->A.
   
   Hypothesis P_id_a__U46 : A ->A.
   
   Hypothesis P_id_U32 : A ->A ->A.
   
   Hypothesis P_id_a__U31 : A ->A ->A.
   
   Hypothesis P_id_U13 : A ->A.
   
   Hypothesis P_id_a__U83 : A ->A ->A ->A.
   
   Hypothesis P_id_isNatIList : A ->A.
   
   Hypothesis P_id_U51_monotonic :
    forall x12 x14 x13 x11, 
     (A0 <= x14)/\ (x14 <= x13) ->
      (A0 <= x12)/\ (x12 <= x11) ->P_id_U51 x12 x14 <= P_id_U51 x11 x13.
   
   Hypothesis P_id_a__U84_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (A0 <= x16)/\ (x16 <= x15) ->
      (A0 <= x14)/\ (x14 <= x13) ->
       (A0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U84 x12 x14 x16 <= P_id_a__U84 x11 x13 x15.
   
   Hypothesis P_id_a__U32_monotonic :
    forall x12 x14 x13 x11, 
     (A0 <= x14)/\ (x14 <= x13) ->
      (A0 <= x12)/\ (x12 <= x11) ->P_id_a__U32 x12 x14 <= P_id_a__U32 x11 x13.
   
   Hypothesis P_id_isNatList_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->P_id_isNatList x12 <= P_id_isNatList x11.
   
   Hypothesis P_id_a__U13_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->P_id_a__U13 x12 <= P_id_a__U13 x11.
   
   Hypothesis P_id_U85_monotonic :
    forall x12 x14 x13 x11, 
     (A0 <= x14)/\ (x14 <= x13) ->
      (A0 <= x12)/\ (x12 <= x11) ->P_id_U85 x12 x14 <= P_id_U85 x11 x13.
   
   Hypothesis P_id_a__length_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->P_id_a__length x12 <= P_id_a__length x11.
   
   Hypothesis P_id_a__isNatIList_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->
      P_id_a__isNatIList x12 <= P_id_a__isNatIList x11.
   
   Hypothesis P_id_U33_monotonic :
    forall x12 x11, (A0 <= x12)/\ (x12 <= x11) ->P_id_U33 x12 <= P_id_U33 x11.
   
   Hypothesis P_id_a__U11_monotonic :
    forall x12 x14 x13 x11, 
     (A0 <= x14)/\ (x14 <= x13) ->
      (A0 <= x12)/\ (x12 <= x11) ->P_id_a__U11 x12 x14 <= P_id_a__U11 x11 x13.
   
   Hypothesis P_id_U81_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (A0 <= x16)/\ (x16 <= x15) ->
      (A0 <= x14)/\ (x14 <= x13) ->
       (A0 <= x12)/\ (x12 <= x11) ->
        P_id_U81 x12 x14 x16 <= P_id_U81 x11 x13 x15.
   
   Hypothesis P_id_a__U92_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (A0 <= x16)/\ (x16 <= x15) ->
      (A0 <= x14)/\ (x14 <= x13) ->
       (A0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U92 x12 x14 x16 <= P_id_a__U92 x11 x13 x15.
   
   Hypothesis P_id_a__U43_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (A0 <= x16)/\ (x16 <= x15) ->
      (A0 <= x14)/\ (x14 <= x13) ->
       (A0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U43 x12 x14 x16 <= P_id_a__U43 x11 x13 x15.
   
   Hypothesis P_id_U23_monotonic :
    forall x12 x11, (A0 <= x12)/\ (x12 <= x11) ->P_id_U23 x12 <= P_id_U23 x11.
   
   Hypothesis P_id_a__isNatKind_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->
      P_id_a__isNatKind x12 <= P_id_a__isNatKind x11.
   
   Hypothesis P_id_U93_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (A0 <= x16)/\ (x16 <= x15) ->
      (A0 <= x14)/\ (x14 <= x13) ->
       (A0 <= x12)/\ (x12 <= x11) ->
        P_id_U93 x12 x14 x16 <= P_id_U93 x11 x13 x15.
   
   Hypothesis P_id_U11_monotonic :
    forall x12 x14 x13 x11, 
     (A0 <= x14)/\ (x14 <= x13) ->
      (A0 <= x12)/\ (x12 <= x11) ->P_id_U11 x12 x14 <= P_id_U11 x11 x13.
   
   Hypothesis P_id_a__U71_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->P_id_a__U71 x12 <= P_id_a__U71 x11.
   
   Hypothesis P_id_U44_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (A0 <= x16)/\ (x16 <= x15) ->
      (A0 <= x14)/\ (x14 <= x13) ->
       (A0 <= x12)/\ (x12 <= x11) ->
        P_id_U44 x12 x14 x16 <= P_id_U44 x11 x13 x15.
   
   Hypothesis P_id_U61_monotonic :
    forall x12 x11, (A0 <= x12)/\ (x12 <= x11) ->P_id_U61 x12 <= P_id_U61 x11.
   
   Hypothesis P_id_a__U86_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->P_id_a__U86 x12 <= P_id_a__U86 x11.
   
   Hypothesis P_id_a__U41_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (A0 <= x16)/\ (x16 <= x15) ->
      (A0 <= x14)/\ (x14 <= x13) ->
       (A0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U41 x12 x14 x16 <= P_id_a__U41 x11 x13 x15.
   
   Hypothesis P_id_U22_monotonic :
    forall x12 x14 x13 x11, 
     (A0 <= x14)/\ (x14 <= x13) ->
      (A0 <= x12)/\ (x12 <= x11) ->P_id_U22 x12 x14 <= P_id_U22 x11 x13.
   
   Hypothesis P_id_a__U21_monotonic :
    forall x12 x14 x13 x11, 
     (A0 <= x14)/\ (x14 <= x13) ->
      (A0 <= x12)/\ (x12 <= x11) ->P_id_a__U21 x12 x14 <= P_id_a__U21 x11 x13.
   
   Hypothesis P_id_U91_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (A0 <= x16)/\ (x16 <= x15) ->
      (A0 <= x14)/\ (x14 <= x13) ->
       (A0 <= x12)/\ (x12 <= x11) ->
        P_id_U91 x12 x14 x16 <= P_id_U91 x11 x13 x15.
   
   Hypothesis P_id_length_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->P_id_length x12 <= P_id_length x11.
   
   Hypothesis P_id_a__U52_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->P_id_a__U52 x12 <= P_id_a__U52 x11.
   
   Hypothesis P_id_U42_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (A0 <= x16)/\ (x16 <= x15) ->
      (A0 <= x14)/\ (x14 <= x13) ->
       (A0 <= x12)/\ (x12 <= x11) ->
        P_id_U42 x12 x14 x16 <= P_id_U42 x11 x13 x15.
   
   Hypothesis P_id_a__U12_monotonic :
    forall x12 x14 x13 x11, 
     (A0 <= x14)/\ (x14 <= x13) ->
      (A0 <= x12)/\ (x12 <= x11) ->P_id_a__U12 x12 x14 <= P_id_a__U12 x11 x13.
   
   Hypothesis P_id_U83_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (A0 <= x16)/\ (x16 <= x15) ->
      (A0 <= x14)/\ (x14 <= x13) ->
       (A0 <= x12)/\ (x12 <= x11) ->
        P_id_U83 x12 x14 x16 <= P_id_U83 x11 x13 x15.
   
   Hypothesis P_id_a__U94_monotonic :
    forall x12 x14 x13 x11, 
     (A0 <= x14)/\ (x14 <= x13) ->
      (A0 <= x12)/\ (x12 <= x11) ->P_id_a__U94 x12 x14 <= P_id_a__U94 x11 x13.
   
   Hypothesis P_id_a__U45_monotonic :
    forall x12 x14 x13 x11, 
     (A0 <= x14)/\ (x14 <= x13) ->
      (A0 <= x12)/\ (x12 <= x11) ->P_id_a__U45 x12 x14 <= P_id_a__U45 x11 x13.
   
   Hypothesis P_id_U31_monotonic :
    forall x12 x14 x13 x11, 
     (A0 <= x14)/\ (x14 <= x13) ->
      (A0 <= x12)/\ (x12 <= x11) ->P_id_U31 x12 x14 <= P_id_U31 x11 x13.
   
   Hypothesis P_id_a__isNat_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->P_id_a__isNat x12 <= P_id_a__isNat x11.
   
   Hypothesis P_id_isNatIListKind_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->
      P_id_isNatIListKind x12 <= P_id_isNatIListKind x11.
   
   Hypothesis P_id_a__U82_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (A0 <= x16)/\ (x16 <= x15) ->
      (A0 <= x14)/\ (x14 <= x13) ->
       (A0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U82 x12 x14 x16 <= P_id_a__U82 x11 x13 x15.
   
   Hypothesis P_id_U46_monotonic :
    forall x12 x11, (A0 <= x12)/\ (x12 <= x11) ->P_id_U46 x12 <= P_id_U46 x11.
   
   Hypothesis P_id_cons_monotonic :
    forall x12 x14 x13 x11, 
     (A0 <= x14)/\ (x14 <= x13) ->
      (A0 <= x12)/\ (x12 <= x11) ->P_id_cons x12 x14 <= P_id_cons x11 x13.
   
   Hypothesis P_id_U52_monotonic :
    forall x12 x11, (A0 <= x12)/\ (x12 <= x11) ->P_id_U52 x12 <= P_id_U52 x11.
   
   Hypothesis P_id_a__U85_monotonic :
    forall x12 x14 x13 x11, 
     (A0 <= x14)/\ (x14 <= x13) ->
      (A0 <= x12)/\ (x12 <= x11) ->P_id_a__U85 x12 x14 <= P_id_a__U85 x11 x13.
   
   Hypothesis P_id_a__U33_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->P_id_a__U33 x12 <= P_id_a__U33 x11.
   
   Hypothesis P_id_U21_monotonic :
    forall x12 x14 x13 x11, 
     (A0 <= x14)/\ (x14 <= x13) ->
      (A0 <= x12)/\ (x12 <= x11) ->P_id_U21 x12 x14 <= P_id_U21 x11 x13.
   
   Hypothesis P_id_a__isNatList_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->
      P_id_a__isNatList x12 <= P_id_a__isNatList x11.
   
   Hypothesis P_id_U86_monotonic :
    forall x12 x11, (A0 <= x12)/\ (x12 <= x11) ->P_id_U86 x12 <= P_id_U86 x11.
   
   Hypothesis P_id_mark_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->P_id_mark x12 <= P_id_mark x11.
   
   Hypothesis P_id_a__U51_monotonic :
    forall x12 x14 x13 x11, 
     (A0 <= x14)/\ (x14 <= x13) ->
      (A0 <= x12)/\ (x12 <= x11) ->P_id_a__U51 x12 x14 <= P_id_a__U51 x11 x13.
   
   Hypothesis P_id_U41_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (A0 <= x16)/\ (x16 <= x15) ->
      (A0 <= x14)/\ (x14 <= x13) ->
       (A0 <= x12)/\ (x12 <= x11) ->
        P_id_U41 x12 x14 x16 <= P_id_U41 x11 x13 x15.
   
   Hypothesis P_id_U82_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (A0 <= x16)/\ (x16 <= x15) ->
      (A0 <= x14)/\ (x14 <= x13) ->
       (A0 <= x12)/\ (x12 <= x11) ->
        P_id_U82 x12 x14 x16 <= P_id_U82 x11 x13 x15.
   
   Hypothesis P_id_a__U93_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (A0 <= x16)/\ (x16 <= x15) ->
      (A0 <= x14)/\ (x14 <= x13) ->
       (A0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U93 x12 x14 x16 <= P_id_a__U93 x11 x13 x15.
   
   Hypothesis P_id_a__U44_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (A0 <= x16)/\ (x16 <= x15) ->
      (A0 <= x14)/\ (x14 <= x13) ->
       (A0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U44 x12 x14 x16 <= P_id_a__U44 x11 x13 x15.
   
   Hypothesis P_id_isNat_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->P_id_isNat x12 <= P_id_isNat x11.
   
   Hypothesis P_id_a__U23_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->P_id_a__U23 x12 <= P_id_a__U23 x11.
   
   Hypothesis P_id_U94_monotonic :
    forall x12 x14 x13 x11, 
     (A0 <= x14)/\ (x14 <= x13) ->
      (A0 <= x12)/\ (x12 <= x11) ->P_id_U94 x12 x14 <= P_id_U94 x11 x13.
   
   Hypothesis P_id_U12_monotonic :
    forall x12 x14 x13 x11, 
     (A0 <= x14)/\ (x14 <= x13) ->
      (A0 <= x12)/\ (x12 <= x11) ->P_id_U12 x12 x14 <= P_id_U12 x11 x13.
   
   Hypothesis P_id_a__U81_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (A0 <= x16)/\ (x16 <= x15) ->
      (A0 <= x14)/\ (x14 <= x13) ->
       (A0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U81 x12 x14 x16 <= P_id_a__U81 x11 x13 x15.
   
   Hypothesis P_id_U45_monotonic :
    forall x12 x14 x13 x11, 
     (A0 <= x14)/\ (x14 <= x13) ->
      (A0 <= x12)/\ (x12 <= x11) ->P_id_U45 x12 x14 <= P_id_U45 x11 x13.
   
   Hypothesis P_id_U71_monotonic :
    forall x12 x11, (A0 <= x12)/\ (x12 <= x11) ->P_id_U71 x12 <= P_id_U71 x11.
   
   Hypothesis P_id_a__U91_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (A0 <= x16)/\ (x16 <= x15) ->
      (A0 <= x14)/\ (x14 <= x13) ->
       (A0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U91 x12 x14 x16 <= P_id_a__U91 x11 x13 x15.
   
   Hypothesis P_id_a__U42_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (A0 <= x16)/\ (x16 <= x15) ->
      (A0 <= x14)/\ (x14 <= x13) ->
       (A0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U42 x12 x14 x16 <= P_id_a__U42 x11 x13 x15.
   
   Hypothesis P_id_isNatKind_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->P_id_isNatKind x12 <= P_id_isNatKind x11.
   
   Hypothesis P_id_a__U22_monotonic :
    forall x12 x14 x13 x11, 
     (A0 <= x14)/\ (x14 <= x13) ->
      (A0 <= x12)/\ (x12 <= x11) ->P_id_a__U22 x12 x14 <= P_id_a__U22 x11 x13.
   
   Hypothesis P_id_U92_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (A0 <= x16)/\ (x16 <= x15) ->
      (A0 <= x14)/\ (x14 <= x13) ->
       (A0 <= x12)/\ (x12 <= x11) ->
        P_id_U92 x12 x14 x16 <= P_id_U92 x11 x13 x15.
   
   Hypothesis P_id_a__U61_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->P_id_a__U61 x12 <= P_id_a__U61 x11.
   
   Hypothesis P_id_U43_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (A0 <= x16)/\ (x16 <= x15) ->
      (A0 <= x14)/\ (x14 <= x13) ->
       (A0 <= x12)/\ (x12 <= x11) ->
        P_id_U43 x12 x14 x16 <= P_id_U43 x11 x13 x15.
   
   Hypothesis P_id_a__isNatIListKind_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->
      P_id_a__isNatIListKind x12 <= P_id_a__isNatIListKind x11.
   
   Hypothesis P_id_U84_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (A0 <= x16)/\ (x16 <= x15) ->
      (A0 <= x14)/\ (x14 <= x13) ->
       (A0 <= x12)/\ (x12 <= x11) ->
        P_id_U84 x12 x14 x16 <= P_id_U84 x11 x13 x15.
   
   Hypothesis P_id_s_monotonic :
    forall x12 x11, (A0 <= x12)/\ (x12 <= x11) ->P_id_s x12 <= P_id_s x11.
   
   Hypothesis P_id_a__U46_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->P_id_a__U46 x12 <= P_id_a__U46 x11.
   
   Hypothesis P_id_U32_monotonic :
    forall x12 x14 x13 x11, 
     (A0 <= x14)/\ (x14 <= x13) ->
      (A0 <= x12)/\ (x12 <= x11) ->P_id_U32 x12 x14 <= P_id_U32 x11 x13.
   
   Hypothesis P_id_a__U31_monotonic :
    forall x12 x14 x13 x11, 
     (A0 <= x14)/\ (x14 <= x13) ->
      (A0 <= x12)/\ (x12 <= x11) ->P_id_a__U31 x12 x14 <= P_id_a__U31 x11 x13.
   
   Hypothesis P_id_U13_monotonic :
    forall x12 x11, (A0 <= x12)/\ (x12 <= x11) ->P_id_U13 x12 <= P_id_U13 x11.
   
   Hypothesis P_id_a__U83_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (A0 <= x16)/\ (x16 <= x15) ->
      (A0 <= x14)/\ (x14 <= x13) ->
       (A0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U83 x12 x14 x16 <= P_id_a__U83 x11 x13 x15.
   
   Hypothesis P_id_isNatIList_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->P_id_isNatIList x12 <= P_id_isNatIList x11.
   
   Hypothesis P_id_a__zeros_bounded : A0 <= P_id_a__zeros .
   
   Hypothesis P_id_U51_bounded :
    forall x12 x11, (A0 <= x11) ->(A0 <= x12) ->A0 <= P_id_U51 x12 x11.
   
   Hypothesis P_id_a__U84_bounded :
    forall x12 x13 x11, 
     (A0 <= x11) ->(A0 <= x12) ->(A0 <= x13) ->A0 <= P_id_a__U84 x13 x12 x11.
   
   Hypothesis P_id_a__U32_bounded :
    forall x12 x11, (A0 <= x11) ->(A0 <= x12) ->A0 <= P_id_a__U32 x12 x11.
   
   Hypothesis P_id_isNatList_bounded :
    forall x11, (A0 <= x11) ->A0 <= P_id_isNatList x11.
   
   Hypothesis P_id_a__U13_bounded :
    forall x11, (A0 <= x11) ->A0 <= P_id_a__U13 x11.
   
   Hypothesis P_id_U85_bounded :
    forall x12 x11, (A0 <= x11) ->(A0 <= x12) ->A0 <= P_id_U85 x12 x11.
   
   Hypothesis P_id_a__length_bounded :
    forall x11, (A0 <= x11) ->A0 <= P_id_a__length x11.
   
   Hypothesis P_id_a__isNatIList_bounded :
    forall x11, (A0 <= x11) ->A0 <= P_id_a__isNatIList x11.
   
   Hypothesis P_id_U33_bounded :
    forall x11, (A0 <= x11) ->A0 <= P_id_U33 x11.
   
   Hypothesis P_id_a__U11_bounded :
    forall x12 x11, (A0 <= x11) ->(A0 <= x12) ->A0 <= P_id_a__U11 x12 x11.
   
   Hypothesis P_id_U81_bounded :
    forall x12 x13 x11, 
     (A0 <= x11) ->(A0 <= x12) ->(A0 <= x13) ->A0 <= P_id_U81 x13 x12 x11.
   
   Hypothesis P_id_a__U92_bounded :
    forall x12 x13 x11, 
     (A0 <= x11) ->(A0 <= x12) ->(A0 <= x13) ->A0 <= P_id_a__U92 x13 x12 x11.
   
   Hypothesis P_id_a__U43_bounded :
    forall x12 x13 x11, 
     (A0 <= x11) ->(A0 <= x12) ->(A0 <= x13) ->A0 <= P_id_a__U43 x13 x12 x11.
   
   Hypothesis P_id_U23_bounded :
    forall x11, (A0 <= x11) ->A0 <= P_id_U23 x11.
   
   Hypothesis P_id_a__isNatKind_bounded :
    forall x11, (A0 <= x11) ->A0 <= P_id_a__isNatKind x11.
   
   Hypothesis P_id_U93_bounded :
    forall x12 x13 x11, 
     (A0 <= x11) ->(A0 <= x12) ->(A0 <= x13) ->A0 <= P_id_U93 x13 x12 x11.
   
   Hypothesis P_id_U11_bounded :
    forall x12 x11, (A0 <= x11) ->(A0 <= x12) ->A0 <= P_id_U11 x12 x11.
   
   Hypothesis P_id_a__U71_bounded :
    forall x11, (A0 <= x11) ->A0 <= P_id_a__U71 x11.
   
   Hypothesis P_id_U44_bounded :
    forall x12 x13 x11, 
     (A0 <= x11) ->(A0 <= x12) ->(A0 <= x13) ->A0 <= P_id_U44 x13 x12 x11.
   
   Hypothesis P_id_0_bounded : A0 <= P_id_0 .
   
   Hypothesis P_id_U61_bounded :
    forall x11, (A0 <= x11) ->A0 <= P_id_U61 x11.
   
   Hypothesis P_id_a__U86_bounded :
    forall x11, (A0 <= x11) ->A0 <= P_id_a__U86 x11.
   
   Hypothesis P_id_a__U41_bounded :
    forall x12 x13 x11, 
     (A0 <= x11) ->(A0 <= x12) ->(A0 <= x13) ->A0 <= P_id_a__U41 x13 x12 x11.
   
   Hypothesis P_id_U22_bounded :
    forall x12 x11, (A0 <= x11) ->(A0 <= x12) ->A0 <= P_id_U22 x12 x11.
   
   Hypothesis P_id_a__U21_bounded :
    forall x12 x11, (A0 <= x11) ->(A0 <= x12) ->A0 <= P_id_a__U21 x12 x11.
   
   Hypothesis P_id_U91_bounded :
    forall x12 x13 x11, 
     (A0 <= x11) ->(A0 <= x12) ->(A0 <= x13) ->A0 <= P_id_U91 x13 x12 x11.
   
   Hypothesis P_id_length_bounded :
    forall x11, (A0 <= x11) ->A0 <= P_id_length x11.
   
   Hypothesis P_id_a__U52_bounded :
    forall x11, (A0 <= x11) ->A0 <= P_id_a__U52 x11.
   
   Hypothesis P_id_U42_bounded :
    forall x12 x13 x11, 
     (A0 <= x11) ->(A0 <= x12) ->(A0 <= x13) ->A0 <= P_id_U42 x13 x12 x11.
   
   Hypothesis P_id_a__U12_bounded :
    forall x12 x11, (A0 <= x11) ->(A0 <= x12) ->A0 <= P_id_a__U12 x12 x11.
   
   Hypothesis P_id_U83_bounded :
    forall x12 x13 x11, 
     (A0 <= x11) ->(A0 <= x12) ->(A0 <= x13) ->A0 <= P_id_U83 x13 x12 x11.
   
   Hypothesis P_id_a__U94_bounded :
    forall x12 x11, (A0 <= x11) ->(A0 <= x12) ->A0 <= P_id_a__U94 x12 x11.
   
   Hypothesis P_id_a__U45_bounded :
    forall x12 x11, (A0 <= x11) ->(A0 <= x12) ->A0 <= P_id_a__U45 x12 x11.
   
   Hypothesis P_id_U31_bounded :
    forall x12 x11, (A0 <= x11) ->(A0 <= x12) ->A0 <= P_id_U31 x12 x11.
   
   Hypothesis P_id_a__isNat_bounded :
    forall x11, (A0 <= x11) ->A0 <= P_id_a__isNat x11.
   
   Hypothesis P_id_isNatIListKind_bounded :
    forall x11, (A0 <= x11) ->A0 <= P_id_isNatIListKind x11.
   
   Hypothesis P_id_a__U82_bounded :
    forall x12 x13 x11, 
     (A0 <= x11) ->(A0 <= x12) ->(A0 <= x13) ->A0 <= P_id_a__U82 x13 x12 x11.
   
   Hypothesis P_id_U46_bounded :
    forall x11, (A0 <= x11) ->A0 <= P_id_U46 x11.
   
   Hypothesis P_id_cons_bounded :
    forall x12 x11, (A0 <= x11) ->(A0 <= x12) ->A0 <= P_id_cons x12 x11.
   
   Hypothesis P_id_U52_bounded :
    forall x11, (A0 <= x11) ->A0 <= P_id_U52 x11.
   
   Hypothesis P_id_a__U85_bounded :
    forall x12 x11, (A0 <= x11) ->(A0 <= x12) ->A0 <= P_id_a__U85 x12 x11.
   
   Hypothesis P_id_a__U33_bounded :
    forall x11, (A0 <= x11) ->A0 <= P_id_a__U33 x11.
   
   Hypothesis P_id_U21_bounded :
    forall x12 x11, (A0 <= x11) ->(A0 <= x12) ->A0 <= P_id_U21 x12 x11.
   
   Hypothesis P_id_a__isNatList_bounded :
    forall x11, (A0 <= x11) ->A0 <= P_id_a__isNatList x11.
   
   Hypothesis P_id_U86_bounded :
    forall x11, (A0 <= x11) ->A0 <= P_id_U86 x11.
   
   Hypothesis P_id_mark_bounded :
    forall x11, (A0 <= x11) ->A0 <= P_id_mark x11.
   
   Hypothesis P_id_a__U51_bounded :
    forall x12 x11, (A0 <= x11) ->(A0 <= x12) ->A0 <= P_id_a__U51 x12 x11.
   
   Hypothesis P_id_U41_bounded :
    forall x12 x13 x11, 
     (A0 <= x11) ->(A0 <= x12) ->(A0 <= x13) ->A0 <= P_id_U41 x13 x12 x11.
   
   Hypothesis P_id_tt_bounded : A0 <= P_id_tt .
   
   Hypothesis P_id_U82_bounded :
    forall x12 x13 x11, 
     (A0 <= x11) ->(A0 <= x12) ->(A0 <= x13) ->A0 <= P_id_U82 x13 x12 x11.
   
   Hypothesis P_id_a__U93_bounded :
    forall x12 x13 x11, 
     (A0 <= x11) ->(A0 <= x12) ->(A0 <= x13) ->A0 <= P_id_a__U93 x13 x12 x11.
   
   Hypothesis P_id_a__U44_bounded :
    forall x12 x13 x11, 
     (A0 <= x11) ->(A0 <= x12) ->(A0 <= x13) ->A0 <= P_id_a__U44 x13 x12 x11.
   
   Hypothesis P_id_isNat_bounded :
    forall x11, (A0 <= x11) ->A0 <= P_id_isNat x11.
   
   Hypothesis P_id_a__U23_bounded :
    forall x11, (A0 <= x11) ->A0 <= P_id_a__U23 x11.
   
   Hypothesis P_id_U94_bounded :
    forall x12 x11, (A0 <= x11) ->(A0 <= x12) ->A0 <= P_id_U94 x12 x11.
   
   Hypothesis P_id_U12_bounded :
    forall x12 x11, (A0 <= x11) ->(A0 <= x12) ->A0 <= P_id_U12 x12 x11.
   
   Hypothesis P_id_a__U81_bounded :
    forall x12 x13 x11, 
     (A0 <= x11) ->(A0 <= x12) ->(A0 <= x13) ->A0 <= P_id_a__U81 x13 x12 x11.
   
   Hypothesis P_id_U45_bounded :
    forall x12 x11, (A0 <= x11) ->(A0 <= x12) ->A0 <= P_id_U45 x12 x11.
   
   Hypothesis P_id_zeros_bounded : A0 <= P_id_zeros .
   
   Hypothesis P_id_U71_bounded :
    forall x11, (A0 <= x11) ->A0 <= P_id_U71 x11.
   
   Hypothesis P_id_a__U91_bounded :
    forall x12 x13 x11, 
     (A0 <= x11) ->(A0 <= x12) ->(A0 <= x13) ->A0 <= P_id_a__U91 x13 x12 x11.
   
   Hypothesis P_id_a__U42_bounded :
    forall x12 x13 x11, 
     (A0 <= x11) ->(A0 <= x12) ->(A0 <= x13) ->A0 <= P_id_a__U42 x13 x12 x11.
   
   Hypothesis P_id_isNatKind_bounded :
    forall x11, (A0 <= x11) ->A0 <= P_id_isNatKind x11.
   
   Hypothesis P_id_a__U22_bounded :
    forall x12 x11, (A0 <= x11) ->(A0 <= x12) ->A0 <= P_id_a__U22 x12 x11.
   
   Hypothesis P_id_U92_bounded :
    forall x12 x13 x11, 
     (A0 <= x11) ->(A0 <= x12) ->(A0 <= x13) ->A0 <= P_id_U92 x13 x12 x11.
   
   Hypothesis P_id_nil_bounded : A0 <= P_id_nil .
   
   Hypothesis P_id_a__U61_bounded :
    forall x11, (A0 <= x11) ->A0 <= P_id_a__U61 x11.
   
   Hypothesis P_id_U43_bounded :
    forall x12 x13 x11, 
     (A0 <= x11) ->(A0 <= x12) ->(A0 <= x13) ->A0 <= P_id_U43 x13 x12 x11.
   
   Hypothesis P_id_a__isNatIListKind_bounded :
    forall x11, (A0 <= x11) ->A0 <= P_id_a__isNatIListKind x11.
   
   Hypothesis P_id_U84_bounded :
    forall x12 x13 x11, 
     (A0 <= x11) ->(A0 <= x12) ->(A0 <= x13) ->A0 <= P_id_U84 x13 x12 x11.
   
   Hypothesis P_id_s_bounded : forall x11, (A0 <= x11) ->A0 <= P_id_s x11.
   
   Hypothesis P_id_a__U46_bounded :
    forall x11, (A0 <= x11) ->A0 <= P_id_a__U46 x11.
   
   Hypothesis P_id_U32_bounded :
    forall x12 x11, (A0 <= x11) ->(A0 <= x12) ->A0 <= P_id_U32 x12 x11.
   
   Hypothesis P_id_a__U31_bounded :
    forall x12 x11, (A0 <= x11) ->(A0 <= x12) ->A0 <= P_id_a__U31 x12 x11.
   
   Hypothesis P_id_U13_bounded :
    forall x11, (A0 <= x11) ->A0 <= P_id_U13 x11.
   
   Hypothesis P_id_a__U83_bounded :
    forall x12 x13 x11, 
     (A0 <= x11) ->(A0 <= x12) ->(A0 <= x13) ->A0 <= P_id_a__U83 x13 x12 x11.
   
   Hypothesis P_id_isNatIList_bounded :
    forall x11, (A0 <= x11) ->A0 <= P_id_isNatIList x11.
   
   Fixpoint measure t { struct t }  := 
     match t with
       | (algebra.Alg.Term algebra.F.id_a__zeros nil) => P_id_a__zeros 
       | (algebra.Alg.Term algebra.F.id_U51 (x12::x11::nil)) =>
        P_id_U51 (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U84 (x13::x12::x11::nil)) =>
        P_id_a__U84 (measure x13) (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U32 (x12::x11::nil)) =>
        P_id_a__U32 (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_isNatList (x11::nil)) =>
        P_id_isNatList (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U13 (x11::nil)) =>
        P_id_a__U13 (measure x11)
       | (algebra.Alg.Term algebra.F.id_U85 (x12::x11::nil)) =>
        P_id_U85 (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__length (x11::nil)) =>
        P_id_a__length (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__isNatIList (x11::nil)) =>
        P_id_a__isNatIList (measure x11)
       | (algebra.Alg.Term algebra.F.id_U33 (x11::nil)) =>
        P_id_U33 (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U11 (x12::x11::nil)) =>
        P_id_a__U11 (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_U81 (x13::x12::x11::nil)) =>
        P_id_U81 (measure x13) (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U92 (x13::x12::x11::nil)) =>
        P_id_a__U92 (measure x13) (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U43 (x13::x12::x11::nil)) =>
        P_id_a__U43 (measure x13) (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_U23 (x11::nil)) =>
        P_id_U23 (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__isNatKind (x11::nil)) =>
        P_id_a__isNatKind (measure x11)
       | (algebra.Alg.Term algebra.F.id_U93 (x13::x12::x11::nil)) =>
        P_id_U93 (measure x13) (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_U11 (x12::x11::nil)) =>
        P_id_U11 (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U71 (x11::nil)) =>
        P_id_a__U71 (measure x11)
       | (algebra.Alg.Term algebra.F.id_U44 (x13::x12::x11::nil)) =>
        P_id_U44 (measure x13) (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_0 nil) => P_id_0 
       | (algebra.Alg.Term algebra.F.id_U61 (x11::nil)) =>
        P_id_U61 (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U86 (x11::nil)) =>
        P_id_a__U86 (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U41 (x13::x12::x11::nil)) =>
        P_id_a__U41 (measure x13) (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_U22 (x12::x11::nil)) =>
        P_id_U22 (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U21 (x12::x11::nil)) =>
        P_id_a__U21 (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_U91 (x13::x12::x11::nil)) =>
        P_id_U91 (measure x13) (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_length (x11::nil)) =>
        P_id_length (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U52 (x11::nil)) =>
        P_id_a__U52 (measure x11)
       | (algebra.Alg.Term algebra.F.id_U42 (x13::x12::x11::nil)) =>
        P_id_U42 (measure x13) (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U12 (x12::x11::nil)) =>
        P_id_a__U12 (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_U83 (x13::x12::x11::nil)) =>
        P_id_U83 (measure x13) (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U94 (x12::x11::nil)) =>
        P_id_a__U94 (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U45 (x12::x11::nil)) =>
        P_id_a__U45 (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_U31 (x12::x11::nil)) =>
        P_id_U31 (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__isNat (x11::nil)) =>
        P_id_a__isNat (measure x11)
       | (algebra.Alg.Term algebra.F.id_isNatIListKind (x11::nil)) =>
        P_id_isNatIListKind (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U82 (x13::x12::x11::nil)) =>
        P_id_a__U82 (measure x13) (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_U46 (x11::nil)) =>
        P_id_U46 (measure x11)
       | (algebra.Alg.Term algebra.F.id_cons (x12::x11::nil)) =>
        P_id_cons (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_U52 (x11::nil)) =>
        P_id_U52 (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U85 (x12::x11::nil)) =>
        P_id_a__U85 (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U33 (x11::nil)) =>
        P_id_a__U33 (measure x11)
       | (algebra.Alg.Term algebra.F.id_U21 (x12::x11::nil)) =>
        P_id_U21 (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__isNatList (x11::nil)) =>
        P_id_a__isNatList (measure x11)
       | (algebra.Alg.Term algebra.F.id_U86 (x11::nil)) =>
        P_id_U86 (measure x11)
       | (algebra.Alg.Term algebra.F.id_mark (x11::nil)) =>
        P_id_mark (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U51 (x12::x11::nil)) =>
        P_id_a__U51 (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_U41 (x13::x12::x11::nil)) =>
        P_id_U41 (measure x13) (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt 
       | (algebra.Alg.Term algebra.F.id_U82 (x13::x12::x11::nil)) =>
        P_id_U82 (measure x13) (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U93 (x13::x12::x11::nil)) =>
        P_id_a__U93 (measure x13) (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U44 (x13::x12::x11::nil)) =>
        P_id_a__U44 (measure x13) (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_isNat (x11::nil)) =>
        P_id_isNat (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U23 (x11::nil)) =>
        P_id_a__U23 (measure x11)
       | (algebra.Alg.Term algebra.F.id_U94 (x12::x11::nil)) =>
        P_id_U94 (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_U12 (x12::x11::nil)) =>
        P_id_U12 (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U81 (x13::x12::x11::nil)) =>
        P_id_a__U81 (measure x13) (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_U45 (x12::x11::nil)) =>
        P_id_U45 (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_zeros nil) => P_id_zeros 
       | (algebra.Alg.Term algebra.F.id_U71 (x11::nil)) =>
        P_id_U71 (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U91 (x13::x12::x11::nil)) =>
        P_id_a__U91 (measure x13) (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U42 (x13::x12::x11::nil)) =>
        P_id_a__U42 (measure x13) (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_isNatKind (x11::nil)) =>
        P_id_isNatKind (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U22 (x12::x11::nil)) =>
        P_id_a__U22 (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_U92 (x13::x12::x11::nil)) =>
        P_id_U92 (measure x13) (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil 
       | (algebra.Alg.Term algebra.F.id_a__U61 (x11::nil)) =>
        P_id_a__U61 (measure x11)
       | (algebra.Alg.Term algebra.F.id_U43 (x13::x12::x11::nil)) =>
        P_id_U43 (measure x13) (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__isNatIListKind (x11::nil)) =>
        P_id_a__isNatIListKind (measure x11)
       | (algebra.Alg.Term algebra.F.id_U84 (x13::x12::x11::nil)) =>
        P_id_U84 (measure x13) (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_s (x11::nil)) => P_id_s (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U46 (x11::nil)) =>
        P_id_a__U46 (measure x11)
       | (algebra.Alg.Term algebra.F.id_U32 (x12::x11::nil)) =>
        P_id_U32 (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U31 (x12::x11::nil)) =>
        P_id_a__U31 (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_U13 (x11::nil)) =>
        P_id_U13 (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U83 (x13::x12::x11::nil)) =>
        P_id_a__U83 (measure x13) (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_isNatIList (x11::nil)) =>
        P_id_isNatIList (measure x11)
       | _ => A0
       end.
   
   Lemma measure_equation :
    forall t, 
     measure t = match t with
                   | (algebra.Alg.Term algebra.F.id_a__zeros nil) =>
                    P_id_a__zeros 
                   | (algebra.Alg.Term algebra.F.id_U51 (x12::x11::nil)) =>
                    P_id_U51 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U84 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U84 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U32 (x12::x11::nil)) =>
                    P_id_a__U32 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_isNatList (x11::nil)) =>
                    P_id_isNatList (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U13 (x11::nil)) =>
                    P_id_a__U13 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U85 (x12::x11::nil)) =>
                    P_id_U85 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__length (x11::nil)) =>
                    P_id_a__length (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__isNatIList (x11::nil)) =>
                    P_id_a__isNatIList (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U33 (x11::nil)) =>
                    P_id_U33 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U11 (x12::x11::nil)) =>
                    P_id_a__U11 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U81 (x13::x12::x11::nil)) =>
                    P_id_U81 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U92 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U92 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U43 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U43 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U23 (x11::nil)) =>
                    P_id_U23 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__isNatKind (x11::nil)) =>
                    P_id_a__isNatKind (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U93 (x13::x12::x11::nil)) =>
                    P_id_U93 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U11 (x12::x11::nil)) =>
                    P_id_U11 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U71 (x11::nil)) =>
                    P_id_a__U71 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U44 (x13::x12::x11::nil)) =>
                    P_id_U44 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_0 nil) => P_id_0 
                   | (algebra.Alg.Term algebra.F.id_U61 (x11::nil)) =>
                    P_id_U61 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U86 (x11::nil)) =>
                    P_id_a__U86 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U41 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U41 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U22 (x12::x11::nil)) =>
                    P_id_U22 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U21 (x12::x11::nil)) =>
                    P_id_a__U21 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U91 (x13::x12::x11::nil)) =>
                    P_id_U91 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_length (x11::nil)) =>
                    P_id_length (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U52 (x11::nil)) =>
                    P_id_a__U52 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U42 (x13::x12::x11::nil)) =>
                    P_id_U42 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U12 (x12::x11::nil)) =>
                    P_id_a__U12 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U83 (x13::x12::x11::nil)) =>
                    P_id_U83 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U94 (x12::x11::nil)) =>
                    P_id_a__U94 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U45 (x12::x11::nil)) =>
                    P_id_a__U45 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U31 (x12::x11::nil)) =>
                    P_id_U31 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__isNat (x11::nil)) =>
                    P_id_a__isNat (measure x11)
                   | (algebra.Alg.Term algebra.F.id_isNatIListKind 
                      (x11::nil)) =>
                    P_id_isNatIListKind (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U82 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U82 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U46 (x11::nil)) =>
                    P_id_U46 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_cons (x12::x11::nil)) =>
                    P_id_cons (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U52 (x11::nil)) =>
                    P_id_U52 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U85 (x12::x11::nil)) =>
                    P_id_a__U85 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U33 (x11::nil)) =>
                    P_id_a__U33 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U21 (x12::x11::nil)) =>
                    P_id_U21 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__isNatList (x11::nil)) =>
                    P_id_a__isNatList (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U86 (x11::nil)) =>
                    P_id_U86 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_mark (x11::nil)) =>
                    P_id_mark (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U51 (x12::x11::nil)) =>
                    P_id_a__U51 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U41 (x13::x12::x11::nil)) =>
                    P_id_U41 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt 
                   | (algebra.Alg.Term algebra.F.id_U82 (x13::x12::x11::nil)) =>
                    P_id_U82 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U93 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U93 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U44 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U44 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_isNat (x11::nil)) =>
                    P_id_isNat (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U23 (x11::nil)) =>
                    P_id_a__U23 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U94 (x12::x11::nil)) =>
                    P_id_U94 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U12 (x12::x11::nil)) =>
                    P_id_U12 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U81 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U81 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U45 (x12::x11::nil)) =>
                    P_id_U45 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_zeros nil) => P_id_zeros 
                   | (algebra.Alg.Term algebra.F.id_U71 (x11::nil)) =>
                    P_id_U71 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U91 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U91 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U42 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U42 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_isNatKind (x11::nil)) =>
                    P_id_isNatKind (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U22 (x12::x11::nil)) =>
                    P_id_a__U22 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U92 (x13::x12::x11::nil)) =>
                    P_id_U92 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil 
                   | (algebra.Alg.Term algebra.F.id_a__U61 (x11::nil)) =>
                    P_id_a__U61 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U43 (x13::x12::x11::nil)) =>
                    P_id_U43 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
                      (x11::nil)) =>
                    P_id_a__isNatIListKind (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U84 (x13::x12::x11::nil)) =>
                    P_id_U84 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_s (x11::nil)) =>
                    P_id_s (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U46 (x11::nil)) =>
                    P_id_a__U46 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U32 (x12::x11::nil)) =>
                    P_id_U32 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U31 (x12::x11::nil)) =>
                    P_id_a__U31 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U13 (x11::nil)) =>
                    P_id_U13 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U83 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U83 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_isNatIList (x11::nil)) =>
                    P_id_isNatIList (measure x11)
                   | _ => A0
                   end.
   Proof.
     intros t;case t;intros ;apply refl_equal.
   Qed.
   
   Definition Pols f : InterpGen.Pol_type A (InterpGen.get_arity f) := 
     match f with
       | algebra.F.id_a__zeros => P_id_a__zeros
       | algebra.F.id_U51 => P_id_U51
       | algebra.F.id_a__U84 => P_id_a__U84
       | algebra.F.id_a__U32 => P_id_a__U32
       | algebra.F.id_isNatList => P_id_isNatList
       | algebra.F.id_a__U13 => P_id_a__U13
       | algebra.F.id_U85 => P_id_U85
       | algebra.F.id_a__length => P_id_a__length
       | algebra.F.id_a__isNatIList => P_id_a__isNatIList
       | algebra.F.id_U33 => P_id_U33
       | algebra.F.id_a__U11 => P_id_a__U11
       | algebra.F.id_U81 => P_id_U81
       | algebra.F.id_a__U92 => P_id_a__U92
       | algebra.F.id_a__U43 => P_id_a__U43
       | algebra.F.id_U23 => P_id_U23
       | algebra.F.id_a__isNatKind => P_id_a__isNatKind
       | algebra.F.id_U93 => P_id_U93
       | algebra.F.id_U11 => P_id_U11
       | algebra.F.id_a__U71 => P_id_a__U71
       | algebra.F.id_U44 => P_id_U44
       | algebra.F.id_0 => P_id_0
       | algebra.F.id_U61 => P_id_U61
       | algebra.F.id_a__U86 => P_id_a__U86
       | algebra.F.id_a__U41 => P_id_a__U41
       | algebra.F.id_U22 => P_id_U22
       | algebra.F.id_a__U21 => P_id_a__U21
       | algebra.F.id_U91 => P_id_U91
       | algebra.F.id_length => P_id_length
       | algebra.F.id_a__U52 => P_id_a__U52
       | algebra.F.id_U42 => P_id_U42
       | algebra.F.id_a__U12 => P_id_a__U12
       | algebra.F.id_U83 => P_id_U83
       | algebra.F.id_a__U94 => P_id_a__U94
       | algebra.F.id_a__U45 => P_id_a__U45
       | algebra.F.id_U31 => P_id_U31
       | algebra.F.id_a__isNat => P_id_a__isNat
       | algebra.F.id_isNatIListKind => P_id_isNatIListKind
       | algebra.F.id_a__U82 => P_id_a__U82
       | algebra.F.id_U46 => P_id_U46
       | algebra.F.id_cons => P_id_cons
       | algebra.F.id_U52 => P_id_U52
       | algebra.F.id_a__U85 => P_id_a__U85
       | algebra.F.id_a__U33 => P_id_a__U33
       | algebra.F.id_U21 => P_id_U21
       | algebra.F.id_a__isNatList => P_id_a__isNatList
       | algebra.F.id_U86 => P_id_U86
       | algebra.F.id_mark => P_id_mark
       | algebra.F.id_a__U51 => P_id_a__U51
       | algebra.F.id_U41 => P_id_U41
       | algebra.F.id_tt => P_id_tt
       | algebra.F.id_U82 => P_id_U82
       | algebra.F.id_a__U93 => P_id_a__U93
       | algebra.F.id_a__U44 => P_id_a__U44
       | algebra.F.id_isNat => P_id_isNat
       | algebra.F.id_a__U23 => P_id_a__U23
       | algebra.F.id_U94 => P_id_U94
       | algebra.F.id_U12 => P_id_U12
       | algebra.F.id_a__U81 => P_id_a__U81
       | algebra.F.id_U45 => P_id_U45
       | algebra.F.id_zeros => P_id_zeros
       | algebra.F.id_U71 => P_id_U71
       | algebra.F.id_a__U91 => P_id_a__U91
       | algebra.F.id_a__U42 => P_id_a__U42
       | algebra.F.id_isNatKind => P_id_isNatKind
       | algebra.F.id_a__U22 => P_id_a__U22
       | algebra.F.id_U92 => P_id_U92
       | algebra.F.id_nil => P_id_nil
       | algebra.F.id_a__U61 => P_id_a__U61
       | algebra.F.id_U43 => P_id_U43
       | algebra.F.id_a__isNatIListKind => P_id_a__isNatIListKind
       | algebra.F.id_U84 => P_id_U84
       | algebra.F.id_s => P_id_s
       | algebra.F.id_a__U46 => P_id_a__U46
       | algebra.F.id_U32 => P_id_U32
       | algebra.F.id_a__U31 => P_id_a__U31
       | algebra.F.id_U13 => P_id_U13
       | algebra.F.id_a__U83 => P_id_a__U83
       | algebra.F.id_isNatIList => P_id_isNatIList
       end.
   
   Lemma same_measure : forall t, measure t = InterpGen.measure A0 Pols t.
   Proof.
     fix 1 .
     intros [a| f l].
     simpl in |-*.
     unfold eq_rect_r, eq_rect, sym_eq in |-*.
     reflexivity .
     
     refine match f with
              | algebra.F.id_a__zeros => match l with
                                           | nil => _
                                           | _::_ => _
                                           end
              | algebra.F.id_U51 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_a__U84 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_a__U32 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_isNatList =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__U13 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_U85 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_a__length =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__isNatIList =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_U33 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__U11 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_U81 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_a__U92 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_a__U43 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_U23 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__isNatKind =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_U93 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_U11 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_a__U71 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_U44 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_0 => match l with
                                    | nil => _
                                    | _::_ => _
                                    end
              | algebra.F.id_U61 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__U86 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__U41 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_U22 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_a__U21 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_U91 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_length =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__U52 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_U42 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_a__U12 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_U83 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_a__U94 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_a__U45 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_U31 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_a__isNat =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_isNatIListKind =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__U82 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_U46 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_cons =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_U52 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__U85 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_a__U33 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_U21 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_a__isNatList =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_U86 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_mark =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__U51 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_U41 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_tt => match l with
                                     | nil => _
                                     | _::_ => _
                                     end
              | algebra.F.id_U82 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_a__U93 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_a__U44 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_isNat =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__U23 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_U94 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_U12 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_a__U81 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_U45 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_zeros => match l with
                                        | nil => _
                                        | _::_ => _
                                        end
              | algebra.F.id_U71 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__U91 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_a__U42 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_isNatKind =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__U22 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_U92 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_nil => match l with
                                      | nil => _
                                      | _::_ => _
                                      end
              | algebra.F.id_a__U61 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_U43 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_a__isNatIListKind =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_U84 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_s =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__U46 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_U32 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_a__U31 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_U13 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__U83 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_isNatIList =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              end;simpl in |-*;unfold eq_rect_r, eq_rect, sym_eq in |-*;
      try (reflexivity );f_equal ;auto.
   Qed.
   
   Lemma measure_bounded : forall t, A0 <= measure t.
   Proof.
     intros t.
     rewrite same_measure in |-*.
     apply (InterpGen.measure_bounded Aop).
     intros f.
     case f.
     vm_compute in |-*;intros ;apply P_id_a__zeros_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U51_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U84_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U32_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_isNatList_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U13_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U85_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__length_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isNatIList_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U33_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U11_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U81_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U92_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U43_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U23_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isNatKind_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U93_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U11_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U71_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U44_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_0_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U61_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U86_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U41_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U22_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U21_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U91_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_length_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U52_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U42_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U12_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U83_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U94_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U45_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U31_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isNat_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_isNatIListKind_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U82_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U46_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_cons_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U52_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U85_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U33_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U21_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isNatList_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U86_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_mark_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U51_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U41_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_tt_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U82_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U93_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U44_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_isNat_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U23_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U94_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U12_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U81_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U45_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_zeros_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U71_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U91_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U42_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_isNatKind_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U22_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U92_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_nil_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U61_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U43_bounded;assumption.
     
     vm_compute in |-*;intros ;apply P_id_a__isNatIListKind_bounded;
      assumption.
     vm_compute in |-*;intros ;apply P_id_U84_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_s_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U46_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U32_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U31_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U13_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U83_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_isNatIList_bounded;assumption.
   Qed.
   
   Ltac generate_pos_hyp  :=
    match goal with
      | H:context [measure ?x] |- _ =>
       let v := fresh "v" in 
        (let H := fresh "h" in 
          (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
            clearbody H;clearbody v))
      |  |- context [measure ?x] =>
       let v := fresh "v" in 
        (let H := fresh "h" in 
          (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
            clearbody H;clearbody v))
      end
    .
   
   Hypothesis rules_monotonic :
    forall l r, 
     (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) ->
      measure r <= measure l.
   
   Lemma measure_star_monotonic :
    forall l r, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                r l) ->measure r <= measure l.
   Proof.
     intros .
     do 2 (rewrite same_measure in |-*).
     
     apply InterpGen.measure_star_monotonic with (1:=Aop) (Pols:=Pols) 
     (rules:=R_xml_0_deep_rew.R_xml_0_rules).
     intros f.
     case f.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id_U51_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U84_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U32_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_isNatList_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U13_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U85_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__length_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isNatIList_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U33_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U11_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U81_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U92_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U43_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U23_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isNatKind_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U93_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U11_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U71_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U44_monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id_U61_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U86_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U41_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U22_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U21_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U91_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_length_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U52_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U42_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U12_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U83_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U94_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U45_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U31_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isNat_monotonic;assumption.
     
     vm_compute in |-*;intros ;apply P_id_isNatIListKind_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U82_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U46_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_cons_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U52_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U85_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U33_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U21_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isNatList_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U86_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_mark_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U51_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U41_monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id_U82_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U93_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U44_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_isNat_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U23_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U94_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U12_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U81_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U45_monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id_U71_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U91_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U42_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_isNatKind_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U22_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U92_monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id_a__U61_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U43_monotonic;assumption.
     
     vm_compute in |-*;intros ;apply P_id_a__isNatIListKind_monotonic;
      assumption.
     vm_compute in |-*;intros ;apply P_id_U84_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_s_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U46_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U32_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U31_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U13_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U83_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_isNatIList_monotonic;assumption.
     intros f.
     case f.
     vm_compute in |-*;intros ;apply P_id_a__zeros_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U51_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U84_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U32_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_isNatList_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U13_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U85_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__length_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isNatIList_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U33_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U11_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U81_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U92_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U43_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U23_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isNatKind_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U93_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U11_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U71_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U44_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_0_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U61_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U86_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U41_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U22_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U21_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U91_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_length_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U52_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U42_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U12_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U83_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U94_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U45_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U31_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isNat_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_isNatIListKind_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U82_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U46_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_cons_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U52_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U85_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U33_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U21_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isNatList_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U86_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_mark_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U51_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U41_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_tt_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U82_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U93_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U44_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_isNat_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U23_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U94_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U12_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U81_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U45_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_zeros_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U71_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U91_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U42_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_isNatKind_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U22_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U92_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_nil_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U61_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U43_bounded;assumption.
     
     vm_compute in |-*;intros ;apply P_id_a__isNatIListKind_bounded;
      assumption.
     vm_compute in |-*;intros ;apply P_id_U84_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_s_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U46_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U32_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U31_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U13_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U83_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_isNatIList_bounded;assumption.
     intros .
     do 2 (rewrite  <- same_measure in |-*).
     apply rules_monotonic;assumption.
     assumption.
   Qed.
   
   Hypothesis P_id_A__U46 : A ->A.
   
   Hypothesis P_id_A__ISNATILISTKIND : A ->A.
   
   Hypothesis P_id_A__U61 : A ->A.
   
   Hypothesis P_id_A__U85 : A ->A ->A.
   
   Hypothesis P_id_A__U31 : A ->A ->A.
   
   Hypothesis P_id_A__U81 : A ->A ->A ->A.
   
   Hypothesis P_id_A__U22 : A ->A ->A.
   
   Hypothesis P_id_A__U93 : A ->A ->A ->A.
   
   Hypothesis P_id_A__U42 : A ->A ->A ->A.
   
   Hypothesis P_id_A__U51 : A ->A ->A.
   
   Hypothesis P_id_A__ISNATLIST : A ->A.
   
   Hypothesis P_id_A__ZEROS : A.
   
   Hypothesis P_id_A__U91 : A ->A ->A ->A.
   
   Hypothesis P_id_A__U33 : A ->A.
   
   Hypothesis P_id_A__U83 : A ->A ->A ->A.
   
   Hypothesis P_id_A__U23 : A ->A.
   
   Hypothesis P_id_A__U11 : A ->A ->A.
   
   Hypothesis P_id_A__LENGTH : A ->A.
   
   Hypothesis P_id_A__U44 : A ->A ->A ->A.
   
   Hypothesis P_id_A__ISNATILIST : A ->A.
   
   Hypothesis P_id_A__U13 : A ->A.
   
   Hypothesis P_id_A__U71 : A ->A.
   
   Hypothesis P_id_A__U86 : A ->A.
   
   Hypothesis P_id_A__U32 : A ->A ->A.
   
   Hypothesis P_id_A__U82 : A ->A ->A ->A.
   
   Hypothesis P_id_A__ISNATKIND : A ->A.
   
   Hypothesis P_id_A__U94 : A ->A ->A.
   
   Hypothesis P_id_A__U43 : A ->A ->A ->A.
   
   Hypothesis P_id_A__U52 : A ->A.
   
   Hypothesis P_id_A__U21 : A ->A ->A.
   
   Hypothesis P_id_A__U92 : A ->A ->A ->A.
   
   Hypothesis P_id_A__U41 : A ->A ->A ->A.
   
   Hypothesis P_id_A__U84 : A ->A ->A ->A.
   
   Hypothesis P_id_A__ISNAT : A ->A.
   
   Hypothesis P_id_A__U12 : A ->A ->A.
   
   Hypothesis P_id_MARK : A ->A.
   
   Hypothesis P_id_A__U45 : A ->A ->A.
   
   Hypothesis P_id_A__U46_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->P_id_A__U46 x12 <= P_id_A__U46 x11.
   
   Hypothesis P_id_A__ISNATILISTKIND_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->
      P_id_A__ISNATILISTKIND x12 <= P_id_A__ISNATILISTKIND x11.
   
   Hypothesis P_id_A__U61_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->P_id_A__U61 x12 <= P_id_A__U61 x11.
   
   Hypothesis P_id_A__U85_monotonic :
    forall x12 x14 x13 x11, 
     (A0 <= x14)/\ (x14 <= x13) ->
      (A0 <= x12)/\ (x12 <= x11) ->P_id_A__U85 x12 x14 <= P_id_A__U85 x11 x13.
   
   Hypothesis P_id_A__U31_monotonic :
    forall x12 x14 x13 x11, 
     (A0 <= x14)/\ (x14 <= x13) ->
      (A0 <= x12)/\ (x12 <= x11) ->P_id_A__U31 x12 x14 <= P_id_A__U31 x11 x13.
   
   Hypothesis P_id_A__U81_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (A0 <= x16)/\ (x16 <= x15) ->
      (A0 <= x14)/\ (x14 <= x13) ->
       (A0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U81 x12 x14 x16 <= P_id_A__U81 x11 x13 x15.
   
   Hypothesis P_id_A__U22_monotonic :
    forall x12 x14 x13 x11, 
     (A0 <= x14)/\ (x14 <= x13) ->
      (A0 <= x12)/\ (x12 <= x11) ->P_id_A__U22 x12 x14 <= P_id_A__U22 x11 x13.
   
   Hypothesis P_id_A__U93_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (A0 <= x16)/\ (x16 <= x15) ->
      (A0 <= x14)/\ (x14 <= x13) ->
       (A0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U93 x12 x14 x16 <= P_id_A__U93 x11 x13 x15.
   
   Hypothesis P_id_A__U42_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (A0 <= x16)/\ (x16 <= x15) ->
      (A0 <= x14)/\ (x14 <= x13) ->
       (A0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U42 x12 x14 x16 <= P_id_A__U42 x11 x13 x15.
   
   Hypothesis P_id_A__U51_monotonic :
    forall x12 x14 x13 x11, 
     (A0 <= x14)/\ (x14 <= x13) ->
      (A0 <= x12)/\ (x12 <= x11) ->P_id_A__U51 x12 x14 <= P_id_A__U51 x11 x13.
   
   Hypothesis P_id_A__ISNATLIST_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->
      P_id_A__ISNATLIST x12 <= P_id_A__ISNATLIST x11.
   
   Hypothesis P_id_A__U91_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (A0 <= x16)/\ (x16 <= x15) ->
      (A0 <= x14)/\ (x14 <= x13) ->
       (A0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U91 x12 x14 x16 <= P_id_A__U91 x11 x13 x15.
   
   Hypothesis P_id_A__U33_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->P_id_A__U33 x12 <= P_id_A__U33 x11.
   
   Hypothesis P_id_A__U83_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (A0 <= x16)/\ (x16 <= x15) ->
      (A0 <= x14)/\ (x14 <= x13) ->
       (A0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U83 x12 x14 x16 <= P_id_A__U83 x11 x13 x15.
   
   Hypothesis P_id_A__U23_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->P_id_A__U23 x12 <= P_id_A__U23 x11.
   
   Hypothesis P_id_A__U11_monotonic :
    forall x12 x14 x13 x11, 
     (A0 <= x14)/\ (x14 <= x13) ->
      (A0 <= x12)/\ (x12 <= x11) ->P_id_A__U11 x12 x14 <= P_id_A__U11 x11 x13.
   
   Hypothesis P_id_A__LENGTH_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->P_id_A__LENGTH x12 <= P_id_A__LENGTH x11.
   
   Hypothesis P_id_A__U44_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (A0 <= x16)/\ (x16 <= x15) ->
      (A0 <= x14)/\ (x14 <= x13) ->
       (A0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U44 x12 x14 x16 <= P_id_A__U44 x11 x13 x15.
   
   Hypothesis P_id_A__ISNATILIST_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->
      P_id_A__ISNATILIST x12 <= P_id_A__ISNATILIST x11.
   
   Hypothesis P_id_A__U13_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->P_id_A__U13 x12 <= P_id_A__U13 x11.
   
   Hypothesis P_id_A__U71_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->P_id_A__U71 x12 <= P_id_A__U71 x11.
   
   Hypothesis P_id_A__U86_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->P_id_A__U86 x12 <= P_id_A__U86 x11.
   
   Hypothesis P_id_A__U32_monotonic :
    forall x12 x14 x13 x11, 
     (A0 <= x14)/\ (x14 <= x13) ->
      (A0 <= x12)/\ (x12 <= x11) ->P_id_A__U32 x12 x14 <= P_id_A__U32 x11 x13.
   
   Hypothesis P_id_A__U82_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (A0 <= x16)/\ (x16 <= x15) ->
      (A0 <= x14)/\ (x14 <= x13) ->
       (A0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U82 x12 x14 x16 <= P_id_A__U82 x11 x13 x15.
   
   Hypothesis P_id_A__ISNATKIND_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->
      P_id_A__ISNATKIND x12 <= P_id_A__ISNATKIND x11.
   
   Hypothesis P_id_A__U94_monotonic :
    forall x12 x14 x13 x11, 
     (A0 <= x14)/\ (x14 <= x13) ->
      (A0 <= x12)/\ (x12 <= x11) ->P_id_A__U94 x12 x14 <= P_id_A__U94 x11 x13.
   
   Hypothesis P_id_A__U43_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (A0 <= x16)/\ (x16 <= x15) ->
      (A0 <= x14)/\ (x14 <= x13) ->
       (A0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U43 x12 x14 x16 <= P_id_A__U43 x11 x13 x15.
   
   Hypothesis P_id_A__U52_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->P_id_A__U52 x12 <= P_id_A__U52 x11.
   
   Hypothesis P_id_A__U21_monotonic :
    forall x12 x14 x13 x11, 
     (A0 <= x14)/\ (x14 <= x13) ->
      (A0 <= x12)/\ (x12 <= x11) ->P_id_A__U21 x12 x14 <= P_id_A__U21 x11 x13.
   
   Hypothesis P_id_A__U92_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (A0 <= x16)/\ (x16 <= x15) ->
      (A0 <= x14)/\ (x14 <= x13) ->
       (A0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U92 x12 x14 x16 <= P_id_A__U92 x11 x13 x15.
   
   Hypothesis P_id_A__U41_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (A0 <= x16)/\ (x16 <= x15) ->
      (A0 <= x14)/\ (x14 <= x13) ->
       (A0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U41 x12 x14 x16 <= P_id_A__U41 x11 x13 x15.
   
   Hypothesis P_id_A__U84_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (A0 <= x16)/\ (x16 <= x15) ->
      (A0 <= x14)/\ (x14 <= x13) ->
       (A0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U84 x12 x14 x16 <= P_id_A__U84 x11 x13 x15.
   
   Hypothesis P_id_A__ISNAT_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->P_id_A__ISNAT x12 <= P_id_A__ISNAT x11.
   
   Hypothesis P_id_A__U12_monotonic :
    forall x12 x14 x13 x11, 
     (A0 <= x14)/\ (x14 <= x13) ->
      (A0 <= x12)/\ (x12 <= x11) ->P_id_A__U12 x12 x14 <= P_id_A__U12 x11 x13.
   
   Hypothesis P_id_MARK_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->P_id_MARK x12 <= P_id_MARK x11.
   
   Hypothesis P_id_A__U45_monotonic :
    forall x12 x14 x13 x11, 
     (A0 <= x14)/\ (x14 <= x13) ->
      (A0 <= x12)/\ (x12 <= x11) ->P_id_A__U45 x12 x14 <= P_id_A__U45 x11 x13.
   
   Definition marked_measure t := 
     match t with
       | (algebra.Alg.Term algebra.F.id_a__U46 (x11::nil)) =>
        P_id_A__U46 (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__isNatIListKind (x11::nil)) =>
        P_id_A__ISNATILISTKIND (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U61 (x11::nil)) =>
        P_id_A__U61 (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U85 (x12::x11::nil)) =>
        P_id_A__U85 (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U31 (x12::x11::nil)) =>
        P_id_A__U31 (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U81 (x13::x12::x11::nil)) =>
        P_id_A__U81 (measure x13) (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U22 (x12::x11::nil)) =>
        P_id_A__U22 (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U93 (x13::x12::x11::nil)) =>
        P_id_A__U93 (measure x13) (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U42 (x13::x12::x11::nil)) =>
        P_id_A__U42 (measure x13) (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U51 (x12::x11::nil)) =>
        P_id_A__U51 (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__isNatList (x11::nil)) =>
        P_id_A__ISNATLIST (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__zeros nil) => P_id_A__ZEROS 
       | (algebra.Alg.Term algebra.F.id_a__U91 (x13::x12::x11::nil)) =>
        P_id_A__U91 (measure x13) (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U33 (x11::nil)) =>
        P_id_A__U33 (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U83 (x13::x12::x11::nil)) =>
        P_id_A__U83 (measure x13) (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U23 (x11::nil)) =>
        P_id_A__U23 (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U11 (x12::x11::nil)) =>
        P_id_A__U11 (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__length (x11::nil)) =>
        P_id_A__LENGTH (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U44 (x13::x12::x11::nil)) =>
        P_id_A__U44 (measure x13) (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__isNatIList (x11::nil)) =>
        P_id_A__ISNATILIST (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U13 (x11::nil)) =>
        P_id_A__U13 (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U71 (x11::nil)) =>
        P_id_A__U71 (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U86 (x11::nil)) =>
        P_id_A__U86 (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U32 (x12::x11::nil)) =>
        P_id_A__U32 (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U82 (x13::x12::x11::nil)) =>
        P_id_A__U82 (measure x13) (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__isNatKind (x11::nil)) =>
        P_id_A__ISNATKIND (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U94 (x12::x11::nil)) =>
        P_id_A__U94 (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U43 (x13::x12::x11::nil)) =>
        P_id_A__U43 (measure x13) (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U52 (x11::nil)) =>
        P_id_A__U52 (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U21 (x12::x11::nil)) =>
        P_id_A__U21 (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U92 (x13::x12::x11::nil)) =>
        P_id_A__U92 (measure x13) (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U41 (x13::x12::x11::nil)) =>
        P_id_A__U41 (measure x13) (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U84 (x13::x12::x11::nil)) =>
        P_id_A__U84 (measure x13) (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__isNat (x11::nil)) =>
        P_id_A__ISNAT (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U12 (x12::x11::nil)) =>
        P_id_A__U12 (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_mark (x11::nil)) =>
        P_id_MARK (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U45 (x12::x11::nil)) =>
        P_id_A__U45 (measure x12) (measure x11)
       | _ => measure t
       end.
   
   Definition  Marked_pols :
    forall f, 
     (algebra.EQT.defined R_xml_0_deep_rew.R_xml_0_rules f) ->
      InterpGen.Pol_type A (InterpGen.get_arity f).
   Proof.
     intros f H.
     
     apply ddp.defined_list_complete with (1:=R_xml_0_deep_rew.R_xml_0_rules_included) in H .
     apply (Symb_more_list.change_in algebra.F.symb_order) in H .
     
     set (u := (Symb_more_list.qs algebra.F.symb_order
           (Symb_more_list.XSet.remove_red
              (ddp.defined_list R_xml_0_deep_rew.R_xml_0_rule_as_list)))) in * .
     vm_compute in u .
     unfold u in * .
     clear u .
     unfold more_list.mem_bool in H .
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros x13 x12 x11;apply (P_id_A__U83 x13 x12 x11).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros x12 x11;apply (P_id_A__U31 x12 x11).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros x11;apply (P_id_A__U46 x11).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros x11;apply (P_id_A__ISNATILISTKIND x11).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros x11;apply (P_id_A__U61 x11).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros x12 x11;apply (P_id_A__U22 x12 x11).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros x13 x12 x11;apply (P_id_A__U42 x13 x12 x11).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros x13 x12 x11;apply (P_id_A__U91 x13 x12 x11).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros x13 x12 x11;apply (P_id_A__U81 x13 x12 x11).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros x11;apply (P_id_A__U23 x11).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros x13 x12 x11;apply (P_id_A__U44 x13 x12 x11).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros x13 x12 x11;apply (P_id_A__U93 x13 x12 x11).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros x12 x11;apply (P_id_A__U51 x12 x11).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros x11;apply (P_id_MARK x11).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros x11;apply (P_id_A__ISNATLIST x11).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros x11;apply (P_id_A__U33 x11).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros x12 x11;apply (P_id_A__U85 x12 x11).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros x13 x12 x11;apply (P_id_A__U82 x13 x12 x11).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros x11;apply (P_id_A__ISNAT x11).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros x12 x11;apply (P_id_A__U45 x12 x11).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros x12 x11;apply (P_id_A__U94 x12 x11).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros x12 x11;apply (P_id_A__U12 x12 x11).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros x11;apply (P_id_A__U52 x11).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros x12 x11;apply (P_id_A__U21 x12 x11).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros x13 x12 x11;apply (P_id_A__U41 x13 x12 x11).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros x11;apply (P_id_A__U86 x11).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros x11;apply (P_id_A__U71 x11).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros x11;apply (P_id_A__ISNATKIND x11).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros x13 x12 x11;apply (P_id_A__U43 x13 x12 x11).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros x13 x12 x11;apply (P_id_A__U92 x13 x12 x11).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros x12 x11;apply (P_id_A__U11 x12 x11).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros x11;apply (P_id_A__ISNATILIST x11).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros x11;apply (P_id_A__LENGTH x11).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros x11;apply (P_id_A__U13 x11).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros x12 x11;apply (P_id_A__U32 x12 x11).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros x13 x12 x11;apply (P_id_A__U84 x13 x12 x11).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply (P_id_A__ZEROS ).
     discriminate H.
   Defined.
   
   Lemma same_marked_measure :
    forall t, 
     marked_measure t = InterpGen.marked_measure A0 Pols Marked_pols 
                         (ddp.defined_dec _ _ 
                           R_xml_0_deep_rew.R_xml_0_rules_included) t.
   Proof.
     intros [a| f l].
     simpl in |-*.
     unfold eq_rect_r, eq_rect, sym_eq in |-*.
     reflexivity .
     
     refine match f with
              | algebra.F.id_a__zeros => match l with
                                           | nil => _
                                           | _::_ => _
                                           end
              | algebra.F.id_U51 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_a__U84 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_a__U32 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_isNatList =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__U13 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_U85 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_a__length =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__isNatIList =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_U33 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__U11 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_U81 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_a__U92 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_a__U43 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_U23 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__isNatKind =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_U93 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_U11 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_a__U71 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_U44 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_0 => match l with
                                    | nil => _
                                    | _::_ => _
                                    end
              | algebra.F.id_U61 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__U86 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__U41 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_U22 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_a__U21 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_U91 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_length =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__U52 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_U42 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_a__U12 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_U83 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_a__U94 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_a__U45 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_U31 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_a__isNat =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_isNatIListKind =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__U82 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_U46 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_cons =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_U52 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__U85 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_a__U33 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_U21 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_a__isNatList =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_U86 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_mark =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__U51 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_U41 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_tt => match l with
                                     | nil => _
                                     | _::_ => _
                                     end
              | algebra.F.id_U82 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_a__U93 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_a__U44 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_isNat =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__U23 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_U94 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_U12 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_a__U81 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_U45 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_zeros => match l with
                                        | nil => _
                                        | _::_ => _
                                        end
              | algebra.F.id_U71 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__U91 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_a__U42 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_isNatKind =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__U22 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_U92 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_nil => match l with
                                      | nil => _
                                      | _::_ => _
                                      end
              | algebra.F.id_a__U61 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_U43 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_a__isNatIListKind =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_U84 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_s =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__U46 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_U32 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_a__U31 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_U13 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__U83 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_isNatIList =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              end.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
   Qed.
   
   Lemma marked_measure_equation :
    forall t, 
     marked_measure t = match t with
                          | (algebra.Alg.Term algebra.F.id_a__U46 (x11::nil)) =>
                           P_id_A__U46 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
                             (x11::nil)) =>
                           P_id_A__ISNATILISTKIND (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U61 (x11::nil)) =>
                           P_id_A__U61 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U85 (x12::
                             x11::nil)) =>
                           P_id_A__U85 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U31 (x12::
                             x11::nil)) =>
                           P_id_A__U31 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U81 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U81 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U22 (x12::
                             x11::nil)) =>
                           P_id_A__U22 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U93 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U93 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U42 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U42 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U51 (x12::
                             x11::nil)) =>
                           P_id_A__U51 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__isNatList 
                             (x11::nil)) =>
                           P_id_A__ISNATLIST (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__zeros nil) =>
                           P_id_A__ZEROS 
                          | (algebra.Alg.Term algebra.F.id_a__U91 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U91 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U33 (x11::nil)) =>
                           P_id_A__U33 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U83 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U83 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U23 (x11::nil)) =>
                           P_id_A__U23 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U11 (x12::
                             x11::nil)) =>
                           P_id_A__U11 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__length 
                             (x11::nil)) =>
                           P_id_A__LENGTH (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U44 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U44 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__isNatIList 
                             (x11::nil)) =>
                           P_id_A__ISNATILIST (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U13 (x11::nil)) =>
                           P_id_A__U13 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U71 (x11::nil)) =>
                           P_id_A__U71 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U86 (x11::nil)) =>
                           P_id_A__U86 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U32 (x12::
                             x11::nil)) =>
                           P_id_A__U32 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U82 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U82 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__isNatKind 
                             (x11::nil)) =>
                           P_id_A__ISNATKIND (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U94 (x12::
                             x11::nil)) =>
                           P_id_A__U94 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U43 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U43 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U52 (x11::nil)) =>
                           P_id_A__U52 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U21 (x12::
                             x11::nil)) =>
                           P_id_A__U21 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U92 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U92 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U41 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U41 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U84 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U84 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__isNat 
                             (x11::nil)) =>
                           P_id_A__ISNAT (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U12 (x12::
                             x11::nil)) =>
                           P_id_A__U12 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_mark (x11::nil)) =>
                           P_id_MARK (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U45 (x12::
                             x11::nil)) =>
                           P_id_A__U45 (measure x12) (measure x11)
                          | _ => measure t
                          end.
   Proof.
     reflexivity .
   Qed.
   
   Lemma marked_measure_star_monotonic :
    forall f l1 l2, 
     (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step 
                                                       R_xml_0_deep_rew.R_xml_0_rules)
                                                      ) l1 l2) ->
      marked_measure (algebra.Alg.Term f l1) <= marked_measure (algebra.Alg.Term 
                                                                 f l2).
   Proof.
     intros .
     do 2 (rewrite same_marked_measure in |-*).
     
     apply InterpGen.marked_measure_star_monotonic with (1:=Aop) (Pols:=
     Pols) (rules:=R_xml_0_deep_rew.R_xml_0_rules).
     clear f.
     intros f.
     case f.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id_U51_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U84_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U32_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_isNatList_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U13_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U85_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__length_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isNatIList_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U33_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U11_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U81_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U92_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U43_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U23_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isNatKind_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U93_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U11_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U71_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U44_monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id_U61_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U86_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U41_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U22_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U21_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U91_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_length_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U52_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U42_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U12_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U83_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U94_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U45_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U31_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isNat_monotonic;assumption.
     
     vm_compute in |-*;intros ;apply P_id_isNatIListKind_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U82_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U46_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_cons_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U52_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U85_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U33_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U21_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isNatList_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U86_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_mark_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U51_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U41_monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id_U82_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U93_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U44_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_isNat_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U23_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U94_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U12_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U81_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U45_monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id_U71_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U91_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U42_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_isNatKind_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U22_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U92_monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id_a__U61_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U43_monotonic;assumption.
     
     vm_compute in |-*;intros ;apply P_id_a__isNatIListKind_monotonic;
      assumption.
     vm_compute in |-*;intros ;apply P_id_U84_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_s_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U46_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U32_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U31_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U13_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U83_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_isNatIList_monotonic;assumption.
     clear f.
     intros f.
     case f.
     vm_compute in |-*;intros ;apply P_id_a__zeros_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U51_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U84_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U32_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_isNatList_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U13_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U85_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__length_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isNatIList_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U33_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U11_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U81_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U92_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U43_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U23_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isNatKind_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U93_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U11_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U71_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U44_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_0_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U61_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U86_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U41_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U22_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U21_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U91_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_length_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U52_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U42_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U12_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U83_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U94_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U45_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U31_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isNat_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_isNatIListKind_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U82_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U46_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_cons_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U52_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U85_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U33_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U21_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isNatList_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U86_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_mark_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U51_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U41_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_tt_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U82_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U93_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U44_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_isNat_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U23_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U94_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U12_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U81_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U45_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_zeros_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U71_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U91_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U42_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_isNatKind_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U22_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U92_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_nil_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U61_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U43_bounded;assumption.
     
     vm_compute in |-*;intros ;apply P_id_a__isNatIListKind_bounded;
      assumption.
     vm_compute in |-*;intros ;apply P_id_U84_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_s_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U46_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U32_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U31_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U13_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U83_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_isNatIList_bounded;assumption.
     intros .
     do 2 (rewrite  <- same_measure in |-*).
     apply rules_monotonic;assumption.
     clear f.
     intros f.
     clear H.
     intros H.
     generalize H.
     
     apply ddp.defined_list_complete with (1:=R_xml_0_deep_rew.R_xml_0_rules_included) in H .
     apply (Symb_more_list.change_in algebra.F.symb_order) in H .
     
     set (u := (Symb_more_list.qs algebra.F.symb_order
           (Symb_more_list.XSet.remove_red
              (ddp.defined_list R_xml_0_deep_rew.R_xml_0_rule_as_list)))) in * .
     vm_compute in u .
     unfold u in * .
     clear u .
     unfold more_list.mem_bool in H .
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_A__U83_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_A__U31_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_A__U46_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     
     vm_compute in |-*;intros ;apply P_id_A__ISNATILISTKIND_monotonic;
      assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_A__U61_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_A__U22_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_A__U42_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_A__U91_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_A__U81_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_A__U23_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_A__U44_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_A__U93_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_A__U51_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_MARK_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_A__ISNATLIST_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_A__U33_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_A__U85_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_A__U82_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_A__ISNAT_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_A__U45_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_A__U94_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_A__U12_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_A__U52_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_A__U21_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_A__U41_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_A__U86_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_A__U71_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_A__ISNATKIND_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_A__U43_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_A__U92_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_A__U11_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_A__ISNATILIST_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_A__LENGTH_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_A__U13_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_A__U32_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_A__U84_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply (Aop.(le_refl)).
     discriminate H.
     assumption.
   Qed.
   
   End S.
End Interp.

Module InterpZ.
 Section S.
   Open Scope Z_scope.
   
   Hypothesis min_value : Z.
   
   Import ring_extention.
   
   Notation Local "'Alt'" := (Zwf.Zwf min_value).
   
   Notation Local "'Ale'" := Zle.
   
   Notation Local "'Aeq'" := (@eq Z).
   
   Notation Local "a <= b" := (Ale a b).
   
   Notation Local "a < b" := (Alt a b).
   
   Hypothesis P_id_a__zeros : Z.
   
   Hypothesis P_id_U51 : Z ->Z ->Z.
   
   Hypothesis P_id_a__U84 : Z ->Z ->Z ->Z.
   
   Hypothesis P_id_a__U32 : Z ->Z ->Z.
   
   Hypothesis P_id_isNatList : Z ->Z.
   
   Hypothesis P_id_a__U13 : Z ->Z.
   
   Hypothesis P_id_U85 : Z ->Z ->Z.
   
   Hypothesis P_id_a__length : Z ->Z.
   
   Hypothesis P_id_a__isNatIList : Z ->Z.
   
   Hypothesis P_id_U33 : Z ->Z.
   
   Hypothesis P_id_a__U11 : Z ->Z ->Z.
   
   Hypothesis P_id_U81 : Z ->Z ->Z ->Z.
   
   Hypothesis P_id_a__U92 : Z ->Z ->Z ->Z.
   
   Hypothesis P_id_a__U43 : Z ->Z ->Z ->Z.
   
   Hypothesis P_id_U23 : Z ->Z.
   
   Hypothesis P_id_a__isNatKind : Z ->Z.
   
   Hypothesis P_id_U93 : Z ->Z ->Z ->Z.
   
   Hypothesis P_id_U11 : Z ->Z ->Z.
   
   Hypothesis P_id_a__U71 : Z ->Z.
   
   Hypothesis P_id_U44 : Z ->Z ->Z ->Z.
   
   Hypothesis P_id_0 : Z.
   
   Hypothesis P_id_U61 : Z ->Z.
   
   Hypothesis P_id_a__U86 : Z ->Z.
   
   Hypothesis P_id_a__U41 : Z ->Z ->Z ->Z.
   
   Hypothesis P_id_U22 : Z ->Z ->Z.
   
   Hypothesis P_id_a__U21 : Z ->Z ->Z.
   
   Hypothesis P_id_U91 : Z ->Z ->Z ->Z.
   
   Hypothesis P_id_length : Z ->Z.
   
   Hypothesis P_id_a__U52 : Z ->Z.
   
   Hypothesis P_id_U42 : Z ->Z ->Z ->Z.
   
   Hypothesis P_id_a__U12 : Z ->Z ->Z.
   
   Hypothesis P_id_U83 : Z ->Z ->Z ->Z.
   
   Hypothesis P_id_a__U94 : Z ->Z ->Z.
   
   Hypothesis P_id_a__U45 : Z ->Z ->Z.
   
   Hypothesis P_id_U31 : Z ->Z ->Z.
   
   Hypothesis P_id_a__isNat : Z ->Z.
   
   Hypothesis P_id_isNatIListKind : Z ->Z.
   
   Hypothesis P_id_a__U82 : Z ->Z ->Z ->Z.
   
   Hypothesis P_id_U46 : Z ->Z.
   
   Hypothesis P_id_cons : Z ->Z ->Z.
   
   Hypothesis P_id_U52 : Z ->Z.
   
   Hypothesis P_id_a__U85 : Z ->Z ->Z.
   
   Hypothesis P_id_a__U33 : Z ->Z.
   
   Hypothesis P_id_U21 : Z ->Z ->Z.
   
   Hypothesis P_id_a__isNatList : Z ->Z.
   
   Hypothesis P_id_U86 : Z ->Z.
   
   Hypothesis P_id_mark : Z ->Z.
   
   Hypothesis P_id_a__U51 : Z ->Z ->Z.
   
   Hypothesis P_id_U41 : Z ->Z ->Z ->Z.
   
   Hypothesis P_id_tt : Z.
   
   Hypothesis P_id_U82 : Z ->Z ->Z ->Z.
   
   Hypothesis P_id_a__U93 : Z ->Z ->Z ->Z.
   
   Hypothesis P_id_a__U44 : Z ->Z ->Z ->Z.
   
   Hypothesis P_id_isNat : Z ->Z.
   
   Hypothesis P_id_a__U23 : Z ->Z.
   
   Hypothesis P_id_U94 : Z ->Z ->Z.
   
   Hypothesis P_id_U12 : Z ->Z ->Z.
   
   Hypothesis P_id_a__U81 : Z ->Z ->Z ->Z.
   
   Hypothesis P_id_U45 : Z ->Z ->Z.
   
   Hypothesis P_id_zeros : Z.
   
   Hypothesis P_id_U71 : Z ->Z.
   
   Hypothesis P_id_a__U91 : Z ->Z ->Z ->Z.
   
   Hypothesis P_id_a__U42 : Z ->Z ->Z ->Z.
   
   Hypothesis P_id_isNatKind : Z ->Z.
   
   Hypothesis P_id_a__U22 : Z ->Z ->Z.
   
   Hypothesis P_id_U92 : Z ->Z ->Z ->Z.
   
   Hypothesis P_id_nil : Z.
   
   Hypothesis P_id_a__U61 : Z ->Z.
   
   Hypothesis P_id_U43 : Z ->Z ->Z ->Z.
   
   Hypothesis P_id_a__isNatIListKind : Z ->Z.
   
   Hypothesis P_id_U84 : Z ->Z ->Z ->Z.
   
   Hypothesis P_id_s : Z ->Z.
   
   Hypothesis P_id_a__U46 : Z ->Z.
   
   Hypothesis P_id_U32 : Z ->Z ->Z.
   
   Hypothesis P_id_a__U31 : Z ->Z ->Z.
   
   Hypothesis P_id_U13 : Z ->Z.
   
   Hypothesis P_id_a__U83 : Z ->Z ->Z ->Z.
   
   Hypothesis P_id_isNatIList : Z ->Z.
   
   Hypothesis P_id_U51_monotonic :
    forall x12 x14 x13 x11, 
     (min_value <= x14)/\ (x14 <= x13) ->
      (min_value <= x12)/\ (x12 <= x11) ->
       P_id_U51 x12 x14 <= P_id_U51 x11 x13.
   
   Hypothesis P_id_a__U84_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (min_value <= x16)/\ (x16 <= x15) ->
      (min_value <= x14)/\ (x14 <= x13) ->
       (min_value <= x12)/\ (x12 <= x11) ->
        P_id_a__U84 x12 x14 x16 <= P_id_a__U84 x11 x13 x15.
   
   Hypothesis P_id_a__U32_monotonic :
    forall x12 x14 x13 x11, 
     (min_value <= x14)/\ (x14 <= x13) ->
      (min_value <= x12)/\ (x12 <= x11) ->
       P_id_a__U32 x12 x14 <= P_id_a__U32 x11 x13.
   
   Hypothesis P_id_isNatList_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->
      P_id_isNatList x12 <= P_id_isNatList x11.
   
   Hypothesis P_id_a__U13_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->P_id_a__U13 x12 <= P_id_a__U13 x11.
   
   Hypothesis P_id_U85_monotonic :
    forall x12 x14 x13 x11, 
     (min_value <= x14)/\ (x14 <= x13) ->
      (min_value <= x12)/\ (x12 <= x11) ->
       P_id_U85 x12 x14 <= P_id_U85 x11 x13.
   
   Hypothesis P_id_a__length_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->
      P_id_a__length x12 <= P_id_a__length x11.
   
   Hypothesis P_id_a__isNatIList_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->
      P_id_a__isNatIList x12 <= P_id_a__isNatIList x11.
   
   Hypothesis P_id_U33_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->P_id_U33 x12 <= P_id_U33 x11.
   
   Hypothesis P_id_a__U11_monotonic :
    forall x12 x14 x13 x11, 
     (min_value <= x14)/\ (x14 <= x13) ->
      (min_value <= x12)/\ (x12 <= x11) ->
       P_id_a__U11 x12 x14 <= P_id_a__U11 x11 x13.
   
   Hypothesis P_id_U81_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (min_value <= x16)/\ (x16 <= x15) ->
      (min_value <= x14)/\ (x14 <= x13) ->
       (min_value <= x12)/\ (x12 <= x11) ->
        P_id_U81 x12 x14 x16 <= P_id_U81 x11 x13 x15.
   
   Hypothesis P_id_a__U92_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (min_value <= x16)/\ (x16 <= x15) ->
      (min_value <= x14)/\ (x14 <= x13) ->
       (min_value <= x12)/\ (x12 <= x11) ->
        P_id_a__U92 x12 x14 x16 <= P_id_a__U92 x11 x13 x15.
   
   Hypothesis P_id_a__U43_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (min_value <= x16)/\ (x16 <= x15) ->
      (min_value <= x14)/\ (x14 <= x13) ->
       (min_value <= x12)/\ (x12 <= x11) ->
        P_id_a__U43 x12 x14 x16 <= P_id_a__U43 x11 x13 x15.
   
   Hypothesis P_id_U23_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->P_id_U23 x12 <= P_id_U23 x11.
   
   Hypothesis P_id_a__isNatKind_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->
      P_id_a__isNatKind x12 <= P_id_a__isNatKind x11.
   
   Hypothesis P_id_U93_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (min_value <= x16)/\ (x16 <= x15) ->
      (min_value <= x14)/\ (x14 <= x13) ->
       (min_value <= x12)/\ (x12 <= x11) ->
        P_id_U93 x12 x14 x16 <= P_id_U93 x11 x13 x15.
   
   Hypothesis P_id_U11_monotonic :
    forall x12 x14 x13 x11, 
     (min_value <= x14)/\ (x14 <= x13) ->
      (min_value <= x12)/\ (x12 <= x11) ->
       P_id_U11 x12 x14 <= P_id_U11 x11 x13.
   
   Hypothesis P_id_a__U71_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->P_id_a__U71 x12 <= P_id_a__U71 x11.
   
   Hypothesis P_id_U44_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (min_value <= x16)/\ (x16 <= x15) ->
      (min_value <= x14)/\ (x14 <= x13) ->
       (min_value <= x12)/\ (x12 <= x11) ->
        P_id_U44 x12 x14 x16 <= P_id_U44 x11 x13 x15.
   
   Hypothesis P_id_U61_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->P_id_U61 x12 <= P_id_U61 x11.
   
   Hypothesis P_id_a__U86_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->P_id_a__U86 x12 <= P_id_a__U86 x11.
   
   Hypothesis P_id_a__U41_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (min_value <= x16)/\ (x16 <= x15) ->
      (min_value <= x14)/\ (x14 <= x13) ->
       (min_value <= x12)/\ (x12 <= x11) ->
        P_id_a__U41 x12 x14 x16 <= P_id_a__U41 x11 x13 x15.
   
   Hypothesis P_id_U22_monotonic :
    forall x12 x14 x13 x11, 
     (min_value <= x14)/\ (x14 <= x13) ->
      (min_value <= x12)/\ (x12 <= x11) ->
       P_id_U22 x12 x14 <= P_id_U22 x11 x13.
   
   Hypothesis P_id_a__U21_monotonic :
    forall x12 x14 x13 x11, 
     (min_value <= x14)/\ (x14 <= x13) ->
      (min_value <= x12)/\ (x12 <= x11) ->
       P_id_a__U21 x12 x14 <= P_id_a__U21 x11 x13.
   
   Hypothesis P_id_U91_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (min_value <= x16)/\ (x16 <= x15) ->
      (min_value <= x14)/\ (x14 <= x13) ->
       (min_value <= x12)/\ (x12 <= x11) ->
        P_id_U91 x12 x14 x16 <= P_id_U91 x11 x13 x15.
   
   Hypothesis P_id_length_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->P_id_length x12 <= P_id_length x11.
   
   Hypothesis P_id_a__U52_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->P_id_a__U52 x12 <= P_id_a__U52 x11.
   
   Hypothesis P_id_U42_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (min_value <= x16)/\ (x16 <= x15) ->
      (min_value <= x14)/\ (x14 <= x13) ->
       (min_value <= x12)/\ (x12 <= x11) ->
        P_id_U42 x12 x14 x16 <= P_id_U42 x11 x13 x15.
   
   Hypothesis P_id_a__U12_monotonic :
    forall x12 x14 x13 x11, 
     (min_value <= x14)/\ (x14 <= x13) ->
      (min_value <= x12)/\ (x12 <= x11) ->
       P_id_a__U12 x12 x14 <= P_id_a__U12 x11 x13.
   
   Hypothesis P_id_U83_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (min_value <= x16)/\ (x16 <= x15) ->
      (min_value <= x14)/\ (x14 <= x13) ->
       (min_value <= x12)/\ (x12 <= x11) ->
        P_id_U83 x12 x14 x16 <= P_id_U83 x11 x13 x15.
   
   Hypothesis P_id_a__U94_monotonic :
    forall x12 x14 x13 x11, 
     (min_value <= x14)/\ (x14 <= x13) ->
      (min_value <= x12)/\ (x12 <= x11) ->
       P_id_a__U94 x12 x14 <= P_id_a__U94 x11 x13.
   
   Hypothesis P_id_a__U45_monotonic :
    forall x12 x14 x13 x11, 
     (min_value <= x14)/\ (x14 <= x13) ->
      (min_value <= x12)/\ (x12 <= x11) ->
       P_id_a__U45 x12 x14 <= P_id_a__U45 x11 x13.
   
   Hypothesis P_id_U31_monotonic :
    forall x12 x14 x13 x11, 
     (min_value <= x14)/\ (x14 <= x13) ->
      (min_value <= x12)/\ (x12 <= x11) ->
       P_id_U31 x12 x14 <= P_id_U31 x11 x13.
   
   Hypothesis P_id_a__isNat_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->
      P_id_a__isNat x12 <= P_id_a__isNat x11.
   
   Hypothesis P_id_isNatIListKind_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->
      P_id_isNatIListKind x12 <= P_id_isNatIListKind x11.
   
   Hypothesis P_id_a__U82_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (min_value <= x16)/\ (x16 <= x15) ->
      (min_value <= x14)/\ (x14 <= x13) ->
       (min_value <= x12)/\ (x12 <= x11) ->
        P_id_a__U82 x12 x14 x16 <= P_id_a__U82 x11 x13 x15.
   
   Hypothesis P_id_U46_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->P_id_U46 x12 <= P_id_U46 x11.
   
   Hypothesis P_id_cons_monotonic :
    forall x12 x14 x13 x11, 
     (min_value <= x14)/\ (x14 <= x13) ->
      (min_value <= x12)/\ (x12 <= x11) ->
       P_id_cons x12 x14 <= P_id_cons x11 x13.
   
   Hypothesis P_id_U52_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->P_id_U52 x12 <= P_id_U52 x11.
   
   Hypothesis P_id_a__U85_monotonic :
    forall x12 x14 x13 x11, 
     (min_value <= x14)/\ (x14 <= x13) ->
      (min_value <= x12)/\ (x12 <= x11) ->
       P_id_a__U85 x12 x14 <= P_id_a__U85 x11 x13.
   
   Hypothesis P_id_a__U33_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->P_id_a__U33 x12 <= P_id_a__U33 x11.
   
   Hypothesis P_id_U21_monotonic :
    forall x12 x14 x13 x11, 
     (min_value <= x14)/\ (x14 <= x13) ->
      (min_value <= x12)/\ (x12 <= x11) ->
       P_id_U21 x12 x14 <= P_id_U21 x11 x13.
   
   Hypothesis P_id_a__isNatList_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->
      P_id_a__isNatList x12 <= P_id_a__isNatList x11.
   
   Hypothesis P_id_U86_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->P_id_U86 x12 <= P_id_U86 x11.
   
   Hypothesis P_id_mark_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->P_id_mark x12 <= P_id_mark x11.
   
   Hypothesis P_id_a__U51_monotonic :
    forall x12 x14 x13 x11, 
     (min_value <= x14)/\ (x14 <= x13) ->
      (min_value <= x12)/\ (x12 <= x11) ->
       P_id_a__U51 x12 x14 <= P_id_a__U51 x11 x13.
   
   Hypothesis P_id_U41_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (min_value <= x16)/\ (x16 <= x15) ->
      (min_value <= x14)/\ (x14 <= x13) ->
       (min_value <= x12)/\ (x12 <= x11) ->
        P_id_U41 x12 x14 x16 <= P_id_U41 x11 x13 x15.
   
   Hypothesis P_id_U82_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (min_value <= x16)/\ (x16 <= x15) ->
      (min_value <= x14)/\ (x14 <= x13) ->
       (min_value <= x12)/\ (x12 <= x11) ->
        P_id_U82 x12 x14 x16 <= P_id_U82 x11 x13 x15.
   
   Hypothesis P_id_a__U93_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (min_value <= x16)/\ (x16 <= x15) ->
      (min_value <= x14)/\ (x14 <= x13) ->
       (min_value <= x12)/\ (x12 <= x11) ->
        P_id_a__U93 x12 x14 x16 <= P_id_a__U93 x11 x13 x15.
   
   Hypothesis P_id_a__U44_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (min_value <= x16)/\ (x16 <= x15) ->
      (min_value <= x14)/\ (x14 <= x13) ->
       (min_value <= x12)/\ (x12 <= x11) ->
        P_id_a__U44 x12 x14 x16 <= P_id_a__U44 x11 x13 x15.
   
   Hypothesis P_id_isNat_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->P_id_isNat x12 <= P_id_isNat x11.
   
   Hypothesis P_id_a__U23_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->P_id_a__U23 x12 <= P_id_a__U23 x11.
   
   Hypothesis P_id_U94_monotonic :
    forall x12 x14 x13 x11, 
     (min_value <= x14)/\ (x14 <= x13) ->
      (min_value <= x12)/\ (x12 <= x11) ->
       P_id_U94 x12 x14 <= P_id_U94 x11 x13.
   
   Hypothesis P_id_U12_monotonic :
    forall x12 x14 x13 x11, 
     (min_value <= x14)/\ (x14 <= x13) ->
      (min_value <= x12)/\ (x12 <= x11) ->
       P_id_U12 x12 x14 <= P_id_U12 x11 x13.
   
   Hypothesis P_id_a__U81_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (min_value <= x16)/\ (x16 <= x15) ->
      (min_value <= x14)/\ (x14 <= x13) ->
       (min_value <= x12)/\ (x12 <= x11) ->
        P_id_a__U81 x12 x14 x16 <= P_id_a__U81 x11 x13 x15.
   
   Hypothesis P_id_U45_monotonic :
    forall x12 x14 x13 x11, 
     (min_value <= x14)/\ (x14 <= x13) ->
      (min_value <= x12)/\ (x12 <= x11) ->
       P_id_U45 x12 x14 <= P_id_U45 x11 x13.
   
   Hypothesis P_id_U71_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->P_id_U71 x12 <= P_id_U71 x11.
   
   Hypothesis P_id_a__U91_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (min_value <= x16)/\ (x16 <= x15) ->
      (min_value <= x14)/\ (x14 <= x13) ->
       (min_value <= x12)/\ (x12 <= x11) ->
        P_id_a__U91 x12 x14 x16 <= P_id_a__U91 x11 x13 x15.
   
   Hypothesis P_id_a__U42_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (min_value <= x16)/\ (x16 <= x15) ->
      (min_value <= x14)/\ (x14 <= x13) ->
       (min_value <= x12)/\ (x12 <= x11) ->
        P_id_a__U42 x12 x14 x16 <= P_id_a__U42 x11 x13 x15.
   
   Hypothesis P_id_isNatKind_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->
      P_id_isNatKind x12 <= P_id_isNatKind x11.
   
   Hypothesis P_id_a__U22_monotonic :
    forall x12 x14 x13 x11, 
     (min_value <= x14)/\ (x14 <= x13) ->
      (min_value <= x12)/\ (x12 <= x11) ->
       P_id_a__U22 x12 x14 <= P_id_a__U22 x11 x13.
   
   Hypothesis P_id_U92_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (min_value <= x16)/\ (x16 <= x15) ->
      (min_value <= x14)/\ (x14 <= x13) ->
       (min_value <= x12)/\ (x12 <= x11) ->
        P_id_U92 x12 x14 x16 <= P_id_U92 x11 x13 x15.
   
   Hypothesis P_id_a__U61_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->P_id_a__U61 x12 <= P_id_a__U61 x11.
   
   Hypothesis P_id_U43_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (min_value <= x16)/\ (x16 <= x15) ->
      (min_value <= x14)/\ (x14 <= x13) ->
       (min_value <= x12)/\ (x12 <= x11) ->
        P_id_U43 x12 x14 x16 <= P_id_U43 x11 x13 x15.
   
   Hypothesis P_id_a__isNatIListKind_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->
      P_id_a__isNatIListKind x12 <= P_id_a__isNatIListKind x11.
   
   Hypothesis P_id_U84_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (min_value <= x16)/\ (x16 <= x15) ->
      (min_value <= x14)/\ (x14 <= x13) ->
       (min_value <= x12)/\ (x12 <= x11) ->
        P_id_U84 x12 x14 x16 <= P_id_U84 x11 x13 x15.
   
   Hypothesis P_id_s_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->P_id_s x12 <= P_id_s x11.
   
   Hypothesis P_id_a__U46_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->P_id_a__U46 x12 <= P_id_a__U46 x11.
   
   Hypothesis P_id_U32_monotonic :
    forall x12 x14 x13 x11, 
     (min_value <= x14)/\ (x14 <= x13) ->
      (min_value <= x12)/\ (x12 <= x11) ->
       P_id_U32 x12 x14 <= P_id_U32 x11 x13.
   
   Hypothesis P_id_a__U31_monotonic :
    forall x12 x14 x13 x11, 
     (min_value <= x14)/\ (x14 <= x13) ->
      (min_value <= x12)/\ (x12 <= x11) ->
       P_id_a__U31 x12 x14 <= P_id_a__U31 x11 x13.
   
   Hypothesis P_id_U13_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->P_id_U13 x12 <= P_id_U13 x11.
   
   Hypothesis P_id_a__U83_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (min_value <= x16)/\ (x16 <= x15) ->
      (min_value <= x14)/\ (x14 <= x13) ->
       (min_value <= x12)/\ (x12 <= x11) ->
        P_id_a__U83 x12 x14 x16 <= P_id_a__U83 x11 x13 x15.
   
   Hypothesis P_id_isNatIList_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->
      P_id_isNatIList x12 <= P_id_isNatIList x11.
   
   Hypothesis P_id_a__zeros_bounded : min_value <= P_id_a__zeros .
   
   Hypothesis P_id_U51_bounded :
    forall x12 x11, 
     (min_value <= x11) ->(min_value <= x12) ->min_value <= P_id_U51 x12 x11.
   
   Hypothesis P_id_a__U84_bounded :
    forall x12 x13 x11, 
     (min_value <= x11) ->
      (min_value <= x12) ->
       (min_value <= x13) ->min_value <= P_id_a__U84 x13 x12 x11.
   
   Hypothesis P_id_a__U32_bounded :
    forall x12 x11, 
     (min_value <= x11) ->
      (min_value <= x12) ->min_value <= P_id_a__U32 x12 x11.
   
   Hypothesis P_id_isNatList_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_isNatList x11.
   
   Hypothesis P_id_a__U13_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_a__U13 x11.
   
   Hypothesis P_id_U85_bounded :
    forall x12 x11, 
     (min_value <= x11) ->(min_value <= x12) ->min_value <= P_id_U85 x12 x11.
   
   Hypothesis P_id_a__length_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_a__length x11.
   
   Hypothesis P_id_a__isNatIList_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_a__isNatIList x11.
   
   Hypothesis P_id_U33_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_U33 x11.
   
   Hypothesis P_id_a__U11_bounded :
    forall x12 x11, 
     (min_value <= x11) ->
      (min_value <= x12) ->min_value <= P_id_a__U11 x12 x11.
   
   Hypothesis P_id_U81_bounded :
    forall x12 x13 x11, 
     (min_value <= x11) ->
      (min_value <= x12) ->
       (min_value <= x13) ->min_value <= P_id_U81 x13 x12 x11.
   
   Hypothesis P_id_a__U92_bounded :
    forall x12 x13 x11, 
     (min_value <= x11) ->
      (min_value <= x12) ->
       (min_value <= x13) ->min_value <= P_id_a__U92 x13 x12 x11.
   
   Hypothesis P_id_a__U43_bounded :
    forall x12 x13 x11, 
     (min_value <= x11) ->
      (min_value <= x12) ->
       (min_value <= x13) ->min_value <= P_id_a__U43 x13 x12 x11.
   
   Hypothesis P_id_U23_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_U23 x11.
   
   Hypothesis P_id_a__isNatKind_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_a__isNatKind x11.
   
   Hypothesis P_id_U93_bounded :
    forall x12 x13 x11, 
     (min_value <= x11) ->
      (min_value <= x12) ->
       (min_value <= x13) ->min_value <= P_id_U93 x13 x12 x11.
   
   Hypothesis P_id_U11_bounded :
    forall x12 x11, 
     (min_value <= x11) ->(min_value <= x12) ->min_value <= P_id_U11 x12 x11.
   
   Hypothesis P_id_a__U71_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_a__U71 x11.
   
   Hypothesis P_id_U44_bounded :
    forall x12 x13 x11, 
     (min_value <= x11) ->
      (min_value <= x12) ->
       (min_value <= x13) ->min_value <= P_id_U44 x13 x12 x11.
   
   Hypothesis P_id_0_bounded : min_value <= P_id_0 .
   
   Hypothesis P_id_U61_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_U61 x11.
   
   Hypothesis P_id_a__U86_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_a__U86 x11.
   
   Hypothesis P_id_a__U41_bounded :
    forall x12 x13 x11, 
     (min_value <= x11) ->
      (min_value <= x12) ->
       (min_value <= x13) ->min_value <= P_id_a__U41 x13 x12 x11.
   
   Hypothesis P_id_U22_bounded :
    forall x12 x11, 
     (min_value <= x11) ->(min_value <= x12) ->min_value <= P_id_U22 x12 x11.
   
   Hypothesis P_id_a__U21_bounded :
    forall x12 x11, 
     (min_value <= x11) ->
      (min_value <= x12) ->min_value <= P_id_a__U21 x12 x11.
   
   Hypothesis P_id_U91_bounded :
    forall x12 x13 x11, 
     (min_value <= x11) ->
      (min_value <= x12) ->
       (min_value <= x13) ->min_value <= P_id_U91 x13 x12 x11.
   
   Hypothesis P_id_length_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_length x11.
   
   Hypothesis P_id_a__U52_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_a__U52 x11.
   
   Hypothesis P_id_U42_bounded :
    forall x12 x13 x11, 
     (min_value <= x11) ->
      (min_value <= x12) ->
       (min_value <= x13) ->min_value <= P_id_U42 x13 x12 x11.
   
   Hypothesis P_id_a__U12_bounded :
    forall x12 x11, 
     (min_value <= x11) ->
      (min_value <= x12) ->min_value <= P_id_a__U12 x12 x11.
   
   Hypothesis P_id_U83_bounded :
    forall x12 x13 x11, 
     (min_value <= x11) ->
      (min_value <= x12) ->
       (min_value <= x13) ->min_value <= P_id_U83 x13 x12 x11.
   
   Hypothesis P_id_a__U94_bounded :
    forall x12 x11, 
     (min_value <= x11) ->
      (min_value <= x12) ->min_value <= P_id_a__U94 x12 x11.
   
   Hypothesis P_id_a__U45_bounded :
    forall x12 x11, 
     (min_value <= x11) ->
      (min_value <= x12) ->min_value <= P_id_a__U45 x12 x11.
   
   Hypothesis P_id_U31_bounded :
    forall x12 x11, 
     (min_value <= x11) ->(min_value <= x12) ->min_value <= P_id_U31 x12 x11.
   
   Hypothesis P_id_a__isNat_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_a__isNat x11.
   
   Hypothesis P_id_isNatIListKind_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_isNatIListKind x11.
   
   Hypothesis P_id_a__U82_bounded :
    forall x12 x13 x11, 
     (min_value <= x11) ->
      (min_value <= x12) ->
       (min_value <= x13) ->min_value <= P_id_a__U82 x13 x12 x11.
   
   Hypothesis P_id_U46_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_U46 x11.
   
   Hypothesis P_id_cons_bounded :
    forall x12 x11, 
     (min_value <= x11) ->(min_value <= x12) ->min_value <= P_id_cons x12 x11.
   
   Hypothesis P_id_U52_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_U52 x11.
   
   Hypothesis P_id_a__U85_bounded :
    forall x12 x11, 
     (min_value <= x11) ->
      (min_value <= x12) ->min_value <= P_id_a__U85 x12 x11.
   
   Hypothesis P_id_a__U33_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_a__U33 x11.
   
   Hypothesis P_id_U21_bounded :
    forall x12 x11, 
     (min_value <= x11) ->(min_value <= x12) ->min_value <= P_id_U21 x12 x11.
   
   Hypothesis P_id_a__isNatList_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_a__isNatList x11.
   
   Hypothesis P_id_U86_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_U86 x11.
   
   Hypothesis P_id_mark_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_mark x11.
   
   Hypothesis P_id_a__U51_bounded :
    forall x12 x11, 
     (min_value <= x11) ->
      (min_value <= x12) ->min_value <= P_id_a__U51 x12 x11.
   
   Hypothesis P_id_U41_bounded :
    forall x12 x13 x11, 
     (min_value <= x11) ->
      (min_value <= x12) ->
       (min_value <= x13) ->min_value <= P_id_U41 x13 x12 x11.
   
   Hypothesis P_id_tt_bounded : min_value <= P_id_tt .
   
   Hypothesis P_id_U82_bounded :
    forall x12 x13 x11, 
     (min_value <= x11) ->
      (min_value <= x12) ->
       (min_value <= x13) ->min_value <= P_id_U82 x13 x12 x11.
   
   Hypothesis P_id_a__U93_bounded :
    forall x12 x13 x11, 
     (min_value <= x11) ->
      (min_value <= x12) ->
       (min_value <= x13) ->min_value <= P_id_a__U93 x13 x12 x11.
   
   Hypothesis P_id_a__U44_bounded :
    forall x12 x13 x11, 
     (min_value <= x11) ->
      (min_value <= x12) ->
       (min_value <= x13) ->min_value <= P_id_a__U44 x13 x12 x11.
   
   Hypothesis P_id_isNat_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_isNat x11.
   
   Hypothesis P_id_a__U23_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_a__U23 x11.
   
   Hypothesis P_id_U94_bounded :
    forall x12 x11, 
     (min_value <= x11) ->(min_value <= x12) ->min_value <= P_id_U94 x12 x11.
   
   Hypothesis P_id_U12_bounded :
    forall x12 x11, 
     (min_value <= x11) ->(min_value <= x12) ->min_value <= P_id_U12 x12 x11.
   
   Hypothesis P_id_a__U81_bounded :
    forall x12 x13 x11, 
     (min_value <= x11) ->
      (min_value <= x12) ->
       (min_value <= x13) ->min_value <= P_id_a__U81 x13 x12 x11.
   
   Hypothesis P_id_U45_bounded :
    forall x12 x11, 
     (min_value <= x11) ->(min_value <= x12) ->min_value <= P_id_U45 x12 x11.
   
   Hypothesis P_id_zeros_bounded : min_value <= P_id_zeros .
   
   Hypothesis P_id_U71_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_U71 x11.
   
   Hypothesis P_id_a__U91_bounded :
    forall x12 x13 x11, 
     (min_value <= x11) ->
      (min_value <= x12) ->
       (min_value <= x13) ->min_value <= P_id_a__U91 x13 x12 x11.
   
   Hypothesis P_id_a__U42_bounded :
    forall x12 x13 x11, 
     (min_value <= x11) ->
      (min_value <= x12) ->
       (min_value <= x13) ->min_value <= P_id_a__U42 x13 x12 x11.
   
   Hypothesis P_id_isNatKind_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_isNatKind x11.
   
   Hypothesis P_id_a__U22_bounded :
    forall x12 x11, 
     (min_value <= x11) ->
      (min_value <= x12) ->min_value <= P_id_a__U22 x12 x11.
   
   Hypothesis P_id_U92_bounded :
    forall x12 x13 x11, 
     (min_value <= x11) ->
      (min_value <= x12) ->
       (min_value <= x13) ->min_value <= P_id_U92 x13 x12 x11.
   
   Hypothesis P_id_nil_bounded : min_value <= P_id_nil .
   
   Hypothesis P_id_a__U61_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_a__U61 x11.
   
   Hypothesis P_id_U43_bounded :
    forall x12 x13 x11, 
     (min_value <= x11) ->
      (min_value <= x12) ->
       (min_value <= x13) ->min_value <= P_id_U43 x13 x12 x11.
   
   Hypothesis P_id_a__isNatIListKind_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_a__isNatIListKind x11.
   
   Hypothesis P_id_U84_bounded :
    forall x12 x13 x11, 
     (min_value <= x11) ->
      (min_value <= x12) ->
       (min_value <= x13) ->min_value <= P_id_U84 x13 x12 x11.
   
   Hypothesis P_id_s_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_s x11.
   
   Hypothesis P_id_a__U46_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_a__U46 x11.
   
   Hypothesis P_id_U32_bounded :
    forall x12 x11, 
     (min_value <= x11) ->(min_value <= x12) ->min_value <= P_id_U32 x12 x11.
   
   Hypothesis P_id_a__U31_bounded :
    forall x12 x11, 
     (min_value <= x11) ->
      (min_value <= x12) ->min_value <= P_id_a__U31 x12 x11.
   
   Hypothesis P_id_U13_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_U13 x11.
   
   Hypothesis P_id_a__U83_bounded :
    forall x12 x13 x11, 
     (min_value <= x11) ->
      (min_value <= x12) ->
       (min_value <= x13) ->min_value <= P_id_a__U83 x13 x12 x11.
   
   Hypothesis P_id_isNatIList_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_isNatIList x11.
   
   Definition measure  := 
     Interp.measure min_value P_id_a__zeros P_id_U51 P_id_a__U84 P_id_a__U32 
      P_id_isNatList P_id_a__U13 P_id_U85 P_id_a__length P_id_a__isNatIList 
      P_id_U33 P_id_a__U11 P_id_U81 P_id_a__U92 P_id_a__U43 P_id_U23 
      P_id_a__isNatKind P_id_U93 P_id_U11 P_id_a__U71 P_id_U44 P_id_0 
      P_id_U61 P_id_a__U86 P_id_a__U41 P_id_U22 P_id_a__U21 P_id_U91 
      P_id_length P_id_a__U52 P_id_U42 P_id_a__U12 P_id_U83 P_id_a__U94 
      P_id_a__U45 P_id_U31 P_id_a__isNat P_id_isNatIListKind P_id_a__U82 
      P_id_U46 P_id_cons P_id_U52 P_id_a__U85 P_id_a__U33 P_id_U21 
      P_id_a__isNatList P_id_U86 P_id_mark P_id_a__U51 P_id_U41 P_id_tt 
      P_id_U82 P_id_a__U93 P_id_a__U44 P_id_isNat P_id_a__U23 P_id_U94 
      P_id_U12 P_id_a__U81 P_id_U45 P_id_zeros P_id_U71 P_id_a__U91 
      P_id_a__U42 P_id_isNatKind P_id_a__U22 P_id_U92 P_id_nil P_id_a__U61 
      P_id_U43 P_id_a__isNatIListKind P_id_U84 P_id_s P_id_a__U46 P_id_U32 
      P_id_a__U31 P_id_U13 P_id_a__U83 P_id_isNatIList.
   
   Lemma measure_equation :
    forall t, 
     measure t = match t with
                   | (algebra.Alg.Term algebra.F.id_a__zeros nil) =>
                    P_id_a__zeros 
                   | (algebra.Alg.Term algebra.F.id_U51 (x12::x11::nil)) =>
                    P_id_U51 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U84 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U84 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U32 (x12::x11::nil)) =>
                    P_id_a__U32 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_isNatList (x11::nil)) =>
                    P_id_isNatList (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U13 (x11::nil)) =>
                    P_id_a__U13 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U85 (x12::x11::nil)) =>
                    P_id_U85 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__length (x11::nil)) =>
                    P_id_a__length (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__isNatIList (x11::nil)) =>
                    P_id_a__isNatIList (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U33 (x11::nil)) =>
                    P_id_U33 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U11 (x12::x11::nil)) =>
                    P_id_a__U11 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U81 (x13::x12::x11::nil)) =>
                    P_id_U81 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U92 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U92 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U43 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U43 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U23 (x11::nil)) =>
                    P_id_U23 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__isNatKind (x11::nil)) =>
                    P_id_a__isNatKind (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U93 (x13::x12::x11::nil)) =>
                    P_id_U93 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U11 (x12::x11::nil)) =>
                    P_id_U11 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U71 (x11::nil)) =>
                    P_id_a__U71 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U44 (x13::x12::x11::nil)) =>
                    P_id_U44 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_0 nil) => P_id_0 
                   | (algebra.Alg.Term algebra.F.id_U61 (x11::nil)) =>
                    P_id_U61 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U86 (x11::nil)) =>
                    P_id_a__U86 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U41 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U41 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U22 (x12::x11::nil)) =>
                    P_id_U22 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U21 (x12::x11::nil)) =>
                    P_id_a__U21 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U91 (x13::x12::x11::nil)) =>
                    P_id_U91 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_length (x11::nil)) =>
                    P_id_length (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U52 (x11::nil)) =>
                    P_id_a__U52 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U42 (x13::x12::x11::nil)) =>
                    P_id_U42 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U12 (x12::x11::nil)) =>
                    P_id_a__U12 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U83 (x13::x12::x11::nil)) =>
                    P_id_U83 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U94 (x12::x11::nil)) =>
                    P_id_a__U94 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U45 (x12::x11::nil)) =>
                    P_id_a__U45 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U31 (x12::x11::nil)) =>
                    P_id_U31 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__isNat (x11::nil)) =>
                    P_id_a__isNat (measure x11)
                   | (algebra.Alg.Term algebra.F.id_isNatIListKind 
                      (x11::nil)) =>
                    P_id_isNatIListKind (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U82 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U82 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U46 (x11::nil)) =>
                    P_id_U46 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_cons (x12::x11::nil)) =>
                    P_id_cons (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U52 (x11::nil)) =>
                    P_id_U52 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U85 (x12::x11::nil)) =>
                    P_id_a__U85 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U33 (x11::nil)) =>
                    P_id_a__U33 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U21 (x12::x11::nil)) =>
                    P_id_U21 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__isNatList (x11::nil)) =>
                    P_id_a__isNatList (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U86 (x11::nil)) =>
                    P_id_U86 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_mark (x11::nil)) =>
                    P_id_mark (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U51 (x12::x11::nil)) =>
                    P_id_a__U51 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U41 (x13::x12::x11::nil)) =>
                    P_id_U41 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt 
                   | (algebra.Alg.Term algebra.F.id_U82 (x13::x12::x11::nil)) =>
                    P_id_U82 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U93 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U93 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U44 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U44 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_isNat (x11::nil)) =>
                    P_id_isNat (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U23 (x11::nil)) =>
                    P_id_a__U23 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U94 (x12::x11::nil)) =>
                    P_id_U94 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U12 (x12::x11::nil)) =>
                    P_id_U12 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U81 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U81 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U45 (x12::x11::nil)) =>
                    P_id_U45 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_zeros nil) => P_id_zeros 
                   | (algebra.Alg.Term algebra.F.id_U71 (x11::nil)) =>
                    P_id_U71 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U91 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U91 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U42 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U42 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_isNatKind (x11::nil)) =>
                    P_id_isNatKind (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U22 (x12::x11::nil)) =>
                    P_id_a__U22 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U92 (x13::x12::x11::nil)) =>
                    P_id_U92 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil 
                   | (algebra.Alg.Term algebra.F.id_a__U61 (x11::nil)) =>
                    P_id_a__U61 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U43 (x13::x12::x11::nil)) =>
                    P_id_U43 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
                      (x11::nil)) =>
                    P_id_a__isNatIListKind (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U84 (x13::x12::x11::nil)) =>
                    P_id_U84 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_s (x11::nil)) =>
                    P_id_s (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U46 (x11::nil)) =>
                    P_id_a__U46 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U32 (x12::x11::nil)) =>
                    P_id_U32 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U31 (x12::x11::nil)) =>
                    P_id_a__U31 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U13 (x11::nil)) =>
                    P_id_U13 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U83 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U83 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_isNatIList (x11::nil)) =>
                    P_id_isNatIList (measure x11)
                   | _ => min_value
                   end.
   Proof.
     intros t;case t;intros ;apply refl_equal.
   Qed.
   
   Lemma measure_bounded : forall t, min_value <= measure t.
   Proof.
     unfold measure in |-*.
     
     apply Interp.measure_bounded with Alt Aeq;
      (apply interp.o_Z)||
      (cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;auto with zarith).
   Qed.
   
   Ltac generate_pos_hyp  :=
    match goal with
      | H:context [measure ?x] |- _ =>
       let v := fresh "v" in 
        (let H := fresh "h" in 
          (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
            clearbody H;clearbody v))
      |  |- context [measure ?x] =>
       let v := fresh "v" in 
        (let H := fresh "h" in 
          (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
            clearbody H;clearbody v))
      end
    .
   
   Hypothesis rules_monotonic :
    forall l r, 
     (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) ->
      measure r <= measure l.
   
   Lemma measure_star_monotonic :
    forall l r, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                r l) ->measure r <= measure l.
   Proof.
     unfold measure in *.
     apply Interp.measure_star_monotonic with Alt Aeq.
     
     (apply interp.o_Z)||
     (cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;auto with zarith).
     intros ;apply P_id_U51_monotonic;assumption.
     intros ;apply P_id_a__U84_monotonic;assumption.
     intros ;apply P_id_a__U32_monotonic;assumption.
     intros ;apply P_id_isNatList_monotonic;assumption.
     intros ;apply P_id_a__U13_monotonic;assumption.
     intros ;apply P_id_U85_monotonic;assumption.
     intros ;apply P_id_a__length_monotonic;assumption.
     intros ;apply P_id_a__isNatIList_monotonic;assumption.
     intros ;apply P_id_U33_monotonic;assumption.
     intros ;apply P_id_a__U11_monotonic;assumption.
     intros ;apply P_id_U81_monotonic;assumption.
     intros ;apply P_id_a__U92_monotonic;assumption.
     intros ;apply P_id_a__U43_monotonic;assumption.
     intros ;apply P_id_U23_monotonic;assumption.
     intros ;apply P_id_a__isNatKind_monotonic;assumption.
     intros ;apply P_id_U93_monotonic;assumption.
     intros ;apply P_id_U11_monotonic;assumption.
     intros ;apply P_id_a__U71_monotonic;assumption.
     intros ;apply P_id_U44_monotonic;assumption.
     intros ;apply P_id_U61_monotonic;assumption.
     intros ;apply P_id_a__U86_monotonic;assumption.
     intros ;apply P_id_a__U41_monotonic;assumption.
     intros ;apply P_id_U22_monotonic;assumption.
     intros ;apply P_id_a__U21_monotonic;assumption.
     intros ;apply P_id_U91_monotonic;assumption.
     intros ;apply P_id_length_monotonic;assumption.
     intros ;apply P_id_a__U52_monotonic;assumption.
     intros ;apply P_id_U42_monotonic;assumption.
     intros ;apply P_id_a__U12_monotonic;assumption.
     intros ;apply P_id_U83_monotonic;assumption.
     intros ;apply P_id_a__U94_monotonic;assumption.
     intros ;apply P_id_a__U45_monotonic;assumption.
     intros ;apply P_id_U31_monotonic;assumption.
     intros ;apply P_id_a__isNat_monotonic;assumption.
     intros ;apply P_id_isNatIListKind_monotonic;assumption.
     intros ;apply P_id_a__U82_monotonic;assumption.
     intros ;apply P_id_U46_monotonic;assumption.
     intros ;apply P_id_cons_monotonic;assumption.
     intros ;apply P_id_U52_monotonic;assumption.
     intros ;apply P_id_a__U85_monotonic;assumption.
     intros ;apply P_id_a__U33_monotonic;assumption.
     intros ;apply P_id_U21_monotonic;assumption.
     intros ;apply P_id_a__isNatList_monotonic;assumption.
     intros ;apply P_id_U86_monotonic;assumption.
     intros ;apply P_id_mark_monotonic;assumption.
     intros ;apply P_id_a__U51_monotonic;assumption.
     intros ;apply P_id_U41_monotonic;assumption.
     intros ;apply P_id_U82_monotonic;assumption.
     intros ;apply P_id_a__U93_monotonic;assumption.
     intros ;apply P_id_a__U44_monotonic;assumption.
     intros ;apply P_id_isNat_monotonic;assumption.
     intros ;apply P_id_a__U23_monotonic;assumption.
     intros ;apply P_id_U94_monotonic;assumption.
     intros ;apply P_id_U12_monotonic;assumption.
     intros ;apply P_id_a__U81_monotonic;assumption.
     intros ;apply P_id_U45_monotonic;assumption.
     intros ;apply P_id_U71_monotonic;assumption.
     intros ;apply P_id_a__U91_monotonic;assumption.
     intros ;apply P_id_a__U42_monotonic;assumption.
     intros ;apply P_id_isNatKind_monotonic;assumption.
     intros ;apply P_id_a__U22_monotonic;assumption.
     intros ;apply P_id_U92_monotonic;assumption.
     intros ;apply P_id_a__U61_monotonic;assumption.
     intros ;apply P_id_U43_monotonic;assumption.
     intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
     intros ;apply P_id_U84_monotonic;assumption.
     intros ;apply P_id_s_monotonic;assumption.
     intros ;apply P_id_a__U46_monotonic;assumption.
     intros ;apply P_id_U32_monotonic;assumption.
     intros ;apply P_id_a__U31_monotonic;assumption.
     intros ;apply P_id_U13_monotonic;assumption.
     intros ;apply P_id_a__U83_monotonic;assumption.
     intros ;apply P_id_isNatIList_monotonic;assumption.
     intros ;apply P_id_a__zeros_bounded;assumption.
     intros ;apply P_id_U51_bounded;assumption.
     intros ;apply P_id_a__U84_bounded;assumption.
     intros ;apply P_id_a__U32_bounded;assumption.
     intros ;apply P_id_isNatList_bounded;assumption.
     intros ;apply P_id_a__U13_bounded;assumption.
     intros ;apply P_id_U85_bounded;assumption.
     intros ;apply P_id_a__length_bounded;assumption.
     intros ;apply P_id_a__isNatIList_bounded;assumption.
     intros ;apply P_id_U33_bounded;assumption.
     intros ;apply P_id_a__U11_bounded;assumption.
     intros ;apply P_id_U81_bounded;assumption.
     intros ;apply P_id_a__U92_bounded;assumption.
     intros ;apply P_id_a__U43_bounded;assumption.
     intros ;apply P_id_U23_bounded;assumption.
     intros ;apply P_id_a__isNatKind_bounded;assumption.
     intros ;apply P_id_U93_bounded;assumption.
     intros ;apply P_id_U11_bounded;assumption.
     intros ;apply P_id_a__U71_bounded;assumption.
     intros ;apply P_id_U44_bounded;assumption.
     intros ;apply P_id_0_bounded;assumption.
     intros ;apply P_id_U61_bounded;assumption.
     intros ;apply P_id_a__U86_bounded;assumption.
     intros ;apply P_id_a__U41_bounded;assumption.
     intros ;apply P_id_U22_bounded;assumption.
     intros ;apply P_id_a__U21_bounded;assumption.
     intros ;apply P_id_U91_bounded;assumption.
     intros ;apply P_id_length_bounded;assumption.
     intros ;apply P_id_a__U52_bounded;assumption.
     intros ;apply P_id_U42_bounded;assumption.
     intros ;apply P_id_a__U12_bounded;assumption.
     intros ;apply P_id_U83_bounded;assumption.
     intros ;apply P_id_a__U94_bounded;assumption.
     intros ;apply P_id_a__U45_bounded;assumption.
     intros ;apply P_id_U31_bounded;assumption.
     intros ;apply P_id_a__isNat_bounded;assumption.
     intros ;apply P_id_isNatIListKind_bounded;assumption.
     intros ;apply P_id_a__U82_bounded;assumption.
     intros ;apply P_id_U46_bounded;assumption.
     intros ;apply P_id_cons_bounded;assumption.
     intros ;apply P_id_U52_bounded;assumption.
     intros ;apply P_id_a__U85_bounded;assumption.
     intros ;apply P_id_a__U33_bounded;assumption.
     intros ;apply P_id_U21_bounded;assumption.
     intros ;apply P_id_a__isNatList_bounded;assumption.
     intros ;apply P_id_U86_bounded;assumption.
     intros ;apply P_id_mark_bounded;assumption.
     intros ;apply P_id_a__U51_bounded;assumption.
     intros ;apply P_id_U41_bounded;assumption.
     intros ;apply P_id_tt_bounded;assumption.
     intros ;apply P_id_U82_bounded;assumption.
     intros ;apply P_id_a__U93_bounded;assumption.
     intros ;apply P_id_a__U44_bounded;assumption.
     intros ;apply P_id_isNat_bounded;assumption.
     intros ;apply P_id_a__U23_bounded;assumption.
     intros ;apply P_id_U94_bounded;assumption.
     intros ;apply P_id_U12_bounded;assumption.
     intros ;apply P_id_a__U81_bounded;assumption.
     intros ;apply P_id_U45_bounded;assumption.
     intros ;apply P_id_zeros_bounded;assumption.
     intros ;apply P_id_U71_bounded;assumption.
     intros ;apply P_id_a__U91_bounded;assumption.
     intros ;apply P_id_a__U42_bounded;assumption.
     intros ;apply P_id_isNatKind_bounded;assumption.
     intros ;apply P_id_a__U22_bounded;assumption.
     intros ;apply P_id_U92_bounded;assumption.
     intros ;apply P_id_nil_bounded;assumption.
     intros ;apply P_id_a__U61_bounded;assumption.
     intros ;apply P_id_U43_bounded;assumption.
     intros ;apply P_id_a__isNatIListKind_bounded;assumption.
     intros ;apply P_id_U84_bounded;assumption.
     intros ;apply P_id_s_bounded;assumption.
     intros ;apply P_id_a__U46_bounded;assumption.
     intros ;apply P_id_U32_bounded;assumption.
     intros ;apply P_id_a__U31_bounded;assumption.
     intros ;apply P_id_U13_bounded;assumption.
     intros ;apply P_id_a__U83_bounded;assumption.
     intros ;apply P_id_isNatIList_bounded;assumption.
     apply rules_monotonic.
   Qed.
   
   Hypothesis P_id_A__U46 : Z ->Z.
   
   Hypothesis P_id_A__ISNATILISTKIND : Z ->Z.
   
   Hypothesis P_id_A__U61 : Z ->Z.
   
   Hypothesis P_id_A__U85 : Z ->Z ->Z.
   
   Hypothesis P_id_A__U31 : Z ->Z ->Z.
   
   Hypothesis P_id_A__U81 : Z ->Z ->Z ->Z.
   
   Hypothesis P_id_A__U22 : Z ->Z ->Z.
   
   Hypothesis P_id_A__U93 : Z ->Z ->Z ->Z.
   
   Hypothesis P_id_A__U42 : Z ->Z ->Z ->Z.
   
   Hypothesis P_id_A__U51 : Z ->Z ->Z.
   
   Hypothesis P_id_A__ISNATLIST : Z ->Z.
   
   Hypothesis P_id_A__ZEROS : Z.
   
   Hypothesis P_id_A__U91 : Z ->Z ->Z ->Z.
   
   Hypothesis P_id_A__U33 : Z ->Z.
   
   Hypothesis P_id_A__U83 : Z ->Z ->Z ->Z.
   
   Hypothesis P_id_A__U23 : Z ->Z.
   
   Hypothesis P_id_A__U11 : Z ->Z ->Z.
   
   Hypothesis P_id_A__LENGTH : Z ->Z.
   
   Hypothesis P_id_A__U44 : Z ->Z ->Z ->Z.
   
   Hypothesis P_id_A__ISNATILIST : Z ->Z.
   
   Hypothesis P_id_A__U13 : Z ->Z.
   
   Hypothesis P_id_A__U71 : Z ->Z.
   
   Hypothesis P_id_A__U86 : Z ->Z.
   
   Hypothesis P_id_A__U32 : Z ->Z ->Z.
   
   Hypothesis P_id_A__U82 : Z ->Z ->Z ->Z.
   
   Hypothesis P_id_A__ISNATKIND : Z ->Z.
   
   Hypothesis P_id_A__U94 : Z ->Z ->Z.
   
   Hypothesis P_id_A__U43 : Z ->Z ->Z ->Z.
   
   Hypothesis P_id_A__U52 : Z ->Z.
   
   Hypothesis P_id_A__U21 : Z ->Z ->Z.
   
   Hypothesis P_id_A__U92 : Z ->Z ->Z ->Z.
   
   Hypothesis P_id_A__U41 : Z ->Z ->Z ->Z.
   
   Hypothesis P_id_A__U84 : Z ->Z ->Z ->Z.
   
   Hypothesis P_id_A__ISNAT : Z ->Z.
   
   Hypothesis P_id_A__U12 : Z ->Z ->Z.
   
   Hypothesis P_id_MARK : Z ->Z.
   
   Hypothesis P_id_A__U45 : Z ->Z ->Z.
   
   Hypothesis P_id_A__U46_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->P_id_A__U46 x12 <= P_id_A__U46 x11.
   
   Hypothesis P_id_A__ISNATILISTKIND_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->
      P_id_A__ISNATILISTKIND x12 <= P_id_A__ISNATILISTKIND x11.
   
   Hypothesis P_id_A__U61_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->P_id_A__U61 x12 <= P_id_A__U61 x11.
   
   Hypothesis P_id_A__U85_monotonic :
    forall x12 x14 x13 x11, 
     (min_value <= x14)/\ (x14 <= x13) ->
      (min_value <= x12)/\ (x12 <= x11) ->
       P_id_A__U85 x12 x14 <= P_id_A__U85 x11 x13.
   
   Hypothesis P_id_A__U31_monotonic :
    forall x12 x14 x13 x11, 
     (min_value <= x14)/\ (x14 <= x13) ->
      (min_value <= x12)/\ (x12 <= x11) ->
       P_id_A__U31 x12 x14 <= P_id_A__U31 x11 x13.
   
   Hypothesis P_id_A__U81_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (min_value <= x16)/\ (x16 <= x15) ->
      (min_value <= x14)/\ (x14 <= x13) ->
       (min_value <= x12)/\ (x12 <= x11) ->
        P_id_A__U81 x12 x14 x16 <= P_id_A__U81 x11 x13 x15.
   
   Hypothesis P_id_A__U22_monotonic :
    forall x12 x14 x13 x11, 
     (min_value <= x14)/\ (x14 <= x13) ->
      (min_value <= x12)/\ (x12 <= x11) ->
       P_id_A__U22 x12 x14 <= P_id_A__U22 x11 x13.
   
   Hypothesis P_id_A__U93_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (min_value <= x16)/\ (x16 <= x15) ->
      (min_value <= x14)/\ (x14 <= x13) ->
       (min_value <= x12)/\ (x12 <= x11) ->
        P_id_A__U93 x12 x14 x16 <= P_id_A__U93 x11 x13 x15.
   
   Hypothesis P_id_A__U42_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (min_value <= x16)/\ (x16 <= x15) ->
      (min_value <= x14)/\ (x14 <= x13) ->
       (min_value <= x12)/\ (x12 <= x11) ->
        P_id_A__U42 x12 x14 x16 <= P_id_A__U42 x11 x13 x15.
   
   Hypothesis P_id_A__U51_monotonic :
    forall x12 x14 x13 x11, 
     (min_value <= x14)/\ (x14 <= x13) ->
      (min_value <= x12)/\ (x12 <= x11) ->
       P_id_A__U51 x12 x14 <= P_id_A__U51 x11 x13.
   
   Hypothesis P_id_A__ISNATLIST_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->
      P_id_A__ISNATLIST x12 <= P_id_A__ISNATLIST x11.
   
   Hypothesis P_id_A__U91_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (min_value <= x16)/\ (x16 <= x15) ->
      (min_value <= x14)/\ (x14 <= x13) ->
       (min_value <= x12)/\ (x12 <= x11) ->
        P_id_A__U91 x12 x14 x16 <= P_id_A__U91 x11 x13 x15.
   
   Hypothesis P_id_A__U33_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->P_id_A__U33 x12 <= P_id_A__U33 x11.
   
   Hypothesis P_id_A__U83_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (min_value <= x16)/\ (x16 <= x15) ->
      (min_value <= x14)/\ (x14 <= x13) ->
       (min_value <= x12)/\ (x12 <= x11) ->
        P_id_A__U83 x12 x14 x16 <= P_id_A__U83 x11 x13 x15.
   
   Hypothesis P_id_A__U23_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->P_id_A__U23 x12 <= P_id_A__U23 x11.
   
   Hypothesis P_id_A__U11_monotonic :
    forall x12 x14 x13 x11, 
     (min_value <= x14)/\ (x14 <= x13) ->
      (min_value <= x12)/\ (x12 <= x11) ->
       P_id_A__U11 x12 x14 <= P_id_A__U11 x11 x13.
   
   Hypothesis P_id_A__LENGTH_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->
      P_id_A__LENGTH x12 <= P_id_A__LENGTH x11.
   
   Hypothesis P_id_A__U44_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (min_value <= x16)/\ (x16 <= x15) ->
      (min_value <= x14)/\ (x14 <= x13) ->
       (min_value <= x12)/\ (x12 <= x11) ->
        P_id_A__U44 x12 x14 x16 <= P_id_A__U44 x11 x13 x15.
   
   Hypothesis P_id_A__ISNATILIST_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->
      P_id_A__ISNATILIST x12 <= P_id_A__ISNATILIST x11.
   
   Hypothesis P_id_A__U13_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->P_id_A__U13 x12 <= P_id_A__U13 x11.
   
   Hypothesis P_id_A__U71_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->P_id_A__U71 x12 <= P_id_A__U71 x11.
   
   Hypothesis P_id_A__U86_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->P_id_A__U86 x12 <= P_id_A__U86 x11.
   
   Hypothesis P_id_A__U32_monotonic :
    forall x12 x14 x13 x11, 
     (min_value <= x14)/\ (x14 <= x13) ->
      (min_value <= x12)/\ (x12 <= x11) ->
       P_id_A__U32 x12 x14 <= P_id_A__U32 x11 x13.
   
   Hypothesis P_id_A__U82_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (min_value <= x16)/\ (x16 <= x15) ->
      (min_value <= x14)/\ (x14 <= x13) ->
       (min_value <= x12)/\ (x12 <= x11) ->
        P_id_A__U82 x12 x14 x16 <= P_id_A__U82 x11 x13 x15.
   
   Hypothesis P_id_A__ISNATKIND_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->
      P_id_A__ISNATKIND x12 <= P_id_A__ISNATKIND x11.
   
   Hypothesis P_id_A__U94_monotonic :
    forall x12 x14 x13 x11, 
     (min_value <= x14)/\ (x14 <= x13) ->
      (min_value <= x12)/\ (x12 <= x11) ->
       P_id_A__U94 x12 x14 <= P_id_A__U94 x11 x13.
   
   Hypothesis P_id_A__U43_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (min_value <= x16)/\ (x16 <= x15) ->
      (min_value <= x14)/\ (x14 <= x13) ->
       (min_value <= x12)/\ (x12 <= x11) ->
        P_id_A__U43 x12 x14 x16 <= P_id_A__U43 x11 x13 x15.
   
   Hypothesis P_id_A__U52_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->P_id_A__U52 x12 <= P_id_A__U52 x11.
   
   Hypothesis P_id_A__U21_monotonic :
    forall x12 x14 x13 x11, 
     (min_value <= x14)/\ (x14 <= x13) ->
      (min_value <= x12)/\ (x12 <= x11) ->
       P_id_A__U21 x12 x14 <= P_id_A__U21 x11 x13.
   
   Hypothesis P_id_A__U92_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (min_value <= x16)/\ (x16 <= x15) ->
      (min_value <= x14)/\ (x14 <= x13) ->
       (min_value <= x12)/\ (x12 <= x11) ->
        P_id_A__U92 x12 x14 x16 <= P_id_A__U92 x11 x13 x15.
   
   Hypothesis P_id_A__U41_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (min_value <= x16)/\ (x16 <= x15) ->
      (min_value <= x14)/\ (x14 <= x13) ->
       (min_value <= x12)/\ (x12 <= x11) ->
        P_id_A__U41 x12 x14 x16 <= P_id_A__U41 x11 x13 x15.
   
   Hypothesis P_id_A__U84_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (min_value <= x16)/\ (x16 <= x15) ->
      (min_value <= x14)/\ (x14 <= x13) ->
       (min_value <= x12)/\ (x12 <= x11) ->
        P_id_A__U84 x12 x14 x16 <= P_id_A__U84 x11 x13 x15.
   
   Hypothesis P_id_A__ISNAT_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->
      P_id_A__ISNAT x12 <= P_id_A__ISNAT x11.
   
   Hypothesis P_id_A__U12_monotonic :
    forall x12 x14 x13 x11, 
     (min_value <= x14)/\ (x14 <= x13) ->
      (min_value <= x12)/\ (x12 <= x11) ->
       P_id_A__U12 x12 x14 <= P_id_A__U12 x11 x13.
   
   Hypothesis P_id_MARK_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->P_id_MARK x12 <= P_id_MARK x11.
   
   Hypothesis P_id_A__U45_monotonic :
    forall x12 x14 x13 x11, 
     (min_value <= x14)/\ (x14 <= x13) ->
      (min_value <= x12)/\ (x12 <= x11) ->
       P_id_A__U45 x12 x14 <= P_id_A__U45 x11 x13.
   
   Definition marked_measure  := 
     Interp.marked_measure min_value P_id_a__zeros P_id_U51 P_id_a__U84 
      P_id_a__U32 P_id_isNatList P_id_a__U13 P_id_U85 P_id_a__length 
      P_id_a__isNatIList P_id_U33 P_id_a__U11 P_id_U81 P_id_a__U92 
      P_id_a__U43 P_id_U23 P_id_a__isNatKind P_id_U93 P_id_U11 P_id_a__U71 
      P_id_U44 P_id_0 P_id_U61 P_id_a__U86 P_id_a__U41 P_id_U22 P_id_a__U21 
      P_id_U91 P_id_length P_id_a__U52 P_id_U42 P_id_a__U12 P_id_U83 
      P_id_a__U94 P_id_a__U45 P_id_U31 P_id_a__isNat P_id_isNatIListKind 
      P_id_a__U82 P_id_U46 P_id_cons P_id_U52 P_id_a__U85 P_id_a__U33 
      P_id_U21 P_id_a__isNatList P_id_U86 P_id_mark P_id_a__U51 P_id_U41 
      P_id_tt P_id_U82 P_id_a__U93 P_id_a__U44 P_id_isNat P_id_a__U23 
      P_id_U94 P_id_U12 P_id_a__U81 P_id_U45 P_id_zeros P_id_U71 P_id_a__U91 
      P_id_a__U42 P_id_isNatKind P_id_a__U22 P_id_U92 P_id_nil P_id_a__U61 
      P_id_U43 P_id_a__isNatIListKind P_id_U84 P_id_s P_id_a__U46 P_id_U32 
      P_id_a__U31 P_id_U13 P_id_a__U83 P_id_isNatIList P_id_A__U46 
      P_id_A__ISNATILISTKIND P_id_A__U61 P_id_A__U85 P_id_A__U31 P_id_A__U81 
      P_id_A__U22 P_id_A__U93 P_id_A__U42 P_id_A__U51 P_id_A__ISNATLIST 
      P_id_A__ZEROS P_id_A__U91 P_id_A__U33 P_id_A__U83 P_id_A__U23 
      P_id_A__U11 P_id_A__LENGTH P_id_A__U44 P_id_A__ISNATILIST P_id_A__U13 
      P_id_A__U71 P_id_A__U86 P_id_A__U32 P_id_A__U82 P_id_A__ISNATKIND 
      P_id_A__U94 P_id_A__U43 P_id_A__U52 P_id_A__U21 P_id_A__U92 
      P_id_A__U41 P_id_A__U84 P_id_A__ISNAT P_id_A__U12 P_id_MARK P_id_A__U45
     .
   
   Lemma marked_measure_equation :
    forall t, 
     marked_measure t = match t with
                          | (algebra.Alg.Term algebra.F.id_a__U46 (x11::nil)) =>
                           P_id_A__U46 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
                             (x11::nil)) =>
                           P_id_A__ISNATILISTKIND (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U61 (x11::nil)) =>
                           P_id_A__U61 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U85 (x12::
                             x11::nil)) =>
                           P_id_A__U85 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U31 (x12::
                             x11::nil)) =>
                           P_id_A__U31 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U81 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U81 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U22 (x12::
                             x11::nil)) =>
                           P_id_A__U22 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U93 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U93 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U42 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U42 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U51 (x12::
                             x11::nil)) =>
                           P_id_A__U51 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__isNatList 
                             (x11::nil)) =>
                           P_id_A__ISNATLIST (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__zeros nil) =>
                           P_id_A__ZEROS 
                          | (algebra.Alg.Term algebra.F.id_a__U91 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U91 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U33 (x11::nil)) =>
                           P_id_A__U33 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U83 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U83 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U23 (x11::nil)) =>
                           P_id_A__U23 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U11 (x12::
                             x11::nil)) =>
                           P_id_A__U11 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__length 
                             (x11::nil)) =>
                           P_id_A__LENGTH (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U44 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U44 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__isNatIList 
                             (x11::nil)) =>
                           P_id_A__ISNATILIST (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U13 (x11::nil)) =>
                           P_id_A__U13 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U71 (x11::nil)) =>
                           P_id_A__U71 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U86 (x11::nil)) =>
                           P_id_A__U86 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U32 (x12::
                             x11::nil)) =>
                           P_id_A__U32 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U82 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U82 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__isNatKind 
                             (x11::nil)) =>
                           P_id_A__ISNATKIND (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U94 (x12::
                             x11::nil)) =>
                           P_id_A__U94 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U43 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U43 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U52 (x11::nil)) =>
                           P_id_A__U52 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U21 (x12::
                             x11::nil)) =>
                           P_id_A__U21 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U92 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U92 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U41 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U41 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U84 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U84 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__isNat 
                             (x11::nil)) =>
                           P_id_A__ISNAT (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U12 (x12::
                             x11::nil)) =>
                           P_id_A__U12 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_mark (x11::nil)) =>
                           P_id_MARK (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U45 (x12::
                             x11::nil)) =>
                           P_id_A__U45 (measure x12) (measure x11)
                          | _ => measure t
                          end.
   Proof.
     reflexivity .
   Qed.
   
   Lemma marked_measure_star_monotonic :
    forall f l1 l2, 
     (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step 
                                                       R_xml_0_deep_rew.R_xml_0_rules)
                                                      ) l1 l2) ->
      marked_measure (algebra.Alg.Term f l1) <= marked_measure (algebra.Alg.Term 
                                                                 f l2).
   Proof.
     unfold marked_measure in *.
     apply Interp.marked_measure_star_monotonic with Alt Aeq.
     
     (apply interp.o_Z)||
     (cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;auto with zarith).
     intros ;apply P_id_U51_monotonic;assumption.
     intros ;apply P_id_a__U84_monotonic;assumption.
     intros ;apply P_id_a__U32_monotonic;assumption.
     intros ;apply P_id_isNatList_monotonic;assumption.
     intros ;apply P_id_a__U13_monotonic;assumption.
     intros ;apply P_id_U85_monotonic;assumption.
     intros ;apply P_id_a__length_monotonic;assumption.
     intros ;apply P_id_a__isNatIList_monotonic;assumption.
     intros ;apply P_id_U33_monotonic;assumption.
     intros ;apply P_id_a__U11_monotonic;assumption.
     intros ;apply P_id_U81_monotonic;assumption.
     intros ;apply P_id_a__U92_monotonic;assumption.
     intros ;apply P_id_a__U43_monotonic;assumption.
     intros ;apply P_id_U23_monotonic;assumption.
     intros ;apply P_id_a__isNatKind_monotonic;assumption.
     intros ;apply P_id_U93_monotonic;assumption.
     intros ;apply P_id_U11_monotonic;assumption.
     intros ;apply P_id_a__U71_monotonic;assumption.
     intros ;apply P_id_U44_monotonic;assumption.
     intros ;apply P_id_U61_monotonic;assumption.
     intros ;apply P_id_a__U86_monotonic;assumption.
     intros ;apply P_id_a__U41_monotonic;assumption.
     intros ;apply P_id_U22_monotonic;assumption.
     intros ;apply P_id_a__U21_monotonic;assumption.
     intros ;apply P_id_U91_monotonic;assumption.
     intros ;apply P_id_length_monotonic;assumption.
     intros ;apply P_id_a__U52_monotonic;assumption.
     intros ;apply P_id_U42_monotonic;assumption.
     intros ;apply P_id_a__U12_monotonic;assumption.
     intros ;apply P_id_U83_monotonic;assumption.
     intros ;apply P_id_a__U94_monotonic;assumption.
     intros ;apply P_id_a__U45_monotonic;assumption.
     intros ;apply P_id_U31_monotonic;assumption.
     intros ;apply P_id_a__isNat_monotonic;assumption.
     intros ;apply P_id_isNatIListKind_monotonic;assumption.
     intros ;apply P_id_a__U82_monotonic;assumption.
     intros ;apply P_id_U46_monotonic;assumption.
     intros ;apply P_id_cons_monotonic;assumption.
     intros ;apply P_id_U52_monotonic;assumption.
     intros ;apply P_id_a__U85_monotonic;assumption.
     intros ;apply P_id_a__U33_monotonic;assumption.
     intros ;apply P_id_U21_monotonic;assumption.
     intros ;apply P_id_a__isNatList_monotonic;assumption.
     intros ;apply P_id_U86_monotonic;assumption.
     intros ;apply P_id_mark_monotonic;assumption.
     intros ;apply P_id_a__U51_monotonic;assumption.
     intros ;apply P_id_U41_monotonic;assumption.
     intros ;apply P_id_U82_monotonic;assumption.
     intros ;apply P_id_a__U93_monotonic;assumption.
     intros ;apply P_id_a__U44_monotonic;assumption.
     intros ;apply P_id_isNat_monotonic;assumption.
     intros ;apply P_id_a__U23_monotonic;assumption.
     intros ;apply P_id_U94_monotonic;assumption.
     intros ;apply P_id_U12_monotonic;assumption.
     intros ;apply P_id_a__U81_monotonic;assumption.
     intros ;apply P_id_U45_monotonic;assumption.
     intros ;apply P_id_U71_monotonic;assumption.
     intros ;apply P_id_a__U91_monotonic;assumption.
     intros ;apply P_id_a__U42_monotonic;assumption.
     intros ;apply P_id_isNatKind_monotonic;assumption.
     intros ;apply P_id_a__U22_monotonic;assumption.
     intros ;apply P_id_U92_monotonic;assumption.
     intros ;apply P_id_a__U61_monotonic;assumption.
     intros ;apply P_id_U43_monotonic;assumption.
     intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
     intros ;apply P_id_U84_monotonic;assumption.
     intros ;apply P_id_s_monotonic;assumption.
     intros ;apply P_id_a__U46_monotonic;assumption.
     intros ;apply P_id_U32_monotonic;assumption.
     intros ;apply P_id_a__U31_monotonic;assumption.
     intros ;apply P_id_U13_monotonic;assumption.
     intros ;apply P_id_a__U83_monotonic;assumption.
     intros ;apply P_id_isNatIList_monotonic;assumption.
     intros ;apply P_id_a__zeros_bounded;assumption.
     intros ;apply P_id_U51_bounded;assumption.
     intros ;apply P_id_a__U84_bounded;assumption.
     intros ;apply P_id_a__U32_bounded;assumption.
     intros ;apply P_id_isNatList_bounded;assumption.
     intros ;apply P_id_a__U13_bounded;assumption.
     intros ;apply P_id_U85_bounded;assumption.
     intros ;apply P_id_a__length_bounded;assumption.
     intros ;apply P_id_a__isNatIList_bounded;assumption.
     intros ;apply P_id_U33_bounded;assumption.
     intros ;apply P_id_a__U11_bounded;assumption.
     intros ;apply P_id_U81_bounded;assumption.
     intros ;apply P_id_a__U92_bounded;assumption.
     intros ;apply P_id_a__U43_bounded;assumption.
     intros ;apply P_id_U23_bounded;assumption.
     intros ;apply P_id_a__isNatKind_bounded;assumption.
     intros ;apply P_id_U93_bounded;assumption.
     intros ;apply P_id_U11_bounded;assumption.
     intros ;apply P_id_a__U71_bounded;assumption.
     intros ;apply P_id_U44_bounded;assumption.
     intros ;apply P_id_0_bounded;assumption.
     intros ;apply P_id_U61_bounded;assumption.
     intros ;apply P_id_a__U86_bounded;assumption.
     intros ;apply P_id_a__U41_bounded;assumption.
     intros ;apply P_id_U22_bounded;assumption.
     intros ;apply P_id_a__U21_bounded;assumption.
     intros ;apply P_id_U91_bounded;assumption.
     intros ;apply P_id_length_bounded;assumption.
     intros ;apply P_id_a__U52_bounded;assumption.
     intros ;apply P_id_U42_bounded;assumption.
     intros ;apply P_id_a__U12_bounded;assumption.
     intros ;apply P_id_U83_bounded;assumption.
     intros ;apply P_id_a__U94_bounded;assumption.
     intros ;apply P_id_a__U45_bounded;assumption.
     intros ;apply P_id_U31_bounded;assumption.
     intros ;apply P_id_a__isNat_bounded;assumption.
     intros ;apply P_id_isNatIListKind_bounded;assumption.
     intros ;apply P_id_a__U82_bounded;assumption.
     intros ;apply P_id_U46_bounded;assumption.
     intros ;apply P_id_cons_bounded;assumption.
     intros ;apply P_id_U52_bounded;assumption.
     intros ;apply P_id_a__U85_bounded;assumption.
     intros ;apply P_id_a__U33_bounded;assumption.
     intros ;apply P_id_U21_bounded;assumption.
     intros ;apply P_id_a__isNatList_bounded;assumption.
     intros ;apply P_id_U86_bounded;assumption.
     intros ;apply P_id_mark_bounded;assumption.
     intros ;apply P_id_a__U51_bounded;assumption.
     intros ;apply P_id_U41_bounded;assumption.
     intros ;apply P_id_tt_bounded;assumption.
     intros ;apply P_id_U82_bounded;assumption.
     intros ;apply P_id_a__U93_bounded;assumption.
     intros ;apply P_id_a__U44_bounded;assumption.
     intros ;apply P_id_isNat_bounded;assumption.
     intros ;apply P_id_a__U23_bounded;assumption.
     intros ;apply P_id_U94_bounded;assumption.
     intros ;apply P_id_U12_bounded;assumption.
     intros ;apply P_id_a__U81_bounded;assumption.
     intros ;apply P_id_U45_bounded;assumption.
     intros ;apply P_id_zeros_bounded;assumption.
     intros ;apply P_id_U71_bounded;assumption.
     intros ;apply P_id_a__U91_bounded;assumption.
     intros ;apply P_id_a__U42_bounded;assumption.
     intros ;apply P_id_isNatKind_bounded;assumption.
     intros ;apply P_id_a__U22_bounded;assumption.
     intros ;apply P_id_U92_bounded;assumption.
     intros ;apply P_id_nil_bounded;assumption.
     intros ;apply P_id_a__U61_bounded;assumption.
     intros ;apply P_id_U43_bounded;assumption.
     intros ;apply P_id_a__isNatIListKind_bounded;assumption.
     intros ;apply P_id_U84_bounded;assumption.
     intros ;apply P_id_s_bounded;assumption.
     intros ;apply P_id_a__U46_bounded;assumption.
     intros ;apply P_id_U32_bounded;assumption.
     intros ;apply P_id_a__U31_bounded;assumption.
     intros ;apply P_id_U13_bounded;assumption.
     intros ;apply P_id_a__U83_bounded;assumption.
     intros ;apply P_id_isNatIList_bounded;assumption.
     apply rules_monotonic.
     intros ;apply P_id_A__U46_monotonic;assumption.
     intros ;apply P_id_A__ISNATILISTKIND_monotonic;assumption.
     intros ;apply P_id_A__U61_monotonic;assumption.
     intros ;apply P_id_A__U85_monotonic;assumption.
     intros ;apply P_id_A__U31_monotonic;assumption.
     intros ;apply P_id_A__U81_monotonic;assumption.
     intros ;apply P_id_A__U22_monotonic;assumption.
     intros ;apply P_id_A__U93_monotonic;assumption.
     intros ;apply P_id_A__U42_monotonic;assumption.
     intros ;apply P_id_A__U51_monotonic;assumption.
     intros ;apply P_id_A__ISNATLIST_monotonic;assumption.
     intros ;apply P_id_A__U91_monotonic;assumption.
     intros ;apply P_id_A__U33_monotonic;assumption.
     intros ;apply P_id_A__U83_monotonic;assumption.
     intros ;apply P_id_A__U23_monotonic;assumption.
     intros ;apply P_id_A__U11_monotonic;assumption.
     intros ;apply P_id_A__LENGTH_monotonic;assumption.
     intros ;apply P_id_A__U44_monotonic;assumption.
     intros ;apply P_id_A__ISNATILIST_monotonic;assumption.
     intros ;apply P_id_A__U13_monotonic;assumption.
     intros ;apply P_id_A__U71_monotonic;assumption.
     intros ;apply P_id_A__U86_monotonic;assumption.
     intros ;apply P_id_A__U32_monotonic;assumption.
     intros ;apply P_id_A__U82_monotonic;assumption.
     intros ;apply P_id_A__ISNATKIND_monotonic;assumption.
     intros ;apply P_id_A__U94_monotonic;assumption.
     intros ;apply P_id_A__U43_monotonic;assumption.
     intros ;apply P_id_A__U52_monotonic;assumption.
     intros ;apply P_id_A__U21_monotonic;assumption.
     intros ;apply P_id_A__U92_monotonic;assumption.
     intros ;apply P_id_A__U41_monotonic;assumption.
     intros ;apply P_id_A__U84_monotonic;assumption.
     intros ;apply P_id_A__ISNAT_monotonic;assumption.
     intros ;apply P_id_A__U12_monotonic;assumption.
     intros ;apply P_id_MARK_monotonic;assumption.
     intros ;apply P_id_A__U45_monotonic;assumption.
   Qed.
   
   End S.
End InterpZ.

Module WF_R_xml_0_deep_rew.
 Inductive DP_R_xml_0  :
  algebra.Alg.term ->algebra.Alg.term ->Prop := 
    (* <a__U11(tt,V1_),a__U12(a__isNatIListKind(V1_),V1_)> *)
   | DP_R_xml_0_0 :
    forall x12 x1 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_tt nil) 
       x12) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x1 x11) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U12 ((algebra.Alg.Term 
                   algebra.F.id_a__isNatIListKind (x1::nil))::x1::nil)) 
        (algebra.Alg.Term algebra.F.id_a__U11 (x12::x11::nil))
    (* <a__U11(tt,V1_),a__isNatIListKind(V1_)> *)
   | DP_R_xml_0_1 :
    forall x12 x1 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_tt nil) 
       x12) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x1 x11) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isNatIListKind (x1::nil)) 
        (algebra.Alg.Term algebra.F.id_a__U11 (x12::x11::nil))
    (* <a__U12(tt,V1_),a__U13(a__isNatList(V1_))> *)
   | DP_R_xml_0_2 :
    forall x12 x1 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_tt nil) 
       x12) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x1 x11) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U13 ((algebra.Alg.Term 
                   algebra.F.id_a__isNatList (x1::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_a__U12 (x12::x11::nil))
    (* <a__U12(tt,V1_),a__isNatList(V1_)> *)
   | DP_R_xml_0_3 :
    forall x12 x1 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_tt nil) 
       x12) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x1 x11) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isNatList (x1::nil)) 
        (algebra.Alg.Term algebra.F.id_a__U12 (x12::x11::nil))
    (* <a__U21(tt,V1_),a__U22(a__isNatKind(V1_),V1_)> *)
   | DP_R_xml_0_4 :
    forall x12 x1 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_tt nil) 
       x12) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x1 x11) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U22 ((algebra.Alg.Term 
                   algebra.F.id_a__isNatKind (x1::nil))::x1::nil)) 
        (algebra.Alg.Term algebra.F.id_a__U21 (x12::x11::nil))
    (* <a__U21(tt,V1_),a__isNatKind(V1_)> *)
   | DP_R_xml_0_5 :
    forall x12 x1 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_tt nil) 
       x12) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x1 x11) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isNatKind (x1::nil)) 
        (algebra.Alg.Term algebra.F.id_a__U21 (x12::x11::nil))
    (* <a__U22(tt,V1_),a__U23(a__isNat(V1_))> *)
   | DP_R_xml_0_6 :
    forall x12 x1 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_tt nil) 
       x12) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x1 x11) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U23 ((algebra.Alg.Term 
                   algebra.F.id_a__isNat (x1::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_a__U22 (x12::x11::nil))
    (* <a__U22(tt,V1_),a__isNat(V1_)> *)
   | DP_R_xml_0_7 :
    forall x12 x1 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_tt nil) 
       x12) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x1 x11) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isNat (x1::nil)) 
        (algebra.Alg.Term algebra.F.id_a__U22 (x12::x11::nil))
    (* <a__U31(tt,V_),a__U32(a__isNatIListKind(V_),V_)> *)
   | DP_R_xml_0_8 :
    forall x12 x2 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_tt nil) 
       x12) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x2 x11) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U32 ((algebra.Alg.Term 
                   algebra.F.id_a__isNatIListKind (x2::nil))::x2::nil)) 
        (algebra.Alg.Term algebra.F.id_a__U31 (x12::x11::nil))
    (* <a__U31(tt,V_),a__isNatIListKind(V_)> *)
   | DP_R_xml_0_9 :
    forall x12 x2 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_tt nil) 
       x12) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x2 x11) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isNatIListKind (x2::nil)) 
        (algebra.Alg.Term algebra.F.id_a__U31 (x12::x11::nil))
    (* <a__U32(tt,V_),a__U33(a__isNatList(V_))> *)
   | DP_R_xml_0_10 :
    forall x12 x2 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_tt nil) 
       x12) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x2 x11) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U33 ((algebra.Alg.Term 
                   algebra.F.id_a__isNatList (x2::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_a__U32 (x12::x11::nil))
    (* <a__U32(tt,V_),a__isNatList(V_)> *)
   | DP_R_xml_0_11 :
    forall x12 x2 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_tt nil) 
       x12) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x2 x11) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isNatList (x2::nil)) 
        (algebra.Alg.Term algebra.F.id_a__U32 (x12::x11::nil))
   
    (* <a__U41(tt,V1_,V2_),a__U42(a__isNatKind(V1_),V1_,V2_)> *)
   | DP_R_xml_0_12 :
    forall x12 x1 x13 x3 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_tt nil) 
       x13) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x1 x12) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x3 x11) ->
        DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U42 ((algebra.Alg.Term 
                    algebra.F.id_a__isNatKind (x1::nil))::x1::x3::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U41 (x13::x12::x11::nil))
    (* <a__U41(tt,V1_,V2_),a__isNatKind(V1_)> *)
   | DP_R_xml_0_13 :
    forall x12 x1 x13 x3 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_tt nil) 
       x13) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x1 x12) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x3 x11) ->
        DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isNatKind (x1::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U41 (x13::x12::x11::nil))
   
    (* <a__U42(tt,V1_,V2_),a__U43(a__isNatIListKind(V2_),V1_,V2_)> *)
   | DP_R_xml_0_14 :
    forall x12 x1 x13 x3 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_tt nil) 
       x13) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x1 x12) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x3 x11) ->
        DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U43 ((algebra.Alg.Term 
                    algebra.F.id_a__isNatIListKind (x3::nil))::x1::x3::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U42 (x13::x12::x11::nil))
    (* <a__U42(tt,V1_,V2_),a__isNatIListKind(V2_)> *)
   | DP_R_xml_0_15 :
    forall x12 x1 x13 x3 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_tt nil) 
       x13) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x1 x12) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x3 x11) ->
        DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
                    (x3::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U42 (x13::x12::x11::nil))
   
    (* <a__U43(tt,V1_,V2_),a__U44(a__isNatIListKind(V2_),V1_,V2_)> *)
   | DP_R_xml_0_16 :
    forall x12 x1 x13 x3 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_tt nil) 
       x13) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x1 x12) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x3 x11) ->
        DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U44 ((algebra.Alg.Term 
                    algebra.F.id_a__isNatIListKind (x3::nil))::x1::x3::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U43 (x13::x12::x11::nil))
    (* <a__U43(tt,V1_,V2_),a__isNatIListKind(V2_)> *)
   | DP_R_xml_0_17 :
    forall x12 x1 x13 x3 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_tt nil) 
       x13) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x1 x12) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x3 x11) ->
        DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
                    (x3::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U43 (x13::x12::x11::nil))
    (* <a__U44(tt,V1_,V2_),a__U45(a__isNat(V1_),V2_)> *)
   | DP_R_xml_0_18 :
    forall x12 x1 x13 x3 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_tt nil) 
       x13) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x1 x12) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x3 x11) ->
        DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U45 ((algebra.Alg.Term 
                    algebra.F.id_a__isNat (x1::nil))::x3::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U44 (x13::x12::x11::nil))
    (* <a__U44(tt,V1_,V2_),a__isNat(V1_)> *)
   | DP_R_xml_0_19 :
    forall x12 x1 x13 x3 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_tt nil) 
       x13) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x1 x12) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x3 x11) ->
        DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isNat (x1::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U44 (x13::x12::x11::nil))
    (* <a__U45(tt,V2_),a__U46(a__isNatIList(V2_))> *)
   | DP_R_xml_0_20 :
    forall x12 x3 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_tt nil) 
       x12) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x3 x11) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U46 ((algebra.Alg.Term 
                   algebra.F.id_a__isNatIList (x3::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_a__U45 (x12::x11::nil))
    (* <a__U45(tt,V2_),a__isNatIList(V2_)> *)
   | DP_R_xml_0_21 :
    forall x12 x3 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_tt nil) 
       x12) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x3 x11) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isNatIList (x3::nil)) 
        (algebra.Alg.Term algebra.F.id_a__U45 (x12::x11::nil))
    (* <a__U51(tt,V2_),a__U52(a__isNatIListKind(V2_))> *)
   | DP_R_xml_0_22 :
    forall x12 x3 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_tt nil) 
       x12) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x3 x11) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U52 ((algebra.Alg.Term 
                   algebra.F.id_a__isNatIListKind (x3::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_a__U51 (x12::x11::nil))
    (* <a__U51(tt,V2_),a__isNatIListKind(V2_)> *)
   | DP_R_xml_0_23 :
    forall x12 x3 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_tt nil) 
       x12) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x3 x11) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isNatIListKind (x3::nil)) 
        (algebra.Alg.Term algebra.F.id_a__U51 (x12::x11::nil))
   
    (* <a__U81(tt,V1_,V2_),a__U82(a__isNatKind(V1_),V1_,V2_)> *)
   | DP_R_xml_0_24 :
    forall x12 x1 x13 x3 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_tt nil) 
       x13) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x1 x12) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x3 x11) ->
        DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U82 ((algebra.Alg.Term 
                    algebra.F.id_a__isNatKind (x1::nil))::x1::x3::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U81 (x13::x12::x11::nil))
    (* <a__U81(tt,V1_,V2_),a__isNatKind(V1_)> *)
   | DP_R_xml_0_25 :
    forall x12 x1 x13 x3 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_tt nil) 
       x13) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x1 x12) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x3 x11) ->
        DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isNatKind (x1::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U81 (x13::x12::x11::nil))
   
    (* <a__U82(tt,V1_,V2_),a__U83(a__isNatIListKind(V2_),V1_,V2_)> *)
   | DP_R_xml_0_26 :
    forall x12 x1 x13 x3 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_tt nil) 
       x13) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x1 x12) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x3 x11) ->
        DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U83 ((algebra.Alg.Term 
                    algebra.F.id_a__isNatIListKind (x3::nil))::x1::x3::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U82 (x13::x12::x11::nil))
    (* <a__U82(tt,V1_,V2_),a__isNatIListKind(V2_)> *)
   | DP_R_xml_0_27 :
    forall x12 x1 x13 x3 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_tt nil) 
       x13) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x1 x12) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x3 x11) ->
        DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
                    (x3::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U82 (x13::x12::x11::nil))
   
    (* <a__U83(tt,V1_,V2_),a__U84(a__isNatIListKind(V2_),V1_,V2_)> *)
   | DP_R_xml_0_28 :
    forall x12 x1 x13 x3 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_tt nil) 
       x13) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x1 x12) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x3 x11) ->
        DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U84 ((algebra.Alg.Term 
                    algebra.F.id_a__isNatIListKind (x3::nil))::x1::x3::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U83 (x13::x12::x11::nil))
    (* <a__U83(tt,V1_,V2_),a__isNatIListKind(V2_)> *)
   | DP_R_xml_0_29 :
    forall x12 x1 x13 x3 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_tt nil) 
       x13) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x1 x12) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x3 x11) ->
        DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
                    (x3::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U83 (x13::x12::x11::nil))
    (* <a__U84(tt,V1_,V2_),a__U85(a__isNat(V1_),V2_)> *)
   | DP_R_xml_0_30 :
    forall x12 x1 x13 x3 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_tt nil) 
       x13) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x1 x12) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x3 x11) ->
        DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U85 ((algebra.Alg.Term 
                    algebra.F.id_a__isNat (x1::nil))::x3::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U84 (x13::x12::x11::nil))
    (* <a__U84(tt,V1_,V2_),a__isNat(V1_)> *)
   | DP_R_xml_0_31 :
    forall x12 x1 x13 x3 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_tt nil) 
       x13) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x1 x12) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x3 x11) ->
        DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isNat (x1::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U84 (x13::x12::x11::nil))
    (* <a__U85(tt,V2_),a__U86(a__isNatList(V2_))> *)
   | DP_R_xml_0_32 :
    forall x12 x3 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_tt nil) 
       x12) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x3 x11) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U86 ((algebra.Alg.Term 
                   algebra.F.id_a__isNatList (x3::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_a__U85 (x12::x11::nil))
    (* <a__U85(tt,V2_),a__isNatList(V2_)> *)
   | DP_R_xml_0_33 :
    forall x12 x3 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_tt nil) 
       x12) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x3 x11) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isNatList (x3::nil)) 
        (algebra.Alg.Term algebra.F.id_a__U85 (x12::x11::nil))
   
    (* <a__U91(tt,L_,N_),a__U92(a__isNatIListKind(L_),L_,N_)> *)
   | DP_R_xml_0_34 :
    forall x4 x12 x5 x13 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_tt nil) 
       x13) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x4 x12) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x5 x11) ->
        DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U92 ((algebra.Alg.Term 
                    algebra.F.id_a__isNatIListKind (x4::nil))::x4::x5::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U91 (x13::x12::x11::nil))
    (* <a__U91(tt,L_,N_),a__isNatIListKind(L_)> *)
   | DP_R_xml_0_35 :
    forall x4 x12 x5 x13 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_tt nil) 
       x13) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x4 x12) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x5 x11) ->
        DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
                    (x4::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U91 (x13::x12::x11::nil))
    (* <a__U92(tt,L_,N_),a__U93(a__isNat(N_),L_,N_)> *)
   | DP_R_xml_0_36 :
    forall x4 x12 x5 x13 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_tt nil) 
       x13) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x4 x12) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x5 x11) ->
        DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U93 ((algebra.Alg.Term 
                    algebra.F.id_a__isNat (x5::nil))::x4::x5::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U92 (x13::x12::x11::nil))
    (* <a__U92(tt,L_,N_),a__isNat(N_)> *)
   | DP_R_xml_0_37 :
    forall x4 x12 x5 x13 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_tt nil) 
       x13) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x4 x12) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x5 x11) ->
        DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isNat (x5::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U92 (x13::x12::x11::nil))
    (* <a__U93(tt,L_,N_),a__U94(a__isNatKind(N_),L_)> *)
   | DP_R_xml_0_38 :
    forall x4 x12 x5 x13 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_tt nil) 
       x13) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x4 x12) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x5 x11) ->
        DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U94 ((algebra.Alg.Term 
                    algebra.F.id_a__isNatKind (x5::nil))::x4::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U93 (x13::x12::x11::nil))
    (* <a__U93(tt,L_,N_),a__isNatKind(N_)> *)
   | DP_R_xml_0_39 :
    forall x4 x12 x5 x13 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_tt nil) 
       x13) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x4 x12) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x5 x11) ->
        DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isNatKind (x5::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U93 (x13::x12::x11::nil))
    (* <a__U94(tt,L_),a__length(mark(L_))> *)
   | DP_R_xml_0_40 :
    forall x4 x12 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_tt nil) 
       x12) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x4 x11) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__length 
                   ((algebra.Alg.Term algebra.F.id_mark (x4::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_a__U94 (x12::x11::nil))
    (* <a__U94(tt,L_),mark(L_)> *)
   | DP_R_xml_0_41 :
    forall x4 x12 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_tt nil) 
       x12) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x4 x11) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x4::nil)) 
        (algebra.Alg.Term algebra.F.id_a__U94 (x12::x11::nil))
   
    (* <a__isNat(length(V1_)),a__U11(a__isNatIListKind(V1_),V1_)> *)
   | DP_R_xml_0_42 :
    forall x1 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_length (x1::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U11 ((algebra.Alg.Term 
                  algebra.F.id_a__isNatIListKind (x1::nil))::x1::nil)) 
       (algebra.Alg.Term algebra.F.id_a__isNat (x11::nil))
    (* <a__isNat(length(V1_)),a__isNatIListKind(V1_)> *)
   | DP_R_xml_0_43 :
    forall x1 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_length (x1::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isNatIListKind (x1::nil)) 
       (algebra.Alg.Term algebra.F.id_a__isNat (x11::nil))
    (* <a__isNat(s(V1_)),a__U21(a__isNatKind(V1_),V1_)> *)
   | DP_R_xml_0_44 :
    forall x1 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_s (x1::nil)) 
       x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U21 ((algebra.Alg.Term 
                  algebra.F.id_a__isNatKind (x1::nil))::x1::nil)) 
       (algebra.Alg.Term algebra.F.id_a__isNat (x11::nil))
    (* <a__isNat(s(V1_)),a__isNatKind(V1_)> *)
   | DP_R_xml_0_45 :
    forall x1 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_s (x1::nil)) 
       x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isNatKind (x1::nil)) 
       (algebra.Alg.Term algebra.F.id_a__isNat (x11::nil))
   
    (* <a__isNatIList(V_),a__U31(a__isNatIListKind(V_),V_)> *)
   | DP_R_xml_0_46 :
    forall x2 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                x2 x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U31 ((algebra.Alg.Term 
                  algebra.F.id_a__isNatIListKind (x2::nil))::x2::nil)) 
       (algebra.Alg.Term algebra.F.id_a__isNatIList (x11::nil))
    (* <a__isNatIList(V_),a__isNatIListKind(V_)> *)
   | DP_R_xml_0_47 :
    forall x2 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                x2 x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isNatIListKind (x2::nil)) 
       (algebra.Alg.Term algebra.F.id_a__isNatIList (x11::nil))
   
    (* <a__isNatIList(cons(V1_,V2_)),a__U41(a__isNatKind(V1_),V1_,V2_)> *)
   | DP_R_xml_0_48 :
    forall x1 x3 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_cons (x1::x3::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U41 ((algebra.Alg.Term 
                  algebra.F.id_a__isNatKind (x1::nil))::x1::x3::nil)) 
       (algebra.Alg.Term algebra.F.id_a__isNatIList (x11::nil))
    (* <a__isNatIList(cons(V1_,V2_)),a__isNatKind(V1_)> *)
   | DP_R_xml_0_49 :
    forall x1 x3 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_cons (x1::x3::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isNatKind (x1::nil)) 
       (algebra.Alg.Term algebra.F.id_a__isNatIList (x11::nil))
   
    (* <a__isNatIListKind(cons(V1_,V2_)),a__U51(a__isNatKind(V1_),V2_)> *)
   | DP_R_xml_0_50 :
    forall x1 x3 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_cons (x1::x3::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U51 ((algebra.Alg.Term 
                  algebra.F.id_a__isNatKind (x1::nil))::x3::nil)) 
       (algebra.Alg.Term algebra.F.id_a__isNatIListKind (x11::nil))
   
    (* <a__isNatIListKind(cons(V1_,V2_)),a__isNatKind(V1_)> *)
   | DP_R_xml_0_51 :
    forall x1 x3 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_cons (x1::x3::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isNatKind (x1::nil)) 
       (algebra.Alg.Term algebra.F.id_a__isNatIListKind (x11::nil))
   
    (* <a__isNatKind(length(V1_)),a__U61(a__isNatIListKind(V1_))> *)
   | DP_R_xml_0_52 :
    forall x1 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_length (x1::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U61 ((algebra.Alg.Term 
                  algebra.F.id_a__isNatIListKind (x1::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_a__isNatKind (x11::nil))
    (* <a__isNatKind(length(V1_)),a__isNatIListKind(V1_)> *)
   | DP_R_xml_0_53 :
    forall x1 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_length (x1::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isNatIListKind (x1::nil)) 
       (algebra.Alg.Term algebra.F.id_a__isNatKind (x11::nil))
    (* <a__isNatKind(s(V1_)),a__U71(a__isNatKind(V1_))> *)
   | DP_R_xml_0_54 :
    forall x1 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_s (x1::nil)) 
       x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U71 ((algebra.Alg.Term 
                  algebra.F.id_a__isNatKind (x1::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_a__isNatKind (x11::nil))
    (* <a__isNatKind(s(V1_)),a__isNatKind(V1_)> *)
   | DP_R_xml_0_55 :
    forall x1 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_s (x1::nil)) 
       x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isNatKind (x1::nil)) 
       (algebra.Alg.Term algebra.F.id_a__isNatKind (x11::nil))
   
    (* <a__isNatList(cons(V1_,V2_)),a__U81(a__isNatKind(V1_),V1_,V2_)> *)
   | DP_R_xml_0_56 :
    forall x1 x3 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_cons (x1::x3::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U81 ((algebra.Alg.Term 
                  algebra.F.id_a__isNatKind (x1::nil))::x1::x3::nil)) 
       (algebra.Alg.Term algebra.F.id_a__isNatList (x11::nil))
    (* <a__isNatList(cons(V1_,V2_)),a__isNatKind(V1_)> *)
   | DP_R_xml_0_57 :
    forall x1 x3 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_cons (x1::x3::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isNatKind (x1::nil)) 
       (algebra.Alg.Term algebra.F.id_a__isNatList (x11::nil))
   
    (* <a__length(cons(N_,L_)),a__U91(a__isNatList(L_),L_,N_)> *)
   | DP_R_xml_0_58 :
    forall x4 x5 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_cons (x5::x4::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U91 ((algebra.Alg.Term 
                  algebra.F.id_a__isNatList (x4::nil))::x4::x5::nil)) 
       (algebra.Alg.Term algebra.F.id_a__length (x11::nil))
    (* <a__length(cons(N_,L_)),a__isNatList(L_)> *)
   | DP_R_xml_0_59 :
    forall x4 x5 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_cons (x5::x4::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isNatList (x4::nil)) 
       (algebra.Alg.Term algebra.F.id_a__length (x11::nil))
    (* <mark(zeros),a__zeros> *)
   | DP_R_xml_0_60 :
    forall x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_zeros nil) 
       x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__zeros nil) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U11(X1_,X2_)),a__U11(mark(X1_),X2_)> *)
   | DP_R_xml_0_61 :
    forall x6 x11 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U11 (x6::x7::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U11 ((algebra.Alg.Term 
                  algebra.F.id_mark (x6::nil))::x7::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U11(X1_,X2_)),mark(X1_)> *)
   | DP_R_xml_0_62 :
    forall x6 x11 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U11 (x6::x7::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U12(X1_,X2_)),a__U12(mark(X1_),X2_)> *)
   | DP_R_xml_0_63 :
    forall x6 x11 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U12 (x6::x7::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U12 ((algebra.Alg.Term 
                  algebra.F.id_mark (x6::nil))::x7::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U12(X1_,X2_)),mark(X1_)> *)
   | DP_R_xml_0_64 :
    forall x6 x11 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U12 (x6::x7::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(isNatIListKind(X_)),a__isNatIListKind(X_)> *)
   | DP_R_xml_0_65 :
    forall x8 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_isNatIListKind (x8::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isNatIListKind (x8::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U13(X_)),a__U13(mark(X_))> *)
   | DP_R_xml_0_66 :
    forall x8 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U13 (x8::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U13 ((algebra.Alg.Term 
                  algebra.F.id_mark (x8::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U13(X_)),mark(X_)> *)
   | DP_R_xml_0_67 :
    forall x8 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U13 (x8::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(isNatList(X_)),a__isNatList(X_)> *)
   | DP_R_xml_0_68 :
    forall x8 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_isNatList (x8::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isNatList (x8::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U21(X1_,X2_)),a__U21(mark(X1_),X2_)> *)
   | DP_R_xml_0_69 :
    forall x6 x11 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U21 (x6::x7::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U21 ((algebra.Alg.Term 
                  algebra.F.id_mark (x6::nil))::x7::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U21(X1_,X2_)),mark(X1_)> *)
   | DP_R_xml_0_70 :
    forall x6 x11 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U21 (x6::x7::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U22(X1_,X2_)),a__U22(mark(X1_),X2_)> *)
   | DP_R_xml_0_71 :
    forall x6 x11 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U22 (x6::x7::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U22 ((algebra.Alg.Term 
                  algebra.F.id_mark (x6::nil))::x7::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U22(X1_,X2_)),mark(X1_)> *)
   | DP_R_xml_0_72 :
    forall x6 x11 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U22 (x6::x7::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(isNatKind(X_)),a__isNatKind(X_)> *)
   | DP_R_xml_0_73 :
    forall x8 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_isNatKind (x8::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isNatKind (x8::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U23(X_)),a__U23(mark(X_))> *)
   | DP_R_xml_0_74 :
    forall x8 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U23 (x8::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U23 ((algebra.Alg.Term 
                  algebra.F.id_mark (x8::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U23(X_)),mark(X_)> *)
   | DP_R_xml_0_75 :
    forall x8 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U23 (x8::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(isNat(X_)),a__isNat(X_)> *)
   | DP_R_xml_0_76 :
    forall x8 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_isNat (x8::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isNat (x8::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U31(X1_,X2_)),a__U31(mark(X1_),X2_)> *)
   | DP_R_xml_0_77 :
    forall x6 x11 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U31 (x6::x7::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U31 ((algebra.Alg.Term 
                  algebra.F.id_mark (x6::nil))::x7::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U31(X1_,X2_)),mark(X1_)> *)
   | DP_R_xml_0_78 :
    forall x6 x11 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U31 (x6::x7::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U32(X1_,X2_)),a__U32(mark(X1_),X2_)> *)
   | DP_R_xml_0_79 :
    forall x6 x11 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U32 (x6::x7::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U32 ((algebra.Alg.Term 
                  algebra.F.id_mark (x6::nil))::x7::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U32(X1_,X2_)),mark(X1_)> *)
   | DP_R_xml_0_80 :
    forall x6 x11 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U32 (x6::x7::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U33(X_)),a__U33(mark(X_))> *)
   | DP_R_xml_0_81 :
    forall x8 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U33 (x8::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U33 ((algebra.Alg.Term 
                  algebra.F.id_mark (x8::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U33(X_)),mark(X_)> *)
   | DP_R_xml_0_82 :
    forall x8 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U33 (x8::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U41(X1_,X2_,X3_)),a__U41(mark(X1_),X2_,X3_)> *)
   | DP_R_xml_0_83 :
    forall x6 x9 x11 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U41 (x6::x7::x9::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U41 ((algebra.Alg.Term 
                  algebra.F.id_mark (x6::nil))::x7::x9::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U41(X1_,X2_,X3_)),mark(X1_)> *)
   | DP_R_xml_0_84 :
    forall x6 x9 x11 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U41 (x6::x7::x9::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U42(X1_,X2_,X3_)),a__U42(mark(X1_),X2_,X3_)> *)
   | DP_R_xml_0_85 :
    forall x6 x9 x11 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U42 (x6::x7::x9::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U42 ((algebra.Alg.Term 
                  algebra.F.id_mark (x6::nil))::x7::x9::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U42(X1_,X2_,X3_)),mark(X1_)> *)
   | DP_R_xml_0_86 :
    forall x6 x9 x11 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U42 (x6::x7::x9::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U43(X1_,X2_,X3_)),a__U43(mark(X1_),X2_,X3_)> *)
   | DP_R_xml_0_87 :
    forall x6 x9 x11 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U43 (x6::x7::x9::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U43 ((algebra.Alg.Term 
                  algebra.F.id_mark (x6::nil))::x7::x9::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U43(X1_,X2_,X3_)),mark(X1_)> *)
   | DP_R_xml_0_88 :
    forall x6 x9 x11 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U43 (x6::x7::x9::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U44(X1_,X2_,X3_)),a__U44(mark(X1_),X2_,X3_)> *)
   | DP_R_xml_0_89 :
    forall x6 x9 x11 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U44 (x6::x7::x9::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U44 ((algebra.Alg.Term 
                  algebra.F.id_mark (x6::nil))::x7::x9::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U44(X1_,X2_,X3_)),mark(X1_)> *)
   | DP_R_xml_0_90 :
    forall x6 x9 x11 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U44 (x6::x7::x9::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U45(X1_,X2_)),a__U45(mark(X1_),X2_)> *)
   | DP_R_xml_0_91 :
    forall x6 x11 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U45 (x6::x7::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U45 ((algebra.Alg.Term 
                  algebra.F.id_mark (x6::nil))::x7::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U45(X1_,X2_)),mark(X1_)> *)
   | DP_R_xml_0_92 :
    forall x6 x11 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U45 (x6::x7::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U46(X_)),a__U46(mark(X_))> *)
   | DP_R_xml_0_93 :
    forall x8 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U46 (x8::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U46 ((algebra.Alg.Term 
                  algebra.F.id_mark (x8::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U46(X_)),mark(X_)> *)
   | DP_R_xml_0_94 :
    forall x8 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U46 (x8::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(isNatIList(X_)),a__isNatIList(X_)> *)
   | DP_R_xml_0_95 :
    forall x8 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_isNatIList (x8::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isNatIList (x8::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U51(X1_,X2_)),a__U51(mark(X1_),X2_)> *)
   | DP_R_xml_0_96 :
    forall x6 x11 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U51 (x6::x7::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U51 ((algebra.Alg.Term 
                  algebra.F.id_mark (x6::nil))::x7::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U51(X1_,X2_)),mark(X1_)> *)
   | DP_R_xml_0_97 :
    forall x6 x11 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U51 (x6::x7::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U52(X_)),a__U52(mark(X_))> *)
   | DP_R_xml_0_98 :
    forall x8 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U52 (x8::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U52 ((algebra.Alg.Term 
                  algebra.F.id_mark (x8::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U52(X_)),mark(X_)> *)
   | DP_R_xml_0_99 :
    forall x8 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U52 (x8::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U61(X_)),a__U61(mark(X_))> *)
   | DP_R_xml_0_100 :
    forall x8 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U61 (x8::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U61 ((algebra.Alg.Term 
                  algebra.F.id_mark (x8::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U61(X_)),mark(X_)> *)
   | DP_R_xml_0_101 :
    forall x8 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U61 (x8::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U71(X_)),a__U71(mark(X_))> *)
   | DP_R_xml_0_102 :
    forall x8 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U71 (x8::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U71 ((algebra.Alg.Term 
                  algebra.F.id_mark (x8::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U71(X_)),mark(X_)> *)
   | DP_R_xml_0_103 :
    forall x8 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U71 (x8::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
   
    (* <mark(U81(X1_,X2_,X3_)),a__U81(mark(X1_),X2_,X3_)> *)
   | DP_R_xml_0_104 :
    forall x6 x9 x11 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U81 (x6::x7::x9::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U81 ((algebra.Alg.Term 
                  algebra.F.id_mark (x6::nil))::x7::x9::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U81(X1_,X2_,X3_)),mark(X1_)> *)
   | DP_R_xml_0_105 :
    forall x6 x9 x11 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U81 (x6::x7::x9::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
   
    (* <mark(U82(X1_,X2_,X3_)),a__U82(mark(X1_),X2_,X3_)> *)
   | DP_R_xml_0_106 :
    forall x6 x9 x11 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U82 (x6::x7::x9::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U82 ((algebra.Alg.Term 
                  algebra.F.id_mark (x6::nil))::x7::x9::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U82(X1_,X2_,X3_)),mark(X1_)> *)
   | DP_R_xml_0_107 :
    forall x6 x9 x11 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U82 (x6::x7::x9::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
   
    (* <mark(U83(X1_,X2_,X3_)),a__U83(mark(X1_),X2_,X3_)> *)
   | DP_R_xml_0_108 :
    forall x6 x9 x11 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U83 (x6::x7::x9::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U83 ((algebra.Alg.Term 
                  algebra.F.id_mark (x6::nil))::x7::x9::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U83(X1_,X2_,X3_)),mark(X1_)> *)
   | DP_R_xml_0_109 :
    forall x6 x9 x11 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U83 (x6::x7::x9::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
   
    (* <mark(U84(X1_,X2_,X3_)),a__U84(mark(X1_),X2_,X3_)> *)
   | DP_R_xml_0_110 :
    forall x6 x9 x11 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U84 (x6::x7::x9::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U84 ((algebra.Alg.Term 
                  algebra.F.id_mark (x6::nil))::x7::x9::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U84(X1_,X2_,X3_)),mark(X1_)> *)
   | DP_R_xml_0_111 :
    forall x6 x9 x11 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U84 (x6::x7::x9::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U85(X1_,X2_)),a__U85(mark(X1_),X2_)> *)
   | DP_R_xml_0_112 :
    forall x6 x11 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U85 (x6::x7::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U85 ((algebra.Alg.Term 
                  algebra.F.id_mark (x6::nil))::x7::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U85(X1_,X2_)),mark(X1_)> *)
   | DP_R_xml_0_113 :
    forall x6 x11 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U85 (x6::x7::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U86(X_)),a__U86(mark(X_))> *)
   | DP_R_xml_0_114 :
    forall x8 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U86 (x8::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U86 ((algebra.Alg.Term 
                  algebra.F.id_mark (x8::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U86(X_)),mark(X_)> *)
   | DP_R_xml_0_115 :
    forall x8 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U86 (x8::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
   
    (* <mark(U91(X1_,X2_,X3_)),a__U91(mark(X1_),X2_,X3_)> *)
   | DP_R_xml_0_116 :
    forall x6 x9 x11 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U91 (x6::x7::x9::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U91 ((algebra.Alg.Term 
                  algebra.F.id_mark (x6::nil))::x7::x9::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U91(X1_,X2_,X3_)),mark(X1_)> *)
   | DP_R_xml_0_117 :
    forall x6 x9 x11 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U91 (x6::x7::x9::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
   
    (* <mark(U92(X1_,X2_,X3_)),a__U92(mark(X1_),X2_,X3_)> *)
   | DP_R_xml_0_118 :
    forall x6 x9 x11 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U92 (x6::x7::x9::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U92 ((algebra.Alg.Term 
                  algebra.F.id_mark (x6::nil))::x7::x9::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U92(X1_,X2_,X3_)),mark(X1_)> *)
   | DP_R_xml_0_119 :
    forall x6 x9 x11 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U92 (x6::x7::x9::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
   
    (* <mark(U93(X1_,X2_,X3_)),a__U93(mark(X1_),X2_,X3_)> *)
   | DP_R_xml_0_120 :
    forall x6 x9 x11 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U93 (x6::x7::x9::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U93 ((algebra.Alg.Term 
                  algebra.F.id_mark (x6::nil))::x7::x9::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U93(X1_,X2_,X3_)),mark(X1_)> *)
   | DP_R_xml_0_121 :
    forall x6 x9 x11 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U93 (x6::x7::x9::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U94(X1_,X2_)),a__U94(mark(X1_),X2_)> *)
   | DP_R_xml_0_122 :
    forall x6 x11 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U94 (x6::x7::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U94 ((algebra.Alg.Term 
                  algebra.F.id_mark (x6::nil))::x7::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U94(X1_,X2_)),mark(X1_)> *)
   | DP_R_xml_0_123 :
    forall x6 x11 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_U94 (x6::x7::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(length(X_)),a__length(mark(X_))> *)
   | DP_R_xml_0_124 :
    forall x8 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_length (x8::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__length ((algebra.Alg.Term 
                  algebra.F.id_mark (x8::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(length(X_)),mark(X_)> *)
   | DP_R_xml_0_125 :
    forall x8 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_length (x8::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(cons(X1_,X2_)),mark(X1_)> *)
   | DP_R_xml_0_126 :
    forall x6 x11 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_cons (x6::x7::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(s(X_)),mark(X_)> *)
   | DP_R_xml_0_127 :
    forall x8 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_s (x8::nil)) 
       x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
 .
 
 Module ddp := dp.MakeDP(algebra.EQT).
 
 
 Lemma R_xml_0_dp_step_spec :
  forall x y, 
   (ddp.dp_step R_xml_0_deep_rew.R_xml_0_rules x y) ->
    exists f,
      exists l1,
        exists l2,
          y = algebra.Alg.Term f l2/\ 
          (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step 
                                                            R_xml_0_deep_rew.R_xml_0_rules)
                                                           ) l1 l2)/\ 
          (ddp.dp R_xml_0_deep_rew.R_xml_0_rules x (algebra.Alg.Term f l1)).
 Proof.
   intros x y H.
   induction H.
   inversion H.
   subst.
   destruct t0.
   refine ((False_ind) _ _).
   refine (R_xml_0_deep_rew.R_xml_0_non_var H0).
   simpl in H|-*.
   exists a.
   exists ((List.map) (algebra.Alg.apply_subst sigma) l).
   exists ((List.map) (algebra.Alg.apply_subst sigma) l).
   repeat (constructor).
   assumption.
   exists f.
   exists l2.
   exists l1.
   constructor.
   constructor.
   constructor.
   constructor.
   rewrite  <- closure.rwr_list_trans_clos_one_step_list.
   assumption.
   assumption.
 Qed.
 
 
 Ltac included_dp_tac H :=
  injection H;clear H;intros;subst;
  repeat (match goal with 
  | H: closure.refl_trans_clos (closure.one_step_list _) (_::_) _ |- _=>           
  let x := fresh "x" in 
  let l := fresh "l" in 
  let h1 := fresh "h" in 
  let h2 := fresh "h" in 
  let h3 := fresh "h" in 
  destruct (@algebra.EQT_ext.one_step_list_star_decompose_cons _ _ _ _  H) as [x [l[h1[h2 h3]]]];clear H;subst
  | H: closure.refl_trans_clos (closure.one_step_list _) nil _ |- _ => 
  rewrite (@algebra.EQT_ext.one_step_list_star_decompose_nil _ _ H) in *;clear H
  end
  );simpl;
  econstructor eassumption
 .
 
 
 Ltac dp_concl_tac h2 h cont_tac 
  t :=
  match t with
    | False => let h' := fresh "a" in 
                (set (h':=t) in *;cont_tac h';
                  repeat (
                  let e := type of h in 
                   (match e with
                      | ?t => unfold t in h|-;
                               (case h;
                                [abstract (clear h;intros h;injection h;
                                            clear h;intros ;subst;
                                            included_dp_tac h2)|
                                clear h;intros h;clear t])
                      | ?t => unfold t in h|-;elim h
                      end
                    )
                  ))
    | or ?a ?b => let cont_tac 
                   h' := let h'' := fresh "a" in 
                          (set (h'':=or a h') in *;cont_tac h'') in 
                   (dp_concl_tac h2 h cont_tac b)
    end
  .
 
 
 Module WF_DP_R_xml_0.
  Inductive DP_R_xml_0_non_scc_1  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <mark(U86(X_)),a__U86(mark(X_))> *)
    | DP_R_xml_0_non_scc_1_0 :
     forall x8 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U86 (x8::nil)) x11) ->
       DP_R_xml_0_non_scc_1 (algebra.Alg.Term algebra.F.id_a__U86 
                             ((algebra.Alg.Term algebra.F.id_mark 
                             (x8::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_1 :
   forall x y, 
    (DP_R_xml_0_non_scc_1 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
      (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_2  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <mark(U71(X_)),a__U71(mark(X_))> *)
    | DP_R_xml_0_non_scc_2_0 :
     forall x8 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U71 (x8::nil)) x11) ->
       DP_R_xml_0_non_scc_2 (algebra.Alg.Term algebra.F.id_a__U71 
                             ((algebra.Alg.Term algebra.F.id_mark 
                             (x8::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_2 :
   forall x y, 
    (DP_R_xml_0_non_scc_2 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
      (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_3  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <mark(U61(X_)),a__U61(mark(X_))> *)
    | DP_R_xml_0_non_scc_3_0 :
     forall x8 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U61 (x8::nil)) x11) ->
       DP_R_xml_0_non_scc_3 (algebra.Alg.Term algebra.F.id_a__U61 
                             ((algebra.Alg.Term algebra.F.id_mark 
                             (x8::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_3 :
   forall x y, 
    (DP_R_xml_0_non_scc_3 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
      (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_4  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <mark(U52(X_)),a__U52(mark(X_))> *)
    | DP_R_xml_0_non_scc_4_0 :
     forall x8 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U52 (x8::nil)) x11) ->
       DP_R_xml_0_non_scc_4 (algebra.Alg.Term algebra.F.id_a__U52 
                             ((algebra.Alg.Term algebra.F.id_mark 
                             (x8::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_4 :
   forall x y, 
    (DP_R_xml_0_non_scc_4 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
      (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_5  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <mark(U46(X_)),a__U46(mark(X_))> *)
    | DP_R_xml_0_non_scc_5_0 :
     forall x8 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U46 (x8::nil)) x11) ->
       DP_R_xml_0_non_scc_5 (algebra.Alg.Term algebra.F.id_a__U46 
                             ((algebra.Alg.Term algebra.F.id_mark 
                             (x8::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_5 :
   forall x y, 
    (DP_R_xml_0_non_scc_5 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
      (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_6  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <mark(U33(X_)),a__U33(mark(X_))> *)
    | DP_R_xml_0_non_scc_6_0 :
     forall x8 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U33 (x8::nil)) x11) ->
       DP_R_xml_0_non_scc_6 (algebra.Alg.Term algebra.F.id_a__U33 
                             ((algebra.Alg.Term algebra.F.id_mark 
                             (x8::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_6 :
   forall x y, 
    (DP_R_xml_0_non_scc_6 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
      (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_7  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <mark(U23(X_)),a__U23(mark(X_))> *)
    | DP_R_xml_0_non_scc_7_0 :
     forall x8 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U23 (x8::nil)) x11) ->
       DP_R_xml_0_non_scc_7 (algebra.Alg.Term algebra.F.id_a__U23 
                             ((algebra.Alg.Term algebra.F.id_mark 
                             (x8::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_7 :
   forall x y, 
    (DP_R_xml_0_non_scc_7 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
      (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_8  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <mark(U13(X_)),a__U13(mark(X_))> *)
    | DP_R_xml_0_non_scc_8_0 :
     forall x8 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U13 (x8::nil)) x11) ->
       DP_R_xml_0_non_scc_8 (algebra.Alg.Term algebra.F.id_a__U13 
                             ((algebra.Alg.Term algebra.F.id_mark 
                             (x8::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_8 :
   forall x y, 
    (DP_R_xml_0_non_scc_8 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
      (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_9  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <mark(zeros),a__zeros> *)
    | DP_R_xml_0_non_scc_9_0 :
     forall x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_zeros nil) 
        x11) ->
       DP_R_xml_0_non_scc_9 (algebra.Alg.Term algebra.F.id_a__zeros nil) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_9 :
   forall x y, 
    (DP_R_xml_0_non_scc_9 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
      (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_10  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <a__isNatKind(s(V1_)),a__U71(a__isNatKind(V1_))> *)
    | DP_R_xml_0_non_scc_10_0 :
     forall x1 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_s (x1::nil)) 
        x11) ->
       DP_R_xml_0_non_scc_10 (algebra.Alg.Term algebra.F.id_a__U71 
                              ((algebra.Alg.Term algebra.F.id_a__isNatKind 
                              (x1::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_a__isNatKind (x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_10 :
   forall x y, 
    (DP_R_xml_0_non_scc_10 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
      (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_11  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <a__isNatKind(length(V1_)),a__U61(a__isNatIListKind(V1_))> *)
    | DP_R_xml_0_non_scc_11_0 :
     forall x1 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_length (x1::nil)) x11) ->
       DP_R_xml_0_non_scc_11 (algebra.Alg.Term algebra.F.id_a__U61 
                              ((algebra.Alg.Term 
                              algebra.F.id_a__isNatIListKind 
                              (x1::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_a__isNatKind (x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_11 :
   forall x y, 
    (DP_R_xml_0_non_scc_11 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
      (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_12  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <a__U85(tt,V2_),a__U86(a__isNatList(V2_))> *)
    | DP_R_xml_0_non_scc_12_0 :
     forall x12 x3 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_tt nil) 
        x12) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x3 x11) ->
        DP_R_xml_0_non_scc_12 (algebra.Alg.Term algebra.F.id_a__U86 
                               ((algebra.Alg.Term algebra.F.id_a__isNatList 
                               (x3::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U85 (x12::x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_12 :
   forall x y, 
    (DP_R_xml_0_non_scc_12 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
      (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_13  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <a__U51(tt,V2_),a__U52(a__isNatIListKind(V2_))> *)
    | DP_R_xml_0_non_scc_13_0 :
     forall x12 x3 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_tt nil) 
        x12) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x3 x11) ->
        DP_R_xml_0_non_scc_13 (algebra.Alg.Term algebra.F.id_a__U52 
                               ((algebra.Alg.Term 
                               algebra.F.id_a__isNatIListKind 
                               (x3::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U51 (x12::x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_13 :
   forall x y, 
    (DP_R_xml_0_non_scc_13 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
      (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )).
  Qed.
  
  
  Inductive DP_R_xml_0_scc_14  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <a__U51(tt,V2_),a__isNatIListKind(V2_)> *)
    | DP_R_xml_0_scc_14_0 :
     forall x12 x3 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_tt nil) 
        x12) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x3 x11) ->
        DP_R_xml_0_scc_14 (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
                           (x3::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U51 (x12::x11::nil))
    
     (* <a__isNatIListKind(cons(V1_,V2_)),a__U51(a__isNatKind(V1_),V2_)> *)
    | DP_R_xml_0_scc_14_1 :
     forall x1 x3 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_cons (x1::x3::nil)) x11) ->
       DP_R_xml_0_scc_14 (algebra.Alg.Term algebra.F.id_a__U51 
                          ((algebra.Alg.Term algebra.F.id_a__isNatKind 
                          (x1::nil))::x3::nil)) 
        (algebra.Alg.Term algebra.F.id_a__isNatIListKind (x11::nil))
    
     (* <a__isNatIListKind(cons(V1_,V2_)),a__isNatKind(V1_)> *)
    | DP_R_xml_0_scc_14_2 :
     forall x1 x3 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_cons (x1::x3::nil)) x11) ->
       DP_R_xml_0_scc_14 (algebra.Alg.Term algebra.F.id_a__isNatKind 
                          (x1::nil)) 
        (algebra.Alg.Term algebra.F.id_a__isNatIListKind (x11::nil))
    
     (* <a__isNatKind(length(V1_)),a__isNatIListKind(V1_)> *)
    | DP_R_xml_0_scc_14_3 :
     forall x1 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_length (x1::nil)) x11) ->
       DP_R_xml_0_scc_14 (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
                          (x1::nil)) 
        (algebra.Alg.Term algebra.F.id_a__isNatKind (x11::nil))
     (* <a__isNatKind(s(V1_)),a__isNatKind(V1_)> *)
    | DP_R_xml_0_scc_14_4 :
     forall x1 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_s (x1::nil)) 
        x11) ->
       DP_R_xml_0_scc_14 (algebra.Alg.Term algebra.F.id_a__isNatKind 
                          (x1::nil)) 
        (algebra.Alg.Term algebra.F.id_a__isNatKind (x11::nil))
  .
  
  
  Module WF_DP_R_xml_0_scc_14.
   Inductive DP_R_xml_0_scc_14_large  :
    algebra.Alg.term ->algebra.Alg.term ->Prop := 
      (* <a__U51(tt,V2_),a__isNatIListKind(V2_)> *)
     | DP_R_xml_0_scc_14_large_0 :
      forall x12 x3 x11, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  (algebra.Alg.Term algebra.F.id_tt nil) 
         x12) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x3 x11) ->
         DP_R_xml_0_scc_14_large (algebra.Alg.Term 
                                  algebra.F.id_a__isNatIListKind (x3::nil)) 
          (algebra.Alg.Term algebra.F.id_a__U51 (x12::x11::nil))
     
      (* <a__isNatIListKind(cons(V1_,V2_)),a__U51(a__isNatKind(V1_),V2_)> *)
     | DP_R_xml_0_scc_14_large_1 :
      forall x1 x3 x11, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_cons (x1::x3::nil)) x11) ->
        DP_R_xml_0_scc_14_large (algebra.Alg.Term algebra.F.id_a__U51 
                                 ((algebra.Alg.Term 
                                 algebra.F.id_a__isNatKind (x1::nil))::
                                 x3::nil)) 
         (algebra.Alg.Term algebra.F.id_a__isNatIListKind (x11::nil))
     
      (* <a__isNatKind(s(V1_)),a__isNatKind(V1_)> *)
     | DP_R_xml_0_scc_14_large_2 :
      forall x1 x11, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_s (x1::nil)) x11) ->
        DP_R_xml_0_scc_14_large (algebra.Alg.Term algebra.F.id_a__isNatKind 
                                 (x1::nil)) 
         (algebra.Alg.Term algebra.F.id_a__isNatKind (x11::nil))
   .
   
   
   Inductive DP_R_xml_0_scc_14_strict  :
    algebra.Alg.term ->algebra.Alg.term ->Prop := 
      (* <a__isNatIListKind(cons(V1_,V2_)),a__isNatKind(V1_)> *)
     | DP_R_xml_0_scc_14_strict_0 :
      forall x1 x3 x11, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_cons (x1::x3::nil)) x11) ->
        DP_R_xml_0_scc_14_strict (algebra.Alg.Term algebra.F.id_a__isNatKind 
                                  (x1::nil)) 
         (algebra.Alg.Term algebra.F.id_a__isNatIListKind (x11::nil))
     
      (* <a__isNatKind(length(V1_)),a__isNatIListKind(V1_)> *)
     | DP_R_xml_0_scc_14_strict_1 :
      forall x1 x11, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_length (x1::nil)) x11) ->
        DP_R_xml_0_scc_14_strict (algebra.Alg.Term 
                                  algebra.F.id_a__isNatIListKind (x1::nil)) 
         (algebra.Alg.Term algebra.F.id_a__isNatKind (x11::nil))
   .
   
   
   Module WF_DP_R_xml_0_scc_14_large.
    Inductive DP_R_xml_0_scc_14_large_scc_1  :
     algebra.Alg.term ->algebra.Alg.term ->Prop := 
       (* <a__isNatKind(s(V1_)),a__isNatKind(V1_)> *)
      | DP_R_xml_0_scc_14_large_scc_1_0 :
       forall x1 x11, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_s (x1::nil)) x11) ->
         DP_R_xml_0_scc_14_large_scc_1 (algebra.Alg.Term 
                                        algebra.F.id_a__isNatKind (x1::nil)) 
          (algebra.Alg.Term algebra.F.id_a__isNatKind (x11::nil))
    .
    
    
    Module WF_DP_R_xml_0_scc_14_large_scc_1.
     Open Scope Z_scope.
     
     Import ring_extention.
     
     Notation Local "a <= b" := (Zle a b).
     
     Notation Local "a < b" := (Zlt a b).
     
     Definition P_id_a__zeros  := 0.
     
     Definition P_id_U51 (x11:Z) (x12:Z) := 1* x11.
     
     Definition P_id_a__U84 (x11:Z) (x12:Z) (x13:Z) := 2* x13.
     
     Definition P_id_a__U32 (x11:Z) (x12:Z) := 1* x11.
     
     Definition P_id_isNatList (x11:Z) := 1* x11.
     
     Definition P_id_a__U13 (x11:Z) := 1.
     
     Definition P_id_U85 (x11:Z) (x12:Z) := 1* x12.
     
     Definition P_id_a__length (x11:Z) := 1* x11.
     
     Definition P_id_a__isNatIList (x11:Z) := 2 + 2* x11.
     
     Definition P_id_U33 (x11:Z) := 1.
     
     Definition P_id_a__U11 (x11:Z) (x12:Z) := 1.
     
     Definition P_id_U81 (x11:Z) (x12:Z) (x13:Z) := 2* x13.
     
     Definition P_id_a__U92 (x11:Z) (x12:Z) (x13:Z) := 1 + 2* x12.
     
     Definition P_id_a__U43 (x11:Z) (x12:Z) (x13:Z) := 1.
     
     Definition P_id_U23 (x11:Z) := 1* x11.
     
     Definition P_id_a__isNatKind (x11:Z) := 1.
     
     Definition P_id_U93 (x11:Z) (x12:Z) (x13:Z) := 1 + 2* x12.
     
     Definition P_id_U11 (x11:Z) (x12:Z) := 1.
     
     Definition P_id_a__U71 (x11:Z) := 1.
     
     Definition P_id_U44 (x11:Z) (x12:Z) (x13:Z) := 1.
     
     Definition P_id_0  := 0.
     
     Definition P_id_U61 (x11:Z) := 1* x11.
     
     Definition P_id_a__U86 (x11:Z) := 1* x11.
     
     Definition P_id_a__U41 (x11:Z) (x12:Z) (x13:Z) := 2* x11 + 3* x13.
     
     Definition P_id_U22 (x11:Z) (x12:Z) := 1.
     
     Definition P_id_a__U21 (x11:Z) (x12:Z) := 1* x11.
     
     Definition P_id_U91 (x11:Z) (x12:Z) (x13:Z) := 1* x11 + 2* x12.
     
     Definition P_id_length (x11:Z) := 1* x11.
     
     Definition P_id_a__U52 (x11:Z) := 1.
     
     Definition P_id_U42 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
     
     Definition P_id_a__U12 (x11:Z) (x12:Z) := 1.
     
     Definition P_id_U83 (x11:Z) (x12:Z) (x13:Z) := 2* x13.
     
     Definition P_id_a__U94 (x11:Z) (x12:Z) := 1 + 2* x12.
     
     Definition P_id_a__U45 (x11:Z) (x12:Z) := 1* x11.
     
     Definition P_id_U31 (x11:Z) (x12:Z) := 1* x11 + 2* x12.
     
     Definition P_id_a__isNat (x11:Z) := 1.
     
     Definition P_id_isNatIListKind (x11:Z) := 1.
     
     Definition P_id_a__U82 (x11:Z) (x12:Z) (x13:Z) := 2* x13.
     
     Definition P_id_U46 (x11:Z) := 1.
     
     Definition P_id_cons (x11:Z) (x12:Z) := 3* x12.
     
     Definition P_id_U52 (x11:Z) := 1.
     
     Definition P_id_a__U85 (x11:Z) (x12:Z) := 1* x12.
     
     Definition P_id_a__U33 (x11:Z) := 1.
     
     Definition P_id_U21 (x11:Z) (x12:Z) := 1* x11.
     
     Definition P_id_a__isNatList (x11:Z) := 1* x11.
     
     Definition P_id_U86 (x11:Z) := 1* x11.
     
     Definition P_id_mark (x11:Z) := 1* x11.
     
     Definition P_id_a__U51 (x11:Z) (x12:Z) := 1* x11.
     
     Definition P_id_U41 (x11:Z) (x12:Z) (x13:Z) := 2* x11 + 3* x13.
     
     Definition P_id_tt  := 1.
     
     Definition P_id_U82 (x11:Z) (x12:Z) (x13:Z) := 2* x13.
     
     Definition P_id_a__U93 (x11:Z) (x12:Z) (x13:Z) := 1 + 2* x12.
     
     Definition P_id_a__U44 (x11:Z) (x12:Z) (x13:Z) := 1.
     
     Definition P_id_isNat (x11:Z) := 1.
     
     Definition P_id_a__U23 (x11:Z) := 1* x11.
     
     Definition P_id_U94 (x11:Z) (x12:Z) := 1 + 2* x12.
     
     Definition P_id_U12 (x11:Z) (x12:Z) := 1.
     
     Definition P_id_a__U81 (x11:Z) (x12:Z) (x13:Z) := 2* x13.
     
     Definition P_id_U45 (x11:Z) (x12:Z) := 1* x11.
     
     Definition P_id_zeros  := 0.
     
     Definition P_id_U71 (x11:Z) := 1.
     
     Definition P_id_a__U91 (x11:Z) (x12:Z) (x13:Z) := 1* x11 + 2* x12.
     
     Definition P_id_a__U42 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
     
     Definition P_id_isNatKind (x11:Z) := 1.
     
     Definition P_id_a__U22 (x11:Z) (x12:Z) := 1.
     
     Definition P_id_U92 (x11:Z) (x12:Z) (x13:Z) := 1 + 2* x12.
     
     Definition P_id_nil  := 1.
     
     Definition P_id_a__U61 (x11:Z) := 1* x11.
     
     Definition P_id_U43 (x11:Z) (x12:Z) (x13:Z) := 1.
     
     Definition P_id_a__isNatIListKind (x11:Z) := 1.
     
     Definition P_id_U84 (x11:Z) (x12:Z) (x13:Z) := 2* x13.
     
     Definition P_id_s (x11:Z) := 1 + 2* x11.
     
     Definition P_id_a__U46 (x11:Z) := 1.
     
     Definition P_id_U32 (x11:Z) (x12:Z) := 1* x11.
     
     Definition P_id_a__U31 (x11:Z) (x12:Z) := 1* x11 + 2* x12.
     
     Definition P_id_U13 (x11:Z) := 1.
     
     Definition P_id_a__U83 (x11:Z) (x12:Z) (x13:Z) := 2* x13.
     
     Definition P_id_isNatIList (x11:Z) := 2 + 2* x11.
     
     Lemma P_id_U51_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U51 x12 x14 <= P_id_U51 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U84_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U84 x12 x14 x16 <= P_id_a__U84 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U32_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U32 x12 x14 <= P_id_a__U32 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNatList_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_isNatList x12 <= P_id_isNatList x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U13_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U13 x12 <= P_id_a__U13 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U85_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U85 x12 x14 <= P_id_U85 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__length_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__length x12 <= P_id_a__length x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNatIList_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__isNatIList x12 <= P_id_a__isNatIList x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U33_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U33 x12 <= P_id_U33 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U11_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U11 x12 x14 <= P_id_a__U11 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U81_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U81 x12 x14 x16 <= P_id_U81 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U92_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U92 x12 x14 x16 <= P_id_a__U92 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U43_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U43 x12 x14 x16 <= P_id_a__U43 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U23_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U23 x12 <= P_id_U23 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNatKind_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__isNatKind x12 <= P_id_a__isNatKind x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U93_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U93 x12 x14 x16 <= P_id_U93 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U11_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U11 x12 x14 <= P_id_U11 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U71_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U71 x12 <= P_id_a__U71 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U44_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U44 x12 x14 x16 <= P_id_U44 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U61_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U61 x12 <= P_id_U61 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U86_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U86 x12 <= P_id_a__U86 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U41_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U41 x12 x14 x16 <= P_id_a__U41 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U22_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U22 x12 x14 <= P_id_U22 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U21_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U21 x12 x14 <= P_id_a__U21 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U91_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U91 x12 x14 x16 <= P_id_U91 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_length_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_length x12 <= P_id_length x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U52_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U52 x12 <= P_id_a__U52 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U42_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U42 x12 x14 x16 <= P_id_U42 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U12_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U12 x12 x14 <= P_id_a__U12 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U83_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U83 x12 x14 x16 <= P_id_U83 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U94_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U94 x12 x14 <= P_id_a__U94 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U45_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U45 x12 x14 <= P_id_a__U45 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U31_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U31 x12 x14 <= P_id_U31 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNat_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__isNat x12 <= P_id_a__isNat x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNatIListKind_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_isNatIListKind x12 <= P_id_isNatIListKind x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U82_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U82 x12 x14 x16 <= P_id_a__U82 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U46_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U46 x12 <= P_id_U46 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_cons_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_cons x12 x14 <= P_id_cons x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U52_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U52 x12 <= P_id_U52 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U85_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U85 x12 x14 <= P_id_a__U85 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U33_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U33 x12 <= P_id_a__U33 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U21_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U21 x12 x14 <= P_id_U21 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNatList_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__isNatList x12 <= P_id_a__isNatList x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U86_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U86 x12 <= P_id_U86 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_mark_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_mark x12 <= P_id_mark x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U51_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U51 x12 x14 <= P_id_a__U51 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U41_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U41 x12 x14 x16 <= P_id_U41 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U82_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U82 x12 x14 x16 <= P_id_U82 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U93_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U93 x12 x14 x16 <= P_id_a__U93 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U44_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U44 x12 x14 x16 <= P_id_a__U44 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNat_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_isNat x12 <= P_id_isNat x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U23_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U23 x12 <= P_id_a__U23 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U94_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U94 x12 x14 <= P_id_U94 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U12_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U12 x12 x14 <= P_id_U12 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U81_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U81 x12 x14 x16 <= P_id_a__U81 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U45_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U45 x12 x14 <= P_id_U45 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U71_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U71 x12 <= P_id_U71 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U91_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U91 x12 x14 x16 <= P_id_a__U91 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U42_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U42 x12 x14 x16 <= P_id_a__U42 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNatKind_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_isNatKind x12 <= P_id_isNatKind x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U22_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U22 x12 x14 <= P_id_a__U22 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U92_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U92 x12 x14 x16 <= P_id_U92 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U61_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U61 x12 <= P_id_a__U61 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U43_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U43 x12 x14 x16 <= P_id_U43 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNatIListKind_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__isNatIListKind x12 <= P_id_a__isNatIListKind x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U84_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U84 x12 x14 x16 <= P_id_U84 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_s_monotonic :
      forall x12 x11, (0 <= x12)/\ (x12 <= x11) ->P_id_s x12 <= P_id_s x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U46_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U46 x12 <= P_id_a__U46 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U32_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U32 x12 x14 <= P_id_U32 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U31_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U31 x12 x14 <= P_id_a__U31 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U13_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U13 x12 <= P_id_U13 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U83_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U83 x12 x14 x16 <= P_id_a__U83 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNatIList_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_isNatIList x12 <= P_id_isNatIList x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__zeros_bounded : 0 <= P_id_a__zeros .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U51_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U51 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U84_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U84 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U32_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U32 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNatList_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_isNatList x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U13_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U13 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U85_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U85 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__length_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__length x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNatIList_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__isNatIList x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U33_bounded : forall x11, (0 <= x11) ->0 <= P_id_U33 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U11_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U11 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U81_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U81 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U92_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U92 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U43_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U43 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U23_bounded : forall x11, (0 <= x11) ->0 <= P_id_U23 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNatKind_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__isNatKind x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U93_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U93 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U11_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U11 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U71_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U71 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U44_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U44 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_0_bounded : 0 <= P_id_0 .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U61_bounded : forall x11, (0 <= x11) ->0 <= P_id_U61 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U86_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U86 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U41_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U41 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U22_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U22 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U21_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U21 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U91_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U91 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_length_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_length x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U52_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U52 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U42_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U42 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U12_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U12 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U83_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U83 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U94_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U94 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U45_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U45 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U31_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U31 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNat_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__isNat x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNatIListKind_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_isNatIListKind x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U82_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U82 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U46_bounded : forall x11, (0 <= x11) ->0 <= P_id_U46 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_cons_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_cons x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U52_bounded : forall x11, (0 <= x11) ->0 <= P_id_U52 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U85_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U85 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U33_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U33 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U21_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U21 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNatList_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__isNatList x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U86_bounded : forall x11, (0 <= x11) ->0 <= P_id_U86 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_mark_bounded : forall x11, (0 <= x11) ->0 <= P_id_mark x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U51_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U51 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U41_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U41 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_tt_bounded : 0 <= P_id_tt .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U82_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U82 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U93_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U93 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U44_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U44 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNat_bounded : forall x11, (0 <= x11) ->0 <= P_id_isNat x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U23_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U23 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U94_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U94 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U12_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U12 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U81_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U81 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U45_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U45 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_zeros_bounded : 0 <= P_id_zeros .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U71_bounded : forall x11, (0 <= x11) ->0 <= P_id_U71 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U91_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U91 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U42_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U42 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNatKind_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_isNatKind x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U22_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U22 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U92_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U92 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_nil_bounded : 0 <= P_id_nil .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U61_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U61 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U43_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U43 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNatIListKind_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__isNatIListKind x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U84_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U84 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_s_bounded : forall x11, (0 <= x11) ->0 <= P_id_s x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U46_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U46 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U32_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U32 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U31_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U31 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U13_bounded : forall x11, (0 <= x11) ->0 <= P_id_U13 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U83_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U83 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNatIList_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_isNatIList x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Definition measure  := 
       InterpZ.measure 0 P_id_a__zeros P_id_U51 P_id_a__U84 P_id_a__U32 
        P_id_isNatList P_id_a__U13 P_id_U85 P_id_a__length 
        P_id_a__isNatIList P_id_U33 P_id_a__U11 P_id_U81 P_id_a__U92 
        P_id_a__U43 P_id_U23 P_id_a__isNatKind P_id_U93 P_id_U11 P_id_a__U71 
        P_id_U44 P_id_0 P_id_U61 P_id_a__U86 P_id_a__U41 P_id_U22 
        P_id_a__U21 P_id_U91 P_id_length P_id_a__U52 P_id_U42 P_id_a__U12 
        P_id_U83 P_id_a__U94 P_id_a__U45 P_id_U31 P_id_a__isNat 
        P_id_isNatIListKind P_id_a__U82 P_id_U46 P_id_cons P_id_U52 
        P_id_a__U85 P_id_a__U33 P_id_U21 P_id_a__isNatList P_id_U86 
        P_id_mark P_id_a__U51 P_id_U41 P_id_tt P_id_U82 P_id_a__U93 
        P_id_a__U44 P_id_isNat P_id_a__U23 P_id_U94 P_id_U12 P_id_a__U81 
        P_id_U45 P_id_zeros P_id_U71 P_id_a__U91 P_id_a__U42 P_id_isNatKind 
        P_id_a__U22 P_id_U92 P_id_nil P_id_a__U61 P_id_U43 
        P_id_a__isNatIListKind P_id_U84 P_id_s P_id_a__U46 P_id_U32 
        P_id_a__U31 P_id_U13 P_id_a__U83 P_id_isNatIList.
     
     Lemma measure_equation :
      forall t, 
       measure t = match t with
                     | (algebra.Alg.Term algebra.F.id_a__zeros nil) =>
                      P_id_a__zeros 
                     | (algebra.Alg.Term algebra.F.id_U51 (x12::x11::nil)) =>
                      P_id_U51 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U84 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U84 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U32 (x12::x11::nil)) =>
                      P_id_a__U32 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_isNatList (x11::nil)) =>
                      P_id_isNatList (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U13 (x11::nil)) =>
                      P_id_a__U13 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U85 (x12::x11::nil)) =>
                      P_id_U85 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__length (x11::nil)) =>
                      P_id_a__length (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__isNatIList 
                        (x11::nil)) =>
                      P_id_a__isNatIList (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U33 (x11::nil)) =>
                      P_id_U33 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U11 (x12::x11::nil)) =>
                      P_id_a__U11 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U81 (x13::x12::
                        x11::nil)) =>
                      P_id_U81 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U92 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U92 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U43 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U43 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U23 (x11::nil)) =>
                      P_id_U23 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__isNatKind 
                        (x11::nil)) =>
                      P_id_a__isNatKind (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U93 (x13::x12::
                        x11::nil)) =>
                      P_id_U93 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U11 (x12::x11::nil)) =>
                      P_id_U11 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U71 (x11::nil)) =>
                      P_id_a__U71 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U44 (x13::x12::
                        x11::nil)) =>
                      P_id_U44 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_0 nil) => P_id_0 
                     | (algebra.Alg.Term algebra.F.id_U61 (x11::nil)) =>
                      P_id_U61 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U86 (x11::nil)) =>
                      P_id_a__U86 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U41 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U41 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U22 (x12::x11::nil)) =>
                      P_id_U22 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U21 (x12::x11::nil)) =>
                      P_id_a__U21 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U91 (x13::x12::
                        x11::nil)) =>
                      P_id_U91 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_length (x11::nil)) =>
                      P_id_length (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U52 (x11::nil)) =>
                      P_id_a__U52 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U42 (x13::x12::
                        x11::nil)) =>
                      P_id_U42 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U12 (x12::x11::nil)) =>
                      P_id_a__U12 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U83 (x13::x12::
                        x11::nil)) =>
                      P_id_U83 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U94 (x12::x11::nil)) =>
                      P_id_a__U94 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U45 (x12::x11::nil)) =>
                      P_id_a__U45 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U31 (x12::x11::nil)) =>
                      P_id_U31 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__isNat (x11::nil)) =>
                      P_id_a__isNat (measure x11)
                     | (algebra.Alg.Term algebra.F.id_isNatIListKind 
                        (x11::nil)) =>
                      P_id_isNatIListKind (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U82 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U82 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U46 (x11::nil)) =>
                      P_id_U46 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_cons (x12::x11::nil)) =>
                      P_id_cons (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U52 (x11::nil)) =>
                      P_id_U52 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U85 (x12::x11::nil)) =>
                      P_id_a__U85 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U33 (x11::nil)) =>
                      P_id_a__U33 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U21 (x12::x11::nil)) =>
                      P_id_U21 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__isNatList 
                        (x11::nil)) =>
                      P_id_a__isNatList (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U86 (x11::nil)) =>
                      P_id_U86 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_mark (x11::nil)) =>
                      P_id_mark (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U51 (x12::x11::nil)) =>
                      P_id_a__U51 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U41 (x13::x12::
                        x11::nil)) =>
                      P_id_U41 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt 
                     | (algebra.Alg.Term algebra.F.id_U82 (x13::x12::
                        x11::nil)) =>
                      P_id_U82 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U93 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U93 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U44 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U44 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_isNat (x11::nil)) =>
                      P_id_isNat (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U23 (x11::nil)) =>
                      P_id_a__U23 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U94 (x12::x11::nil)) =>
                      P_id_U94 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U12 (x12::x11::nil)) =>
                      P_id_U12 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U81 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U81 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U45 (x12::x11::nil)) =>
                      P_id_U45 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_zeros nil) =>
                      P_id_zeros 
                     | (algebra.Alg.Term algebra.F.id_U71 (x11::nil)) =>
                      P_id_U71 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U91 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U91 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U42 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U42 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_isNatKind (x11::nil)) =>
                      P_id_isNatKind (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U22 (x12::x11::nil)) =>
                      P_id_a__U22 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U92 (x13::x12::
                        x11::nil)) =>
                      P_id_U92 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil 
                     | (algebra.Alg.Term algebra.F.id_a__U61 (x11::nil)) =>
                      P_id_a__U61 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U43 (x13::x12::
                        x11::nil)) =>
                      P_id_U43 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
                        (x11::nil)) =>
                      P_id_a__isNatIListKind (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U84 (x13::x12::
                        x11::nil)) =>
                      P_id_U84 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_s (x11::nil)) =>
                      P_id_s (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U46 (x11::nil)) =>
                      P_id_a__U46 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U32 (x12::x11::nil)) =>
                      P_id_U32 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U31 (x12::x11::nil)) =>
                      P_id_a__U31 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U13 (x11::nil)) =>
                      P_id_U13 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U83 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U83 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_isNatIList (x11::nil)) =>
                      P_id_isNatIList (measure x11)
                     | _ => 0
                     end.
     Proof.
       intros t;case t;intros ;apply refl_equal.
     Qed.
     
     Lemma measure_bounded : forall t, 0 <= measure t.
     Proof.
       unfold measure in |-*.
       
       apply InterpZ.measure_bounded;
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Ltac generate_pos_hyp  :=
      match goal with
        | H:context [measure ?x] |- _ =>
         let v := fresh "v" in 
          (let H := fresh "h" in 
            (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
              clearbody H;clearbody v))
        |  |- context [measure ?x] =>
         let v := fresh "v" in 
          (let H := fresh "h" in 
            (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
              clearbody H;clearbody v))
        end
      .
     
     Lemma rules_monotonic :
      forall l r, 
       (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) ->
        measure r <= measure l.
     Proof.
       intros l r H.
       fold measure in |-*.
       
       inversion H;clear H;subst;inversion H0;clear H0;subst;
        simpl algebra.EQT.T.apply_subst in |-*;
        repeat (
        match goal with
          |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
           rewrite (measure_equation (algebra.Alg.Term f t))
          end
        );repeat (generate_pos_hyp );
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma measure_star_monotonic :
      forall l r, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  r l) ->measure r <= measure l.
     Proof.
       unfold measure in *.
       apply InterpZ.measure_star_monotonic.
       intros ;apply P_id_U51_monotonic;assumption.
       intros ;apply P_id_a__U84_monotonic;assumption.
       intros ;apply P_id_a__U32_monotonic;assumption.
       intros ;apply P_id_isNatList_monotonic;assumption.
       intros ;apply P_id_a__U13_monotonic;assumption.
       intros ;apply P_id_U85_monotonic;assumption.
       intros ;apply P_id_a__length_monotonic;assumption.
       intros ;apply P_id_a__isNatIList_monotonic;assumption.
       intros ;apply P_id_U33_monotonic;assumption.
       intros ;apply P_id_a__U11_monotonic;assumption.
       intros ;apply P_id_U81_monotonic;assumption.
       intros ;apply P_id_a__U92_monotonic;assumption.
       intros ;apply P_id_a__U43_monotonic;assumption.
       intros ;apply P_id_U23_monotonic;assumption.
       intros ;apply P_id_a__isNatKind_monotonic;assumption.
       intros ;apply P_id_U93_monotonic;assumption.
       intros ;apply P_id_U11_monotonic;assumption.
       intros ;apply P_id_a__U71_monotonic;assumption.
       intros ;apply P_id_U44_monotonic;assumption.
       intros ;apply P_id_U61_monotonic;assumption.
       intros ;apply P_id_a__U86_monotonic;assumption.
       intros ;apply P_id_a__U41_monotonic;assumption.
       intros ;apply P_id_U22_monotonic;assumption.
       intros ;apply P_id_a__U21_monotonic;assumption.
       intros ;apply P_id_U91_monotonic;assumption.
       intros ;apply P_id_length_monotonic;assumption.
       intros ;apply P_id_a__U52_monotonic;assumption.
       intros ;apply P_id_U42_monotonic;assumption.
       intros ;apply P_id_a__U12_monotonic;assumption.
       intros ;apply P_id_U83_monotonic;assumption.
       intros ;apply P_id_a__U94_monotonic;assumption.
       intros ;apply P_id_a__U45_monotonic;assumption.
       intros ;apply P_id_U31_monotonic;assumption.
       intros ;apply P_id_a__isNat_monotonic;assumption.
       intros ;apply P_id_isNatIListKind_monotonic;assumption.
       intros ;apply P_id_a__U82_monotonic;assumption.
       intros ;apply P_id_U46_monotonic;assumption.
       intros ;apply P_id_cons_monotonic;assumption.
       intros ;apply P_id_U52_monotonic;assumption.
       intros ;apply P_id_a__U85_monotonic;assumption.
       intros ;apply P_id_a__U33_monotonic;assumption.
       intros ;apply P_id_U21_monotonic;assumption.
       intros ;apply P_id_a__isNatList_monotonic;assumption.
       intros ;apply P_id_U86_monotonic;assumption.
       intros ;apply P_id_mark_monotonic;assumption.
       intros ;apply P_id_a__U51_monotonic;assumption.
       intros ;apply P_id_U41_monotonic;assumption.
       intros ;apply P_id_U82_monotonic;assumption.
       intros ;apply P_id_a__U93_monotonic;assumption.
       intros ;apply P_id_a__U44_monotonic;assumption.
       intros ;apply P_id_isNat_monotonic;assumption.
       intros ;apply P_id_a__U23_monotonic;assumption.
       intros ;apply P_id_U94_monotonic;assumption.
       intros ;apply P_id_U12_monotonic;assumption.
       intros ;apply P_id_a__U81_monotonic;assumption.
       intros ;apply P_id_U45_monotonic;assumption.
       intros ;apply P_id_U71_monotonic;assumption.
       intros ;apply P_id_a__U91_monotonic;assumption.
       intros ;apply P_id_a__U42_monotonic;assumption.
       intros ;apply P_id_isNatKind_monotonic;assumption.
       intros ;apply P_id_a__U22_monotonic;assumption.
       intros ;apply P_id_U92_monotonic;assumption.
       intros ;apply P_id_a__U61_monotonic;assumption.
       intros ;apply P_id_U43_monotonic;assumption.
       intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
       intros ;apply P_id_U84_monotonic;assumption.
       intros ;apply P_id_s_monotonic;assumption.
       intros ;apply P_id_a__U46_monotonic;assumption.
       intros ;apply P_id_U32_monotonic;assumption.
       intros ;apply P_id_a__U31_monotonic;assumption.
       intros ;apply P_id_U13_monotonic;assumption.
       intros ;apply P_id_a__U83_monotonic;assumption.
       intros ;apply P_id_isNatIList_monotonic;assumption.
       intros ;apply P_id_a__zeros_bounded;assumption.
       intros ;apply P_id_U51_bounded;assumption.
       intros ;apply P_id_a__U84_bounded;assumption.
       intros ;apply P_id_a__U32_bounded;assumption.
       intros ;apply P_id_isNatList_bounded;assumption.
       intros ;apply P_id_a__U13_bounded;assumption.
       intros ;apply P_id_U85_bounded;assumption.
       intros ;apply P_id_a__length_bounded;assumption.
       intros ;apply P_id_a__isNatIList_bounded;assumption.
       intros ;apply P_id_U33_bounded;assumption.
       intros ;apply P_id_a__U11_bounded;assumption.
       intros ;apply P_id_U81_bounded;assumption.
       intros ;apply P_id_a__U92_bounded;assumption.
       intros ;apply P_id_a__U43_bounded;assumption.
       intros ;apply P_id_U23_bounded;assumption.
       intros ;apply P_id_a__isNatKind_bounded;assumption.
       intros ;apply P_id_U93_bounded;assumption.
       intros ;apply P_id_U11_bounded;assumption.
       intros ;apply P_id_a__U71_bounded;assumption.
       intros ;apply P_id_U44_bounded;assumption.
       intros ;apply P_id_0_bounded;assumption.
       intros ;apply P_id_U61_bounded;assumption.
       intros ;apply P_id_a__U86_bounded;assumption.
       intros ;apply P_id_a__U41_bounded;assumption.
       intros ;apply P_id_U22_bounded;assumption.
       intros ;apply P_id_a__U21_bounded;assumption.
       intros ;apply P_id_U91_bounded;assumption.
       intros ;apply P_id_length_bounded;assumption.
       intros ;apply P_id_a__U52_bounded;assumption.
       intros ;apply P_id_U42_bounded;assumption.
       intros ;apply P_id_a__U12_bounded;assumption.
       intros ;apply P_id_U83_bounded;assumption.
       intros ;apply P_id_a__U94_bounded;assumption.
       intros ;apply P_id_a__U45_bounded;assumption.
       intros ;apply P_id_U31_bounded;assumption.
       intros ;apply P_id_a__isNat_bounded;assumption.
       intros ;apply P_id_isNatIListKind_bounded;assumption.
       intros ;apply P_id_a__U82_bounded;assumption.
       intros ;apply P_id_U46_bounded;assumption.
       intros ;apply P_id_cons_bounded;assumption.
       intros ;apply P_id_U52_bounded;assumption.
       intros ;apply P_id_a__U85_bounded;assumption.
       intros ;apply P_id_a__U33_bounded;assumption.
       intros ;apply P_id_U21_bounded;assumption.
       intros ;apply P_id_a__isNatList_bounded;assumption.
       intros ;apply P_id_U86_bounded;assumption.
       intros ;apply P_id_mark_bounded;assumption.
       intros ;apply P_id_a__U51_bounded;assumption.
       intros ;apply P_id_U41_bounded;assumption.
       intros ;apply P_id_tt_bounded;assumption.
       intros ;apply P_id_U82_bounded;assumption.
       intros ;apply P_id_a__U93_bounded;assumption.
       intros ;apply P_id_a__U44_bounded;assumption.
       intros ;apply P_id_isNat_bounded;assumption.
       intros ;apply P_id_a__U23_bounded;assumption.
       intros ;apply P_id_U94_bounded;assumption.
       intros ;apply P_id_U12_bounded;assumption.
       intros ;apply P_id_a__U81_bounded;assumption.
       intros ;apply P_id_U45_bounded;assumption.
       intros ;apply P_id_zeros_bounded;assumption.
       intros ;apply P_id_U71_bounded;assumption.
       intros ;apply P_id_a__U91_bounded;assumption.
       intros ;apply P_id_a__U42_bounded;assumption.
       intros ;apply P_id_isNatKind_bounded;assumption.
       intros ;apply P_id_a__U22_bounded;assumption.
       intros ;apply P_id_U92_bounded;assumption.
       intros ;apply P_id_nil_bounded;assumption.
       intros ;apply P_id_a__U61_bounded;assumption.
       intros ;apply P_id_U43_bounded;assumption.
       intros ;apply P_id_a__isNatIListKind_bounded;assumption.
       intros ;apply P_id_U84_bounded;assumption.
       intros ;apply P_id_s_bounded;assumption.
       intros ;apply P_id_a__U46_bounded;assumption.
       intros ;apply P_id_U32_bounded;assumption.
       intros ;apply P_id_a__U31_bounded;assumption.
       intros ;apply P_id_U13_bounded;assumption.
       intros ;apply P_id_a__U83_bounded;assumption.
       intros ;apply P_id_isNatIList_bounded;assumption.
       apply rules_monotonic.
     Qed.
     
     Definition P_id_A__U46 (x11:Z) := 0.
     
     Definition P_id_A__ISNATILISTKIND (x11:Z) := 0.
     
     Definition P_id_A__U61 (x11:Z) := 0.
     
     Definition P_id_A__U85 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__U31 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__U81 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__U22 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__U93 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__U42 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__U51 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__ISNATLIST (x11:Z) := 0.
     
     Definition P_id_A__ZEROS  := 0.
     
     Definition P_id_A__U91 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__U33 (x11:Z) := 0.
     
     Definition P_id_A__U83 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__U23 (x11:Z) := 0.
     
     Definition P_id_A__U11 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__LENGTH (x11:Z) := 0.
     
     Definition P_id_A__U44 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__ISNATILIST (x11:Z) := 0.
     
     Definition P_id_A__U13 (x11:Z) := 0.
     
     Definition P_id_A__U71 (x11:Z) := 0.
     
     Definition P_id_A__U86 (x11:Z) := 0.
     
     Definition P_id_A__U32 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__U82 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__ISNATKIND (x11:Z) := 2* x11.
     
     Definition P_id_A__U94 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__U43 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__U52 (x11:Z) := 0.
     
     Definition P_id_A__U21 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__U92 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__U41 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__U84 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__ISNAT (x11:Z) := 0.
     
     Definition P_id_A__U12 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_MARK (x11:Z) := 0.
     
     Definition P_id_A__U45 (x11:Z) (x12:Z) := 0.
     
     Lemma P_id_A__U46_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U46 x12 <= P_id_A__U46 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__ISNATILISTKIND_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__ISNATILISTKIND x12 <= P_id_A__ISNATILISTKIND x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U61_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U61 x12 <= P_id_A__U61 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U85_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U85 x12 x14 <= P_id_A__U85 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U31_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U31 x12 x14 <= P_id_A__U31 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U81_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U81 x12 x14 x16 <= P_id_A__U81 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U22_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U22 x12 x14 <= P_id_A__U22 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U93_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U93 x12 x14 x16 <= P_id_A__U93 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U42_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U42 x12 x14 x16 <= P_id_A__U42 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U51_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U51 x12 x14 <= P_id_A__U51 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__ISNATLIST_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__ISNATLIST x12 <= P_id_A__ISNATLIST x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U91_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U91 x12 x14 x16 <= P_id_A__U91 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U33_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U33 x12 <= P_id_A__U33 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U83_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U83 x12 x14 x16 <= P_id_A__U83 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U23_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U23 x12 <= P_id_A__U23 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U11_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U11 x12 x14 <= P_id_A__U11 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__LENGTH_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__LENGTH x12 <= P_id_A__LENGTH x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U44_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U44 x12 x14 x16 <= P_id_A__U44 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__ISNATILIST_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__ISNATILIST x12 <= P_id_A__ISNATILIST x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U13_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U13 x12 <= P_id_A__U13 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U71_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U71 x12 <= P_id_A__U71 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U86_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U86 x12 <= P_id_A__U86 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U32_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U32 x12 x14 <= P_id_A__U32 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U82_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U82 x12 x14 x16 <= P_id_A__U82 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__ISNATKIND_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__ISNATKIND x12 <= P_id_A__ISNATKIND x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U94_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U94 x12 x14 <= P_id_A__U94 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U43_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U43 x12 x14 x16 <= P_id_A__U43 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U52_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U52 x12 <= P_id_A__U52 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U21_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U21 x12 x14 <= P_id_A__U21 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U92_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U92 x12 x14 x16 <= P_id_A__U92 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U41_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U41 x12 x14 x16 <= P_id_A__U41 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U84_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U84 x12 x14 x16 <= P_id_A__U84 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__ISNAT_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__ISNAT x12 <= P_id_A__ISNAT x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U12_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U12 x12 x14 <= P_id_A__U12 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_MARK_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_MARK x12 <= P_id_MARK x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U45_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U45 x12 x14 <= P_id_A__U45 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Definition marked_measure  := 
       InterpZ.marked_measure 0 P_id_a__zeros P_id_U51 P_id_a__U84 
        P_id_a__U32 P_id_isNatList P_id_a__U13 P_id_U85 P_id_a__length 
        P_id_a__isNatIList P_id_U33 P_id_a__U11 P_id_U81 P_id_a__U92 
        P_id_a__U43 P_id_U23 P_id_a__isNatKind P_id_U93 P_id_U11 P_id_a__U71 
        P_id_U44 P_id_0 P_id_U61 P_id_a__U86 P_id_a__U41 P_id_U22 
        P_id_a__U21 P_id_U91 P_id_length P_id_a__U52 P_id_U42 P_id_a__U12 
        P_id_U83 P_id_a__U94 P_id_a__U45 P_id_U31 P_id_a__isNat 
        P_id_isNatIListKind P_id_a__U82 P_id_U46 P_id_cons P_id_U52 
        P_id_a__U85 P_id_a__U33 P_id_U21 P_id_a__isNatList P_id_U86 
        P_id_mark P_id_a__U51 P_id_U41 P_id_tt P_id_U82 P_id_a__U93 
        P_id_a__U44 P_id_isNat P_id_a__U23 P_id_U94 P_id_U12 P_id_a__U81 
        P_id_U45 P_id_zeros P_id_U71 P_id_a__U91 P_id_a__U42 P_id_isNatKind 
        P_id_a__U22 P_id_U92 P_id_nil P_id_a__U61 P_id_U43 
        P_id_a__isNatIListKind P_id_U84 P_id_s P_id_a__U46 P_id_U32 
        P_id_a__U31 P_id_U13 P_id_a__U83 P_id_isNatIList P_id_A__U46 
        P_id_A__ISNATILISTKIND P_id_A__U61 P_id_A__U85 P_id_A__U31 
        P_id_A__U81 P_id_A__U22 P_id_A__U93 P_id_A__U42 P_id_A__U51 
        P_id_A__ISNATLIST P_id_A__ZEROS P_id_A__U91 P_id_A__U33 P_id_A__U83 
        P_id_A__U23 P_id_A__U11 P_id_A__LENGTH P_id_A__U44 
        P_id_A__ISNATILIST P_id_A__U13 P_id_A__U71 P_id_A__U86 P_id_A__U32 
        P_id_A__U82 P_id_A__ISNATKIND P_id_A__U94 P_id_A__U43 P_id_A__U52 
        P_id_A__U21 P_id_A__U92 P_id_A__U41 P_id_A__U84 P_id_A__ISNAT 
        P_id_A__U12 P_id_MARK P_id_A__U45.
     
     Lemma marked_measure_equation :
      forall t, 
       marked_measure t = match t with
                            | (algebra.Alg.Term algebra.F.id_a__U46 
                               (x11::nil)) =>
                             P_id_A__U46 (measure x11)
                            | (algebra.Alg.Term 
                               algebra.F.id_a__isNatIListKind (x11::nil)) =>
                             P_id_A__ISNATILISTKIND (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U61 
                               (x11::nil)) =>
                             P_id_A__U61 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U85 (x12::
                               x11::nil)) =>
                             P_id_A__U85 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U31 (x12::
                               x11::nil)) =>
                             P_id_A__U31 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U81 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U81 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U22 (x12::
                               x11::nil)) =>
                             P_id_A__U22 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U93 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U93 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U42 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U42 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U51 (x12::
                               x11::nil)) =>
                             P_id_A__U51 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__isNatList 
                               (x11::nil)) =>
                             P_id_A__ISNATLIST (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__zeros nil) =>
                             P_id_A__ZEROS 
                            | (algebra.Alg.Term algebra.F.id_a__U91 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U91 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U33 
                               (x11::nil)) =>
                             P_id_A__U33 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U83 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U83 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U23 
                               (x11::nil)) =>
                             P_id_A__U23 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U11 (x12::
                               x11::nil)) =>
                             P_id_A__U11 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__length 
                               (x11::nil)) =>
                             P_id_A__LENGTH (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U44 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U44 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__isNatIList 
                               (x11::nil)) =>
                             P_id_A__ISNATILIST (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U13 
                               (x11::nil)) =>
                             P_id_A__U13 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U71 
                               (x11::nil)) =>
                             P_id_A__U71 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U86 
                               (x11::nil)) =>
                             P_id_A__U86 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U32 (x12::
                               x11::nil)) =>
                             P_id_A__U32 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U82 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U82 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__isNatKind 
                               (x11::nil)) =>
                             P_id_A__ISNATKIND (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U94 (x12::
                               x11::nil)) =>
                             P_id_A__U94 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U43 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U43 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U52 
                               (x11::nil)) =>
                             P_id_A__U52 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U21 (x12::
                               x11::nil)) =>
                             P_id_A__U21 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U92 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U92 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U41 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U41 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U84 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U84 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__isNat 
                               (x11::nil)) =>
                             P_id_A__ISNAT (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U12 (x12::
                               x11::nil)) =>
                             P_id_A__U12 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_mark (x11::nil)) =>
                             P_id_MARK (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U45 (x12::
                               x11::nil)) =>
                             P_id_A__U45 (measure x12) (measure x11)
                            | _ => measure t
                            end.
     Proof.
       reflexivity .
     Qed.
     
     Lemma marked_measure_star_monotonic :
      forall f l1 l2, 
       (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step 
                                                         R_xml_0_deep_rew.R_xml_0_rules)
                                                        ) l1 l2) ->
        marked_measure (algebra.Alg.Term f l1) <= marked_measure (algebra.Alg.Term 
                                                                   f 
                                                                   l2).
     Proof.
       unfold marked_measure in *.
       apply InterpZ.marked_measure_star_monotonic.
       intros ;apply P_id_U51_monotonic;assumption.
       intros ;apply P_id_a__U84_monotonic;assumption.
       intros ;apply P_id_a__U32_monotonic;assumption.
       intros ;apply P_id_isNatList_monotonic;assumption.
       intros ;apply P_id_a__U13_monotonic;assumption.
       intros ;apply P_id_U85_monotonic;assumption.
       intros ;apply P_id_a__length_monotonic;assumption.
       intros ;apply P_id_a__isNatIList_monotonic;assumption.
       intros ;apply P_id_U33_monotonic;assumption.
       intros ;apply P_id_a__U11_monotonic;assumption.
       intros ;apply P_id_U81_monotonic;assumption.
       intros ;apply P_id_a__U92_monotonic;assumption.
       intros ;apply P_id_a__U43_monotonic;assumption.
       intros ;apply P_id_U23_monotonic;assumption.
       intros ;apply P_id_a__isNatKind_monotonic;assumption.
       intros ;apply P_id_U93_monotonic;assumption.
       intros ;apply P_id_U11_monotonic;assumption.
       intros ;apply P_id_a__U71_monotonic;assumption.
       intros ;apply P_id_U44_monotonic;assumption.
       intros ;apply P_id_U61_monotonic;assumption.
       intros ;apply P_id_a__U86_monotonic;assumption.
       intros ;apply P_id_a__U41_monotonic;assumption.
       intros ;apply P_id_U22_monotonic;assumption.
       intros ;apply P_id_a__U21_monotonic;assumption.
       intros ;apply P_id_U91_monotonic;assumption.
       intros ;apply P_id_length_monotonic;assumption.
       intros ;apply P_id_a__U52_monotonic;assumption.
       intros ;apply P_id_U42_monotonic;assumption.
       intros ;apply P_id_a__U12_monotonic;assumption.
       intros ;apply P_id_U83_monotonic;assumption.
       intros ;apply P_id_a__U94_monotonic;assumption.
       intros ;apply P_id_a__U45_monotonic;assumption.
       intros ;apply P_id_U31_monotonic;assumption.
       intros ;apply P_id_a__isNat_monotonic;assumption.
       intros ;apply P_id_isNatIListKind_monotonic;assumption.
       intros ;apply P_id_a__U82_monotonic;assumption.
       intros ;apply P_id_U46_monotonic;assumption.
       intros ;apply P_id_cons_monotonic;assumption.
       intros ;apply P_id_U52_monotonic;assumption.
       intros ;apply P_id_a__U85_monotonic;assumption.
       intros ;apply P_id_a__U33_monotonic;assumption.
       intros ;apply P_id_U21_monotonic;assumption.
       intros ;apply P_id_a__isNatList_monotonic;assumption.
       intros ;apply P_id_U86_monotonic;assumption.
       intros ;apply P_id_mark_monotonic;assumption.
       intros ;apply P_id_a__U51_monotonic;assumption.
       intros ;apply P_id_U41_monotonic;assumption.
       intros ;apply P_id_U82_monotonic;assumption.
       intros ;apply P_id_a__U93_monotonic;assumption.
       intros ;apply P_id_a__U44_monotonic;assumption.
       intros ;apply P_id_isNat_monotonic;assumption.
       intros ;apply P_id_a__U23_monotonic;assumption.
       intros ;apply P_id_U94_monotonic;assumption.
       intros ;apply P_id_U12_monotonic;assumption.
       intros ;apply P_id_a__U81_monotonic;assumption.
       intros ;apply P_id_U45_monotonic;assumption.
       intros ;apply P_id_U71_monotonic;assumption.
       intros ;apply P_id_a__U91_monotonic;assumption.
       intros ;apply P_id_a__U42_monotonic;assumption.
       intros ;apply P_id_isNatKind_monotonic;assumption.
       intros ;apply P_id_a__U22_monotonic;assumption.
       intros ;apply P_id_U92_monotonic;assumption.
       intros ;apply P_id_a__U61_monotonic;assumption.
       intros ;apply P_id_U43_monotonic;assumption.
       intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
       intros ;apply P_id_U84_monotonic;assumption.
       intros ;apply P_id_s_monotonic;assumption.
       intros ;apply P_id_a__U46_monotonic;assumption.
       intros ;apply P_id_U32_monotonic;assumption.
       intros ;apply P_id_a__U31_monotonic;assumption.
       intros ;apply P_id_U13_monotonic;assumption.
       intros ;apply P_id_a__U83_monotonic;assumption.
       intros ;apply P_id_isNatIList_monotonic;assumption.
       intros ;apply P_id_a__zeros_bounded;assumption.
       intros ;apply P_id_U51_bounded;assumption.
       intros ;apply P_id_a__U84_bounded;assumption.
       intros ;apply P_id_a__U32_bounded;assumption.
       intros ;apply P_id_isNatList_bounded;assumption.
       intros ;apply P_id_a__U13_bounded;assumption.
       intros ;apply P_id_U85_bounded;assumption.
       intros ;apply P_id_a__length_bounded;assumption.
       intros ;apply P_id_a__isNatIList_bounded;assumption.
       intros ;apply P_id_U33_bounded;assumption.
       intros ;apply P_id_a__U11_bounded;assumption.
       intros ;apply P_id_U81_bounded;assumption.
       intros ;apply P_id_a__U92_bounded;assumption.
       intros ;apply P_id_a__U43_bounded;assumption.
       intros ;apply P_id_U23_bounded;assumption.
       intros ;apply P_id_a__isNatKind_bounded;assumption.
       intros ;apply P_id_U93_bounded;assumption.
       intros ;apply P_id_U11_bounded;assumption.
       intros ;apply P_id_a__U71_bounded;assumption.
       intros ;apply P_id_U44_bounded;assumption.
       intros ;apply P_id_0_bounded;assumption.
       intros ;apply P_id_U61_bounded;assumption.
       intros ;apply P_id_a__U86_bounded;assumption.
       intros ;apply P_id_a__U41_bounded;assumption.
       intros ;apply P_id_U22_bounded;assumption.
       intros ;apply P_id_a__U21_bounded;assumption.
       intros ;apply P_id_U91_bounded;assumption.
       intros ;apply P_id_length_bounded;assumption.
       intros ;apply P_id_a__U52_bounded;assumption.
       intros ;apply P_id_U42_bounded;assumption.
       intros ;apply P_id_a__U12_bounded;assumption.
       intros ;apply P_id_U83_bounded;assumption.
       intros ;apply P_id_a__U94_bounded;assumption.
       intros ;apply P_id_a__U45_bounded;assumption.
       intros ;apply P_id_U31_bounded;assumption.
       intros ;apply P_id_a__isNat_bounded;assumption.
       intros ;apply P_id_isNatIListKind_bounded;assumption.
       intros ;apply P_id_a__U82_bounded;assumption.
       intros ;apply P_id_U46_bounded;assumption.
       intros ;apply P_id_cons_bounded;assumption.
       intros ;apply P_id_U52_bounded;assumption.
       intros ;apply P_id_a__U85_bounded;assumption.
       intros ;apply P_id_a__U33_bounded;assumption.
       intros ;apply P_id_U21_bounded;assumption.
       intros ;apply P_id_a__isNatList_bounded;assumption.
       intros ;apply P_id_U86_bounded;assumption.
       intros ;apply P_id_mark_bounded;assumption.
       intros ;apply P_id_a__U51_bounded;assumption.
       intros ;apply P_id_U41_bounded;assumption.
       intros ;apply P_id_tt_bounded;assumption.
       intros ;apply P_id_U82_bounded;assumption.
       intros ;apply P_id_a__U93_bounded;assumption.
       intros ;apply P_id_a__U44_bounded;assumption.
       intros ;apply P_id_isNat_bounded;assumption.
       intros ;apply P_id_a__U23_bounded;assumption.
       intros ;apply P_id_U94_bounded;assumption.
       intros ;apply P_id_U12_bounded;assumption.
       intros ;apply P_id_a__U81_bounded;assumption.
       intros ;apply P_id_U45_bounded;assumption.
       intros ;apply P_id_zeros_bounded;assumption.
       intros ;apply P_id_U71_bounded;assumption.
       intros ;apply P_id_a__U91_bounded;assumption.
       intros ;apply P_id_a__U42_bounded;assumption.
       intros ;apply P_id_isNatKind_bounded;assumption.
       intros ;apply P_id_a__U22_bounded;assumption.
       intros ;apply P_id_U92_bounded;assumption.
       intros ;apply P_id_nil_bounded;assumption.
       intros ;apply P_id_a__U61_bounded;assumption.
       intros ;apply P_id_U43_bounded;assumption.
       intros ;apply P_id_a__isNatIListKind_bounded;assumption.
       intros ;apply P_id_U84_bounded;assumption.
       intros ;apply P_id_s_bounded;assumption.
       intros ;apply P_id_a__U46_bounded;assumption.
       intros ;apply P_id_U32_bounded;assumption.
       intros ;apply P_id_a__U31_bounded;assumption.
       intros ;apply P_id_U13_bounded;assumption.
       intros ;apply P_id_a__U83_bounded;assumption.
       intros ;apply P_id_isNatIList_bounded;assumption.
       apply rules_monotonic.
       intros ;apply P_id_A__U46_monotonic;assumption.
       intros ;apply P_id_A__ISNATILISTKIND_monotonic;assumption.
       intros ;apply P_id_A__U61_monotonic;assumption.
       intros ;apply P_id_A__U85_monotonic;assumption.
       intros ;apply P_id_A__U31_monotonic;assumption.
       intros ;apply P_id_A__U81_monotonic;assumption.
       intros ;apply P_id_A__U22_monotonic;assumption.
       intros ;apply P_id_A__U93_monotonic;assumption.
       intros ;apply P_id_A__U42_monotonic;assumption.
       intros ;apply P_id_A__U51_monotonic;assumption.
       intros ;apply P_id_A__ISNATLIST_monotonic;assumption.
       intros ;apply P_id_A__U91_monotonic;assumption.
       intros ;apply P_id_A__U33_monotonic;assumption.
       intros ;apply P_id_A__U83_monotonic;assumption.
       intros ;apply P_id_A__U23_monotonic;assumption.
       intros ;apply P_id_A__U11_monotonic;assumption.
       intros ;apply P_id_A__LENGTH_monotonic;assumption.
       intros ;apply P_id_A__U44_monotonic;assumption.
       intros ;apply P_id_A__ISNATILIST_monotonic;assumption.
       intros ;apply P_id_A__U13_monotonic;assumption.
       intros ;apply P_id_A__U71_monotonic;assumption.
       intros ;apply P_id_A__U86_monotonic;assumption.
       intros ;apply P_id_A__U32_monotonic;assumption.
       intros ;apply P_id_A__U82_monotonic;assumption.
       intros ;apply P_id_A__ISNATKIND_monotonic;assumption.
       intros ;apply P_id_A__U94_monotonic;assumption.
       intros ;apply P_id_A__U43_monotonic;assumption.
       intros ;apply P_id_A__U52_monotonic;assumption.
       intros ;apply P_id_A__U21_monotonic;assumption.
       intros ;apply P_id_A__U92_monotonic;assumption.
       intros ;apply P_id_A__U41_monotonic;assumption.
       intros ;apply P_id_A__U84_monotonic;assumption.
       intros ;apply P_id_A__ISNAT_monotonic;assumption.
       intros ;apply P_id_A__U12_monotonic;assumption.
       intros ;apply P_id_MARK_monotonic;assumption.
       intros ;apply P_id_A__U45_monotonic;assumption.
     Qed.
     
     Ltac rewrite_and_unfold  :=
      do 2 (rewrite marked_measure_equation);
       repeat (
       match goal with
         |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
          rewrite (measure_equation (algebra.Alg.Term f t))
         | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ =>
          rewrite (measure_equation (algebra.Alg.Term f t)) in H|-
         end
       ).
     
     
     Lemma wf :
      well_founded WF_DP_R_xml_0_scc_14_large.DP_R_xml_0_scc_14_large_scc_1.
     Proof.
       intros x.
       
       apply well_founded_ind with
         (R:=fun x y => (Zwf.Zwf 0) (marked_measure x) (marked_measure y)).
       apply Inverse_Image.wf_inverse_image with  (B:=Z).
       apply Zwf.Zwf_well_founded.
       clear x.
       intros x IHx.
       
       repeat (
       constructor;inversion 1;subst;
        full_prove_ineq algebra.Alg.Term 
        ltac:(algebra.Alg_ext.find_replacement ) 
        algebra.EQT_ext.one_step_list_refl_trans_clos marked_measure 
        marked_measure_star_monotonic (Zwf.Zwf 0) (interp.o_Z 0) 
        ltac:(fun _ => R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 ) 
        ltac:(fun _ => rewrite_and_unfold ) ltac:(fun _ => generate_pos_hyp )
         
        ltac:(fun _ => cbv beta iota zeta delta - [Zplus Zmult Zle Zlt] in * ;
                        try (constructor))
         IHx
       ).
     Qed.
    End WF_DP_R_xml_0_scc_14_large_scc_1.
    
    Definition wf_DP_R_xml_0_scc_14_large_scc_1  := 
      WF_DP_R_xml_0_scc_14_large_scc_1.wf.
    
    
    Lemma acc_DP_R_xml_0_scc_14_large_scc_1 :
     forall x y, 
      (DP_R_xml_0_scc_14_large_scc_1 x y) ->
       Acc WF_DP_R_xml_0_scc_14.DP_R_xml_0_scc_14_large x.
    Proof.
      intros x.
      pattern x.
      apply (@Acc_ind _ DP_R_xml_0_scc_14_large_scc_1).
      intros x' _ Hrec y h.
      
      inversion h;clear h;subst;
       constructor;intros _y _h;inversion _h;clear _h;subst;
        (eapply Hrec;econstructor eassumption)||
        ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
         (eapply Hrec;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))).
      apply wf_DP_R_xml_0_scc_14_large_scc_1.
    Qed.
    
    
    Inductive DP_R_xml_0_scc_14_large_scc_2  :
     algebra.Alg.term ->algebra.Alg.term ->Prop := 
       (* <a__isNatIListKind(cons(V1_,V2_)),a__U51(a__isNatKind(V1_),V2_)> *)
      | DP_R_xml_0_scc_14_large_scc_2_0 :
       forall x1 x3 x11, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_cons (x1::x3::nil)) x11) ->
         DP_R_xml_0_scc_14_large_scc_2 (algebra.Alg.Term algebra.F.id_a__U51 
                                        ((algebra.Alg.Term 
                                        algebra.F.id_a__isNatKind 
                                        (x1::nil))::x3::nil)) 
          (algebra.Alg.Term algebra.F.id_a__isNatIListKind (x11::nil))
      
       (* <a__U51(tt,V2_),a__isNatIListKind(V2_)> *)
      | DP_R_xml_0_scc_14_large_scc_2_1 :
       forall x12 x3 x11, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   (algebra.Alg.Term algebra.F.id_tt nil) 
          x12) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x3 x11) ->
          DP_R_xml_0_scc_14_large_scc_2 (algebra.Alg.Term 
                                         algebra.F.id_a__isNatIListKind 
                                         (x3::nil)) 
           (algebra.Alg.Term algebra.F.id_a__U51 (x12::x11::nil))
    .
    
    
    Module WF_DP_R_xml_0_scc_14_large_scc_2.
     Inductive DP_R_xml_0_scc_14_large_scc_2_large  :
      algebra.Alg.term ->algebra.Alg.term ->Prop := 
        (* <a__U51(tt,V2_),a__isNatIListKind(V2_)> *)
       | DP_R_xml_0_scc_14_large_scc_2_large_0 :
        forall x12 x3 x11, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    (algebra.Alg.Term algebra.F.id_tt nil) 
           x12) ->
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     x3 x11) ->
           DP_R_xml_0_scc_14_large_scc_2_large (algebra.Alg.Term 
                                                algebra.F.id_a__isNatIListKind 
                                                (x3::nil)) 
            (algebra.Alg.Term algebra.F.id_a__U51 (x12::x11::nil))
     .
     
     
     Inductive DP_R_xml_0_scc_14_large_scc_2_strict  :
      algebra.Alg.term ->algebra.Alg.term ->Prop := 
        (* <a__isNatIListKind(cons(V1_,V2_)),a__U51(a__isNatKind(V1_),V2_)> *)
       | DP_R_xml_0_scc_14_large_scc_2_strict_0 :
        forall x1 x3 x11, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    
           (algebra.Alg.Term algebra.F.id_cons (x1::x3::nil)) x11) ->
          DP_R_xml_0_scc_14_large_scc_2_strict (algebra.Alg.Term 
                                                algebra.F.id_a__U51 
                                                ((algebra.Alg.Term 
                                                algebra.F.id_a__isNatKind 
                                                (x1::nil))::x3::nil)) 
           (algebra.Alg.Term algebra.F.id_a__isNatIListKind (x11::nil))
     .
     
     
     Module WF_DP_R_xml_0_scc_14_large_scc_2_large.
      Inductive DP_R_xml_0_scc_14_large_scc_2_large_non_scc_1  :
       algebra.Alg.term ->algebra.Alg.term ->Prop := 
         (* <a__U51(tt,V2_),a__isNatIListKind(V2_)> *)
        | DP_R_xml_0_scc_14_large_scc_2_large_non_scc_1_0 :
         forall x12 x3 x11, 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     (algebra.Alg.Term algebra.F.id_tt nil) 
            x12) ->
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                      x3 x11) ->
            DP_R_xml_0_scc_14_large_scc_2_large_non_scc_1 (algebra.Alg.Term 
                                                           algebra.F.id_a__isNatIListKind 
                                                           (x3::nil)) 
             (algebra.Alg.Term algebra.F.id_a__U51 (x12::x11::nil))
      .
      
      
      Lemma acc_DP_R_xml_0_scc_14_large_scc_2_large_non_scc_1 :
       forall x y, 
        (DP_R_xml_0_scc_14_large_scc_2_large_non_scc_1 x y) ->
         Acc WF_DP_R_xml_0_scc_14_large_scc_2.DP_R_xml_0_scc_14_large_scc_2_large 
          x.
      Proof.
        intros x y h.
        
        inversion h;clear h;subst;
         constructor;intros _y _h;inversion _h;clear _h;subst;
          (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
          (eapply Hrec;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' )).
      Qed.
      
      
      Lemma wf :
       well_founded WF_DP_R_xml_0_scc_14_large_scc_2.DP_R_xml_0_scc_14_large_scc_2_large
        .
      Proof.
        constructor;intros _y _h;inversion _h;clear _h;subst;
         (eapply acc_DP_R_xml_0_scc_14_large_scc_2_large_non_scc_1;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
         ((eapply acc_DP_R_xml_0_scc_14_large_scc_2_large_non_scc_0;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
          ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||(fail))).
      Qed.
     End WF_DP_R_xml_0_scc_14_large_scc_2_large.
     
     Open Scope Z_scope.
     
     Import ring_extention.
     
     Notation Local "a <= b" := (Zle a b).
     
     Notation Local "a < b" := (Zlt a b).
     
     Definition P_id_a__zeros  := 3.
     
     Definition P_id_U51 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_a__U84 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_a__U32 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_isNatList (x11:Z) := 0.
     
     Definition P_id_a__U13 (x11:Z) := 0.
     
     Definition P_id_U85 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_a__length (x11:Z) := 0.
     
     Definition P_id_a__isNatIList (x11:Z) := 2.
     
     Definition P_id_U33 (x11:Z) := 0.
     
     Definition P_id_a__U11 (x11:Z) (x12:Z) := 2.
     
     Definition P_id_U81 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_a__U92 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_a__U43 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_U23 (x11:Z) := 0.
     
     Definition P_id_a__isNatKind (x11:Z) := 0.
     
     Definition P_id_U93 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_U11 (x11:Z) (x12:Z) := 1.
     
     Definition P_id_a__U71 (x11:Z) := 0.
     
     Definition P_id_U44 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_0  := 0.
     
     Definition P_id_U61 (x11:Z) := 0.
     
     Definition P_id_a__U86 (x11:Z) := 0.
     
     Definition P_id_a__U41 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_U22 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_a__U21 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_U91 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_length (x11:Z) := 0.
     
     Definition P_id_a__U52 (x11:Z) := 0.
     
     Definition P_id_U42 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_a__U12 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_U83 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_a__U94 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_a__U45 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_U31 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_a__isNat (x11:Z) := 2.
     
     Definition P_id_isNatIListKind (x11:Z) := 0.
     
     Definition P_id_a__U82 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_U46 (x11:Z) := 0.
     
     Definition P_id_cons (x11:Z) (x12:Z) := 1 + 1* x12.
     
     Definition P_id_U52 (x11:Z) := 0.
     
     Definition P_id_a__U85 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_a__U33 (x11:Z) := 0.
     
     Definition P_id_U21 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_a__isNatList (x11:Z) := 0.
     
     Definition P_id_U86 (x11:Z) := 0.
     
     Definition P_id_mark (x11:Z) := 2* x11.
     
     Definition P_id_a__U51 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_U41 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_tt  := 0.
     
     Definition P_id_U82 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_a__U93 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_a__U44 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_isNat (x11:Z) := 1.
     
     Definition P_id_a__U23 (x11:Z) := 0.
     
     Definition P_id_U94 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_U12 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_a__U81 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_U45 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_zeros  := 2.
     
     Definition P_id_U71 (x11:Z) := 0.
     
     Definition P_id_a__U91 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_a__U42 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_isNatKind (x11:Z) := 0.
     
     Definition P_id_a__U22 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_U92 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_nil  := 1.
     
     Definition P_id_a__U61 (x11:Z) := 0.
     
     Definition P_id_U43 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_a__isNatIListKind (x11:Z) := 0.
     
     Definition P_id_U84 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_s (x11:Z) := 0.
     
     Definition P_id_a__U46 (x11:Z) := 0.
     
     Definition P_id_U32 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_a__U31 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_U13 (x11:Z) := 0.
     
     Definition P_id_a__U83 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_isNatIList (x11:Z) := 2.
     
     Lemma P_id_U51_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U51 x12 x14 <= P_id_U51 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U84_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U84 x12 x14 x16 <= P_id_a__U84 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U32_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U32 x12 x14 <= P_id_a__U32 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNatList_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_isNatList x12 <= P_id_isNatList x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U13_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U13 x12 <= P_id_a__U13 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U85_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U85 x12 x14 <= P_id_U85 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__length_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__length x12 <= P_id_a__length x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNatIList_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__isNatIList x12 <= P_id_a__isNatIList x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U33_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U33 x12 <= P_id_U33 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U11_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U11 x12 x14 <= P_id_a__U11 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U81_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U81 x12 x14 x16 <= P_id_U81 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U92_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U92 x12 x14 x16 <= P_id_a__U92 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U43_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U43 x12 x14 x16 <= P_id_a__U43 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U23_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U23 x12 <= P_id_U23 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNatKind_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__isNatKind x12 <= P_id_a__isNatKind x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U93_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U93 x12 x14 x16 <= P_id_U93 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U11_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U11 x12 x14 <= P_id_U11 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U71_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U71 x12 <= P_id_a__U71 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U44_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U44 x12 x14 x16 <= P_id_U44 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U61_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U61 x12 <= P_id_U61 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U86_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U86 x12 <= P_id_a__U86 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U41_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U41 x12 x14 x16 <= P_id_a__U41 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U22_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U22 x12 x14 <= P_id_U22 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U21_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U21 x12 x14 <= P_id_a__U21 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U91_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U91 x12 x14 x16 <= P_id_U91 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_length_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_length x12 <= P_id_length x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U52_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U52 x12 <= P_id_a__U52 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U42_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U42 x12 x14 x16 <= P_id_U42 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U12_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U12 x12 x14 <= P_id_a__U12 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U83_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U83 x12 x14 x16 <= P_id_U83 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U94_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U94 x12 x14 <= P_id_a__U94 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U45_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U45 x12 x14 <= P_id_a__U45 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U31_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U31 x12 x14 <= P_id_U31 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNat_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__isNat x12 <= P_id_a__isNat x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNatIListKind_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_isNatIListKind x12 <= P_id_isNatIListKind x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U82_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U82 x12 x14 x16 <= P_id_a__U82 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U46_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U46 x12 <= P_id_U46 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_cons_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_cons x12 x14 <= P_id_cons x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U52_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U52 x12 <= P_id_U52 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U85_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U85 x12 x14 <= P_id_a__U85 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U33_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U33 x12 <= P_id_a__U33 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U21_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U21 x12 x14 <= P_id_U21 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNatList_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__isNatList x12 <= P_id_a__isNatList x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U86_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U86 x12 <= P_id_U86 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_mark_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_mark x12 <= P_id_mark x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U51_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U51 x12 x14 <= P_id_a__U51 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U41_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U41 x12 x14 x16 <= P_id_U41 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U82_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U82 x12 x14 x16 <= P_id_U82 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U93_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U93 x12 x14 x16 <= P_id_a__U93 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U44_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U44 x12 x14 x16 <= P_id_a__U44 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNat_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_isNat x12 <= P_id_isNat x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U23_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U23 x12 <= P_id_a__U23 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U94_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U94 x12 x14 <= P_id_U94 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U12_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U12 x12 x14 <= P_id_U12 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U81_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U81 x12 x14 x16 <= P_id_a__U81 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U45_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U45 x12 x14 <= P_id_U45 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U71_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U71 x12 <= P_id_U71 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U91_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U91 x12 x14 x16 <= P_id_a__U91 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U42_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U42 x12 x14 x16 <= P_id_a__U42 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNatKind_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_isNatKind x12 <= P_id_isNatKind x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U22_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U22 x12 x14 <= P_id_a__U22 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U92_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U92 x12 x14 x16 <= P_id_U92 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U61_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U61 x12 <= P_id_a__U61 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U43_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U43 x12 x14 x16 <= P_id_U43 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNatIListKind_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__isNatIListKind x12 <= P_id_a__isNatIListKind x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U84_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U84 x12 x14 x16 <= P_id_U84 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_s_monotonic :
      forall x12 x11, (0 <= x12)/\ (x12 <= x11) ->P_id_s x12 <= P_id_s x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U46_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U46 x12 <= P_id_a__U46 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U32_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U32 x12 x14 <= P_id_U32 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U31_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U31 x12 x14 <= P_id_a__U31 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U13_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U13 x12 <= P_id_U13 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U83_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U83 x12 x14 x16 <= P_id_a__U83 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNatIList_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_isNatIList x12 <= P_id_isNatIList x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__zeros_bounded : 0 <= P_id_a__zeros .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U51_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U51 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U84_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U84 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U32_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U32 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNatList_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_isNatList x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U13_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U13 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U85_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U85 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__length_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__length x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNatIList_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__isNatIList x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U33_bounded : forall x11, (0 <= x11) ->0 <= P_id_U33 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U11_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U11 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U81_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U81 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U92_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U92 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U43_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U43 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U23_bounded : forall x11, (0 <= x11) ->0 <= P_id_U23 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNatKind_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__isNatKind x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U93_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U93 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U11_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U11 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U71_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U71 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U44_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U44 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_0_bounded : 0 <= P_id_0 .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U61_bounded : forall x11, (0 <= x11) ->0 <= P_id_U61 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U86_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U86 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U41_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U41 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U22_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U22 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U21_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U21 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U91_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U91 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_length_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_length x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U52_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U52 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U42_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U42 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U12_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U12 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U83_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U83 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U94_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U94 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U45_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U45 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U31_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U31 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNat_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__isNat x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNatIListKind_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_isNatIListKind x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U82_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U82 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U46_bounded : forall x11, (0 <= x11) ->0 <= P_id_U46 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_cons_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_cons x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U52_bounded : forall x11, (0 <= x11) ->0 <= P_id_U52 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U85_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U85 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U33_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U33 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U21_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U21 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNatList_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__isNatList x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U86_bounded : forall x11, (0 <= x11) ->0 <= P_id_U86 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_mark_bounded : forall x11, (0 <= x11) ->0 <= P_id_mark x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U51_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U51 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U41_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U41 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_tt_bounded : 0 <= P_id_tt .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U82_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U82 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U93_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U93 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U44_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U44 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNat_bounded : forall x11, (0 <= x11) ->0 <= P_id_isNat x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U23_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U23 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U94_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U94 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U12_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U12 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U81_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U81 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U45_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U45 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_zeros_bounded : 0 <= P_id_zeros .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U71_bounded : forall x11, (0 <= x11) ->0 <= P_id_U71 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U91_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U91 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U42_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U42 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNatKind_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_isNatKind x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U22_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U22 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U92_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U92 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_nil_bounded : 0 <= P_id_nil .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U61_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U61 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U43_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U43 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNatIListKind_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__isNatIListKind x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U84_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U84 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_s_bounded : forall x11, (0 <= x11) ->0 <= P_id_s x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U46_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U46 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U32_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U32 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U31_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U31 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U13_bounded : forall x11, (0 <= x11) ->0 <= P_id_U13 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U83_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U83 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNatIList_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_isNatIList x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Definition measure  := 
       InterpZ.measure 0 P_id_a__zeros P_id_U51 P_id_a__U84 P_id_a__U32 
        P_id_isNatList P_id_a__U13 P_id_U85 P_id_a__length 
        P_id_a__isNatIList P_id_U33 P_id_a__U11 P_id_U81 P_id_a__U92 
        P_id_a__U43 P_id_U23 P_id_a__isNatKind P_id_U93 P_id_U11 P_id_a__U71 
        P_id_U44 P_id_0 P_id_U61 P_id_a__U86 P_id_a__U41 P_id_U22 
        P_id_a__U21 P_id_U91 P_id_length P_id_a__U52 P_id_U42 P_id_a__U12 
        P_id_U83 P_id_a__U94 P_id_a__U45 P_id_U31 P_id_a__isNat 
        P_id_isNatIListKind P_id_a__U82 P_id_U46 P_id_cons P_id_U52 
        P_id_a__U85 P_id_a__U33 P_id_U21 P_id_a__isNatList P_id_U86 
        P_id_mark P_id_a__U51 P_id_U41 P_id_tt P_id_U82 P_id_a__U93 
        P_id_a__U44 P_id_isNat P_id_a__U23 P_id_U94 P_id_U12 P_id_a__U81 
        P_id_U45 P_id_zeros P_id_U71 P_id_a__U91 P_id_a__U42 P_id_isNatKind 
        P_id_a__U22 P_id_U92 P_id_nil P_id_a__U61 P_id_U43 
        P_id_a__isNatIListKind P_id_U84 P_id_s P_id_a__U46 P_id_U32 
        P_id_a__U31 P_id_U13 P_id_a__U83 P_id_isNatIList.
     
     Lemma measure_equation :
      forall t, 
       measure t = match t with
                     | (algebra.Alg.Term algebra.F.id_a__zeros nil) =>
                      P_id_a__zeros 
                     | (algebra.Alg.Term algebra.F.id_U51 (x12::x11::nil)) =>
                      P_id_U51 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U84 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U84 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U32 (x12::x11::nil)) =>
                      P_id_a__U32 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_isNatList (x11::nil)) =>
                      P_id_isNatList (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U13 (x11::nil)) =>
                      P_id_a__U13 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U85 (x12::x11::nil)) =>
                      P_id_U85 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__length (x11::nil)) =>
                      P_id_a__length (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__isNatIList 
                        (x11::nil)) =>
                      P_id_a__isNatIList (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U33 (x11::nil)) =>
                      P_id_U33 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U11 (x12::x11::nil)) =>
                      P_id_a__U11 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U81 (x13::x12::
                        x11::nil)) =>
                      P_id_U81 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U92 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U92 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U43 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U43 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U23 (x11::nil)) =>
                      P_id_U23 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__isNatKind 
                        (x11::nil)) =>
                      P_id_a__isNatKind (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U93 (x13::x12::
                        x11::nil)) =>
                      P_id_U93 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U11 (x12::x11::nil)) =>
                      P_id_U11 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U71 (x11::nil)) =>
                      P_id_a__U71 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U44 (x13::x12::
                        x11::nil)) =>
                      P_id_U44 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_0 nil) => P_id_0 
                     | (algebra.Alg.Term algebra.F.id_U61 (x11::nil)) =>
                      P_id_U61 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U86 (x11::nil)) =>
                      P_id_a__U86 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U41 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U41 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U22 (x12::x11::nil)) =>
                      P_id_U22 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U21 (x12::x11::nil)) =>
                      P_id_a__U21 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U91 (x13::x12::
                        x11::nil)) =>
                      P_id_U91 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_length (x11::nil)) =>
                      P_id_length (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U52 (x11::nil)) =>
                      P_id_a__U52 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U42 (x13::x12::
                        x11::nil)) =>
                      P_id_U42 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U12 (x12::x11::nil)) =>
                      P_id_a__U12 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U83 (x13::x12::
                        x11::nil)) =>
                      P_id_U83 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U94 (x12::x11::nil)) =>
                      P_id_a__U94 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U45 (x12::x11::nil)) =>
                      P_id_a__U45 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U31 (x12::x11::nil)) =>
                      P_id_U31 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__isNat (x11::nil)) =>
                      P_id_a__isNat (measure x11)
                     | (algebra.Alg.Term algebra.F.id_isNatIListKind 
                        (x11::nil)) =>
                      P_id_isNatIListKind (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U82 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U82 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U46 (x11::nil)) =>
                      P_id_U46 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_cons (x12::x11::nil)) =>
                      P_id_cons (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U52 (x11::nil)) =>
                      P_id_U52 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U85 (x12::x11::nil)) =>
                      P_id_a__U85 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U33 (x11::nil)) =>
                      P_id_a__U33 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U21 (x12::x11::nil)) =>
                      P_id_U21 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__isNatList 
                        (x11::nil)) =>
                      P_id_a__isNatList (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U86 (x11::nil)) =>
                      P_id_U86 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_mark (x11::nil)) =>
                      P_id_mark (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U51 (x12::x11::nil)) =>
                      P_id_a__U51 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U41 (x13::x12::
                        x11::nil)) =>
                      P_id_U41 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt 
                     | (algebra.Alg.Term algebra.F.id_U82 (x13::x12::
                        x11::nil)) =>
                      P_id_U82 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U93 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U93 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U44 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U44 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_isNat (x11::nil)) =>
                      P_id_isNat (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U23 (x11::nil)) =>
                      P_id_a__U23 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U94 (x12::x11::nil)) =>
                      P_id_U94 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U12 (x12::x11::nil)) =>
                      P_id_U12 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U81 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U81 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U45 (x12::x11::nil)) =>
                      P_id_U45 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_zeros nil) =>
                      P_id_zeros 
                     | (algebra.Alg.Term algebra.F.id_U71 (x11::nil)) =>
                      P_id_U71 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U91 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U91 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U42 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U42 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_isNatKind (x11::nil)) =>
                      P_id_isNatKind (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U22 (x12::x11::nil)) =>
                      P_id_a__U22 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U92 (x13::x12::
                        x11::nil)) =>
                      P_id_U92 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil 
                     | (algebra.Alg.Term algebra.F.id_a__U61 (x11::nil)) =>
                      P_id_a__U61 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U43 (x13::x12::
                        x11::nil)) =>
                      P_id_U43 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
                        (x11::nil)) =>
                      P_id_a__isNatIListKind (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U84 (x13::x12::
                        x11::nil)) =>
                      P_id_U84 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_s (x11::nil)) =>
                      P_id_s (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U46 (x11::nil)) =>
                      P_id_a__U46 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U32 (x12::x11::nil)) =>
                      P_id_U32 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U31 (x12::x11::nil)) =>
                      P_id_a__U31 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U13 (x11::nil)) =>
                      P_id_U13 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U83 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U83 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_isNatIList (x11::nil)) =>
                      P_id_isNatIList (measure x11)
                     | _ => 0
                     end.
     Proof.
       intros t;case t;intros ;apply refl_equal.
     Qed.
     
     Lemma measure_bounded : forall t, 0 <= measure t.
     Proof.
       unfold measure in |-*.
       
       apply InterpZ.measure_bounded;
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Ltac generate_pos_hyp  :=
      match goal with
        | H:context [measure ?x] |- _ =>
         let v := fresh "v" in 
          (let H := fresh "h" in 
            (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
              clearbody H;clearbody v))
        |  |- context [measure ?x] =>
         let v := fresh "v" in 
          (let H := fresh "h" in 
            (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
              clearbody H;clearbody v))
        end
      .
     
     Lemma rules_monotonic :
      forall l r, 
       (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) ->
        measure r <= measure l.
     Proof.
       intros l r H.
       fold measure in |-*.
       
       inversion H;clear H;subst;inversion H0;clear H0;subst;
        simpl algebra.EQT.T.apply_subst in |-*;
        repeat (
        match goal with
          |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
           rewrite (measure_equation (algebra.Alg.Term f t))
          end
        );repeat (generate_pos_hyp );
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma measure_star_monotonic :
      forall l r, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  r l) ->measure r <= measure l.
     Proof.
       unfold measure in *.
       apply InterpZ.measure_star_monotonic.
       intros ;apply P_id_U51_monotonic;assumption.
       intros ;apply P_id_a__U84_monotonic;assumption.
       intros ;apply P_id_a__U32_monotonic;assumption.
       intros ;apply P_id_isNatList_monotonic;assumption.
       intros ;apply P_id_a__U13_monotonic;assumption.
       intros ;apply P_id_U85_monotonic;assumption.
       intros ;apply P_id_a__length_monotonic;assumption.
       intros ;apply P_id_a__isNatIList_monotonic;assumption.
       intros ;apply P_id_U33_monotonic;assumption.
       intros ;apply P_id_a__U11_monotonic;assumption.
       intros ;apply P_id_U81_monotonic;assumption.
       intros ;apply P_id_a__U92_monotonic;assumption.
       intros ;apply P_id_a__U43_monotonic;assumption.
       intros ;apply P_id_U23_monotonic;assumption.
       intros ;apply P_id_a__isNatKind_monotonic;assumption.
       intros ;apply P_id_U93_monotonic;assumption.
       intros ;apply P_id_U11_monotonic;assumption.
       intros ;apply P_id_a__U71_monotonic;assumption.
       intros ;apply P_id_U44_monotonic;assumption.
       intros ;apply P_id_U61_monotonic;assumption.
       intros ;apply P_id_a__U86_monotonic;assumption.
       intros ;apply P_id_a__U41_monotonic;assumption.
       intros ;apply P_id_U22_monotonic;assumption.
       intros ;apply P_id_a__U21_monotonic;assumption.
       intros ;apply P_id_U91_monotonic;assumption.
       intros ;apply P_id_length_monotonic;assumption.
       intros ;apply P_id_a__U52_monotonic;assumption.
       intros ;apply P_id_U42_monotonic;assumption.
       intros ;apply P_id_a__U12_monotonic;assumption.
       intros ;apply P_id_U83_monotonic;assumption.
       intros ;apply P_id_a__U94_monotonic;assumption.
       intros ;apply P_id_a__U45_monotonic;assumption.
       intros ;apply P_id_U31_monotonic;assumption.
       intros ;apply P_id_a__isNat_monotonic;assumption.
       intros ;apply P_id_isNatIListKind_monotonic;assumption.
       intros ;apply P_id_a__U82_monotonic;assumption.
       intros ;apply P_id_U46_monotonic;assumption.
       intros ;apply P_id_cons_monotonic;assumption.
       intros ;apply P_id_U52_monotonic;assumption.
       intros ;apply P_id_a__U85_monotonic;assumption.
       intros ;apply P_id_a__U33_monotonic;assumption.
       intros ;apply P_id_U21_monotonic;assumption.
       intros ;apply P_id_a__isNatList_monotonic;assumption.
       intros ;apply P_id_U86_monotonic;assumption.
       intros ;apply P_id_mark_monotonic;assumption.
       intros ;apply P_id_a__U51_monotonic;assumption.
       intros ;apply P_id_U41_monotonic;assumption.
       intros ;apply P_id_U82_monotonic;assumption.
       intros ;apply P_id_a__U93_monotonic;assumption.
       intros ;apply P_id_a__U44_monotonic;assumption.
       intros ;apply P_id_isNat_monotonic;assumption.
       intros ;apply P_id_a__U23_monotonic;assumption.
       intros ;apply P_id_U94_monotonic;assumption.
       intros ;apply P_id_U12_monotonic;assumption.
       intros ;apply P_id_a__U81_monotonic;assumption.
       intros ;apply P_id_U45_monotonic;assumption.
       intros ;apply P_id_U71_monotonic;assumption.
       intros ;apply P_id_a__U91_monotonic;assumption.
       intros ;apply P_id_a__U42_monotonic;assumption.
       intros ;apply P_id_isNatKind_monotonic;assumption.
       intros ;apply P_id_a__U22_monotonic;assumption.
       intros ;apply P_id_U92_monotonic;assumption.
       intros ;apply P_id_a__U61_monotonic;assumption.
       intros ;apply P_id_U43_monotonic;assumption.
       intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
       intros ;apply P_id_U84_monotonic;assumption.
       intros ;apply P_id_s_monotonic;assumption.
       intros ;apply P_id_a__U46_monotonic;assumption.
       intros ;apply P_id_U32_monotonic;assumption.
       intros ;apply P_id_a__U31_monotonic;assumption.
       intros ;apply P_id_U13_monotonic;assumption.
       intros ;apply P_id_a__U83_monotonic;assumption.
       intros ;apply P_id_isNatIList_monotonic;assumption.
       intros ;apply P_id_a__zeros_bounded;assumption.
       intros ;apply P_id_U51_bounded;assumption.
       intros ;apply P_id_a__U84_bounded;assumption.
       intros ;apply P_id_a__U32_bounded;assumption.
       intros ;apply P_id_isNatList_bounded;assumption.
       intros ;apply P_id_a__U13_bounded;assumption.
       intros ;apply P_id_U85_bounded;assumption.
       intros ;apply P_id_a__length_bounded;assumption.
       intros ;apply P_id_a__isNatIList_bounded;assumption.
       intros ;apply P_id_U33_bounded;assumption.
       intros ;apply P_id_a__U11_bounded;assumption.
       intros ;apply P_id_U81_bounded;assumption.
       intros ;apply P_id_a__U92_bounded;assumption.
       intros ;apply P_id_a__U43_bounded;assumption.
       intros ;apply P_id_U23_bounded;assumption.
       intros ;apply P_id_a__isNatKind_bounded;assumption.
       intros ;apply P_id_U93_bounded;assumption.
       intros ;apply P_id_U11_bounded;assumption.
       intros ;apply P_id_a__U71_bounded;assumption.
       intros ;apply P_id_U44_bounded;assumption.
       intros ;apply P_id_0_bounded;assumption.
       intros ;apply P_id_U61_bounded;assumption.
       intros ;apply P_id_a__U86_bounded;assumption.
       intros ;apply P_id_a__U41_bounded;assumption.
       intros ;apply P_id_U22_bounded;assumption.
       intros ;apply P_id_a__U21_bounded;assumption.
       intros ;apply P_id_U91_bounded;assumption.
       intros ;apply P_id_length_bounded;assumption.
       intros ;apply P_id_a__U52_bounded;assumption.
       intros ;apply P_id_U42_bounded;assumption.
       intros ;apply P_id_a__U12_bounded;assumption.
       intros ;apply P_id_U83_bounded;assumption.
       intros ;apply P_id_a__U94_bounded;assumption.
       intros ;apply P_id_a__U45_bounded;assumption.
       intros ;apply P_id_U31_bounded;assumption.
       intros ;apply P_id_a__isNat_bounded;assumption.
       intros ;apply P_id_isNatIListKind_bounded;assumption.
       intros ;apply P_id_a__U82_bounded;assumption.
       intros ;apply P_id_U46_bounded;assumption.
       intros ;apply P_id_cons_bounded;assumption.
       intros ;apply P_id_U52_bounded;assumption.
       intros ;apply P_id_a__U85_bounded;assumption.
       intros ;apply P_id_a__U33_bounded;assumption.
       intros ;apply P_id_U21_bounded;assumption.
       intros ;apply P_id_a__isNatList_bounded;assumption.
       intros ;apply P_id_U86_bounded;assumption.
       intros ;apply P_id_mark_bounded;assumption.
       intros ;apply P_id_a__U51_bounded;assumption.
       intros ;apply P_id_U41_bounded;assumption.
       intros ;apply P_id_tt_bounded;assumption.
       intros ;apply P_id_U82_bounded;assumption.
       intros ;apply P_id_a__U93_bounded;assumption.
       intros ;apply P_id_a__U44_bounded;assumption.
       intros ;apply P_id_isNat_bounded;assumption.
       intros ;apply P_id_a__U23_bounded;assumption.
       intros ;apply P_id_U94_bounded;assumption.
       intros ;apply P_id_U12_bounded;assumption.
       intros ;apply P_id_a__U81_bounded;assumption.
       intros ;apply P_id_U45_bounded;assumption.
       intros ;apply P_id_zeros_bounded;assumption.
       intros ;apply P_id_U71_bounded;assumption.
       intros ;apply P_id_a__U91_bounded;assumption.
       intros ;apply P_id_a__U42_bounded;assumption.
       intros ;apply P_id_isNatKind_bounded;assumption.
       intros ;apply P_id_a__U22_bounded;assumption.
       intros ;apply P_id_U92_bounded;assumption.
       intros ;apply P_id_nil_bounded;assumption.
       intros ;apply P_id_a__U61_bounded;assumption.
       intros ;apply P_id_U43_bounded;assumption.
       intros ;apply P_id_a__isNatIListKind_bounded;assumption.
       intros ;apply P_id_U84_bounded;assumption.
       intros ;apply P_id_s_bounded;assumption.
       intros ;apply P_id_a__U46_bounded;assumption.
       intros ;apply P_id_U32_bounded;assumption.
       intros ;apply P_id_a__U31_bounded;assumption.
       intros ;apply P_id_U13_bounded;assumption.
       intros ;apply P_id_a__U83_bounded;assumption.
       intros ;apply P_id_isNatIList_bounded;assumption.
       apply rules_monotonic.
     Qed.
     
     Definition P_id_A__U46 (x11:Z) := 0.
     
     Definition P_id_A__ISNATILISTKIND (x11:Z) := 2* x11.
     
     Definition P_id_A__U61 (x11:Z) := 0.
     
     Definition P_id_A__U85 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__U31 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__U81 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__U22 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__U93 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__U42 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__U51 (x11:Z) (x12:Z) := 2* x12.
     
     Definition P_id_A__ISNATLIST (x11:Z) := 0.
     
     Definition P_id_A__ZEROS  := 0.
     
     Definition P_id_A__U91 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__U33 (x11:Z) := 0.
     
     Definition P_id_A__U83 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__U23 (x11:Z) := 0.
     
     Definition P_id_A__U11 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__LENGTH (x11:Z) := 0.
     
     Definition P_id_A__U44 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__ISNATILIST (x11:Z) := 0.
     
     Definition P_id_A__U13 (x11:Z) := 0.
     
     Definition P_id_A__U71 (x11:Z) := 0.
     
     Definition P_id_A__U86 (x11:Z) := 0.
     
     Definition P_id_A__U32 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__U82 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__ISNATKIND (x11:Z) := 0.
     
     Definition P_id_A__U94 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__U43 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__U52 (x11:Z) := 0.
     
     Definition P_id_A__U21 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__U92 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__U41 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__U84 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__ISNAT (x11:Z) := 0.
     
     Definition P_id_A__U12 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_MARK (x11:Z) := 0.
     
     Definition P_id_A__U45 (x11:Z) (x12:Z) := 0.
     
     Lemma P_id_A__U46_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U46 x12 <= P_id_A__U46 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__ISNATILISTKIND_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__ISNATILISTKIND x12 <= P_id_A__ISNATILISTKIND x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U61_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U61 x12 <= P_id_A__U61 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U85_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U85 x12 x14 <= P_id_A__U85 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U31_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U31 x12 x14 <= P_id_A__U31 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U81_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U81 x12 x14 x16 <= P_id_A__U81 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U22_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U22 x12 x14 <= P_id_A__U22 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U93_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U93 x12 x14 x16 <= P_id_A__U93 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U42_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U42 x12 x14 x16 <= P_id_A__U42 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U51_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U51 x12 x14 <= P_id_A__U51 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__ISNATLIST_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__ISNATLIST x12 <= P_id_A__ISNATLIST x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U91_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U91 x12 x14 x16 <= P_id_A__U91 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U33_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U33 x12 <= P_id_A__U33 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U83_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U83 x12 x14 x16 <= P_id_A__U83 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U23_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U23 x12 <= P_id_A__U23 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U11_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U11 x12 x14 <= P_id_A__U11 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__LENGTH_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__LENGTH x12 <= P_id_A__LENGTH x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U44_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U44 x12 x14 x16 <= P_id_A__U44 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__ISNATILIST_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__ISNATILIST x12 <= P_id_A__ISNATILIST x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U13_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U13 x12 <= P_id_A__U13 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U71_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U71 x12 <= P_id_A__U71 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U86_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U86 x12 <= P_id_A__U86 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U32_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U32 x12 x14 <= P_id_A__U32 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U82_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U82 x12 x14 x16 <= P_id_A__U82 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__ISNATKIND_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__ISNATKIND x12 <= P_id_A__ISNATKIND x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U94_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U94 x12 x14 <= P_id_A__U94 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U43_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U43 x12 x14 x16 <= P_id_A__U43 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U52_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U52 x12 <= P_id_A__U52 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U21_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U21 x12 x14 <= P_id_A__U21 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U92_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U92 x12 x14 x16 <= P_id_A__U92 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U41_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U41 x12 x14 x16 <= P_id_A__U41 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U84_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U84 x12 x14 x16 <= P_id_A__U84 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__ISNAT_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__ISNAT x12 <= P_id_A__ISNAT x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U12_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U12 x12 x14 <= P_id_A__U12 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_MARK_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_MARK x12 <= P_id_MARK x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U45_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U45 x12 x14 <= P_id_A__U45 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Definition marked_measure  := 
       InterpZ.marked_measure 0 P_id_a__zeros P_id_U51 P_id_a__U84 
        P_id_a__U32 P_id_isNatList P_id_a__U13 P_id_U85 P_id_a__length 
        P_id_a__isNatIList P_id_U33 P_id_a__U11 P_id_U81 P_id_a__U92 
        P_id_a__U43 P_id_U23 P_id_a__isNatKind P_id_U93 P_id_U11 P_id_a__U71 
        P_id_U44 P_id_0 P_id_U61 P_id_a__U86 P_id_a__U41 P_id_U22 
        P_id_a__U21 P_id_U91 P_id_length P_id_a__U52 P_id_U42 P_id_a__U12 
        P_id_U83 P_id_a__U94 P_id_a__U45 P_id_U31 P_id_a__isNat 
        P_id_isNatIListKind P_id_a__U82 P_id_U46 P_id_cons P_id_U52 
        P_id_a__U85 P_id_a__U33 P_id_U21 P_id_a__isNatList P_id_U86 
        P_id_mark P_id_a__U51 P_id_U41 P_id_tt P_id_U82 P_id_a__U93 
        P_id_a__U44 P_id_isNat P_id_a__U23 P_id_U94 P_id_U12 P_id_a__U81 
        P_id_U45 P_id_zeros P_id_U71 P_id_a__U91 P_id_a__U42 P_id_isNatKind 
        P_id_a__U22 P_id_U92 P_id_nil P_id_a__U61 P_id_U43 
        P_id_a__isNatIListKind P_id_U84 P_id_s P_id_a__U46 P_id_U32 
        P_id_a__U31 P_id_U13 P_id_a__U83 P_id_isNatIList P_id_A__U46 
        P_id_A__ISNATILISTKIND P_id_A__U61 P_id_A__U85 P_id_A__U31 
        P_id_A__U81 P_id_A__U22 P_id_A__U93 P_id_A__U42 P_id_A__U51 
        P_id_A__ISNATLIST P_id_A__ZEROS P_id_A__U91 P_id_A__U33 P_id_A__U83 
        P_id_A__U23 P_id_A__U11 P_id_A__LENGTH P_id_A__U44 
        P_id_A__ISNATILIST P_id_A__U13 P_id_A__U71 P_id_A__U86 P_id_A__U32 
        P_id_A__U82 P_id_A__ISNATKIND P_id_A__U94 P_id_A__U43 P_id_A__U52 
        P_id_A__U21 P_id_A__U92 P_id_A__U41 P_id_A__U84 P_id_A__ISNAT 
        P_id_A__U12 P_id_MARK P_id_A__U45.
     
     Lemma marked_measure_equation :
      forall t, 
       marked_measure t = match t with
                            | (algebra.Alg.Term algebra.F.id_a__U46 
                               (x11::nil)) =>
                             P_id_A__U46 (measure x11)
                            | (algebra.Alg.Term 
                               algebra.F.id_a__isNatIListKind (x11::nil)) =>
                             P_id_A__ISNATILISTKIND (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U61 
                               (x11::nil)) =>
                             P_id_A__U61 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U85 (x12::
                               x11::nil)) =>
                             P_id_A__U85 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U31 (x12::
                               x11::nil)) =>
                             P_id_A__U31 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U81 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U81 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U22 (x12::
                               x11::nil)) =>
                             P_id_A__U22 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U93 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U93 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U42 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U42 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U51 (x12::
                               x11::nil)) =>
                             P_id_A__U51 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__isNatList 
                               (x11::nil)) =>
                             P_id_A__ISNATLIST (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__zeros nil) =>
                             P_id_A__ZEROS 
                            | (algebra.Alg.Term algebra.F.id_a__U91 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U91 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U33 
                               (x11::nil)) =>
                             P_id_A__U33 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U83 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U83 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U23 
                               (x11::nil)) =>
                             P_id_A__U23 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U11 (x12::
                               x11::nil)) =>
                             P_id_A__U11 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__length 
                               (x11::nil)) =>
                             P_id_A__LENGTH (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U44 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U44 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__isNatIList 
                               (x11::nil)) =>
                             P_id_A__ISNATILIST (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U13 
                               (x11::nil)) =>
                             P_id_A__U13 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U71 
                               (x11::nil)) =>
                             P_id_A__U71 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U86 
                               (x11::nil)) =>
                             P_id_A__U86 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U32 (x12::
                               x11::nil)) =>
                             P_id_A__U32 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U82 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U82 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__isNatKind 
                               (x11::nil)) =>
                             P_id_A__ISNATKIND (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U94 (x12::
                               x11::nil)) =>
                             P_id_A__U94 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U43 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U43 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U52 
                               (x11::nil)) =>
                             P_id_A__U52 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U21 (x12::
                               x11::nil)) =>
                             P_id_A__U21 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U92 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U92 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U41 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U41 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U84 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U84 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__isNat 
                               (x11::nil)) =>
                             P_id_A__ISNAT (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U12 (x12::
                               x11::nil)) =>
                             P_id_A__U12 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_mark (x11::nil)) =>
                             P_id_MARK (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U45 (x12::
                               x11::nil)) =>
                             P_id_A__U45 (measure x12) (measure x11)
                            | _ => measure t
                            end.
     Proof.
       reflexivity .
     Qed.
     
     Lemma marked_measure_star_monotonic :
      forall f l1 l2, 
       (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step 
                                                         R_xml_0_deep_rew.R_xml_0_rules)
                                                        ) l1 l2) ->
        marked_measure (algebra.Alg.Term f l1) <= marked_measure (algebra.Alg.Term 
                                                                   f 
                                                                   l2).
     Proof.
       unfold marked_measure in *.
       apply InterpZ.marked_measure_star_monotonic.
       intros ;apply P_id_U51_monotonic;assumption.
       intros ;apply P_id_a__U84_monotonic;assumption.
       intros ;apply P_id_a__U32_monotonic;assumption.
       intros ;apply P_id_isNatList_monotonic;assumption.
       intros ;apply P_id_a__U13_monotonic;assumption.
       intros ;apply P_id_U85_monotonic;assumption.
       intros ;apply P_id_a__length_monotonic;assumption.
       intros ;apply P_id_a__isNatIList_monotonic;assumption.
       intros ;apply P_id_U33_monotonic;assumption.
       intros ;apply P_id_a__U11_monotonic;assumption.
       intros ;apply P_id_U81_monotonic;assumption.
       intros ;apply P_id_a__U92_monotonic;assumption.
       intros ;apply P_id_a__U43_monotonic;assumption.
       intros ;apply P_id_U23_monotonic;assumption.
       intros ;apply P_id_a__isNatKind_monotonic;assumption.
       intros ;apply P_id_U93_monotonic;assumption.
       intros ;apply P_id_U11_monotonic;assumption.
       intros ;apply P_id_a__U71_monotonic;assumption.
       intros ;apply P_id_U44_monotonic;assumption.
       intros ;apply P_id_U61_monotonic;assumption.
       intros ;apply P_id_a__U86_monotonic;assumption.
       intros ;apply P_id_a__U41_monotonic;assumption.
       intros ;apply P_id_U22_monotonic;assumption.
       intros ;apply P_id_a__U21_monotonic;assumption.
       intros ;apply P_id_U91_monotonic;assumption.
       intros ;apply P_id_length_monotonic;assumption.
       intros ;apply P_id_a__U52_monotonic;assumption.
       intros ;apply P_id_U42_monotonic;assumption.
       intros ;apply P_id_a__U12_monotonic;assumption.
       intros ;apply P_id_U83_monotonic;assumption.
       intros ;apply P_id_a__U94_monotonic;assumption.
       intros ;apply P_id_a__U45_monotonic;assumption.
       intros ;apply P_id_U31_monotonic;assumption.
       intros ;apply P_id_a__isNat_monotonic;assumption.
       intros ;apply P_id_isNatIListKind_monotonic;assumption.
       intros ;apply P_id_a__U82_monotonic;assumption.
       intros ;apply P_id_U46_monotonic;assumption.
       intros ;apply P_id_cons_monotonic;assumption.
       intros ;apply P_id_U52_monotonic;assumption.
       intros ;apply P_id_a__U85_monotonic;assumption.
       intros ;apply P_id_a__U33_monotonic;assumption.
       intros ;apply P_id_U21_monotonic;assumption.
       intros ;apply P_id_a__isNatList_monotonic;assumption.
       intros ;apply P_id_U86_monotonic;assumption.
       intros ;apply P_id_mark_monotonic;assumption.
       intros ;apply P_id_a__U51_monotonic;assumption.
       intros ;apply P_id_U41_monotonic;assumption.
       intros ;apply P_id_U82_monotonic;assumption.
       intros ;apply P_id_a__U93_monotonic;assumption.
       intros ;apply P_id_a__U44_monotonic;assumption.
       intros ;apply P_id_isNat_monotonic;assumption.
       intros ;apply P_id_a__U23_monotonic;assumption.
       intros ;apply P_id_U94_monotonic;assumption.
       intros ;apply P_id_U12_monotonic;assumption.
       intros ;apply P_id_a__U81_monotonic;assumption.
       intros ;apply P_id_U45_monotonic;assumption.
       intros ;apply P_id_U71_monotonic;assumption.
       intros ;apply P_id_a__U91_monotonic;assumption.
       intros ;apply P_id_a__U42_monotonic;assumption.
       intros ;apply P_id_isNatKind_monotonic;assumption.
       intros ;apply P_id_a__U22_monotonic;assumption.
       intros ;apply P_id_U92_monotonic;assumption.
       intros ;apply P_id_a__U61_monotonic;assumption.
       intros ;apply P_id_U43_monotonic;assumption.
       intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
       intros ;apply P_id_U84_monotonic;assumption.
       intros ;apply P_id_s_monotonic;assumption.
       intros ;apply P_id_a__U46_monotonic;assumption.
       intros ;apply P_id_U32_monotonic;assumption.
       intros ;apply P_id_a__U31_monotonic;assumption.
       intros ;apply P_id_U13_monotonic;assumption.
       intros ;apply P_id_a__U83_monotonic;assumption.
       intros ;apply P_id_isNatIList_monotonic;assumption.
       intros ;apply P_id_a__zeros_bounded;assumption.
       intros ;apply P_id_U51_bounded;assumption.
       intros ;apply P_id_a__U84_bounded;assumption.
       intros ;apply P_id_a__U32_bounded;assumption.
       intros ;apply P_id_isNatList_bounded;assumption.
       intros ;apply P_id_a__U13_bounded;assumption.
       intros ;apply P_id_U85_bounded;assumption.
       intros ;apply P_id_a__length_bounded;assumption.
       intros ;apply P_id_a__isNatIList_bounded;assumption.
       intros ;apply P_id_U33_bounded;assumption.
       intros ;apply P_id_a__U11_bounded;assumption.
       intros ;apply P_id_U81_bounded;assumption.
       intros ;apply P_id_a__U92_bounded;assumption.
       intros ;apply P_id_a__U43_bounded;assumption.
       intros ;apply P_id_U23_bounded;assumption.
       intros ;apply P_id_a__isNatKind_bounded;assumption.
       intros ;apply P_id_U93_bounded;assumption.
       intros ;apply P_id_U11_bounded;assumption.
       intros ;apply P_id_a__U71_bounded;assumption.
       intros ;apply P_id_U44_bounded;assumption.
       intros ;apply P_id_0_bounded;assumption.
       intros ;apply P_id_U61_bounded;assumption.
       intros ;apply P_id_a__U86_bounded;assumption.
       intros ;apply P_id_a__U41_bounded;assumption.
       intros ;apply P_id_U22_bounded;assumption.
       intros ;apply P_id_a__U21_bounded;assumption.
       intros ;apply P_id_U91_bounded;assumption.
       intros ;apply P_id_length_bounded;assumption.
       intros ;apply P_id_a__U52_bounded;assumption.
       intros ;apply P_id_U42_bounded;assumption.
       intros ;apply P_id_a__U12_bounded;assumption.
       intros ;apply P_id_U83_bounded;assumption.
       intros ;apply P_id_a__U94_bounded;assumption.
       intros ;apply P_id_a__U45_bounded;assumption.
       intros ;apply P_id_U31_bounded;assumption.
       intros ;apply P_id_a__isNat_bounded;assumption.
       intros ;apply P_id_isNatIListKind_bounded;assumption.
       intros ;apply P_id_a__U82_bounded;assumption.
       intros ;apply P_id_U46_bounded;assumption.
       intros ;apply P_id_cons_bounded;assumption.
       intros ;apply P_id_U52_bounded;assumption.
       intros ;apply P_id_a__U85_bounded;assumption.
       intros ;apply P_id_a__U33_bounded;assumption.
       intros ;apply P_id_U21_bounded;assumption.
       intros ;apply P_id_a__isNatList_bounded;assumption.
       intros ;apply P_id_U86_bounded;assumption.
       intros ;apply P_id_mark_bounded;assumption.
       intros ;apply P_id_a__U51_bounded;assumption.
       intros ;apply P_id_U41_bounded;assumption.
       intros ;apply P_id_tt_bounded;assumption.
       intros ;apply P_id_U82_bounded;assumption.
       intros ;apply P_id_a__U93_bounded;assumption.
       intros ;apply P_id_a__U44_bounded;assumption.
       intros ;apply P_id_isNat_bounded;assumption.
       intros ;apply P_id_a__U23_bounded;assumption.
       intros ;apply P_id_U94_bounded;assumption.
       intros ;apply P_id_U12_bounded;assumption.
       intros ;apply P_id_a__U81_bounded;assumption.
       intros ;apply P_id_U45_bounded;assumption.
       intros ;apply P_id_zeros_bounded;assumption.
       intros ;apply P_id_U71_bounded;assumption.
       intros ;apply P_id_a__U91_bounded;assumption.
       intros ;apply P_id_a__U42_bounded;assumption.
       intros ;apply P_id_isNatKind_bounded;assumption.
       intros ;apply P_id_a__U22_bounded;assumption.
       intros ;apply P_id_U92_bounded;assumption.
       intros ;apply P_id_nil_bounded;assumption.
       intros ;apply P_id_a__U61_bounded;assumption.
       intros ;apply P_id_U43_bounded;assumption.
       intros ;apply P_id_a__isNatIListKind_bounded;assumption.
       intros ;apply P_id_U84_bounded;assumption.
       intros ;apply P_id_s_bounded;assumption.
       intros ;apply P_id_a__U46_bounded;assumption.
       intros ;apply P_id_U32_bounded;assumption.
       intros ;apply P_id_a__U31_bounded;assumption.
       intros ;apply P_id_U13_bounded;assumption.
       intros ;apply P_id_a__U83_bounded;assumption.
       intros ;apply P_id_isNatIList_bounded;assumption.
       apply rules_monotonic.
       intros ;apply P_id_A__U46_monotonic;assumption.
       intros ;apply P_id_A__ISNATILISTKIND_monotonic;assumption.
       intros ;apply P_id_A__U61_monotonic;assumption.
       intros ;apply P_id_A__U85_monotonic;assumption.
       intros ;apply P_id_A__U31_monotonic;assumption.
       intros ;apply P_id_A__U81_monotonic;assumption.
       intros ;apply P_id_A__U22_monotonic;assumption.
       intros ;apply P_id_A__U93_monotonic;assumption.
       intros ;apply P_id_A__U42_monotonic;assumption.
       intros ;apply P_id_A__U51_monotonic;assumption.
       intros ;apply P_id_A__ISNATLIST_monotonic;assumption.
       intros ;apply P_id_A__U91_monotonic;assumption.
       intros ;apply P_id_A__U33_monotonic;assumption.
       intros ;apply P_id_A__U83_monotonic;assumption.
       intros ;apply P_id_A__U23_monotonic;assumption.
       intros ;apply P_id_A__U11_monotonic;assumption.
       intros ;apply P_id_A__LENGTH_monotonic;assumption.
       intros ;apply P_id_A__U44_monotonic;assumption.
       intros ;apply P_id_A__ISNATILIST_monotonic;assumption.
       intros ;apply P_id_A__U13_monotonic;assumption.
       intros ;apply P_id_A__U71_monotonic;assumption.
       intros ;apply P_id_A__U86_monotonic;assumption.
       intros ;apply P_id_A__U32_monotonic;assumption.
       intros ;apply P_id_A__U82_monotonic;assumption.
       intros ;apply P_id_A__ISNATKIND_monotonic;assumption.
       intros ;apply P_id_A__U94_monotonic;assumption.
       intros ;apply P_id_A__U43_monotonic;assumption.
       intros ;apply P_id_A__U52_monotonic;assumption.
       intros ;apply P_id_A__U21_monotonic;assumption.
       intros ;apply P_id_A__U92_monotonic;assumption.
       intros ;apply P_id_A__U41_monotonic;assumption.
       intros ;apply P_id_A__U84_monotonic;assumption.
       intros ;apply P_id_A__ISNAT_monotonic;assumption.
       intros ;apply P_id_A__U12_monotonic;assumption.
       intros ;apply P_id_MARK_monotonic;assumption.
       intros ;apply P_id_A__U45_monotonic;assumption.
     Qed.
     
     Ltac rewrite_and_unfold  :=
      do 2 (rewrite marked_measure_equation);
       repeat (
       match goal with
         |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
          rewrite (measure_equation (algebra.Alg.Term f t))
         | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ =>
          rewrite (measure_equation (algebra.Alg.Term f t)) in H|-
         end
       ).
     
     Definition lt a b := (Zwf.Zwf 0) (marked_measure a) (marked_measure b).
     
     Definition le a b := marked_measure a <= marked_measure b.
     
     Lemma lt_le_compat : forall a b c, (lt a b) ->(le b c) ->lt a c.
     Proof.
       unfold lt, le in *.
       intros a b c.
       apply (interp.le_lt_compat_right (interp.o_Z 0)).
     Qed.
     
     Lemma wf_lt : well_founded lt.
     Proof.
       unfold lt in *.
       apply Inverse_Image.wf_inverse_image with  (B:=Z).
       apply Zwf.Zwf_well_founded.
     Qed.
     
     Lemma DP_R_xml_0_scc_14_large_scc_2_strict_in_lt :
      Relation_Definitions.inclusion _ DP_R_xml_0_scc_14_large_scc_2_strict 
       lt.
     Proof.
       unfold Relation_Definitions.inclusion, lt in *.
       
       intros a b H;destruct H;
        match goal with
          |  |- (Zwf.Zwf 0) _ (marked_measure (algebra.Alg.Term ?f ?l)) =>
           let l'' := algebra.Alg_ext.find_replacement l  in 
            ((apply (interp.le_lt_compat_right (interp.o_Z 0)) with
               (marked_measure (algebra.Alg.Term f l''));[idtac|
              apply marked_measure_star_monotonic;
               repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos);
               (assumption)||(constructor 1)]))
          end
        ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma DP_R_xml_0_scc_14_large_scc_2_large_in_le :
      Relation_Definitions.inclusion _ DP_R_xml_0_scc_14_large_scc_2_large le.
     Proof.
       unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *.
       
       intros a b H;destruct H;
        match goal with
          |  |- _ <= marked_measure (algebra.Alg.Term ?f ?l) =>
           let l'' := algebra.Alg_ext.find_replacement l  in 
            ((apply (interp.le_trans (interp.o_Z 0)) with
               (marked_measure (algebra.Alg.Term f l''));[idtac|
              apply marked_measure_star_monotonic;
               repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos);
               (assumption)||(constructor 1)]))
          end
        ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Definition wf_DP_R_xml_0_scc_14_large_scc_2_large  := 
       WF_DP_R_xml_0_scc_14_large_scc_2_large.wf.
     
     
     Lemma wf :
      well_founded WF_DP_R_xml_0_scc_14_large.DP_R_xml_0_scc_14_large_scc_2.
     Proof.
       intros x.
       apply (well_founded_ind wf_lt).
       clear x.
       intros x.
       pattern x.
       apply (@Acc_ind _ DP_R_xml_0_scc_14_large_scc_2_large).
       clear x.
       intros x _ IHx IHx'.
       constructor.
       intros y H.
       
       destruct H;
        (apply IHx';apply DP_R_xml_0_scc_14_large_scc_2_strict_in_lt;
          econstructor eassumption)||
        ((apply IHx;[econstructor eassumption|
          intros y' Hlt;apply IHx';apply lt_le_compat with (1:=Hlt) ;
           apply DP_R_xml_0_scc_14_large_scc_2_large_in_le;
           econstructor eassumption])).
       apply wf_DP_R_xml_0_scc_14_large_scc_2_large.
     Qed.
    End WF_DP_R_xml_0_scc_14_large_scc_2.
    
    Definition wf_DP_R_xml_0_scc_14_large_scc_2  := 
      WF_DP_R_xml_0_scc_14_large_scc_2.wf.
    
    
    Lemma acc_DP_R_xml_0_scc_14_large_scc_2 :
     forall x y, 
      (DP_R_xml_0_scc_14_large_scc_2 x y) ->
       Acc WF_DP_R_xml_0_scc_14.DP_R_xml_0_scc_14_large x.
    Proof.
      intros x.
      pattern x.
      apply (@Acc_ind _ DP_R_xml_0_scc_14_large_scc_2).
      intros x' _ Hrec y h.
      
      inversion h;clear h;subst;
       constructor;intros _y _h;inversion _h;clear _h;subst;
        (eapply Hrec;econstructor eassumption)||
        ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
         (eapply Hrec;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))).
      apply wf_DP_R_xml_0_scc_14_large_scc_2.
    Qed.
    
    
    Lemma wf : well_founded WF_DP_R_xml_0_scc_14.DP_R_xml_0_scc_14_large.
    Proof.
      constructor;intros _y _h;inversion _h;clear _h;subst;
       (eapply acc_DP_R_xml_0_scc_14_large_scc_2;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_scc_14_large_scc_1;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((eapply acc_DP_R_xml_0_scc_14_large_scc_0;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
         ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||(fail)))).
    Qed.
   End WF_DP_R_xml_0_scc_14_large.
   
   Open Scope Z_scope.
   
   Import ring_extention.
   
   Notation Local "a <= b" := (Zle a b).
   
   Notation Local "a < b" := (Zlt a b).
   
   Definition P_id_a__zeros  := 0.
   
   Definition P_id_U51 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_a__U84 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_a__U32 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_isNatList (x11:Z) := 0.
   
   Definition P_id_a__U13 (x11:Z) := 0.
   
   Definition P_id_U85 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_a__length (x11:Z) := 1 + 1* x11.
   
   Definition P_id_a__isNatIList (x11:Z) := 0.
   
   Definition P_id_U33 (x11:Z) := 0.
   
   Definition P_id_a__U11 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_U81 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_a__U92 (x11:Z) (x12:Z) (x13:Z) := 1 + 1* x12.
   
   Definition P_id_a__U43 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_U23 (x11:Z) := 0.
   
   Definition P_id_a__isNatKind (x11:Z) := 0.
   
   Definition P_id_U93 (x11:Z) (x12:Z) (x13:Z) := 1 + 1* x12.
   
   Definition P_id_U11 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_a__U71 (x11:Z) := 0.
   
   Definition P_id_U44 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_0  := 0.
   
   Definition P_id_U61 (x11:Z) := 0.
   
   Definition P_id_a__U86 (x11:Z) := 0.
   
   Definition P_id_a__U41 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_U22 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_a__U21 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_U91 (x11:Z) (x12:Z) (x13:Z) := 1 + 1* x12.
   
   Definition P_id_length (x11:Z) := 1 + 1* x11.
   
   Definition P_id_a__U52 (x11:Z) := 0.
   
   Definition P_id_U42 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_a__U12 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_U83 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_a__U94 (x11:Z) (x12:Z) := 1 + 1* x12.
   
   Definition P_id_a__U45 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_U31 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_a__isNat (x11:Z) := 2 + 2* x11.
   
   Definition P_id_isNatIListKind (x11:Z) := 0.
   
   Definition P_id_a__U82 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_U46 (x11:Z) := 0.
   
   Definition P_id_cons (x11:Z) (x12:Z) := 2* x11 + 1* x12.
   
   Definition P_id_U52 (x11:Z) := 0.
   
   Definition P_id_a__U85 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_a__U33 (x11:Z) := 0.
   
   Definition P_id_U21 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_a__isNatList (x11:Z) := 0.
   
   Definition P_id_U86 (x11:Z) := 0.
   
   Definition P_id_mark (x11:Z) := 1* x11.
   
   Definition P_id_a__U51 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_U41 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_tt  := 0.
   
   Definition P_id_U82 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_a__U93 (x11:Z) (x12:Z) (x13:Z) := 1 + 1* x12.
   
   Definition P_id_a__U44 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_isNat (x11:Z) := 2 + 2* x11.
   
   Definition P_id_a__U23 (x11:Z) := 0.
   
   Definition P_id_U94 (x11:Z) (x12:Z) := 1 + 1* x12.
   
   Definition P_id_U12 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_a__U81 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_U45 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_zeros  := 0.
   
   Definition P_id_U71 (x11:Z) := 0.
   
   Definition P_id_a__U91 (x11:Z) (x12:Z) (x13:Z) := 1 + 1* x12.
   
   Definition P_id_a__U42 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_isNatKind (x11:Z) := 0.
   
   Definition P_id_a__U22 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_U92 (x11:Z) (x12:Z) (x13:Z) := 1 + 1* x12.
   
   Definition P_id_nil  := 0.
   
   Definition P_id_a__U61 (x11:Z) := 0.
   
   Definition P_id_U43 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_a__isNatIListKind (x11:Z) := 0.
   
   Definition P_id_U84 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_s (x11:Z) := 1* x11.
   
   Definition P_id_a__U46 (x11:Z) := 0.
   
   Definition P_id_U32 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_a__U31 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_U13 (x11:Z) := 0.
   
   Definition P_id_a__U83 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_isNatIList (x11:Z) := 0.
   
   Lemma P_id_U51_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_U51 x12 x14 <= P_id_U51 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U84_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U84 x12 x14 x16 <= P_id_a__U84 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U32_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_a__U32 x12 x14 <= P_id_a__U32 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_isNatList_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_isNatList x12 <= P_id_isNatList x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U13_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_a__U13 x12 <= P_id_a__U13 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U85_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_U85 x12 x14 <= P_id_U85 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__length_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_a__length x12 <= P_id_a__length x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__isNatIList_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->
      P_id_a__isNatIList x12 <= P_id_a__isNatIList x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U33_monotonic :
    forall x12 x11, (0 <= x12)/\ (x12 <= x11) ->P_id_U33 x12 <= P_id_U33 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U11_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_a__U11 x12 x14 <= P_id_a__U11 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U81_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_U81 x12 x14 x16 <= P_id_U81 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U92_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U92 x12 x14 x16 <= P_id_a__U92 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U43_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U43 x12 x14 x16 <= P_id_a__U43 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U23_monotonic :
    forall x12 x11, (0 <= x12)/\ (x12 <= x11) ->P_id_U23 x12 <= P_id_U23 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__isNatKind_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->
      P_id_a__isNatKind x12 <= P_id_a__isNatKind x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U93_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_U93 x12 x14 x16 <= P_id_U93 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U11_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_U11 x12 x14 <= P_id_U11 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U71_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_a__U71 x12 <= P_id_a__U71 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U44_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_U44 x12 x14 x16 <= P_id_U44 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U61_monotonic :
    forall x12 x11, (0 <= x12)/\ (x12 <= x11) ->P_id_U61 x12 <= P_id_U61 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U86_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_a__U86 x12 <= P_id_a__U86 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U41_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U41 x12 x14 x16 <= P_id_a__U41 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U22_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_U22 x12 x14 <= P_id_U22 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U21_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_a__U21 x12 x14 <= P_id_a__U21 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U91_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_U91 x12 x14 x16 <= P_id_U91 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_length_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_length x12 <= P_id_length x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U52_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_a__U52 x12 <= P_id_a__U52 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U42_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_U42 x12 x14 x16 <= P_id_U42 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U12_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_a__U12 x12 x14 <= P_id_a__U12 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U83_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_U83 x12 x14 x16 <= P_id_U83 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U94_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_a__U94 x12 x14 <= P_id_a__U94 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U45_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_a__U45 x12 x14 <= P_id_a__U45 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U31_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_U31 x12 x14 <= P_id_U31 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__isNat_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_a__isNat x12 <= P_id_a__isNat x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_isNatIListKind_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->
      P_id_isNatIListKind x12 <= P_id_isNatIListKind x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U82_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U82 x12 x14 x16 <= P_id_a__U82 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U46_monotonic :
    forall x12 x11, (0 <= x12)/\ (x12 <= x11) ->P_id_U46 x12 <= P_id_U46 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_cons_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_cons x12 x14 <= P_id_cons x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U52_monotonic :
    forall x12 x11, (0 <= x12)/\ (x12 <= x11) ->P_id_U52 x12 <= P_id_U52 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U85_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_a__U85 x12 x14 <= P_id_a__U85 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U33_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_a__U33 x12 <= P_id_a__U33 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U21_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_U21 x12 x14 <= P_id_U21 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__isNatList_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->
      P_id_a__isNatList x12 <= P_id_a__isNatList x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U86_monotonic :
    forall x12 x11, (0 <= x12)/\ (x12 <= x11) ->P_id_U86 x12 <= P_id_U86 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_mark_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_mark x12 <= P_id_mark x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U51_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_a__U51 x12 x14 <= P_id_a__U51 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U41_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_U41 x12 x14 x16 <= P_id_U41 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U82_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_U82 x12 x14 x16 <= P_id_U82 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U93_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U93 x12 x14 x16 <= P_id_a__U93 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U44_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U44 x12 x14 x16 <= P_id_a__U44 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_isNat_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_isNat x12 <= P_id_isNat x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U23_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_a__U23 x12 <= P_id_a__U23 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U94_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_U94 x12 x14 <= P_id_U94 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U12_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_U12 x12 x14 <= P_id_U12 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U81_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U81 x12 x14 x16 <= P_id_a__U81 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U45_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_U45 x12 x14 <= P_id_U45 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U71_monotonic :
    forall x12 x11, (0 <= x12)/\ (x12 <= x11) ->P_id_U71 x12 <= P_id_U71 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U91_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U91 x12 x14 x16 <= P_id_a__U91 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U42_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U42 x12 x14 x16 <= P_id_a__U42 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_isNatKind_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_isNatKind x12 <= P_id_isNatKind x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U22_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_a__U22 x12 x14 <= P_id_a__U22 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U92_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_U92 x12 x14 x16 <= P_id_U92 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U61_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_a__U61 x12 <= P_id_a__U61 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U43_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_U43 x12 x14 x16 <= P_id_U43 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__isNatIListKind_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->
      P_id_a__isNatIListKind x12 <= P_id_a__isNatIListKind x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U84_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_U84 x12 x14 x16 <= P_id_U84 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_s_monotonic :
    forall x12 x11, (0 <= x12)/\ (x12 <= x11) ->P_id_s x12 <= P_id_s x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U46_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_a__U46 x12 <= P_id_a__U46 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U32_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_U32 x12 x14 <= P_id_U32 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U31_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_a__U31 x12 x14 <= P_id_a__U31 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U13_monotonic :
    forall x12 x11, (0 <= x12)/\ (x12 <= x11) ->P_id_U13 x12 <= P_id_U13 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U83_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U83 x12 x14 x16 <= P_id_a__U83 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_isNatIList_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_isNatIList x12 <= P_id_isNatIList x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__zeros_bounded : 0 <= P_id_a__zeros .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U51_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U51 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U84_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U84 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U32_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U32 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_isNatList_bounded :
    forall x11, (0 <= x11) ->0 <= P_id_isNatList x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U13_bounded : forall x11, (0 <= x11) ->0 <= P_id_a__U13 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U85_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U85 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__length_bounded :
    forall x11, (0 <= x11) ->0 <= P_id_a__length x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__isNatIList_bounded :
    forall x11, (0 <= x11) ->0 <= P_id_a__isNatIList x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U33_bounded : forall x11, (0 <= x11) ->0 <= P_id_U33 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U11_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U11 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U81_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U81 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U92_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U92 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U43_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U43 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U23_bounded : forall x11, (0 <= x11) ->0 <= P_id_U23 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__isNatKind_bounded :
    forall x11, (0 <= x11) ->0 <= P_id_a__isNatKind x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U93_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U93 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U11_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U11 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U71_bounded : forall x11, (0 <= x11) ->0 <= P_id_a__U71 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U44_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U44 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_0_bounded : 0 <= P_id_0 .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U61_bounded : forall x11, (0 <= x11) ->0 <= P_id_U61 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U86_bounded : forall x11, (0 <= x11) ->0 <= P_id_a__U86 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U41_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U41 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U22_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U22 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U21_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U21 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U91_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U91 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_length_bounded : forall x11, (0 <= x11) ->0 <= P_id_length x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U52_bounded : forall x11, (0 <= x11) ->0 <= P_id_a__U52 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U42_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U42 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U12_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U12 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U83_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U83 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U94_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U94 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U45_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U45 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U31_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U31 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__isNat_bounded :
    forall x11, (0 <= x11) ->0 <= P_id_a__isNat x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_isNatIListKind_bounded :
    forall x11, (0 <= x11) ->0 <= P_id_isNatIListKind x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U82_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U82 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U46_bounded : forall x11, (0 <= x11) ->0 <= P_id_U46 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_cons_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_cons x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U52_bounded : forall x11, (0 <= x11) ->0 <= P_id_U52 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U85_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U85 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U33_bounded : forall x11, (0 <= x11) ->0 <= P_id_a__U33 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U21_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U21 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__isNatList_bounded :
    forall x11, (0 <= x11) ->0 <= P_id_a__isNatList x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U86_bounded : forall x11, (0 <= x11) ->0 <= P_id_U86 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_mark_bounded : forall x11, (0 <= x11) ->0 <= P_id_mark x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U51_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U51 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U41_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U41 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_tt_bounded : 0 <= P_id_tt .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U82_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U82 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U93_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U93 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U44_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U44 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_isNat_bounded : forall x11, (0 <= x11) ->0 <= P_id_isNat x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U23_bounded : forall x11, (0 <= x11) ->0 <= P_id_a__U23 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U94_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U94 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U12_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U12 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U81_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U81 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U45_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U45 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_zeros_bounded : 0 <= P_id_zeros .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U71_bounded : forall x11, (0 <= x11) ->0 <= P_id_U71 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U91_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U91 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U42_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U42 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_isNatKind_bounded :
    forall x11, (0 <= x11) ->0 <= P_id_isNatKind x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U22_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U22 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U92_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U92 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_nil_bounded : 0 <= P_id_nil .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U61_bounded : forall x11, (0 <= x11) ->0 <= P_id_a__U61 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U43_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U43 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__isNatIListKind_bounded :
    forall x11, (0 <= x11) ->0 <= P_id_a__isNatIListKind x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U84_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U84 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_s_bounded : forall x11, (0 <= x11) ->0 <= P_id_s x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U46_bounded : forall x11, (0 <= x11) ->0 <= P_id_a__U46 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U32_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U32 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U31_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U31 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U13_bounded : forall x11, (0 <= x11) ->0 <= P_id_U13 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U83_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U83 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_isNatIList_bounded :
    forall x11, (0 <= x11) ->0 <= P_id_isNatIList x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Definition measure  := 
     InterpZ.measure 0 P_id_a__zeros P_id_U51 P_id_a__U84 P_id_a__U32 
      P_id_isNatList P_id_a__U13 P_id_U85 P_id_a__length P_id_a__isNatIList 
      P_id_U33 P_id_a__U11 P_id_U81 P_id_a__U92 P_id_a__U43 P_id_U23 
      P_id_a__isNatKind P_id_U93 P_id_U11 P_id_a__U71 P_id_U44 P_id_0 
      P_id_U61 P_id_a__U86 P_id_a__U41 P_id_U22 P_id_a__U21 P_id_U91 
      P_id_length P_id_a__U52 P_id_U42 P_id_a__U12 P_id_U83 P_id_a__U94 
      P_id_a__U45 P_id_U31 P_id_a__isNat P_id_isNatIListKind P_id_a__U82 
      P_id_U46 P_id_cons P_id_U52 P_id_a__U85 P_id_a__U33 P_id_U21 
      P_id_a__isNatList P_id_U86 P_id_mark P_id_a__U51 P_id_U41 P_id_tt 
      P_id_U82 P_id_a__U93 P_id_a__U44 P_id_isNat P_id_a__U23 P_id_U94 
      P_id_U12 P_id_a__U81 P_id_U45 P_id_zeros P_id_U71 P_id_a__U91 
      P_id_a__U42 P_id_isNatKind P_id_a__U22 P_id_U92 P_id_nil P_id_a__U61 
      P_id_U43 P_id_a__isNatIListKind P_id_U84 P_id_s P_id_a__U46 P_id_U32 
      P_id_a__U31 P_id_U13 P_id_a__U83 P_id_isNatIList.
   
   Lemma measure_equation :
    forall t, 
     measure t = match t with
                   | (algebra.Alg.Term algebra.F.id_a__zeros nil) =>
                    P_id_a__zeros 
                   | (algebra.Alg.Term algebra.F.id_U51 (x12::x11::nil)) =>
                    P_id_U51 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U84 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U84 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U32 (x12::x11::nil)) =>
                    P_id_a__U32 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_isNatList (x11::nil)) =>
                    P_id_isNatList (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U13 (x11::nil)) =>
                    P_id_a__U13 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U85 (x12::x11::nil)) =>
                    P_id_U85 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__length (x11::nil)) =>
                    P_id_a__length (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__isNatIList (x11::nil)) =>
                    P_id_a__isNatIList (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U33 (x11::nil)) =>
                    P_id_U33 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U11 (x12::x11::nil)) =>
                    P_id_a__U11 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U81 (x13::x12::x11::nil)) =>
                    P_id_U81 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U92 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U92 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U43 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U43 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U23 (x11::nil)) =>
                    P_id_U23 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__isNatKind (x11::nil)) =>
                    P_id_a__isNatKind (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U93 (x13::x12::x11::nil)) =>
                    P_id_U93 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U11 (x12::x11::nil)) =>
                    P_id_U11 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U71 (x11::nil)) =>
                    P_id_a__U71 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U44 (x13::x12::x11::nil)) =>
                    P_id_U44 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_0 nil) => P_id_0 
                   | (algebra.Alg.Term algebra.F.id_U61 (x11::nil)) =>
                    P_id_U61 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U86 (x11::nil)) =>
                    P_id_a__U86 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U41 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U41 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U22 (x12::x11::nil)) =>
                    P_id_U22 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U21 (x12::x11::nil)) =>
                    P_id_a__U21 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U91 (x13::x12::x11::nil)) =>
                    P_id_U91 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_length (x11::nil)) =>
                    P_id_length (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U52 (x11::nil)) =>
                    P_id_a__U52 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U42 (x13::x12::x11::nil)) =>
                    P_id_U42 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U12 (x12::x11::nil)) =>
                    P_id_a__U12 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U83 (x13::x12::x11::nil)) =>
                    P_id_U83 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U94 (x12::x11::nil)) =>
                    P_id_a__U94 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U45 (x12::x11::nil)) =>
                    P_id_a__U45 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U31 (x12::x11::nil)) =>
                    P_id_U31 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__isNat (x11::nil)) =>
                    P_id_a__isNat (measure x11)
                   | (algebra.Alg.Term algebra.F.id_isNatIListKind 
                      (x11::nil)) =>
                    P_id_isNatIListKind (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U82 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U82 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U46 (x11::nil)) =>
                    P_id_U46 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_cons (x12::x11::nil)) =>
                    P_id_cons (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U52 (x11::nil)) =>
                    P_id_U52 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U85 (x12::x11::nil)) =>
                    P_id_a__U85 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U33 (x11::nil)) =>
                    P_id_a__U33 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U21 (x12::x11::nil)) =>
                    P_id_U21 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__isNatList (x11::nil)) =>
                    P_id_a__isNatList (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U86 (x11::nil)) =>
                    P_id_U86 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_mark (x11::nil)) =>
                    P_id_mark (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U51 (x12::x11::nil)) =>
                    P_id_a__U51 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U41 (x13::x12::x11::nil)) =>
                    P_id_U41 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt 
                   | (algebra.Alg.Term algebra.F.id_U82 (x13::x12::x11::nil)) =>
                    P_id_U82 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U93 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U93 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U44 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U44 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_isNat (x11::nil)) =>
                    P_id_isNat (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U23 (x11::nil)) =>
                    P_id_a__U23 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U94 (x12::x11::nil)) =>
                    P_id_U94 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U12 (x12::x11::nil)) =>
                    P_id_U12 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U81 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U81 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U45 (x12::x11::nil)) =>
                    P_id_U45 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_zeros nil) => P_id_zeros 
                   | (algebra.Alg.Term algebra.F.id_U71 (x11::nil)) =>
                    P_id_U71 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U91 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U91 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U42 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U42 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_isNatKind (x11::nil)) =>
                    P_id_isNatKind (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U22 (x12::x11::nil)) =>
                    P_id_a__U22 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U92 (x13::x12::x11::nil)) =>
                    P_id_U92 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil 
                   | (algebra.Alg.Term algebra.F.id_a__U61 (x11::nil)) =>
                    P_id_a__U61 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U43 (x13::x12::x11::nil)) =>
                    P_id_U43 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
                      (x11::nil)) =>
                    P_id_a__isNatIListKind (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U84 (x13::x12::x11::nil)) =>
                    P_id_U84 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_s (x11::nil)) =>
                    P_id_s (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U46 (x11::nil)) =>
                    P_id_a__U46 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U32 (x12::x11::nil)) =>
                    P_id_U32 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U31 (x12::x11::nil)) =>
                    P_id_a__U31 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U13 (x11::nil)) =>
                    P_id_U13 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U83 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U83 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_isNatIList (x11::nil)) =>
                    P_id_isNatIList (measure x11)
                   | _ => 0
                   end.
   Proof.
     intros t;case t;intros ;apply refl_equal.
   Qed.
   
   Lemma measure_bounded : forall t, 0 <= measure t.
   Proof.
     unfold measure in |-*.
     
     apply InterpZ.measure_bounded;
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Ltac generate_pos_hyp  :=
    match goal with
      | H:context [measure ?x] |- _ =>
       let v := fresh "v" in 
        (let H := fresh "h" in 
          (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
            clearbody H;clearbody v))
      |  |- context [measure ?x] =>
       let v := fresh "v" in 
        (let H := fresh "h" in 
          (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
            clearbody H;clearbody v))
      end
    .
   
   Lemma rules_monotonic :
    forall l r, 
     (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) ->
      measure r <= measure l.
   Proof.
     intros l r H.
     fold measure in |-*.
     
     inversion H;clear H;subst;inversion H0;clear H0;subst;
      simpl algebra.EQT.T.apply_subst in |-*;
      repeat (
      match goal with
        |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
         rewrite (measure_equation (algebra.Alg.Term f t))
        end
      );repeat (generate_pos_hyp );
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma measure_star_monotonic :
    forall l r, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                r l) ->measure r <= measure l.
   Proof.
     unfold measure in *.
     apply InterpZ.measure_star_monotonic.
     intros ;apply P_id_U51_monotonic;assumption.
     intros ;apply P_id_a__U84_monotonic;assumption.
     intros ;apply P_id_a__U32_monotonic;assumption.
     intros ;apply P_id_isNatList_monotonic;assumption.
     intros ;apply P_id_a__U13_monotonic;assumption.
     intros ;apply P_id_U85_monotonic;assumption.
     intros ;apply P_id_a__length_monotonic;assumption.
     intros ;apply P_id_a__isNatIList_monotonic;assumption.
     intros ;apply P_id_U33_monotonic;assumption.
     intros ;apply P_id_a__U11_monotonic;assumption.
     intros ;apply P_id_U81_monotonic;assumption.
     intros ;apply P_id_a__U92_monotonic;assumption.
     intros ;apply P_id_a__U43_monotonic;assumption.
     intros ;apply P_id_U23_monotonic;assumption.
     intros ;apply P_id_a__isNatKind_monotonic;assumption.
     intros ;apply P_id_U93_monotonic;assumption.
     intros ;apply P_id_U11_monotonic;assumption.
     intros ;apply P_id_a__U71_monotonic;assumption.
     intros ;apply P_id_U44_monotonic;assumption.
     intros ;apply P_id_U61_monotonic;assumption.
     intros ;apply P_id_a__U86_monotonic;assumption.
     intros ;apply P_id_a__U41_monotonic;assumption.
     intros ;apply P_id_U22_monotonic;assumption.
     intros ;apply P_id_a__U21_monotonic;assumption.
     intros ;apply P_id_U91_monotonic;assumption.
     intros ;apply P_id_length_monotonic;assumption.
     intros ;apply P_id_a__U52_monotonic;assumption.
     intros ;apply P_id_U42_monotonic;assumption.
     intros ;apply P_id_a__U12_monotonic;assumption.
     intros ;apply P_id_U83_monotonic;assumption.
     intros ;apply P_id_a__U94_monotonic;assumption.
     intros ;apply P_id_a__U45_monotonic;assumption.
     intros ;apply P_id_U31_monotonic;assumption.
     intros ;apply P_id_a__isNat_monotonic;assumption.
     intros ;apply P_id_isNatIListKind_monotonic;assumption.
     intros ;apply P_id_a__U82_monotonic;assumption.
     intros ;apply P_id_U46_monotonic;assumption.
     intros ;apply P_id_cons_monotonic;assumption.
     intros ;apply P_id_U52_monotonic;assumption.
     intros ;apply P_id_a__U85_monotonic;assumption.
     intros ;apply P_id_a__U33_monotonic;assumption.
     intros ;apply P_id_U21_monotonic;assumption.
     intros ;apply P_id_a__isNatList_monotonic;assumption.
     intros ;apply P_id_U86_monotonic;assumption.
     intros ;apply P_id_mark_monotonic;assumption.
     intros ;apply P_id_a__U51_monotonic;assumption.
     intros ;apply P_id_U41_monotonic;assumption.
     intros ;apply P_id_U82_monotonic;assumption.
     intros ;apply P_id_a__U93_monotonic;assumption.
     intros ;apply P_id_a__U44_monotonic;assumption.
     intros ;apply P_id_isNat_monotonic;assumption.
     intros ;apply P_id_a__U23_monotonic;assumption.
     intros ;apply P_id_U94_monotonic;assumption.
     intros ;apply P_id_U12_monotonic;assumption.
     intros ;apply P_id_a__U81_monotonic;assumption.
     intros ;apply P_id_U45_monotonic;assumption.
     intros ;apply P_id_U71_monotonic;assumption.
     intros ;apply P_id_a__U91_monotonic;assumption.
     intros ;apply P_id_a__U42_monotonic;assumption.
     intros ;apply P_id_isNatKind_monotonic;assumption.
     intros ;apply P_id_a__U22_monotonic;assumption.
     intros ;apply P_id_U92_monotonic;assumption.
     intros ;apply P_id_a__U61_monotonic;assumption.
     intros ;apply P_id_U43_monotonic;assumption.
     intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
     intros ;apply P_id_U84_monotonic;assumption.
     intros ;apply P_id_s_monotonic;assumption.
     intros ;apply P_id_a__U46_monotonic;assumption.
     intros ;apply P_id_U32_monotonic;assumption.
     intros ;apply P_id_a__U31_monotonic;assumption.
     intros ;apply P_id_U13_monotonic;assumption.
     intros ;apply P_id_a__U83_monotonic;assumption.
     intros ;apply P_id_isNatIList_monotonic;assumption.
     intros ;apply P_id_a__zeros_bounded;assumption.
     intros ;apply P_id_U51_bounded;assumption.
     intros ;apply P_id_a__U84_bounded;assumption.
     intros ;apply P_id_a__U32_bounded;assumption.
     intros ;apply P_id_isNatList_bounded;assumption.
     intros ;apply P_id_a__U13_bounded;assumption.
     intros ;apply P_id_U85_bounded;assumption.
     intros ;apply P_id_a__length_bounded;assumption.
     intros ;apply P_id_a__isNatIList_bounded;assumption.
     intros ;apply P_id_U33_bounded;assumption.
     intros ;apply P_id_a__U11_bounded;assumption.
     intros ;apply P_id_U81_bounded;assumption.
     intros ;apply P_id_a__U92_bounded;assumption.
     intros ;apply P_id_a__U43_bounded;assumption.
     intros ;apply P_id_U23_bounded;assumption.
     intros ;apply P_id_a__isNatKind_bounded;assumption.
     intros ;apply P_id_U93_bounded;assumption.
     intros ;apply P_id_U11_bounded;assumption.
     intros ;apply P_id_a__U71_bounded;assumption.
     intros ;apply P_id_U44_bounded;assumption.
     intros ;apply P_id_0_bounded;assumption.
     intros ;apply P_id_U61_bounded;assumption.
     intros ;apply P_id_a__U86_bounded;assumption.
     intros ;apply P_id_a__U41_bounded;assumption.
     intros ;apply P_id_U22_bounded;assumption.
     intros ;apply P_id_a__U21_bounded;assumption.
     intros ;apply P_id_U91_bounded;assumption.
     intros ;apply P_id_length_bounded;assumption.
     intros ;apply P_id_a__U52_bounded;assumption.
     intros ;apply P_id_U42_bounded;assumption.
     intros ;apply P_id_a__U12_bounded;assumption.
     intros ;apply P_id_U83_bounded;assumption.
     intros ;apply P_id_a__U94_bounded;assumption.
     intros ;apply P_id_a__U45_bounded;assumption.
     intros ;apply P_id_U31_bounded;assumption.
     intros ;apply P_id_a__isNat_bounded;assumption.
     intros ;apply P_id_isNatIListKind_bounded;assumption.
     intros ;apply P_id_a__U82_bounded;assumption.
     intros ;apply P_id_U46_bounded;assumption.
     intros ;apply P_id_cons_bounded;assumption.
     intros ;apply P_id_U52_bounded;assumption.
     intros ;apply P_id_a__U85_bounded;assumption.
     intros ;apply P_id_a__U33_bounded;assumption.
     intros ;apply P_id_U21_bounded;assumption.
     intros ;apply P_id_a__isNatList_bounded;assumption.
     intros ;apply P_id_U86_bounded;assumption.
     intros ;apply P_id_mark_bounded;assumption.
     intros ;apply P_id_a__U51_bounded;assumption.
     intros ;apply P_id_U41_bounded;assumption.
     intros ;apply P_id_tt_bounded;assumption.
     intros ;apply P_id_U82_bounded;assumption.
     intros ;apply P_id_a__U93_bounded;assumption.
     intros ;apply P_id_a__U44_bounded;assumption.
     intros ;apply P_id_isNat_bounded;assumption.
     intros ;apply P_id_a__U23_bounded;assumption.
     intros ;apply P_id_U94_bounded;assumption.
     intros ;apply P_id_U12_bounded;assumption.
     intros ;apply P_id_a__U81_bounded;assumption.
     intros ;apply P_id_U45_bounded;assumption.
     intros ;apply P_id_zeros_bounded;assumption.
     intros ;apply P_id_U71_bounded;assumption.
     intros ;apply P_id_a__U91_bounded;assumption.
     intros ;apply P_id_a__U42_bounded;assumption.
     intros ;apply P_id_isNatKind_bounded;assumption.
     intros ;apply P_id_a__U22_bounded;assumption.
     intros ;apply P_id_U92_bounded;assumption.
     intros ;apply P_id_nil_bounded;assumption.
     intros ;apply P_id_a__U61_bounded;assumption.
     intros ;apply P_id_U43_bounded;assumption.
     intros ;apply P_id_a__isNatIListKind_bounded;assumption.
     intros ;apply P_id_U84_bounded;assumption.
     intros ;apply P_id_s_bounded;assumption.
     intros ;apply P_id_a__U46_bounded;assumption.
     intros ;apply P_id_U32_bounded;assumption.
     intros ;apply P_id_a__U31_bounded;assumption.
     intros ;apply P_id_U13_bounded;assumption.
     intros ;apply P_id_a__U83_bounded;assumption.
     intros ;apply P_id_isNatIList_bounded;assumption.
     apply rules_monotonic.
   Qed.
   
   Definition P_id_A__U46 (x11:Z) := 0.
   
   Definition P_id_A__ISNATILISTKIND (x11:Z) := 1 + 2* x11.
   
   Definition P_id_A__U61 (x11:Z) := 0.
   
   Definition P_id_A__U85 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_A__U31 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_A__U81 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_A__U22 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_A__U93 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_A__U42 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_A__U51 (x11:Z) (x12:Z) := 1 + 2* x12.
   
   Definition P_id_A__ISNATLIST (x11:Z) := 0.
   
   Definition P_id_A__ZEROS  := 0.
   
   Definition P_id_A__U91 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_A__U33 (x11:Z) := 0.
   
   Definition P_id_A__U83 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_A__U23 (x11:Z) := 0.
   
   Definition P_id_A__U11 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_A__LENGTH (x11:Z) := 0.
   
   Definition P_id_A__U44 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_A__ISNATILIST (x11:Z) := 0.
   
   Definition P_id_A__U13 (x11:Z) := 0.
   
   Definition P_id_A__U71 (x11:Z) := 0.
   
   Definition P_id_A__U86 (x11:Z) := 0.
   
   Definition P_id_A__U32 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_A__U82 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_A__ISNATKIND (x11:Z) := 2* x11.
   
   Definition P_id_A__U94 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_A__U43 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_A__U52 (x11:Z) := 0.
   
   Definition P_id_A__U21 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_A__U92 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_A__U41 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_A__U84 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_A__ISNAT (x11:Z) := 0.
   
   Definition P_id_A__U12 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_MARK (x11:Z) := 0.
   
   Definition P_id_A__U45 (x11:Z) (x12:Z) := 0.
   
   Lemma P_id_A__U46_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_A__U46 x12 <= P_id_A__U46 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__ISNATILISTKIND_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->
      P_id_A__ISNATILISTKIND x12 <= P_id_A__ISNATILISTKIND x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U61_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_A__U61 x12 <= P_id_A__U61 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U85_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_A__U85 x12 x14 <= P_id_A__U85 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U31_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_A__U31 x12 x14 <= P_id_A__U31 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U81_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U81 x12 x14 x16 <= P_id_A__U81 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U22_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_A__U22 x12 x14 <= P_id_A__U22 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U93_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U93 x12 x14 x16 <= P_id_A__U93 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U42_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U42 x12 x14 x16 <= P_id_A__U42 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U51_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_A__U51 x12 x14 <= P_id_A__U51 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__ISNATLIST_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->
      P_id_A__ISNATLIST x12 <= P_id_A__ISNATLIST x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U91_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U91 x12 x14 x16 <= P_id_A__U91 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U33_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_A__U33 x12 <= P_id_A__U33 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U83_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U83 x12 x14 x16 <= P_id_A__U83 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U23_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_A__U23 x12 <= P_id_A__U23 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U11_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_A__U11 x12 x14 <= P_id_A__U11 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__LENGTH_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_A__LENGTH x12 <= P_id_A__LENGTH x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U44_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U44 x12 x14 x16 <= P_id_A__U44 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__ISNATILIST_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->
      P_id_A__ISNATILIST x12 <= P_id_A__ISNATILIST x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U13_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_A__U13 x12 <= P_id_A__U13 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U71_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_A__U71 x12 <= P_id_A__U71 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U86_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_A__U86 x12 <= P_id_A__U86 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U32_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_A__U32 x12 x14 <= P_id_A__U32 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U82_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U82 x12 x14 x16 <= P_id_A__U82 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__ISNATKIND_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->
      P_id_A__ISNATKIND x12 <= P_id_A__ISNATKIND x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U94_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_A__U94 x12 x14 <= P_id_A__U94 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U43_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U43 x12 x14 x16 <= P_id_A__U43 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U52_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_A__U52 x12 <= P_id_A__U52 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U21_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_A__U21 x12 x14 <= P_id_A__U21 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U92_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U92 x12 x14 x16 <= P_id_A__U92 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U41_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U41 x12 x14 x16 <= P_id_A__U41 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U84_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U84 x12 x14 x16 <= P_id_A__U84 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__ISNAT_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_A__ISNAT x12 <= P_id_A__ISNAT x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U12_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_A__U12 x12 x14 <= P_id_A__U12 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_MARK_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_MARK x12 <= P_id_MARK x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U45_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_A__U45 x12 x14 <= P_id_A__U45 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Definition marked_measure  := 
     InterpZ.marked_measure 0 P_id_a__zeros P_id_U51 P_id_a__U84 P_id_a__U32 
      P_id_isNatList P_id_a__U13 P_id_U85 P_id_a__length P_id_a__isNatIList 
      P_id_U33 P_id_a__U11 P_id_U81 P_id_a__U92 P_id_a__U43 P_id_U23 
      P_id_a__isNatKind P_id_U93 P_id_U11 P_id_a__U71 P_id_U44 P_id_0 
      P_id_U61 P_id_a__U86 P_id_a__U41 P_id_U22 P_id_a__U21 P_id_U91 
      P_id_length P_id_a__U52 P_id_U42 P_id_a__U12 P_id_U83 P_id_a__U94 
      P_id_a__U45 P_id_U31 P_id_a__isNat P_id_isNatIListKind P_id_a__U82 
      P_id_U46 P_id_cons P_id_U52 P_id_a__U85 P_id_a__U33 P_id_U21 
      P_id_a__isNatList P_id_U86 P_id_mark P_id_a__U51 P_id_U41 P_id_tt 
      P_id_U82 P_id_a__U93 P_id_a__U44 P_id_isNat P_id_a__U23 P_id_U94 
      P_id_U12 P_id_a__U81 P_id_U45 P_id_zeros P_id_U71 P_id_a__U91 
      P_id_a__U42 P_id_isNatKind P_id_a__U22 P_id_U92 P_id_nil P_id_a__U61 
      P_id_U43 P_id_a__isNatIListKind P_id_U84 P_id_s P_id_a__U46 P_id_U32 
      P_id_a__U31 P_id_U13 P_id_a__U83 P_id_isNatIList P_id_A__U46 
      P_id_A__ISNATILISTKIND P_id_A__U61 P_id_A__U85 P_id_A__U31 P_id_A__U81 
      P_id_A__U22 P_id_A__U93 P_id_A__U42 P_id_A__U51 P_id_A__ISNATLIST 
      P_id_A__ZEROS P_id_A__U91 P_id_A__U33 P_id_A__U83 P_id_A__U23 
      P_id_A__U11 P_id_A__LENGTH P_id_A__U44 P_id_A__ISNATILIST P_id_A__U13 
      P_id_A__U71 P_id_A__U86 P_id_A__U32 P_id_A__U82 P_id_A__ISNATKIND 
      P_id_A__U94 P_id_A__U43 P_id_A__U52 P_id_A__U21 P_id_A__U92 
      P_id_A__U41 P_id_A__U84 P_id_A__ISNAT P_id_A__U12 P_id_MARK P_id_A__U45
     .
   
   Lemma marked_measure_equation :
    forall t, 
     marked_measure t = match t with
                          | (algebra.Alg.Term algebra.F.id_a__U46 (x11::nil)) =>
                           P_id_A__U46 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
                             (x11::nil)) =>
                           P_id_A__ISNATILISTKIND (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U61 (x11::nil)) =>
                           P_id_A__U61 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U85 (x12::
                             x11::nil)) =>
                           P_id_A__U85 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U31 (x12::
                             x11::nil)) =>
                           P_id_A__U31 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U81 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U81 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U22 (x12::
                             x11::nil)) =>
                           P_id_A__U22 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U93 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U93 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U42 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U42 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U51 (x12::
                             x11::nil)) =>
                           P_id_A__U51 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__isNatList 
                             (x11::nil)) =>
                           P_id_A__ISNATLIST (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__zeros nil) =>
                           P_id_A__ZEROS 
                          | (algebra.Alg.Term algebra.F.id_a__U91 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U91 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U33 (x11::nil)) =>
                           P_id_A__U33 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U83 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U83 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U23 (x11::nil)) =>
                           P_id_A__U23 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U11 (x12::
                             x11::nil)) =>
                           P_id_A__U11 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__length 
                             (x11::nil)) =>
                           P_id_A__LENGTH (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U44 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U44 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__isNatIList 
                             (x11::nil)) =>
                           P_id_A__ISNATILIST (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U13 (x11::nil)) =>
                           P_id_A__U13 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U71 (x11::nil)) =>
                           P_id_A__U71 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U86 (x11::nil)) =>
                           P_id_A__U86 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U32 (x12::
                             x11::nil)) =>
                           P_id_A__U32 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U82 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U82 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__isNatKind 
                             (x11::nil)) =>
                           P_id_A__ISNATKIND (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U94 (x12::
                             x11::nil)) =>
                           P_id_A__U94 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U43 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U43 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U52 (x11::nil)) =>
                           P_id_A__U52 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U21 (x12::
                             x11::nil)) =>
                           P_id_A__U21 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U92 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U92 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U41 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U41 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U84 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U84 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__isNat 
                             (x11::nil)) =>
                           P_id_A__ISNAT (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U12 (x12::
                             x11::nil)) =>
                           P_id_A__U12 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_mark (x11::nil)) =>
                           P_id_MARK (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U45 (x12::
                             x11::nil)) =>
                           P_id_A__U45 (measure x12) (measure x11)
                          | _ => measure t
                          end.
   Proof.
     reflexivity .
   Qed.
   
   Lemma marked_measure_star_monotonic :
    forall f l1 l2, 
     (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step 
                                                       R_xml_0_deep_rew.R_xml_0_rules)
                                                      ) l1 l2) ->
      marked_measure (algebra.Alg.Term f l1) <= marked_measure (algebra.Alg.Term 
                                                                 f l2).
   Proof.
     unfold marked_measure in *.
     apply InterpZ.marked_measure_star_monotonic.
     intros ;apply P_id_U51_monotonic;assumption.
     intros ;apply P_id_a__U84_monotonic;assumption.
     intros ;apply P_id_a__U32_monotonic;assumption.
     intros ;apply P_id_isNatList_monotonic;assumption.
     intros ;apply P_id_a__U13_monotonic;assumption.
     intros ;apply P_id_U85_monotonic;assumption.
     intros ;apply P_id_a__length_monotonic;assumption.
     intros ;apply P_id_a__isNatIList_monotonic;assumption.
     intros ;apply P_id_U33_monotonic;assumption.
     intros ;apply P_id_a__U11_monotonic;assumption.
     intros ;apply P_id_U81_monotonic;assumption.
     intros ;apply P_id_a__U92_monotonic;assumption.
     intros ;apply P_id_a__U43_monotonic;assumption.
     intros ;apply P_id_U23_monotonic;assumption.
     intros ;apply P_id_a__isNatKind_monotonic;assumption.
     intros ;apply P_id_U93_monotonic;assumption.
     intros ;apply P_id_U11_monotonic;assumption.
     intros ;apply P_id_a__U71_monotonic;assumption.
     intros ;apply P_id_U44_monotonic;assumption.
     intros ;apply P_id_U61_monotonic;assumption.
     intros ;apply P_id_a__U86_monotonic;assumption.
     intros ;apply P_id_a__U41_monotonic;assumption.
     intros ;apply P_id_U22_monotonic;assumption.
     intros ;apply P_id_a__U21_monotonic;assumption.
     intros ;apply P_id_U91_monotonic;assumption.
     intros ;apply P_id_length_monotonic;assumption.
     intros ;apply P_id_a__U52_monotonic;assumption.
     intros ;apply P_id_U42_monotonic;assumption.
     intros ;apply P_id_a__U12_monotonic;assumption.
     intros ;apply P_id_U83_monotonic;assumption.
     intros ;apply P_id_a__U94_monotonic;assumption.
     intros ;apply P_id_a__U45_monotonic;assumption.
     intros ;apply P_id_U31_monotonic;assumption.
     intros ;apply P_id_a__isNat_monotonic;assumption.
     intros ;apply P_id_isNatIListKind_monotonic;assumption.
     intros ;apply P_id_a__U82_monotonic;assumption.
     intros ;apply P_id_U46_monotonic;assumption.
     intros ;apply P_id_cons_monotonic;assumption.
     intros ;apply P_id_U52_monotonic;assumption.
     intros ;apply P_id_a__U85_monotonic;assumption.
     intros ;apply P_id_a__U33_monotonic;assumption.
     intros ;apply P_id_U21_monotonic;assumption.
     intros ;apply P_id_a__isNatList_monotonic;assumption.
     intros ;apply P_id_U86_monotonic;assumption.
     intros ;apply P_id_mark_monotonic;assumption.
     intros ;apply P_id_a__U51_monotonic;assumption.
     intros ;apply P_id_U41_monotonic;assumption.
     intros ;apply P_id_U82_monotonic;assumption.
     intros ;apply P_id_a__U93_monotonic;assumption.
     intros ;apply P_id_a__U44_monotonic;assumption.
     intros ;apply P_id_isNat_monotonic;assumption.
     intros ;apply P_id_a__U23_monotonic;assumption.
     intros ;apply P_id_U94_monotonic;assumption.
     intros ;apply P_id_U12_monotonic;assumption.
     intros ;apply P_id_a__U81_monotonic;assumption.
     intros ;apply P_id_U45_monotonic;assumption.
     intros ;apply P_id_U71_monotonic;assumption.
     intros ;apply P_id_a__U91_monotonic;assumption.
     intros ;apply P_id_a__U42_monotonic;assumption.
     intros ;apply P_id_isNatKind_monotonic;assumption.
     intros ;apply P_id_a__U22_monotonic;assumption.
     intros ;apply P_id_U92_monotonic;assumption.
     intros ;apply P_id_a__U61_monotonic;assumption.
     intros ;apply P_id_U43_monotonic;assumption.
     intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
     intros ;apply P_id_U84_monotonic;assumption.
     intros ;apply P_id_s_monotonic;assumption.
     intros ;apply P_id_a__U46_monotonic;assumption.
     intros ;apply P_id_U32_monotonic;assumption.
     intros ;apply P_id_a__U31_monotonic;assumption.
     intros ;apply P_id_U13_monotonic;assumption.
     intros ;apply P_id_a__U83_monotonic;assumption.
     intros ;apply P_id_isNatIList_monotonic;assumption.
     intros ;apply P_id_a__zeros_bounded;assumption.
     intros ;apply P_id_U51_bounded;assumption.
     intros ;apply P_id_a__U84_bounded;assumption.
     intros ;apply P_id_a__U32_bounded;assumption.
     intros ;apply P_id_isNatList_bounded;assumption.
     intros ;apply P_id_a__U13_bounded;assumption.
     intros ;apply P_id_U85_bounded;assumption.
     intros ;apply P_id_a__length_bounded;assumption.
     intros ;apply P_id_a__isNatIList_bounded;assumption.
     intros ;apply P_id_U33_bounded;assumption.
     intros ;apply P_id_a__U11_bounded;assumption.
     intros ;apply P_id_U81_bounded;assumption.
     intros ;apply P_id_a__U92_bounded;assumption.
     intros ;apply P_id_a__U43_bounded;assumption.
     intros ;apply P_id_U23_bounded;assumption.
     intros ;apply P_id_a__isNatKind_bounded;assumption.
     intros ;apply P_id_U93_bounded;assumption.
     intros ;apply P_id_U11_bounded;assumption.
     intros ;apply P_id_a__U71_bounded;assumption.
     intros ;apply P_id_U44_bounded;assumption.
     intros ;apply P_id_0_bounded;assumption.
     intros ;apply P_id_U61_bounded;assumption.
     intros ;apply P_id_a__U86_bounded;assumption.
     intros ;apply P_id_a__U41_bounded;assumption.
     intros ;apply P_id_U22_bounded;assumption.
     intros ;apply P_id_a__U21_bounded;assumption.
     intros ;apply P_id_U91_bounded;assumption.
     intros ;apply P_id_length_bounded;assumption.
     intros ;apply P_id_a__U52_bounded;assumption.
     intros ;apply P_id_U42_bounded;assumption.
     intros ;apply P_id_a__U12_bounded;assumption.
     intros ;apply P_id_U83_bounded;assumption.
     intros ;apply P_id_a__U94_bounded;assumption.
     intros ;apply P_id_a__U45_bounded;assumption.
     intros ;apply P_id_U31_bounded;assumption.
     intros ;apply P_id_a__isNat_bounded;assumption.
     intros ;apply P_id_isNatIListKind_bounded;assumption.
     intros ;apply P_id_a__U82_bounded;assumption.
     intros ;apply P_id_U46_bounded;assumption.
     intros ;apply P_id_cons_bounded;assumption.
     intros ;apply P_id_U52_bounded;assumption.
     intros ;apply P_id_a__U85_bounded;assumption.
     intros ;apply P_id_a__U33_bounded;assumption.
     intros ;apply P_id_U21_bounded;assumption.
     intros ;apply P_id_a__isNatList_bounded;assumption.
     intros ;apply P_id_U86_bounded;assumption.
     intros ;apply P_id_mark_bounded;assumption.
     intros ;apply P_id_a__U51_bounded;assumption.
     intros ;apply P_id_U41_bounded;assumption.
     intros ;apply P_id_tt_bounded;assumption.
     intros ;apply P_id_U82_bounded;assumption.
     intros ;apply P_id_a__U93_bounded;assumption.
     intros ;apply P_id_a__U44_bounded;assumption.
     intros ;apply P_id_isNat_bounded;assumption.
     intros ;apply P_id_a__U23_bounded;assumption.
     intros ;apply P_id_U94_bounded;assumption.
     intros ;apply P_id_U12_bounded;assumption.
     intros ;apply P_id_a__U81_bounded;assumption.
     intros ;apply P_id_U45_bounded;assumption.
     intros ;apply P_id_zeros_bounded;assumption.
     intros ;apply P_id_U71_bounded;assumption.
     intros ;apply P_id_a__U91_bounded;assumption.
     intros ;apply P_id_a__U42_bounded;assumption.
     intros ;apply P_id_isNatKind_bounded;assumption.
     intros ;apply P_id_a__U22_bounded;assumption.
     intros ;apply P_id_U92_bounded;assumption.
     intros ;apply P_id_nil_bounded;assumption.
     intros ;apply P_id_a__U61_bounded;assumption.
     intros ;apply P_id_U43_bounded;assumption.
     intros ;apply P_id_a__isNatIListKind_bounded;assumption.
     intros ;apply P_id_U84_bounded;assumption.
     intros ;apply P_id_s_bounded;assumption.
     intros ;apply P_id_a__U46_bounded;assumption.
     intros ;apply P_id_U32_bounded;assumption.
     intros ;apply P_id_a__U31_bounded;assumption.
     intros ;apply P_id_U13_bounded;assumption.
     intros ;apply P_id_a__U83_bounded;assumption.
     intros ;apply P_id_isNatIList_bounded;assumption.
     apply rules_monotonic.
     intros ;apply P_id_A__U46_monotonic;assumption.
     intros ;apply P_id_A__ISNATILISTKIND_monotonic;assumption.
     intros ;apply P_id_A__U61_monotonic;assumption.
     intros ;apply P_id_A__U85_monotonic;assumption.
     intros ;apply P_id_A__U31_monotonic;assumption.
     intros ;apply P_id_A__U81_monotonic;assumption.
     intros ;apply P_id_A__U22_monotonic;assumption.
     intros ;apply P_id_A__U93_monotonic;assumption.
     intros ;apply P_id_A__U42_monotonic;assumption.
     intros ;apply P_id_A__U51_monotonic;assumption.
     intros ;apply P_id_A__ISNATLIST_monotonic;assumption.
     intros ;apply P_id_A__U91_monotonic;assumption.
     intros ;apply P_id_A__U33_monotonic;assumption.
     intros ;apply P_id_A__U83_monotonic;assumption.
     intros ;apply P_id_A__U23_monotonic;assumption.
     intros ;apply P_id_A__U11_monotonic;assumption.
     intros ;apply P_id_A__LENGTH_monotonic;assumption.
     intros ;apply P_id_A__U44_monotonic;assumption.
     intros ;apply P_id_A__ISNATILIST_monotonic;assumption.
     intros ;apply P_id_A__U13_monotonic;assumption.
     intros ;apply P_id_A__U71_monotonic;assumption.
     intros ;apply P_id_A__U86_monotonic;assumption.
     intros ;apply P_id_A__U32_monotonic;assumption.
     intros ;apply P_id_A__U82_monotonic;assumption.
     intros ;apply P_id_A__ISNATKIND_monotonic;assumption.
     intros ;apply P_id_A__U94_monotonic;assumption.
     intros ;apply P_id_A__U43_monotonic;assumption.
     intros ;apply P_id_A__U52_monotonic;assumption.
     intros ;apply P_id_A__U21_monotonic;assumption.
     intros ;apply P_id_A__U92_monotonic;assumption.
     intros ;apply P_id_A__U41_monotonic;assumption.
     intros ;apply P_id_A__U84_monotonic;assumption.
     intros ;apply P_id_A__ISNAT_monotonic;assumption.
     intros ;apply P_id_A__U12_monotonic;assumption.
     intros ;apply P_id_MARK_monotonic;assumption.
     intros ;apply P_id_A__U45_monotonic;assumption.
   Qed.
   
   Ltac rewrite_and_unfold  :=
    do 2 (rewrite marked_measure_equation);
     repeat (
     match goal with
       |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
        rewrite (measure_equation (algebra.Alg.Term f t))
       | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ =>
        rewrite (measure_equation (algebra.Alg.Term f t)) in H|-
       end
     ).
   
   Definition lt a b := (Zwf.Zwf 0) (marked_measure a) (marked_measure b).
   
   Definition le a b := marked_measure a <= marked_measure b.
   
   Lemma lt_le_compat : forall a b c, (lt a b) ->(le b c) ->lt a c.
   Proof.
     unfold lt, le in *.
     intros a b c.
     apply (interp.le_lt_compat_right (interp.o_Z 0)).
   Qed.
   
   Lemma wf_lt : well_founded lt.
   Proof.
     unfold lt in *.
     apply Inverse_Image.wf_inverse_image with  (B:=Z).
     apply Zwf.Zwf_well_founded.
   Qed.
   
   Lemma DP_R_xml_0_scc_14_strict_in_lt :
    Relation_Definitions.inclusion _ DP_R_xml_0_scc_14_strict lt.
   Proof.
     unfold Relation_Definitions.inclusion, lt in *.
     
     intros a b H;destruct H;
      match goal with
        |  |- (Zwf.Zwf 0) _ (marked_measure (algebra.Alg.Term ?f ?l)) =>
         let l'' := algebra.Alg_ext.find_replacement l  in 
          ((apply (interp.le_lt_compat_right (interp.o_Z 0)) with
             (marked_measure (algebra.Alg.Term f l''));[idtac|
            apply marked_measure_star_monotonic;
             repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos);
             (assumption)||(constructor 1)]))
        end
      ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma DP_R_xml_0_scc_14_large_in_le :
    Relation_Definitions.inclusion _ DP_R_xml_0_scc_14_large le.
   Proof.
     unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *.
     
     intros a b H;destruct H;
      match goal with
        |  |- _ <= marked_measure (algebra.Alg.Term ?f ?l) =>
         let l'' := algebra.Alg_ext.find_replacement l  in 
          ((apply (interp.le_trans (interp.o_Z 0)) with
             (marked_measure (algebra.Alg.Term f l''));[idtac|
            apply marked_measure_star_monotonic;
             repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos);
             (assumption)||(constructor 1)]))
        end
      ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Definition wf_DP_R_xml_0_scc_14_large  := WF_DP_R_xml_0_scc_14_large.wf.
   
   
   Lemma wf : well_founded WF_DP_R_xml_0.DP_R_xml_0_scc_14.
   Proof.
     intros x.
     apply (well_founded_ind wf_lt).
     clear x.
     intros x.
     pattern x.
     apply (@Acc_ind _ DP_R_xml_0_scc_14_large).
     clear x.
     intros x _ IHx IHx'.
     constructor.
     intros y H.
     
     destruct H;
      (apply IHx';apply DP_R_xml_0_scc_14_strict_in_lt;
        econstructor eassumption)||
      ((apply IHx;[econstructor eassumption|
        intros y' Hlt;apply IHx';apply lt_le_compat with (1:=Hlt) ;
         apply DP_R_xml_0_scc_14_large_in_le;econstructor eassumption])).
     apply wf_DP_R_xml_0_scc_14_large.
   Qed.
  End WF_DP_R_xml_0_scc_14.
  
  Definition wf_DP_R_xml_0_scc_14  := WF_DP_R_xml_0_scc_14.wf.
  
  
  Lemma acc_DP_R_xml_0_scc_14 :
   forall x y, (DP_R_xml_0_scc_14 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x.
    pattern x.
    apply (@Acc_ind _ DP_R_xml_0_scc_14).
    intros x' _ Hrec y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply Hrec;econstructor eassumption)||
      ((eapply acc_DP_R_xml_0_non_scc_13;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_11;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((eapply acc_DP_R_xml_0_non_scc_10;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
         ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
          (eapply Hrec;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))))).
    apply wf_DP_R_xml_0_scc_14.
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_15  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <mark(isNatKind(X_)),a__isNatKind(X_)> *)
    | DP_R_xml_0_non_scc_15_0 :
     forall x8 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_isNatKind (x8::nil)) x11) ->
       DP_R_xml_0_non_scc_15 (algebra.Alg.Term algebra.F.id_a__isNatKind 
                              (x8::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_15 :
   forall x y, 
    (DP_R_xml_0_non_scc_15 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_14;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_11;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_10;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
         (eapply Hrec;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))))).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_16  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <a__isNatList(cons(V1_,V2_)),a__isNatKind(V1_)> *)
    | DP_R_xml_0_non_scc_16_0 :
     forall x1 x3 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_cons (x1::x3::nil)) x11) ->
       DP_R_xml_0_non_scc_16 (algebra.Alg.Term algebra.F.id_a__isNatKind 
                              (x1::nil)) 
        (algebra.Alg.Term algebra.F.id_a__isNatList (x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_16 :
   forall x y, 
    (DP_R_xml_0_non_scc_16 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_14;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_11;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_10;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
         (eapply Hrec;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))))).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_17  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <mark(isNatIListKind(X_)),a__isNatIListKind(X_)> *)
    | DP_R_xml_0_non_scc_17_0 :
     forall x8 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_isNatIListKind (x8::nil)) x11) ->
       DP_R_xml_0_non_scc_17 (algebra.Alg.Term 
                              algebra.F.id_a__isNatIListKind (x8::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_17 :
   forall x y, 
    (DP_R_xml_0_non_scc_17 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_14;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
       (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_18  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <a__isNatIList(cons(V1_,V2_)),a__isNatKind(V1_)> *)
    | DP_R_xml_0_non_scc_18_0 :
     forall x1 x3 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_cons (x1::x3::nil)) x11) ->
       DP_R_xml_0_non_scc_18 (algebra.Alg.Term algebra.F.id_a__isNatKind 
                              (x1::nil)) 
        (algebra.Alg.Term algebra.F.id_a__isNatIList (x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_18 :
   forall x y, 
    (DP_R_xml_0_non_scc_18 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_14;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_11;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_10;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
         (eapply Hrec;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))))).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_19  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <a__isNat(s(V1_)),a__isNatKind(V1_)> *)
    | DP_R_xml_0_non_scc_19_0 :
     forall x1 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_s (x1::nil)) 
        x11) ->
       DP_R_xml_0_non_scc_19 (algebra.Alg.Term algebra.F.id_a__isNatKind 
                              (x1::nil)) 
        (algebra.Alg.Term algebra.F.id_a__isNat (x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_19 :
   forall x y, 
    (DP_R_xml_0_non_scc_19 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_14;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_11;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_10;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
         (eapply Hrec;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))))).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_20  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <a__U93(tt,L_,N_),a__isNatKind(N_)> *)
    | DP_R_xml_0_non_scc_20_0 :
     forall x4 x12 x5 x13 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_tt nil) 
        x13) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x4 x12) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x5 x11) ->
         DP_R_xml_0_non_scc_20 (algebra.Alg.Term algebra.F.id_a__isNatKind 
                                (x5::nil)) 
          (algebra.Alg.Term algebra.F.id_a__U93 (x13::x12::x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_20 :
   forall x y, 
    (DP_R_xml_0_non_scc_20 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_14;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_11;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_10;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
         (eapply Hrec;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))))).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_21  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <a__U81(tt,V1_,V2_),a__isNatKind(V1_)> *)
    | DP_R_xml_0_non_scc_21_0 :
     forall x12 x1 x13 x3 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_tt nil) 
        x13) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x1 x12) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x3 x11) ->
         DP_R_xml_0_non_scc_21 (algebra.Alg.Term algebra.F.id_a__isNatKind 
                                (x1::nil)) 
          (algebra.Alg.Term algebra.F.id_a__U81 (x13::x12::x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_21 :
   forall x y, 
    (DP_R_xml_0_non_scc_21 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_14;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_11;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_10;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
         (eapply Hrec;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))))).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_22  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <a__isNatIList(V_),a__isNatIListKind(V_)> *)
    | DP_R_xml_0_non_scc_22_0 :
     forall x2 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x2 x11) ->
       DP_R_xml_0_non_scc_22 (algebra.Alg.Term 
                              algebra.F.id_a__isNatIListKind (x2::nil)) 
        (algebra.Alg.Term algebra.F.id_a__isNatIList (x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_22 :
   forall x y, 
    (DP_R_xml_0_non_scc_22 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_14;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
       (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_23  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <a__isNat(length(V1_)),a__isNatIListKind(V1_)> *)
    | DP_R_xml_0_non_scc_23_0 :
     forall x1 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_length (x1::nil)) x11) ->
       DP_R_xml_0_non_scc_23 (algebra.Alg.Term 
                              algebra.F.id_a__isNatIListKind (x1::nil)) 
        (algebra.Alg.Term algebra.F.id_a__isNat (x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_23 :
   forall x y, 
    (DP_R_xml_0_non_scc_23 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_14;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
       (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_24  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <a__U91(tt,L_,N_),a__isNatIListKind(L_)> *)
    | DP_R_xml_0_non_scc_24_0 :
     forall x4 x12 x5 x13 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_tt nil) 
        x13) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x4 x12) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x5 x11) ->
         DP_R_xml_0_non_scc_24 (algebra.Alg.Term 
                                algebra.F.id_a__isNatIListKind (x4::nil)) 
          (algebra.Alg.Term algebra.F.id_a__U91 (x13::x12::x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_24 :
   forall x y, 
    (DP_R_xml_0_non_scc_24 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_14;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
       (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_25  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <a__U83(tt,V1_,V2_),a__isNatIListKind(V2_)> *)
    | DP_R_xml_0_non_scc_25_0 :
     forall x12 x1 x13 x3 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_tt nil) 
        x13) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x1 x12) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x3 x11) ->
         DP_R_xml_0_non_scc_25 (algebra.Alg.Term 
                                algebra.F.id_a__isNatIListKind (x3::nil)) 
          (algebra.Alg.Term algebra.F.id_a__U83 (x13::x12::x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_25 :
   forall x y, 
    (DP_R_xml_0_non_scc_25 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_14;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
       (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_26  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <a__U82(tt,V1_,V2_),a__isNatIListKind(V2_)> *)
    | DP_R_xml_0_non_scc_26_0 :
     forall x12 x1 x13 x3 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_tt nil) 
        x13) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x1 x12) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x3 x11) ->
         DP_R_xml_0_non_scc_26 (algebra.Alg.Term 
                                algebra.F.id_a__isNatIListKind (x3::nil)) 
          (algebra.Alg.Term algebra.F.id_a__U82 (x13::x12::x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_26 :
   forall x y, 
    (DP_R_xml_0_non_scc_26 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_14;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
       (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_27  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <mark(U51(X1_,X2_)),a__U51(mark(X1_),X2_)> *)
    | DP_R_xml_0_non_scc_27_0 :
     forall x6 x11 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U51 (x6::x7::nil)) x11) ->
       DP_R_xml_0_non_scc_27 (algebra.Alg.Term algebra.F.id_a__U51 
                              ((algebra.Alg.Term algebra.F.id_mark 
                              (x6::nil))::x7::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_27 :
   forall x y, 
    (DP_R_xml_0_non_scc_27 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_14;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_13;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
        (eapply Hrec;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_28  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <a__U45(tt,V2_),a__U46(a__isNatIList(V2_))> *)
    | DP_R_xml_0_non_scc_28_0 :
     forall x12 x3 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_tt nil) 
        x12) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x3 x11) ->
        DP_R_xml_0_non_scc_28 (algebra.Alg.Term algebra.F.id_a__U46 
                               ((algebra.Alg.Term algebra.F.id_a__isNatIList 
                               (x3::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U45 (x12::x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_28 :
   forall x y, 
    (DP_R_xml_0_non_scc_28 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
      (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_29  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <a__U43(tt,V1_,V2_),a__isNatIListKind(V2_)> *)
    | DP_R_xml_0_non_scc_29_0 :
     forall x12 x1 x13 x3 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_tt nil) 
        x13) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x1 x12) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x3 x11) ->
         DP_R_xml_0_non_scc_29 (algebra.Alg.Term 
                                algebra.F.id_a__isNatIListKind (x3::nil)) 
          (algebra.Alg.Term algebra.F.id_a__U43 (x13::x12::x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_29 :
   forall x y, 
    (DP_R_xml_0_non_scc_29 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_14;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
       (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_30  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <a__U42(tt,V1_,V2_),a__isNatIListKind(V2_)> *)
    | DP_R_xml_0_non_scc_30_0 :
     forall x12 x1 x13 x3 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_tt nil) 
        x13) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x1 x12) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x3 x11) ->
         DP_R_xml_0_non_scc_30 (algebra.Alg.Term 
                                algebra.F.id_a__isNatIListKind (x3::nil)) 
          (algebra.Alg.Term algebra.F.id_a__U42 (x13::x12::x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_30 :
   forall x y, 
    (DP_R_xml_0_non_scc_30 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_14;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
       (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_31  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <a__U41(tt,V1_,V2_),a__isNatKind(V1_)> *)
    | DP_R_xml_0_non_scc_31_0 :
     forall x12 x1 x13 x3 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_tt nil) 
        x13) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x1 x12) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x3 x11) ->
         DP_R_xml_0_non_scc_31 (algebra.Alg.Term algebra.F.id_a__isNatKind 
                                (x1::nil)) 
          (algebra.Alg.Term algebra.F.id_a__U41 (x13::x12::x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_31 :
   forall x y, 
    (DP_R_xml_0_non_scc_31 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_14;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_11;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_10;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
         (eapply Hrec;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))))).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_32  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <a__U32(tt,V_),a__U33(a__isNatList(V_))> *)
    | DP_R_xml_0_non_scc_32_0 :
     forall x12 x2 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_tt nil) 
        x12) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x2 x11) ->
        DP_R_xml_0_non_scc_32 (algebra.Alg.Term algebra.F.id_a__U33 
                               ((algebra.Alg.Term algebra.F.id_a__isNatList 
                               (x2::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U32 (x12::x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_32 :
   forall x y, 
    (DP_R_xml_0_non_scc_32 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
      (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_33  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <a__U31(tt,V_),a__isNatIListKind(V_)> *)
    | DP_R_xml_0_non_scc_33_0 :
     forall x12 x2 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_tt nil) 
        x12) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x2 x11) ->
        DP_R_xml_0_non_scc_33 (algebra.Alg.Term 
                               algebra.F.id_a__isNatIListKind (x2::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U31 (x12::x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_33 :
   forall x y, 
    (DP_R_xml_0_non_scc_33 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_14;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
       (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_34  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <a__U22(tt,V1_),a__U23(a__isNat(V1_))> *)
    | DP_R_xml_0_non_scc_34_0 :
     forall x12 x1 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_tt nil) 
        x12) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x1 x11) ->
        DP_R_xml_0_non_scc_34 (algebra.Alg.Term algebra.F.id_a__U23 
                               ((algebra.Alg.Term algebra.F.id_a__isNat 
                               (x1::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U22 (x12::x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_34 :
   forall x y, 
    (DP_R_xml_0_non_scc_34 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
      (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_35  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <a__U21(tt,V1_),a__isNatKind(V1_)> *)
    | DP_R_xml_0_non_scc_35_0 :
     forall x12 x1 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_tt nil) 
        x12) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x1 x11) ->
        DP_R_xml_0_non_scc_35 (algebra.Alg.Term algebra.F.id_a__isNatKind 
                               (x1::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U21 (x12::x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_35 :
   forall x y, 
    (DP_R_xml_0_non_scc_35 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_14;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_11;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_10;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
         (eapply Hrec;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))))).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_36  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <a__U12(tt,V1_),a__U13(a__isNatList(V1_))> *)
    | DP_R_xml_0_non_scc_36_0 :
     forall x12 x1 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_tt nil) 
        x12) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x1 x11) ->
        DP_R_xml_0_non_scc_36 (algebra.Alg.Term algebra.F.id_a__U13 
                               ((algebra.Alg.Term algebra.F.id_a__isNatList 
                               (x1::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U12 (x12::x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_36 :
   forall x y, 
    (DP_R_xml_0_non_scc_36 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
      (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_37  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <a__U11(tt,V1_),a__isNatIListKind(V1_)> *)
    | DP_R_xml_0_non_scc_37_0 :
     forall x12 x1 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_tt nil) 
        x12) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x1 x11) ->
        DP_R_xml_0_non_scc_37 (algebra.Alg.Term 
                               algebra.F.id_a__isNatIListKind (x1::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U11 (x12::x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_37 :
   forall x y, 
    (DP_R_xml_0_non_scc_37 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_14;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
       (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))).
  Qed.
  
  
  Inductive DP_R_xml_0_scc_38  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <a__U12(tt,V1_),a__isNatList(V1_)> *)
    | DP_R_xml_0_scc_38_0 :
     forall x12 x1 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_tt nil) 
        x12) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x1 x11) ->
        DP_R_xml_0_scc_38 (algebra.Alg.Term algebra.F.id_a__isNatList 
                           (x1::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U12 (x12::x11::nil))
    
     (* <a__isNatList(cons(V1_,V2_)),a__U81(a__isNatKind(V1_),V1_,V2_)> *)
    | DP_R_xml_0_scc_38_1 :
     forall x1 x3 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_cons (x1::x3::nil)) x11) ->
       DP_R_xml_0_scc_38 (algebra.Alg.Term algebra.F.id_a__U81 
                          ((algebra.Alg.Term algebra.F.id_a__isNatKind 
                          (x1::nil))::x1::x3::nil)) 
        (algebra.Alg.Term algebra.F.id_a__isNatList (x11::nil))
    
     (* <a__U81(tt,V1_,V2_),a__U82(a__isNatKind(V1_),V1_,V2_)> *)
    | DP_R_xml_0_scc_38_2 :
     forall x12 x1 x13 x3 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_tt nil) 
        x13) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x1 x12) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x3 x11) ->
         DP_R_xml_0_scc_38 (algebra.Alg.Term algebra.F.id_a__U82 
                            ((algebra.Alg.Term algebra.F.id_a__isNatKind 
                            (x1::nil))::x1::x3::nil)) 
          (algebra.Alg.Term algebra.F.id_a__U81 (x13::x12::x11::nil))
    
     (* <a__U82(tt,V1_,V2_),a__U83(a__isNatIListKind(V2_),V1_,V2_)> *)
    | DP_R_xml_0_scc_38_3 :
     forall x12 x1 x13 x3 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_tt nil) 
        x13) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x1 x12) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x3 x11) ->
         DP_R_xml_0_scc_38 (algebra.Alg.Term algebra.F.id_a__U83 
                            ((algebra.Alg.Term 
                            algebra.F.id_a__isNatIListKind (x3::nil))::x1::
                            x3::nil)) 
          (algebra.Alg.Term algebra.F.id_a__U82 (x13::x12::x11::nil))
    
     (* <a__U83(tt,V1_,V2_),a__U84(a__isNatIListKind(V2_),V1_,V2_)> *)
    | DP_R_xml_0_scc_38_4 :
     forall x12 x1 x13 x3 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_tt nil) 
        x13) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x1 x12) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x3 x11) ->
         DP_R_xml_0_scc_38 (algebra.Alg.Term algebra.F.id_a__U84 
                            ((algebra.Alg.Term 
                            algebra.F.id_a__isNatIListKind (x3::nil))::x1::
                            x3::nil)) 
          (algebra.Alg.Term algebra.F.id_a__U83 (x13::x12::x11::nil))
    
     (* <a__U84(tt,V1_,V2_),a__U85(a__isNat(V1_),V2_)> *)
    | DP_R_xml_0_scc_38_5 :
     forall x12 x1 x13 x3 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_tt nil) 
        x13) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x1 x12) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x3 x11) ->
         DP_R_xml_0_scc_38 (algebra.Alg.Term algebra.F.id_a__U85 
                            ((algebra.Alg.Term algebra.F.id_a__isNat 
                            (x1::nil))::x3::nil)) 
          (algebra.Alg.Term algebra.F.id_a__U84 (x13::x12::x11::nil))
     (* <a__U85(tt,V2_),a__isNatList(V2_)> *)
    | DP_R_xml_0_scc_38_6 :
     forall x12 x3 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_tt nil) 
        x12) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x3 x11) ->
        DP_R_xml_0_scc_38 (algebra.Alg.Term algebra.F.id_a__isNatList 
                           (x3::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U85 (x12::x11::nil))
     (* <a__U84(tt,V1_,V2_),a__isNat(V1_)> *)
    | DP_R_xml_0_scc_38_7 :
     forall x12 x1 x13 x3 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_tt nil) 
        x13) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x1 x12) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x3 x11) ->
         DP_R_xml_0_scc_38 (algebra.Alg.Term algebra.F.id_a__isNat (x1::nil)) 
          (algebra.Alg.Term algebra.F.id_a__U84 (x13::x12::x11::nil))
    
     (* <a__isNat(length(V1_)),a__U11(a__isNatIListKind(V1_),V1_)> *)
    | DP_R_xml_0_scc_38_8 :
     forall x1 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_length (x1::nil)) x11) ->
       DP_R_xml_0_scc_38 (algebra.Alg.Term algebra.F.id_a__U11 
                          ((algebra.Alg.Term algebra.F.id_a__isNatIListKind 
                          (x1::nil))::x1::nil)) 
        (algebra.Alg.Term algebra.F.id_a__isNat (x11::nil))
    
     (* <a__U11(tt,V1_),a__U12(a__isNatIListKind(V1_),V1_)> *)
    | DP_R_xml_0_scc_38_9 :
     forall x12 x1 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_tt nil) 
        x12) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x1 x11) ->
        DP_R_xml_0_scc_38 (algebra.Alg.Term algebra.F.id_a__U12 
                           ((algebra.Alg.Term algebra.F.id_a__isNatIListKind 
                           (x1::nil))::x1::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U11 (x12::x11::nil))
    
     (* <a__isNat(s(V1_)),a__U21(a__isNatKind(V1_),V1_)> *)
    | DP_R_xml_0_scc_38_10 :
     forall x1 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_s (x1::nil)) 
        x11) ->
       DP_R_xml_0_scc_38 (algebra.Alg.Term algebra.F.id_a__U21 
                          ((algebra.Alg.Term algebra.F.id_a__isNatKind 
                          (x1::nil))::x1::nil)) 
        (algebra.Alg.Term algebra.F.id_a__isNat (x11::nil))
    
     (* <a__U21(tt,V1_),a__U22(a__isNatKind(V1_),V1_)> *)
    | DP_R_xml_0_scc_38_11 :
     forall x12 x1 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_tt nil) 
        x12) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x1 x11) ->
        DP_R_xml_0_scc_38 (algebra.Alg.Term algebra.F.id_a__U22 
                           ((algebra.Alg.Term algebra.F.id_a__isNatKind 
                           (x1::nil))::x1::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U21 (x12::x11::nil))
     (* <a__U22(tt,V1_),a__isNat(V1_)> *)
    | DP_R_xml_0_scc_38_12 :
     forall x12 x1 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_tt nil) 
        x12) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x1 x11) ->
        DP_R_xml_0_scc_38 (algebra.Alg.Term algebra.F.id_a__isNat (x1::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U22 (x12::x11::nil))
  .
  
  
  Module WF_DP_R_xml_0_scc_38.
   Inductive DP_R_xml_0_scc_38_large  :
    algebra.Alg.term ->algebra.Alg.term ->Prop := 
      (* <a__isNatList(cons(V1_,V2_)),a__U81(a__isNatKind(V1_),V1_,V2_)> *)
     | DP_R_xml_0_scc_38_large_0 :
      forall x1 x3 x11, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_cons (x1::x3::nil)) x11) ->
        DP_R_xml_0_scc_38_large (algebra.Alg.Term algebra.F.id_a__U81 
                                 ((algebra.Alg.Term 
                                 algebra.F.id_a__isNatKind (x1::nil))::x1::
                                 x3::nil)) 
         (algebra.Alg.Term algebra.F.id_a__isNatList (x11::nil))
     
      (* <a__U81(tt,V1_,V2_),a__U82(a__isNatKind(V1_),V1_,V2_)> *)
     | DP_R_xml_0_scc_38_large_1 :
      forall x12 x1 x13 x3 x11, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  (algebra.Alg.Term algebra.F.id_tt nil) 
         x13) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x1 x12) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x3 x11) ->
          DP_R_xml_0_scc_38_large (algebra.Alg.Term algebra.F.id_a__U82 
                                   ((algebra.Alg.Term 
                                   algebra.F.id_a__isNatKind (x1::nil))::x1::
                                   x3::nil)) 
           (algebra.Alg.Term algebra.F.id_a__U81 (x13::x12::x11::nil))
     
      (* <a__U82(tt,V1_,V2_),a__U83(a__isNatIListKind(V2_),V1_,V2_)> *)
     | DP_R_xml_0_scc_38_large_2 :
      forall x12 x1 x13 x3 x11, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  (algebra.Alg.Term algebra.F.id_tt nil) 
         x13) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x1 x12) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x3 x11) ->
          DP_R_xml_0_scc_38_large (algebra.Alg.Term algebra.F.id_a__U83 
                                   ((algebra.Alg.Term 
                                   algebra.F.id_a__isNatIListKind 
                                   (x3::nil))::x1::x3::nil)) 
           (algebra.Alg.Term algebra.F.id_a__U82 (x13::x12::x11::nil))
     
      (* <a__U83(tt,V1_,V2_),a__U84(a__isNatIListKind(V2_),V1_,V2_)> *)
     | DP_R_xml_0_scc_38_large_3 :
      forall x12 x1 x13 x3 x11, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  (algebra.Alg.Term algebra.F.id_tt nil) 
         x13) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x1 x12) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x3 x11) ->
          DP_R_xml_0_scc_38_large (algebra.Alg.Term algebra.F.id_a__U84 
                                   ((algebra.Alg.Term 
                                   algebra.F.id_a__isNatIListKind 
                                   (x3::nil))::x1::x3::nil)) 
           (algebra.Alg.Term algebra.F.id_a__U83 (x13::x12::x11::nil))
     
      (* <a__U84(tt,V1_,V2_),a__U85(a__isNat(V1_),V2_)> *)
     | DP_R_xml_0_scc_38_large_4 :
      forall x12 x1 x13 x3 x11, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  (algebra.Alg.Term algebra.F.id_tt nil) 
         x13) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x1 x12) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x3 x11) ->
          DP_R_xml_0_scc_38_large (algebra.Alg.Term algebra.F.id_a__U85 
                                   ((algebra.Alg.Term algebra.F.id_a__isNat 
                                   (x1::nil))::x3::nil)) 
           (algebra.Alg.Term algebra.F.id_a__U84 (x13::x12::x11::nil))
      (* <a__U85(tt,V2_),a__isNatList(V2_)> *)
     | DP_R_xml_0_scc_38_large_5 :
      forall x12 x3 x11, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  (algebra.Alg.Term algebra.F.id_tt nil) 
         x12) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x3 x11) ->
         DP_R_xml_0_scc_38_large (algebra.Alg.Term algebra.F.id_a__isNatList 
                                  (x3::nil)) 
          (algebra.Alg.Term algebra.F.id_a__U85 (x12::x11::nil))
      (* <a__U84(tt,V1_,V2_),a__isNat(V1_)> *)
     | DP_R_xml_0_scc_38_large_6 :
      forall x12 x1 x13 x3 x11, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  (algebra.Alg.Term algebra.F.id_tt nil) 
         x13) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x1 x12) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x3 x11) ->
          DP_R_xml_0_scc_38_large (algebra.Alg.Term algebra.F.id_a__isNat 
                                   (x1::nil)) 
           (algebra.Alg.Term algebra.F.id_a__U84 (x13::x12::x11::nil))
     
      (* <a__isNat(length(V1_)),a__U11(a__isNatIListKind(V1_),V1_)> *)
     | DP_R_xml_0_scc_38_large_7 :
      forall x1 x11, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_length (x1::nil)) x11) ->
        DP_R_xml_0_scc_38_large (algebra.Alg.Term algebra.F.id_a__U11 
                                 ((algebra.Alg.Term 
                                 algebra.F.id_a__isNatIListKind (x1::nil))::
                                 x1::nil)) 
         (algebra.Alg.Term algebra.F.id_a__isNat (x11::nil))
     
      (* <a__U11(tt,V1_),a__U12(a__isNatIListKind(V1_),V1_)> *)
     | DP_R_xml_0_scc_38_large_8 :
      forall x12 x1 x11, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  (algebra.Alg.Term algebra.F.id_tt nil) 
         x12) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x1 x11) ->
         DP_R_xml_0_scc_38_large (algebra.Alg.Term algebra.F.id_a__U12 
                                  ((algebra.Alg.Term 
                                  algebra.F.id_a__isNatIListKind (x1::nil))::
                                  x1::nil)) 
          (algebra.Alg.Term algebra.F.id_a__U11 (x12::x11::nil))
     
      (* <a__isNat(s(V1_)),a__U21(a__isNatKind(V1_),V1_)> *)
     | DP_R_xml_0_scc_38_large_9 :
      forall x1 x11, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_s (x1::nil)) x11) ->
        DP_R_xml_0_scc_38_large (algebra.Alg.Term algebra.F.id_a__U21 
                                 ((algebra.Alg.Term 
                                 algebra.F.id_a__isNatKind (x1::nil))::
                                 x1::nil)) 
         (algebra.Alg.Term algebra.F.id_a__isNat (x11::nil))
     
      (* <a__U21(tt,V1_),a__U22(a__isNatKind(V1_),V1_)> *)
     | DP_R_xml_0_scc_38_large_10 :
      forall x12 x1 x11, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  (algebra.Alg.Term algebra.F.id_tt nil) 
         x12) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x1 x11) ->
         DP_R_xml_0_scc_38_large (algebra.Alg.Term algebra.F.id_a__U22 
                                  ((algebra.Alg.Term 
                                  algebra.F.id_a__isNatKind (x1::nil))::
                                  x1::nil)) 
          (algebra.Alg.Term algebra.F.id_a__U21 (x12::x11::nil))
      (* <a__U22(tt,V1_),a__isNat(V1_)> *)
     | DP_R_xml_0_scc_38_large_11 :
      forall x12 x1 x11, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  (algebra.Alg.Term algebra.F.id_tt nil) 
         x12) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x1 x11) ->
         DP_R_xml_0_scc_38_large (algebra.Alg.Term algebra.F.id_a__isNat 
                                  (x1::nil)) 
          (algebra.Alg.Term algebra.F.id_a__U22 (x12::x11::nil))
   .
   
   
   Inductive DP_R_xml_0_scc_38_strict  :
    algebra.Alg.term ->algebra.Alg.term ->Prop := 
      (* <a__U12(tt,V1_),a__isNatList(V1_)> *)
     | DP_R_xml_0_scc_38_strict_0 :
      forall x12 x1 x11, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  (algebra.Alg.Term algebra.F.id_tt nil) 
         x12) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x1 x11) ->
         DP_R_xml_0_scc_38_strict (algebra.Alg.Term 
                                   algebra.F.id_a__isNatList (x1::nil)) 
          (algebra.Alg.Term algebra.F.id_a__U12 (x12::x11::nil))
   .
   
   
   Module WF_DP_R_xml_0_scc_38_large.
    Inductive DP_R_xml_0_scc_38_large_non_scc_1  :
     algebra.Alg.term ->algebra.Alg.term ->Prop := 
       (* <a__U11(tt,V1_),a__U12(a__isNatIListKind(V1_),V1_)> *)
      | DP_R_xml_0_scc_38_large_non_scc_1_0 :
       forall x12 x1 x11, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   (algebra.Alg.Term algebra.F.id_tt nil) 
          x12) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x1 x11) ->
          DP_R_xml_0_scc_38_large_non_scc_1 (algebra.Alg.Term 
                                             algebra.F.id_a__U12 
                                             ((algebra.Alg.Term 
                                             algebra.F.id_a__isNatIListKind 
                                             (x1::nil))::x1::nil)) 
           (algebra.Alg.Term algebra.F.id_a__U11 (x12::x11::nil))
    .
    
    
    Lemma acc_DP_R_xml_0_scc_38_large_non_scc_1 :
     forall x y, 
      (DP_R_xml_0_scc_38_large_non_scc_1 x y) ->
       Acc WF_DP_R_xml_0_scc_38.DP_R_xml_0_scc_38_large x.
    Proof.
      intros x y h.
      
      inversion h;clear h;subst;
       constructor;intros _y _h;inversion _h;clear _h;subst;
        (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
        (eapply Hrec;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' )).
    Qed.
    
    
    Inductive DP_R_xml_0_scc_38_large_non_scc_2  :
     algebra.Alg.term ->algebra.Alg.term ->Prop := 
       (* <a__isNat(length(V1_)),a__U11(a__isNatIListKind(V1_),V1_)> *)
      | DP_R_xml_0_scc_38_large_non_scc_2_0 :
       forall x1 x11, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_length (x1::nil)) x11) ->
         DP_R_xml_0_scc_38_large_non_scc_2 (algebra.Alg.Term 
                                            algebra.F.id_a__U11 
                                            ((algebra.Alg.Term 
                                            algebra.F.id_a__isNatIListKind 
                                            (x1::nil))::x1::nil)) 
          (algebra.Alg.Term algebra.F.id_a__isNat (x11::nil))
    .
    
    
    Lemma acc_DP_R_xml_0_scc_38_large_non_scc_2 :
     forall x y, 
      (DP_R_xml_0_scc_38_large_non_scc_2 x y) ->
       Acc WF_DP_R_xml_0_scc_38.DP_R_xml_0_scc_38_large x.
    Proof.
      intros x y h.
      
      inversion h;clear h;subst;
       constructor;intros _y _h;inversion _h;clear _h;subst;
        (eapply acc_DP_R_xml_0_scc_38_large_non_scc_1;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
         (eapply Hrec;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))).
    Qed.
    
    
    Inductive DP_R_xml_0_scc_38_large_scc_3  :
     algebra.Alg.term ->algebra.Alg.term ->Prop := 
       (* <a__isNat(s(V1_)),a__U21(a__isNatKind(V1_),V1_)> *)
      | DP_R_xml_0_scc_38_large_scc_3_0 :
       forall x1 x11, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_s (x1::nil)) x11) ->
         DP_R_xml_0_scc_38_large_scc_3 (algebra.Alg.Term algebra.F.id_a__U21 
                                        ((algebra.Alg.Term 
                                        algebra.F.id_a__isNatKind 
                                        (x1::nil))::x1::nil)) 
          (algebra.Alg.Term algebra.F.id_a__isNat (x11::nil))
      
       (* <a__U21(tt,V1_),a__U22(a__isNatKind(V1_),V1_)> *)
      | DP_R_xml_0_scc_38_large_scc_3_1 :
       forall x12 x1 x11, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   (algebra.Alg.Term algebra.F.id_tt nil) 
          x12) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x1 x11) ->
          DP_R_xml_0_scc_38_large_scc_3 (algebra.Alg.Term 
                                         algebra.F.id_a__U22 
                                         ((algebra.Alg.Term 
                                         algebra.F.id_a__isNatKind 
                                         (x1::nil))::x1::nil)) 
           (algebra.Alg.Term algebra.F.id_a__U21 (x12::x11::nil))
      
       (* <a__U22(tt,V1_),a__isNat(V1_)> *)
      | DP_R_xml_0_scc_38_large_scc_3_2 :
       forall x12 x1 x11, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   (algebra.Alg.Term algebra.F.id_tt nil) 
          x12) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x1 x11) ->
          DP_R_xml_0_scc_38_large_scc_3 (algebra.Alg.Term 
                                         algebra.F.id_a__isNat (x1::nil)) 
           (algebra.Alg.Term algebra.F.id_a__U22 (x12::x11::nil))
    .
    
    
    Module WF_DP_R_xml_0_scc_38_large_scc_3.
     Inductive DP_R_xml_0_scc_38_large_scc_3_large  :
      algebra.Alg.term ->algebra.Alg.term ->Prop := 
        (* <a__U22(tt,V1_),a__isNat(V1_)> *)
       | DP_R_xml_0_scc_38_large_scc_3_large_0 :
        forall x12 x1 x11, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    (algebra.Alg.Term algebra.F.id_tt nil) 
           x12) ->
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     x1 x11) ->
           DP_R_xml_0_scc_38_large_scc_3_large (algebra.Alg.Term 
                                                algebra.F.id_a__isNat 
                                                (x1::nil)) 
            (algebra.Alg.Term algebra.F.id_a__U22 (x12::x11::nil))
     .
     
     
     Inductive DP_R_xml_0_scc_38_large_scc_3_strict  :
      algebra.Alg.term ->algebra.Alg.term ->Prop := 
        (* <a__isNat(s(V1_)),a__U21(a__isNatKind(V1_),V1_)> *)
       | DP_R_xml_0_scc_38_large_scc_3_strict_0 :
        forall x1 x11, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    
           (algebra.Alg.Term algebra.F.id_s (x1::nil)) x11) ->
          DP_R_xml_0_scc_38_large_scc_3_strict (algebra.Alg.Term 
                                                algebra.F.id_a__U21 
                                                ((algebra.Alg.Term 
                                                algebra.F.id_a__isNatKind 
                                                (x1::nil))::x1::nil)) 
           (algebra.Alg.Term algebra.F.id_a__isNat (x11::nil))
       
        (* <a__U21(tt,V1_),a__U22(a__isNatKind(V1_),V1_)> *)
       | DP_R_xml_0_scc_38_large_scc_3_strict_1 :
        forall x12 x1 x11, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    (algebra.Alg.Term algebra.F.id_tt nil) 
           x12) ->
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     x1 x11) ->
           DP_R_xml_0_scc_38_large_scc_3_strict (algebra.Alg.Term 
                                                 algebra.F.id_a__U22 
                                                 ((algebra.Alg.Term 
                                                 algebra.F.id_a__isNatKind 
                                                 (x1::nil))::x1::nil)) 
            (algebra.Alg.Term algebra.F.id_a__U21 (x12::x11::nil))
     .
     
     
     Module WF_DP_R_xml_0_scc_38_large_scc_3_large.
      Inductive DP_R_xml_0_scc_38_large_scc_3_large_non_scc_1  :
       algebra.Alg.term ->algebra.Alg.term ->Prop := 
         (* <a__U22(tt,V1_),a__isNat(V1_)> *)
        | DP_R_xml_0_scc_38_large_scc_3_large_non_scc_1_0 :
         forall x12 x1 x11, 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     (algebra.Alg.Term algebra.F.id_tt nil) 
            x12) ->
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                      x1 x11) ->
            DP_R_xml_0_scc_38_large_scc_3_large_non_scc_1 (algebra.Alg.Term 
                                                           algebra.F.id_a__isNat 
                                                           (x1::nil)) 
             (algebra.Alg.Term algebra.F.id_a__U22 (x12::x11::nil))
      .
      
      
      Lemma acc_DP_R_xml_0_scc_38_large_scc_3_large_non_scc_1 :
       forall x y, 
        (DP_R_xml_0_scc_38_large_scc_3_large_non_scc_1 x y) ->
         Acc WF_DP_R_xml_0_scc_38_large_scc_3.DP_R_xml_0_scc_38_large_scc_3_large 
          x.
      Proof.
        intros x y h.
        
        inversion h;clear h;subst;
         constructor;intros _y _h;inversion _h;clear _h;subst;
          (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
          (eapply Hrec;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' )).
      Qed.
      
      
      Lemma wf :
       well_founded WF_DP_R_xml_0_scc_38_large_scc_3.DP_R_xml_0_scc_38_large_scc_3_large
        .
      Proof.
        constructor;intros _y _h;inversion _h;clear _h;subst;
         (eapply acc_DP_R_xml_0_scc_38_large_scc_3_large_non_scc_1;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
         ((eapply acc_DP_R_xml_0_scc_38_large_scc_3_large_non_scc_0;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
          ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||(fail))).
      Qed.
     End WF_DP_R_xml_0_scc_38_large_scc_3_large.
     
     Open Scope Z_scope.
     
     Import ring_extention.
     
     Notation Local "a <= b" := (Zle a b).
     
     Notation Local "a < b" := (Zlt a b).
     
     Definition P_id_a__zeros  := 0.
     
     Definition P_id_U51 (x11:Z) (x12:Z) := 1.
     
     Definition P_id_a__U84 (x11:Z) (x12:Z) (x13:Z) := 2* x13.
     
     Definition P_id_a__U32 (x11:Z) (x12:Z) := 2* x12.
     
     Definition P_id_isNatList (x11:Z) := 2* x11.
     
     Definition P_id_a__U13 (x11:Z) := 1.
     
     Definition P_id_U85 (x11:Z) (x12:Z) := 2* x12.
     
     Definition P_id_a__length (x11:Z) := 1 + 1* x11.
     
     Definition P_id_a__isNatIList (x11:Z) := 1 + 3* x11.
     
     Definition P_id_U33 (x11:Z) := 1* x11.
     
     Definition P_id_a__U11 (x11:Z) (x12:Z) := 1* x11.
     
     Definition P_id_U81 (x11:Z) (x12:Z) (x13:Z) := 2* x13.
     
     Definition P_id_a__U92 (x11:Z) (x12:Z) (x13:Z) := 2* x11 + 1* x12.
     
     Definition P_id_a__U43 (x11:Z) (x12:Z) (x13:Z) := 1.
     
     Definition P_id_U23 (x11:Z) := 1.
     
     Definition P_id_a__isNatKind (x11:Z) := 1* x11.
     
     Definition P_id_U93 (x11:Z) (x12:Z) (x13:Z) := 2 + 1* x12.
     
     Definition P_id_U11 (x11:Z) (x12:Z) := 1* x11.
     
     Definition P_id_a__U71 (x11:Z) := 1* x11.
     
     Definition P_id_U44 (x11:Z) (x12:Z) (x13:Z) := 1.
     
     Definition P_id_0  := 1.
     
     Definition P_id_U61 (x11:Z) := 1.
     
     Definition P_id_a__U86 (x11:Z) := 1* x11.
     
     Definition P_id_a__U41 (x11:Z) (x12:Z) (x13:Z) := 1.
     
     Definition P_id_U22 (x11:Z) (x12:Z) := 1.
     
     Definition P_id_a__U21 (x11:Z) (x12:Z) := 2* x11.
     
     Definition P_id_U91 (x11:Z) (x12:Z) (x13:Z) := 1 + 1* x11 + 1* x12.
     
     Definition P_id_length (x11:Z) := 1 + 1* x11.
     
     Definition P_id_a__U52 (x11:Z) := 1.
     
     Definition P_id_U42 (x11:Z) (x12:Z) (x13:Z) := 1.
     
     Definition P_id_a__U12 (x11:Z) (x12:Z) := 1.
     
     Definition P_id_U83 (x11:Z) (x12:Z) (x13:Z) := 2* x13.
     
     Definition P_id_a__U94 (x11:Z) (x12:Z) := 2 + 1* x12.
     
     Definition P_id_a__U45 (x11:Z) (x12:Z) := 1.
     
     Definition P_id_U31 (x11:Z) (x12:Z) := 1 + 3* x12.
     
     Definition P_id_a__isNat (x11:Z) := 2* x11.
     
     Definition P_id_isNatIListKind (x11:Z) := 1.
     
     Definition P_id_a__U82 (x11:Z) (x12:Z) (x13:Z) := 2* x13.
     
     Definition P_id_U46 (x11:Z) := 1.
     
     Definition P_id_cons (x11:Z) (x12:Z) := 3* x12.
     
     Definition P_id_U52 (x11:Z) := 1.
     
     Definition P_id_a__U85 (x11:Z) (x12:Z) := 2* x12.
     
     Definition P_id_a__U33 (x11:Z) := 1* x11.
     
     Definition P_id_U21 (x11:Z) (x12:Z) := 2* x11.
     
     Definition P_id_a__isNatList (x11:Z) := 2* x11.
     
     Definition P_id_U86 (x11:Z) := 1* x11.
     
     Definition P_id_mark (x11:Z) := 1* x11.
     
     Definition P_id_a__U51 (x11:Z) (x12:Z) := 1.
     
     Definition P_id_U41 (x11:Z) (x12:Z) (x13:Z) := 1.
     
     Definition P_id_tt  := 1.
     
     Definition P_id_U82 (x11:Z) (x12:Z) (x13:Z) := 2* x13.
     
     Definition P_id_a__U93 (x11:Z) (x12:Z) (x13:Z) := 2 + 1* x12.
     
     Definition P_id_a__U44 (x11:Z) (x12:Z) (x13:Z) := 1.
     
     Definition P_id_isNat (x11:Z) := 2* x11.
     
     Definition P_id_a__U23 (x11:Z) := 1.
     
     Definition P_id_U94 (x11:Z) (x12:Z) := 2 + 1* x12.
     
     Definition P_id_U12 (x11:Z) (x12:Z) := 1.
     
     Definition P_id_a__U81 (x11:Z) (x12:Z) (x13:Z) := 2* x13.
     
     Definition P_id_U45 (x11:Z) (x12:Z) := 1.
     
     Definition P_id_zeros  := 0.
     
     Definition P_id_U71 (x11:Z) := 1* x11.
     
     Definition P_id_a__U91 (x11:Z) (x12:Z) (x13:Z) := 1 + 1* x11 + 1* x12.
     
     Definition P_id_a__U42 (x11:Z) (x12:Z) (x13:Z) := 1.
     
     Definition P_id_isNatKind (x11:Z) := 1* x11.
     
     Definition P_id_a__U22 (x11:Z) (x12:Z) := 1.
     
     Definition P_id_U92 (x11:Z) (x12:Z) (x13:Z) := 2* x11 + 1* x12.
     
     Definition P_id_nil  := 2.
     
     Definition P_id_a__U61 (x11:Z) := 1.
     
     Definition P_id_U43 (x11:Z) (x12:Z) (x13:Z) := 1.
     
     Definition P_id_a__isNatIListKind (x11:Z) := 1.
     
     Definition P_id_U84 (x11:Z) (x12:Z) (x13:Z) := 2* x13.
     
     Definition P_id_s (x11:Z) := 1 + 1* x11.
     
     Definition P_id_a__U46 (x11:Z) := 1.
     
     Definition P_id_U32 (x11:Z) (x12:Z) := 2* x12.
     
     Definition P_id_a__U31 (x11:Z) (x12:Z) := 1 + 3* x12.
     
     Definition P_id_U13 (x11:Z) := 1.
     
     Definition P_id_a__U83 (x11:Z) (x12:Z) (x13:Z) := 2* x13.
     
     Definition P_id_isNatIList (x11:Z) := 1 + 3* x11.
     
     Lemma P_id_U51_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U51 x12 x14 <= P_id_U51 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U84_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U84 x12 x14 x16 <= P_id_a__U84 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U32_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U32 x12 x14 <= P_id_a__U32 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNatList_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_isNatList x12 <= P_id_isNatList x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U13_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U13 x12 <= P_id_a__U13 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U85_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U85 x12 x14 <= P_id_U85 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__length_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__length x12 <= P_id_a__length x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNatIList_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__isNatIList x12 <= P_id_a__isNatIList x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U33_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U33 x12 <= P_id_U33 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U11_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U11 x12 x14 <= P_id_a__U11 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U81_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U81 x12 x14 x16 <= P_id_U81 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U92_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U92 x12 x14 x16 <= P_id_a__U92 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U43_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U43 x12 x14 x16 <= P_id_a__U43 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U23_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U23 x12 <= P_id_U23 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNatKind_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__isNatKind x12 <= P_id_a__isNatKind x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U93_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U93 x12 x14 x16 <= P_id_U93 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U11_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U11 x12 x14 <= P_id_U11 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U71_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U71 x12 <= P_id_a__U71 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U44_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U44 x12 x14 x16 <= P_id_U44 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U61_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U61 x12 <= P_id_U61 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U86_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U86 x12 <= P_id_a__U86 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U41_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U41 x12 x14 x16 <= P_id_a__U41 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U22_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U22 x12 x14 <= P_id_U22 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U21_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U21 x12 x14 <= P_id_a__U21 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U91_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U91 x12 x14 x16 <= P_id_U91 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_length_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_length x12 <= P_id_length x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U52_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U52 x12 <= P_id_a__U52 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U42_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U42 x12 x14 x16 <= P_id_U42 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U12_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U12 x12 x14 <= P_id_a__U12 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U83_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U83 x12 x14 x16 <= P_id_U83 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U94_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U94 x12 x14 <= P_id_a__U94 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U45_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U45 x12 x14 <= P_id_a__U45 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U31_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U31 x12 x14 <= P_id_U31 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNat_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__isNat x12 <= P_id_a__isNat x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNatIListKind_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_isNatIListKind x12 <= P_id_isNatIListKind x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U82_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U82 x12 x14 x16 <= P_id_a__U82 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U46_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U46 x12 <= P_id_U46 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_cons_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_cons x12 x14 <= P_id_cons x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U52_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U52 x12 <= P_id_U52 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U85_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U85 x12 x14 <= P_id_a__U85 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U33_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U33 x12 <= P_id_a__U33 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U21_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U21 x12 x14 <= P_id_U21 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNatList_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__isNatList x12 <= P_id_a__isNatList x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U86_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U86 x12 <= P_id_U86 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_mark_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_mark x12 <= P_id_mark x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U51_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U51 x12 x14 <= P_id_a__U51 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U41_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U41 x12 x14 x16 <= P_id_U41 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U82_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U82 x12 x14 x16 <= P_id_U82 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U93_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U93 x12 x14 x16 <= P_id_a__U93 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U44_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U44 x12 x14 x16 <= P_id_a__U44 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNat_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_isNat x12 <= P_id_isNat x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U23_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U23 x12 <= P_id_a__U23 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U94_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U94 x12 x14 <= P_id_U94 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U12_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U12 x12 x14 <= P_id_U12 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U81_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U81 x12 x14 x16 <= P_id_a__U81 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U45_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U45 x12 x14 <= P_id_U45 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U71_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U71 x12 <= P_id_U71 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U91_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U91 x12 x14 x16 <= P_id_a__U91 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U42_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U42 x12 x14 x16 <= P_id_a__U42 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNatKind_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_isNatKind x12 <= P_id_isNatKind x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U22_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U22 x12 x14 <= P_id_a__U22 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U92_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U92 x12 x14 x16 <= P_id_U92 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U61_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U61 x12 <= P_id_a__U61 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U43_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U43 x12 x14 x16 <= P_id_U43 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNatIListKind_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__isNatIListKind x12 <= P_id_a__isNatIListKind x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U84_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U84 x12 x14 x16 <= P_id_U84 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_s_monotonic :
      forall x12 x11, (0 <= x12)/\ (x12 <= x11) ->P_id_s x12 <= P_id_s x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U46_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U46 x12 <= P_id_a__U46 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U32_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U32 x12 x14 <= P_id_U32 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U31_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U31 x12 x14 <= P_id_a__U31 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U13_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U13 x12 <= P_id_U13 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U83_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U83 x12 x14 x16 <= P_id_a__U83 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNatIList_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_isNatIList x12 <= P_id_isNatIList x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__zeros_bounded : 0 <= P_id_a__zeros .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U51_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U51 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U84_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U84 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U32_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U32 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNatList_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_isNatList x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U13_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U13 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U85_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U85 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__length_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__length x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNatIList_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__isNatIList x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U33_bounded : forall x11, (0 <= x11) ->0 <= P_id_U33 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U11_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U11 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U81_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U81 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U92_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U92 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U43_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U43 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U23_bounded : forall x11, (0 <= x11) ->0 <= P_id_U23 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNatKind_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__isNatKind x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U93_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U93 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U11_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U11 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U71_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U71 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U44_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U44 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_0_bounded : 0 <= P_id_0 .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U61_bounded : forall x11, (0 <= x11) ->0 <= P_id_U61 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U86_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U86 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U41_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U41 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U22_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U22 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U21_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U21 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U91_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U91 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_length_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_length x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U52_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U52 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U42_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U42 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U12_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U12 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U83_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U83 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U94_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U94 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U45_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U45 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U31_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U31 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNat_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__isNat x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNatIListKind_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_isNatIListKind x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U82_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U82 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U46_bounded : forall x11, (0 <= x11) ->0 <= P_id_U46 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_cons_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_cons x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U52_bounded : forall x11, (0 <= x11) ->0 <= P_id_U52 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U85_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U85 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U33_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U33 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U21_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U21 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNatList_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__isNatList x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U86_bounded : forall x11, (0 <= x11) ->0 <= P_id_U86 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_mark_bounded : forall x11, (0 <= x11) ->0 <= P_id_mark x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U51_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U51 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U41_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U41 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_tt_bounded : 0 <= P_id_tt .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U82_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U82 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U93_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U93 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U44_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U44 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNat_bounded : forall x11, (0 <= x11) ->0 <= P_id_isNat x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U23_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U23 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U94_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U94 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U12_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U12 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U81_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U81 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U45_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U45 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_zeros_bounded : 0 <= P_id_zeros .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U71_bounded : forall x11, (0 <= x11) ->0 <= P_id_U71 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U91_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U91 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U42_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U42 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNatKind_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_isNatKind x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U22_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U22 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U92_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U92 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_nil_bounded : 0 <= P_id_nil .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U61_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U61 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U43_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U43 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNatIListKind_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__isNatIListKind x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U84_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U84 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_s_bounded : forall x11, (0 <= x11) ->0 <= P_id_s x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U46_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U46 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U32_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U32 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U31_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U31 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U13_bounded : forall x11, (0 <= x11) ->0 <= P_id_U13 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U83_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U83 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNatIList_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_isNatIList x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Definition measure  := 
       InterpZ.measure 0 P_id_a__zeros P_id_U51 P_id_a__U84 P_id_a__U32 
        P_id_isNatList P_id_a__U13 P_id_U85 P_id_a__length 
        P_id_a__isNatIList P_id_U33 P_id_a__U11 P_id_U81 P_id_a__U92 
        P_id_a__U43 P_id_U23 P_id_a__isNatKind P_id_U93 P_id_U11 P_id_a__U71 
        P_id_U44 P_id_0 P_id_U61 P_id_a__U86 P_id_a__U41 P_id_U22 
        P_id_a__U21 P_id_U91 P_id_length P_id_a__U52 P_id_U42 P_id_a__U12 
        P_id_U83 P_id_a__U94 P_id_a__U45 P_id_U31 P_id_a__isNat 
        P_id_isNatIListKind P_id_a__U82 P_id_U46 P_id_cons P_id_U52 
        P_id_a__U85 P_id_a__U33 P_id_U21 P_id_a__isNatList P_id_U86 
        P_id_mark P_id_a__U51 P_id_U41 P_id_tt P_id_U82 P_id_a__U93 
        P_id_a__U44 P_id_isNat P_id_a__U23 P_id_U94 P_id_U12 P_id_a__U81 
        P_id_U45 P_id_zeros P_id_U71 P_id_a__U91 P_id_a__U42 P_id_isNatKind 
        P_id_a__U22 P_id_U92 P_id_nil P_id_a__U61 P_id_U43 
        P_id_a__isNatIListKind P_id_U84 P_id_s P_id_a__U46 P_id_U32 
        P_id_a__U31 P_id_U13 P_id_a__U83 P_id_isNatIList.
     
     Lemma measure_equation :
      forall t, 
       measure t = match t with
                     | (algebra.Alg.Term algebra.F.id_a__zeros nil) =>
                      P_id_a__zeros 
                     | (algebra.Alg.Term algebra.F.id_U51 (x12::x11::nil)) =>
                      P_id_U51 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U84 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U84 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U32 (x12::x11::nil)) =>
                      P_id_a__U32 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_isNatList (x11::nil)) =>
                      P_id_isNatList (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U13 (x11::nil)) =>
                      P_id_a__U13 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U85 (x12::x11::nil)) =>
                      P_id_U85 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__length (x11::nil)) =>
                      P_id_a__length (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__isNatIList 
                        (x11::nil)) =>
                      P_id_a__isNatIList (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U33 (x11::nil)) =>
                      P_id_U33 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U11 (x12::x11::nil)) =>
                      P_id_a__U11 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U81 (x13::x12::
                        x11::nil)) =>
                      P_id_U81 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U92 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U92 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U43 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U43 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U23 (x11::nil)) =>
                      P_id_U23 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__isNatKind 
                        (x11::nil)) =>
                      P_id_a__isNatKind (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U93 (x13::x12::
                        x11::nil)) =>
                      P_id_U93 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U11 (x12::x11::nil)) =>
                      P_id_U11 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U71 (x11::nil)) =>
                      P_id_a__U71 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U44 (x13::x12::
                        x11::nil)) =>
                      P_id_U44 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_0 nil) => P_id_0 
                     | (algebra.Alg.Term algebra.F.id_U61 (x11::nil)) =>
                      P_id_U61 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U86 (x11::nil)) =>
                      P_id_a__U86 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U41 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U41 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U22 (x12::x11::nil)) =>
                      P_id_U22 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U21 (x12::x11::nil)) =>
                      P_id_a__U21 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U91 (x13::x12::
                        x11::nil)) =>
                      P_id_U91 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_length (x11::nil)) =>
                      P_id_length (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U52 (x11::nil)) =>
                      P_id_a__U52 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U42 (x13::x12::
                        x11::nil)) =>
                      P_id_U42 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U12 (x12::x11::nil)) =>
                      P_id_a__U12 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U83 (x13::x12::
                        x11::nil)) =>
                      P_id_U83 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U94 (x12::x11::nil)) =>
                      P_id_a__U94 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U45 (x12::x11::nil)) =>
                      P_id_a__U45 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U31 (x12::x11::nil)) =>
                      P_id_U31 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__isNat (x11::nil)) =>
                      P_id_a__isNat (measure x11)
                     | (algebra.Alg.Term algebra.F.id_isNatIListKind 
                        (x11::nil)) =>
                      P_id_isNatIListKind (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U82 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U82 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U46 (x11::nil)) =>
                      P_id_U46 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_cons (x12::x11::nil)) =>
                      P_id_cons (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U52 (x11::nil)) =>
                      P_id_U52 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U85 (x12::x11::nil)) =>
                      P_id_a__U85 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U33 (x11::nil)) =>
                      P_id_a__U33 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U21 (x12::x11::nil)) =>
                      P_id_U21 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__isNatList 
                        (x11::nil)) =>
                      P_id_a__isNatList (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U86 (x11::nil)) =>
                      P_id_U86 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_mark (x11::nil)) =>
                      P_id_mark (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U51 (x12::x11::nil)) =>
                      P_id_a__U51 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U41 (x13::x12::
                        x11::nil)) =>
                      P_id_U41 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt 
                     | (algebra.Alg.Term algebra.F.id_U82 (x13::x12::
                        x11::nil)) =>
                      P_id_U82 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U93 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U93 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U44 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U44 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_isNat (x11::nil)) =>
                      P_id_isNat (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U23 (x11::nil)) =>
                      P_id_a__U23 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U94 (x12::x11::nil)) =>
                      P_id_U94 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U12 (x12::x11::nil)) =>
                      P_id_U12 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U81 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U81 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U45 (x12::x11::nil)) =>
                      P_id_U45 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_zeros nil) =>
                      P_id_zeros 
                     | (algebra.Alg.Term algebra.F.id_U71 (x11::nil)) =>
                      P_id_U71 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U91 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U91 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U42 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U42 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_isNatKind (x11::nil)) =>
                      P_id_isNatKind (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U22 (x12::x11::nil)) =>
                      P_id_a__U22 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U92 (x13::x12::
                        x11::nil)) =>
                      P_id_U92 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil 
                     | (algebra.Alg.Term algebra.F.id_a__U61 (x11::nil)) =>
                      P_id_a__U61 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U43 (x13::x12::
                        x11::nil)) =>
                      P_id_U43 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
                        (x11::nil)) =>
                      P_id_a__isNatIListKind (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U84 (x13::x12::
                        x11::nil)) =>
                      P_id_U84 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_s (x11::nil)) =>
                      P_id_s (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U46 (x11::nil)) =>
                      P_id_a__U46 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U32 (x12::x11::nil)) =>
                      P_id_U32 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U31 (x12::x11::nil)) =>
                      P_id_a__U31 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U13 (x11::nil)) =>
                      P_id_U13 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U83 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U83 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_isNatIList (x11::nil)) =>
                      P_id_isNatIList (measure x11)
                     | _ => 0
                     end.
     Proof.
       intros t;case t;intros ;apply refl_equal.
     Qed.
     
     Lemma measure_bounded : forall t, 0 <= measure t.
     Proof.
       unfold measure in |-*.
       
       apply InterpZ.measure_bounded;
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Ltac generate_pos_hyp  :=
      match goal with
        | H:context [measure ?x] |- _ =>
         let v := fresh "v" in 
          (let H := fresh "h" in 
            (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
              clearbody H;clearbody v))
        |  |- context [measure ?x] =>
         let v := fresh "v" in 
          (let H := fresh "h" in 
            (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
              clearbody H;clearbody v))
        end
      .
     
     Lemma rules_monotonic :
      forall l r, 
       (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) ->
        measure r <= measure l.
     Proof.
       intros l r H.
       fold measure in |-*.
       
       inversion H;clear H;subst;inversion H0;clear H0;subst;
        simpl algebra.EQT.T.apply_subst in |-*;
        repeat (
        match goal with
          |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
           rewrite (measure_equation (algebra.Alg.Term f t))
          end
        );repeat (generate_pos_hyp );
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma measure_star_monotonic :
      forall l r, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  r l) ->measure r <= measure l.
     Proof.
       unfold measure in *.
       apply InterpZ.measure_star_monotonic.
       intros ;apply P_id_U51_monotonic;assumption.
       intros ;apply P_id_a__U84_monotonic;assumption.
       intros ;apply P_id_a__U32_monotonic;assumption.
       intros ;apply P_id_isNatList_monotonic;assumption.
       intros ;apply P_id_a__U13_monotonic;assumption.
       intros ;apply P_id_U85_monotonic;assumption.
       intros ;apply P_id_a__length_monotonic;assumption.
       intros ;apply P_id_a__isNatIList_monotonic;assumption.
       intros ;apply P_id_U33_monotonic;assumption.
       intros ;apply P_id_a__U11_monotonic;assumption.
       intros ;apply P_id_U81_monotonic;assumption.
       intros ;apply P_id_a__U92_monotonic;assumption.
       intros ;apply P_id_a__U43_monotonic;assumption.
       intros ;apply P_id_U23_monotonic;assumption.
       intros ;apply P_id_a__isNatKind_monotonic;assumption.
       intros ;apply P_id_U93_monotonic;assumption.
       intros ;apply P_id_U11_monotonic;assumption.
       intros ;apply P_id_a__U71_monotonic;assumption.
       intros ;apply P_id_U44_monotonic;assumption.
       intros ;apply P_id_U61_monotonic;assumption.
       intros ;apply P_id_a__U86_monotonic;assumption.
       intros ;apply P_id_a__U41_monotonic;assumption.
       intros ;apply P_id_U22_monotonic;assumption.
       intros ;apply P_id_a__U21_monotonic;assumption.
       intros ;apply P_id_U91_monotonic;assumption.
       intros ;apply P_id_length_monotonic;assumption.
       intros ;apply P_id_a__U52_monotonic;assumption.
       intros ;apply P_id_U42_monotonic;assumption.
       intros ;apply P_id_a__U12_monotonic;assumption.
       intros ;apply P_id_U83_monotonic;assumption.
       intros ;apply P_id_a__U94_monotonic;assumption.
       intros ;apply P_id_a__U45_monotonic;assumption.
       intros ;apply P_id_U31_monotonic;assumption.
       intros ;apply P_id_a__isNat_monotonic;assumption.
       intros ;apply P_id_isNatIListKind_monotonic;assumption.
       intros ;apply P_id_a__U82_monotonic;assumption.
       intros ;apply P_id_U46_monotonic;assumption.
       intros ;apply P_id_cons_monotonic;assumption.
       intros ;apply P_id_U52_monotonic;assumption.
       intros ;apply P_id_a__U85_monotonic;assumption.
       intros ;apply P_id_a__U33_monotonic;assumption.
       intros ;apply P_id_U21_monotonic;assumption.
       intros ;apply P_id_a__isNatList_monotonic;assumption.
       intros ;apply P_id_U86_monotonic;assumption.
       intros ;apply P_id_mark_monotonic;assumption.
       intros ;apply P_id_a__U51_monotonic;assumption.
       intros ;apply P_id_U41_monotonic;assumption.
       intros ;apply P_id_U82_monotonic;assumption.
       intros ;apply P_id_a__U93_monotonic;assumption.
       intros ;apply P_id_a__U44_monotonic;assumption.
       intros ;apply P_id_isNat_monotonic;assumption.
       intros ;apply P_id_a__U23_monotonic;assumption.
       intros ;apply P_id_U94_monotonic;assumption.
       intros ;apply P_id_U12_monotonic;assumption.
       intros ;apply P_id_a__U81_monotonic;assumption.
       intros ;apply P_id_U45_monotonic;assumption.
       intros ;apply P_id_U71_monotonic;assumption.
       intros ;apply P_id_a__U91_monotonic;assumption.
       intros ;apply P_id_a__U42_monotonic;assumption.
       intros ;apply P_id_isNatKind_monotonic;assumption.
       intros ;apply P_id_a__U22_monotonic;assumption.
       intros ;apply P_id_U92_monotonic;assumption.
       intros ;apply P_id_a__U61_monotonic;assumption.
       intros ;apply P_id_U43_monotonic;assumption.
       intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
       intros ;apply P_id_U84_monotonic;assumption.
       intros ;apply P_id_s_monotonic;assumption.
       intros ;apply P_id_a__U46_monotonic;assumption.
       intros ;apply P_id_U32_monotonic;assumption.
       intros ;apply P_id_a__U31_monotonic;assumption.
       intros ;apply P_id_U13_monotonic;assumption.
       intros ;apply P_id_a__U83_monotonic;assumption.
       intros ;apply P_id_isNatIList_monotonic;assumption.
       intros ;apply P_id_a__zeros_bounded;assumption.
       intros ;apply P_id_U51_bounded;assumption.
       intros ;apply P_id_a__U84_bounded;assumption.
       intros ;apply P_id_a__U32_bounded;assumption.
       intros ;apply P_id_isNatList_bounded;assumption.
       intros ;apply P_id_a__U13_bounded;assumption.
       intros ;apply P_id_U85_bounded;assumption.
       intros ;apply P_id_a__length_bounded;assumption.
       intros ;apply P_id_a__isNatIList_bounded;assumption.
       intros ;apply P_id_U33_bounded;assumption.
       intros ;apply P_id_a__U11_bounded;assumption.
       intros ;apply P_id_U81_bounded;assumption.
       intros ;apply P_id_a__U92_bounded;assumption.
       intros ;apply P_id_a__U43_bounded;assumption.
       intros ;apply P_id_U23_bounded;assumption.
       intros ;apply P_id_a__isNatKind_bounded;assumption.
       intros ;apply P_id_U93_bounded;assumption.
       intros ;apply P_id_U11_bounded;assumption.
       intros ;apply P_id_a__U71_bounded;assumption.
       intros ;apply P_id_U44_bounded;assumption.
       intros ;apply P_id_0_bounded;assumption.
       intros ;apply P_id_U61_bounded;assumption.
       intros ;apply P_id_a__U86_bounded;assumption.
       intros ;apply P_id_a__U41_bounded;assumption.
       intros ;apply P_id_U22_bounded;assumption.
       intros ;apply P_id_a__U21_bounded;assumption.
       intros ;apply P_id_U91_bounded;assumption.
       intros ;apply P_id_length_bounded;assumption.
       intros ;apply P_id_a__U52_bounded;assumption.
       intros ;apply P_id_U42_bounded;assumption.
       intros ;apply P_id_a__U12_bounded;assumption.
       intros ;apply P_id_U83_bounded;assumption.
       intros ;apply P_id_a__U94_bounded;assumption.
       intros ;apply P_id_a__U45_bounded;assumption.
       intros ;apply P_id_U31_bounded;assumption.
       intros ;apply P_id_a__isNat_bounded;assumption.
       intros ;apply P_id_isNatIListKind_bounded;assumption.
       intros ;apply P_id_a__U82_bounded;assumption.
       intros ;apply P_id_U46_bounded;assumption.
       intros ;apply P_id_cons_bounded;assumption.
       intros ;apply P_id_U52_bounded;assumption.
       intros ;apply P_id_a__U85_bounded;assumption.
       intros ;apply P_id_a__U33_bounded;assumption.
       intros ;apply P_id_U21_bounded;assumption.
       intros ;apply P_id_a__isNatList_bounded;assumption.
       intros ;apply P_id_U86_bounded;assumption.
       intros ;apply P_id_mark_bounded;assumption.
       intros ;apply P_id_a__U51_bounded;assumption.
       intros ;apply P_id_U41_bounded;assumption.
       intros ;apply P_id_tt_bounded;assumption.
       intros ;apply P_id_U82_bounded;assumption.
       intros ;apply P_id_a__U93_bounded;assumption.
       intros ;apply P_id_a__U44_bounded;assumption.
       intros ;apply P_id_isNat_bounded;assumption.
       intros ;apply P_id_a__U23_bounded;assumption.
       intros ;apply P_id_U94_bounded;assumption.
       intros ;apply P_id_U12_bounded;assumption.
       intros ;apply P_id_a__U81_bounded;assumption.
       intros ;apply P_id_U45_bounded;assumption.
       intros ;apply P_id_zeros_bounded;assumption.
       intros ;apply P_id_U71_bounded;assumption.
       intros ;apply P_id_a__U91_bounded;assumption.
       intros ;apply P_id_a__U42_bounded;assumption.
       intros ;apply P_id_isNatKind_bounded;assumption.
       intros ;apply P_id_a__U22_bounded;assumption.
       intros ;apply P_id_U92_bounded;assumption.
       intros ;apply P_id_nil_bounded;assumption.
       intros ;apply P_id_a__U61_bounded;assumption.
       intros ;apply P_id_U43_bounded;assumption.
       intros ;apply P_id_a__isNatIListKind_bounded;assumption.
       intros ;apply P_id_U84_bounded;assumption.
       intros ;apply P_id_s_bounded;assumption.
       intros ;apply P_id_a__U46_bounded;assumption.
       intros ;apply P_id_U32_bounded;assumption.
       intros ;apply P_id_a__U31_bounded;assumption.
       intros ;apply P_id_U13_bounded;assumption.
       intros ;apply P_id_a__U83_bounded;assumption.
       intros ;apply P_id_isNatIList_bounded;assumption.
       apply rules_monotonic.
     Qed.
     
     Definition P_id_A__U46 (x11:Z) := 0.
     
     Definition P_id_A__ISNATILISTKIND (x11:Z) := 0.
     
     Definition P_id_A__U61 (x11:Z) := 0.
     
     Definition P_id_A__U85 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__U31 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__U81 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__U22 (x11:Z) (x12:Z) := 2* x12.
     
     Definition P_id_A__U93 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__U42 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__U51 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__ISNATLIST (x11:Z) := 0.
     
     Definition P_id_A__ZEROS  := 0.
     
     Definition P_id_A__U91 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__U33 (x11:Z) := 0.
     
     Definition P_id_A__U83 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__U23 (x11:Z) := 0.
     
     Definition P_id_A__U11 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__LENGTH (x11:Z) := 0.
     
     Definition P_id_A__U44 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__ISNATILIST (x11:Z) := 0.
     
     Definition P_id_A__U13 (x11:Z) := 0.
     
     Definition P_id_A__U71 (x11:Z) := 0.
     
     Definition P_id_A__U86 (x11:Z) := 0.
     
     Definition P_id_A__U32 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__U82 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__ISNATKIND (x11:Z) := 0.
     
     Definition P_id_A__U94 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__U43 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__U52 (x11:Z) := 0.
     
     Definition P_id_A__U21 (x11:Z) (x12:Z) := 1 + 2* x12.
     
     Definition P_id_A__U92 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__U41 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__U84 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__ISNAT (x11:Z) := 2* x11.
     
     Definition P_id_A__U12 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_MARK (x11:Z) := 0.
     
     Definition P_id_A__U45 (x11:Z) (x12:Z) := 0.
     
     Lemma P_id_A__U46_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U46 x12 <= P_id_A__U46 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__ISNATILISTKIND_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__ISNATILISTKIND x12 <= P_id_A__ISNATILISTKIND x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U61_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U61 x12 <= P_id_A__U61 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U85_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U85 x12 x14 <= P_id_A__U85 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U31_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U31 x12 x14 <= P_id_A__U31 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U81_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U81 x12 x14 x16 <= P_id_A__U81 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U22_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U22 x12 x14 <= P_id_A__U22 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U93_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U93 x12 x14 x16 <= P_id_A__U93 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U42_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U42 x12 x14 x16 <= P_id_A__U42 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U51_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U51 x12 x14 <= P_id_A__U51 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__ISNATLIST_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__ISNATLIST x12 <= P_id_A__ISNATLIST x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U91_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U91 x12 x14 x16 <= P_id_A__U91 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U33_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U33 x12 <= P_id_A__U33 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U83_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U83 x12 x14 x16 <= P_id_A__U83 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U23_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U23 x12 <= P_id_A__U23 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U11_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U11 x12 x14 <= P_id_A__U11 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__LENGTH_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__LENGTH x12 <= P_id_A__LENGTH x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U44_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U44 x12 x14 x16 <= P_id_A__U44 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__ISNATILIST_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__ISNATILIST x12 <= P_id_A__ISNATILIST x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U13_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U13 x12 <= P_id_A__U13 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U71_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U71 x12 <= P_id_A__U71 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U86_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U86 x12 <= P_id_A__U86 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U32_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U32 x12 x14 <= P_id_A__U32 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U82_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U82 x12 x14 x16 <= P_id_A__U82 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__ISNATKIND_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__ISNATKIND x12 <= P_id_A__ISNATKIND x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U94_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U94 x12 x14 <= P_id_A__U94 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U43_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U43 x12 x14 x16 <= P_id_A__U43 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U52_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U52 x12 <= P_id_A__U52 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U21_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U21 x12 x14 <= P_id_A__U21 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U92_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U92 x12 x14 x16 <= P_id_A__U92 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U41_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U41 x12 x14 x16 <= P_id_A__U41 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U84_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U84 x12 x14 x16 <= P_id_A__U84 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__ISNAT_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__ISNAT x12 <= P_id_A__ISNAT x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U12_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U12 x12 x14 <= P_id_A__U12 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_MARK_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_MARK x12 <= P_id_MARK x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U45_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U45 x12 x14 <= P_id_A__U45 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Definition marked_measure  := 
       InterpZ.marked_measure 0 P_id_a__zeros P_id_U51 P_id_a__U84 
        P_id_a__U32 P_id_isNatList P_id_a__U13 P_id_U85 P_id_a__length 
        P_id_a__isNatIList P_id_U33 P_id_a__U11 P_id_U81 P_id_a__U92 
        P_id_a__U43 P_id_U23 P_id_a__isNatKind P_id_U93 P_id_U11 P_id_a__U71 
        P_id_U44 P_id_0 P_id_U61 P_id_a__U86 P_id_a__U41 P_id_U22 
        P_id_a__U21 P_id_U91 P_id_length P_id_a__U52 P_id_U42 P_id_a__U12 
        P_id_U83 P_id_a__U94 P_id_a__U45 P_id_U31 P_id_a__isNat 
        P_id_isNatIListKind P_id_a__U82 P_id_U46 P_id_cons P_id_U52 
        P_id_a__U85 P_id_a__U33 P_id_U21 P_id_a__isNatList P_id_U86 
        P_id_mark P_id_a__U51 P_id_U41 P_id_tt P_id_U82 P_id_a__U93 
        P_id_a__U44 P_id_isNat P_id_a__U23 P_id_U94 P_id_U12 P_id_a__U81 
        P_id_U45 P_id_zeros P_id_U71 P_id_a__U91 P_id_a__U42 P_id_isNatKind 
        P_id_a__U22 P_id_U92 P_id_nil P_id_a__U61 P_id_U43 
        P_id_a__isNatIListKind P_id_U84 P_id_s P_id_a__U46 P_id_U32 
        P_id_a__U31 P_id_U13 P_id_a__U83 P_id_isNatIList P_id_A__U46 
        P_id_A__ISNATILISTKIND P_id_A__U61 P_id_A__U85 P_id_A__U31 
        P_id_A__U81 P_id_A__U22 P_id_A__U93 P_id_A__U42 P_id_A__U51 
        P_id_A__ISNATLIST P_id_A__ZEROS P_id_A__U91 P_id_A__U33 P_id_A__U83 
        P_id_A__U23 P_id_A__U11 P_id_A__LENGTH P_id_A__U44 
        P_id_A__ISNATILIST P_id_A__U13 P_id_A__U71 P_id_A__U86 P_id_A__U32 
        P_id_A__U82 P_id_A__ISNATKIND P_id_A__U94 P_id_A__U43 P_id_A__U52 
        P_id_A__U21 P_id_A__U92 P_id_A__U41 P_id_A__U84 P_id_A__ISNAT 
        P_id_A__U12 P_id_MARK P_id_A__U45.
     
     Lemma marked_measure_equation :
      forall t, 
       marked_measure t = match t with
                            | (algebra.Alg.Term algebra.F.id_a__U46 
                               (x11::nil)) =>
                             P_id_A__U46 (measure x11)
                            | (algebra.Alg.Term 
                               algebra.F.id_a__isNatIListKind (x11::nil)) =>
                             P_id_A__ISNATILISTKIND (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U61 
                               (x11::nil)) =>
                             P_id_A__U61 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U85 (x12::
                               x11::nil)) =>
                             P_id_A__U85 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U31 (x12::
                               x11::nil)) =>
                             P_id_A__U31 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U81 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U81 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U22 (x12::
                               x11::nil)) =>
                             P_id_A__U22 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U93 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U93 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U42 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U42 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U51 (x12::
                               x11::nil)) =>
                             P_id_A__U51 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__isNatList 
                               (x11::nil)) =>
                             P_id_A__ISNATLIST (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__zeros nil) =>
                             P_id_A__ZEROS 
                            | (algebra.Alg.Term algebra.F.id_a__U91 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U91 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U33 
                               (x11::nil)) =>
                             P_id_A__U33 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U83 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U83 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U23 
                               (x11::nil)) =>
                             P_id_A__U23 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U11 (x12::
                               x11::nil)) =>
                             P_id_A__U11 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__length 
                               (x11::nil)) =>
                             P_id_A__LENGTH (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U44 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U44 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__isNatIList 
                               (x11::nil)) =>
                             P_id_A__ISNATILIST (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U13 
                               (x11::nil)) =>
                             P_id_A__U13 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U71 
                               (x11::nil)) =>
                             P_id_A__U71 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U86 
                               (x11::nil)) =>
                             P_id_A__U86 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U32 (x12::
                               x11::nil)) =>
                             P_id_A__U32 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U82 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U82 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__isNatKind 
                               (x11::nil)) =>
                             P_id_A__ISNATKIND (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U94 (x12::
                               x11::nil)) =>
                             P_id_A__U94 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U43 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U43 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U52 
                               (x11::nil)) =>
                             P_id_A__U52 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U21 (x12::
                               x11::nil)) =>
                             P_id_A__U21 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U92 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U92 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U41 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U41 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U84 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U84 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__isNat 
                               (x11::nil)) =>
                             P_id_A__ISNAT (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U12 (x12::
                               x11::nil)) =>
                             P_id_A__U12 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_mark (x11::nil)) =>
                             P_id_MARK (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U45 (x12::
                               x11::nil)) =>
                             P_id_A__U45 (measure x12) (measure x11)
                            | _ => measure t
                            end.
     Proof.
       reflexivity .
     Qed.
     
     Lemma marked_measure_star_monotonic :
      forall f l1 l2, 
       (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step 
                                                         R_xml_0_deep_rew.R_xml_0_rules)
                                                        ) l1 l2) ->
        marked_measure (algebra.Alg.Term f l1) <= marked_measure (algebra.Alg.Term 
                                                                   f 
                                                                   l2).
     Proof.
       unfold marked_measure in *.
       apply InterpZ.marked_measure_star_monotonic.
       intros ;apply P_id_U51_monotonic;assumption.
       intros ;apply P_id_a__U84_monotonic;assumption.
       intros ;apply P_id_a__U32_monotonic;assumption.
       intros ;apply P_id_isNatList_monotonic;assumption.
       intros ;apply P_id_a__U13_monotonic;assumption.
       intros ;apply P_id_U85_monotonic;assumption.
       intros ;apply P_id_a__length_monotonic;assumption.
       intros ;apply P_id_a__isNatIList_monotonic;assumption.
       intros ;apply P_id_U33_monotonic;assumption.
       intros ;apply P_id_a__U11_monotonic;assumption.
       intros ;apply P_id_U81_monotonic;assumption.
       intros ;apply P_id_a__U92_monotonic;assumption.
       intros ;apply P_id_a__U43_monotonic;assumption.
       intros ;apply P_id_U23_monotonic;assumption.
       intros ;apply P_id_a__isNatKind_monotonic;assumption.
       intros ;apply P_id_U93_monotonic;assumption.
       intros ;apply P_id_U11_monotonic;assumption.
       intros ;apply P_id_a__U71_monotonic;assumption.
       intros ;apply P_id_U44_monotonic;assumption.
       intros ;apply P_id_U61_monotonic;assumption.
       intros ;apply P_id_a__U86_monotonic;assumption.
       intros ;apply P_id_a__U41_monotonic;assumption.
       intros ;apply P_id_U22_monotonic;assumption.
       intros ;apply P_id_a__U21_monotonic;assumption.
       intros ;apply P_id_U91_monotonic;assumption.
       intros ;apply P_id_length_monotonic;assumption.
       intros ;apply P_id_a__U52_monotonic;assumption.
       intros ;apply P_id_U42_monotonic;assumption.
       intros ;apply P_id_a__U12_monotonic;assumption.
       intros ;apply P_id_U83_monotonic;assumption.
       intros ;apply P_id_a__U94_monotonic;assumption.
       intros ;apply P_id_a__U45_monotonic;assumption.
       intros ;apply P_id_U31_monotonic;assumption.
       intros ;apply P_id_a__isNat_monotonic;assumption.
       intros ;apply P_id_isNatIListKind_monotonic;assumption.
       intros ;apply P_id_a__U82_monotonic;assumption.
       intros ;apply P_id_U46_monotonic;assumption.
       intros ;apply P_id_cons_monotonic;assumption.
       intros ;apply P_id_U52_monotonic;assumption.
       intros ;apply P_id_a__U85_monotonic;assumption.
       intros ;apply P_id_a__U33_monotonic;assumption.
       intros ;apply P_id_U21_monotonic;assumption.
       intros ;apply P_id_a__isNatList_monotonic;assumption.
       intros ;apply P_id_U86_monotonic;assumption.
       intros ;apply P_id_mark_monotonic;assumption.
       intros ;apply P_id_a__U51_monotonic;assumption.
       intros ;apply P_id_U41_monotonic;assumption.
       intros ;apply P_id_U82_monotonic;assumption.
       intros ;apply P_id_a__U93_monotonic;assumption.
       intros ;apply P_id_a__U44_monotonic;assumption.
       intros ;apply P_id_isNat_monotonic;assumption.
       intros ;apply P_id_a__U23_monotonic;assumption.
       intros ;apply P_id_U94_monotonic;assumption.
       intros ;apply P_id_U12_monotonic;assumption.
       intros ;apply P_id_a__U81_monotonic;assumption.
       intros ;apply P_id_U45_monotonic;assumption.
       intros ;apply P_id_U71_monotonic;assumption.
       intros ;apply P_id_a__U91_monotonic;assumption.
       intros ;apply P_id_a__U42_monotonic;assumption.
       intros ;apply P_id_isNatKind_monotonic;assumption.
       intros ;apply P_id_a__U22_monotonic;assumption.
       intros ;apply P_id_U92_monotonic;assumption.
       intros ;apply P_id_a__U61_monotonic;assumption.
       intros ;apply P_id_U43_monotonic;assumption.
       intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
       intros ;apply P_id_U84_monotonic;assumption.
       intros ;apply P_id_s_monotonic;assumption.
       intros ;apply P_id_a__U46_monotonic;assumption.
       intros ;apply P_id_U32_monotonic;assumption.
       intros ;apply P_id_a__U31_monotonic;assumption.
       intros ;apply P_id_U13_monotonic;assumption.
       intros ;apply P_id_a__U83_monotonic;assumption.
       intros ;apply P_id_isNatIList_monotonic;assumption.
       intros ;apply P_id_a__zeros_bounded;assumption.
       intros ;apply P_id_U51_bounded;assumption.
       intros ;apply P_id_a__U84_bounded;assumption.
       intros ;apply P_id_a__U32_bounded;assumption.
       intros ;apply P_id_isNatList_bounded;assumption.
       intros ;apply P_id_a__U13_bounded;assumption.
       intros ;apply P_id_U85_bounded;assumption.
       intros ;apply P_id_a__length_bounded;assumption.
       intros ;apply P_id_a__isNatIList_bounded;assumption.
       intros ;apply P_id_U33_bounded;assumption.
       intros ;apply P_id_a__U11_bounded;assumption.
       intros ;apply P_id_U81_bounded;assumption.
       intros ;apply P_id_a__U92_bounded;assumption.
       intros ;apply P_id_a__U43_bounded;assumption.
       intros ;apply P_id_U23_bounded;assumption.
       intros ;apply P_id_a__isNatKind_bounded;assumption.
       intros ;apply P_id_U93_bounded;assumption.
       intros ;apply P_id_U11_bounded;assumption.
       intros ;apply P_id_a__U71_bounded;assumption.
       intros ;apply P_id_U44_bounded;assumption.
       intros ;apply P_id_0_bounded;assumption.
       intros ;apply P_id_U61_bounded;assumption.
       intros ;apply P_id_a__U86_bounded;assumption.
       intros ;apply P_id_a__U41_bounded;assumption.
       intros ;apply P_id_U22_bounded;assumption.
       intros ;apply P_id_a__U21_bounded;assumption.
       intros ;apply P_id_U91_bounded;assumption.
       intros ;apply P_id_length_bounded;assumption.
       intros ;apply P_id_a__U52_bounded;assumption.
       intros ;apply P_id_U42_bounded;assumption.
       intros ;apply P_id_a__U12_bounded;assumption.
       intros ;apply P_id_U83_bounded;assumption.
       intros ;apply P_id_a__U94_bounded;assumption.
       intros ;apply P_id_a__U45_bounded;assumption.
       intros ;apply P_id_U31_bounded;assumption.
       intros ;apply P_id_a__isNat_bounded;assumption.
       intros ;apply P_id_isNatIListKind_bounded;assumption.
       intros ;apply P_id_a__U82_bounded;assumption.
       intros ;apply P_id_U46_bounded;assumption.
       intros ;apply P_id_cons_bounded;assumption.
       intros ;apply P_id_U52_bounded;assumption.
       intros ;apply P_id_a__U85_bounded;assumption.
       intros ;apply P_id_a__U33_bounded;assumption.
       intros ;apply P_id_U21_bounded;assumption.
       intros ;apply P_id_a__isNatList_bounded;assumption.
       intros ;apply P_id_U86_bounded;assumption.
       intros ;apply P_id_mark_bounded;assumption.
       intros ;apply P_id_a__U51_bounded;assumption.
       intros ;apply P_id_U41_bounded;assumption.
       intros ;apply P_id_tt_bounded;assumption.
       intros ;apply P_id_U82_bounded;assumption.
       intros ;apply P_id_a__U93_bounded;assumption.
       intros ;apply P_id_a__U44_bounded;assumption.
       intros ;apply P_id_isNat_bounded;assumption.
       intros ;apply P_id_a__U23_bounded;assumption.
       intros ;apply P_id_U94_bounded;assumption.
       intros ;apply P_id_U12_bounded;assumption.
       intros ;apply P_id_a__U81_bounded;assumption.
       intros ;apply P_id_U45_bounded;assumption.
       intros ;apply P_id_zeros_bounded;assumption.
       intros ;apply P_id_U71_bounded;assumption.
       intros ;apply P_id_a__U91_bounded;assumption.
       intros ;apply P_id_a__U42_bounded;assumption.
       intros ;apply P_id_isNatKind_bounded;assumption.
       intros ;apply P_id_a__U22_bounded;assumption.
       intros ;apply P_id_U92_bounded;assumption.
       intros ;apply P_id_nil_bounded;assumption.
       intros ;apply P_id_a__U61_bounded;assumption.
       intros ;apply P_id_U43_bounded;assumption.
       intros ;apply P_id_a__isNatIListKind_bounded;assumption.
       intros ;apply P_id_U84_bounded;assumption.
       intros ;apply P_id_s_bounded;assumption.
       intros ;apply P_id_a__U46_bounded;assumption.
       intros ;apply P_id_U32_bounded;assumption.
       intros ;apply P_id_a__U31_bounded;assumption.
       intros ;apply P_id_U13_bounded;assumption.
       intros ;apply P_id_a__U83_bounded;assumption.
       intros ;apply P_id_isNatIList_bounded;assumption.
       apply rules_monotonic.
       intros ;apply P_id_A__U46_monotonic;assumption.
       intros ;apply P_id_A__ISNATILISTKIND_monotonic;assumption.
       intros ;apply P_id_A__U61_monotonic;assumption.
       intros ;apply P_id_A__U85_monotonic;assumption.
       intros ;apply P_id_A__U31_monotonic;assumption.
       intros ;apply P_id_A__U81_monotonic;assumption.
       intros ;apply P_id_A__U22_monotonic;assumption.
       intros ;apply P_id_A__U93_monotonic;assumption.
       intros ;apply P_id_A__U42_monotonic;assumption.
       intros ;apply P_id_A__U51_monotonic;assumption.
       intros ;apply P_id_A__ISNATLIST_monotonic;assumption.
       intros ;apply P_id_A__U91_monotonic;assumption.
       intros ;apply P_id_A__U33_monotonic;assumption.
       intros ;apply P_id_A__U83_monotonic;assumption.
       intros ;apply P_id_A__U23_monotonic;assumption.
       intros ;apply P_id_A__U11_monotonic;assumption.
       intros ;apply P_id_A__LENGTH_monotonic;assumption.
       intros ;apply P_id_A__U44_monotonic;assumption.
       intros ;apply P_id_A__ISNATILIST_monotonic;assumption.
       intros ;apply P_id_A__U13_monotonic;assumption.
       intros ;apply P_id_A__U71_monotonic;assumption.
       intros ;apply P_id_A__U86_monotonic;assumption.
       intros ;apply P_id_A__U32_monotonic;assumption.
       intros ;apply P_id_A__U82_monotonic;assumption.
       intros ;apply P_id_A__ISNATKIND_monotonic;assumption.
       intros ;apply P_id_A__U94_monotonic;assumption.
       intros ;apply P_id_A__U43_monotonic;assumption.
       intros ;apply P_id_A__U52_monotonic;assumption.
       intros ;apply P_id_A__U21_monotonic;assumption.
       intros ;apply P_id_A__U92_monotonic;assumption.
       intros ;apply P_id_A__U41_monotonic;assumption.
       intros ;apply P_id_A__U84_monotonic;assumption.
       intros ;apply P_id_A__ISNAT_monotonic;assumption.
       intros ;apply P_id_A__U12_monotonic;assumption.
       intros ;apply P_id_MARK_monotonic;assumption.
       intros ;apply P_id_A__U45_monotonic;assumption.
     Qed.
     
     Ltac rewrite_and_unfold  :=
      do 2 (rewrite marked_measure_equation);
       repeat (
       match goal with
         |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
          rewrite (measure_equation (algebra.Alg.Term f t))
         | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ =>
          rewrite (measure_equation (algebra.Alg.Term f t)) in H|-
         end
       ).
     
     Definition lt a b := (Zwf.Zwf 0) (marked_measure a) (marked_measure b).
     
     Definition le a b := marked_measure a <= marked_measure b.
     
     Lemma lt_le_compat : forall a b c, (lt a b) ->(le b c) ->lt a c.
     Proof.
       unfold lt, le in *.
       intros a b c.
       apply (interp.le_lt_compat_right (interp.o_Z 0)).
     Qed.
     
     Lemma wf_lt : well_founded lt.
     Proof.
       unfold lt in *.
       apply Inverse_Image.wf_inverse_image with  (B:=Z).
       apply Zwf.Zwf_well_founded.
     Qed.
     
     Lemma DP_R_xml_0_scc_38_large_scc_3_strict_in_lt :
      Relation_Definitions.inclusion _ DP_R_xml_0_scc_38_large_scc_3_strict 
       lt.
     Proof.
       unfold Relation_Definitions.inclusion, lt in *.
       
       intros a b H;destruct H;
        match goal with
          |  |- (Zwf.Zwf 0) _ (marked_measure (algebra.Alg.Term ?f ?l)) =>
           let l'' := algebra.Alg_ext.find_replacement l  in 
            ((apply (interp.le_lt_compat_right (interp.o_Z 0)) with
               (marked_measure (algebra.Alg.Term f l''));[idtac|
              apply marked_measure_star_monotonic;
               repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos);
               (assumption)||(constructor 1)]))
          end
        ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma DP_R_xml_0_scc_38_large_scc_3_large_in_le :
      Relation_Definitions.inclusion _ DP_R_xml_0_scc_38_large_scc_3_large le.
     Proof.
       unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *.
       
       intros a b H;destruct H;
        match goal with
          |  |- _ <= marked_measure (algebra.Alg.Term ?f ?l) =>
           let l'' := algebra.Alg_ext.find_replacement l  in 
            ((apply (interp.le_trans (interp.o_Z 0)) with
               (marked_measure (algebra.Alg.Term f l''));[idtac|
              apply marked_measure_star_monotonic;
               repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos);
               (assumption)||(constructor 1)]))
          end
        ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Definition wf_DP_R_xml_0_scc_38_large_scc_3_large  := 
       WF_DP_R_xml_0_scc_38_large_scc_3_large.wf.
     
     
     Lemma wf :
      well_founded WF_DP_R_xml_0_scc_38_large.DP_R_xml_0_scc_38_large_scc_3.
     Proof.
       intros x.
       apply (well_founded_ind wf_lt).
       clear x.
       intros x.
       pattern x.
       apply (@Acc_ind _ DP_R_xml_0_scc_38_large_scc_3_large).
       clear x.
       intros x _ IHx IHx'.
       constructor.
       intros y H.
       
       destruct H;
        (apply IHx';apply DP_R_xml_0_scc_38_large_scc_3_strict_in_lt;
          econstructor eassumption)||
        ((apply IHx;[econstructor eassumption|
          intros y' Hlt;apply IHx';apply lt_le_compat with (1:=Hlt) ;
           apply DP_R_xml_0_scc_38_large_scc_3_large_in_le;
           econstructor eassumption])).
       apply wf_DP_R_xml_0_scc_38_large_scc_3_large.
     Qed.
    End WF_DP_R_xml_0_scc_38_large_scc_3.
    
    Definition wf_DP_R_xml_0_scc_38_large_scc_3  := 
      WF_DP_R_xml_0_scc_38_large_scc_3.wf.
    
    
    Lemma acc_DP_R_xml_0_scc_38_large_scc_3 :
     forall x y, 
      (DP_R_xml_0_scc_38_large_scc_3 x y) ->
       Acc WF_DP_R_xml_0_scc_38.DP_R_xml_0_scc_38_large x.
    Proof.
      intros x.
      pattern x.
      apply (@Acc_ind _ DP_R_xml_0_scc_38_large_scc_3).
      intros x' _ Hrec y h.
      
      inversion h;clear h;subst;
       constructor;intros _y _h;inversion _h;clear _h;subst;
        (eapply Hrec;econstructor eassumption)||
        ((eapply acc_DP_R_xml_0_scc_38_large_non_scc_2;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
         ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
          (eapply Hrec;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))).
      apply wf_DP_R_xml_0_scc_38_large_scc_3.
    Qed.
    
    
    Inductive DP_R_xml_0_scc_38_large_non_scc_4  :
     algebra.Alg.term ->algebra.Alg.term ->Prop := 
       (* <a__U84(tt,V1_,V2_),a__isNat(V1_)> *)
      | DP_R_xml_0_scc_38_large_non_scc_4_0 :
       forall x12 x1 x13 x3 x11, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   (algebra.Alg.Term algebra.F.id_tt nil) 
          x13) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x1 x12) ->
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     x3 x11) ->
           DP_R_xml_0_scc_38_large_non_scc_4 (algebra.Alg.Term 
                                              algebra.F.id_a__isNat 
                                              (x1::nil)) 
            (algebra.Alg.Term algebra.F.id_a__U84 (x13::x12::x11::nil))
    .
    
    
    Lemma acc_DP_R_xml_0_scc_38_large_non_scc_4 :
     forall x y, 
      (DP_R_xml_0_scc_38_large_non_scc_4 x y) ->
       Acc WF_DP_R_xml_0_scc_38.DP_R_xml_0_scc_38_large x.
    Proof.
      intros x y h.
      
      inversion h;clear h;subst;
       constructor;intros _y _h;inversion _h;clear _h;subst;
        (eapply acc_DP_R_xml_0_scc_38_large_scc_3;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((eapply acc_DP_R_xml_0_scc_38_large_non_scc_2;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
         ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
          (eapply Hrec;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))).
    Qed.
    
    
    Inductive DP_R_xml_0_scc_38_large_scc_5  :
     algebra.Alg.term ->algebra.Alg.term ->Prop := 
       (* <a__U81(tt,V1_,V2_),a__U82(a__isNatKind(V1_),V1_,V2_)> *)
      | DP_R_xml_0_scc_38_large_scc_5_0 :
       forall x12 x1 x13 x3 x11, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   (algebra.Alg.Term algebra.F.id_tt nil) 
          x13) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x1 x12) ->
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     x3 x11) ->
           DP_R_xml_0_scc_38_large_scc_5 (algebra.Alg.Term 
                                          algebra.F.id_a__U82 
                                          ((algebra.Alg.Term 
                                          algebra.F.id_a__isNatKind 
                                          (x1::nil))::x1::x3::nil)) 
            (algebra.Alg.Term algebra.F.id_a__U81 (x13::x12::x11::nil))
      
       (* <a__U82(tt,V1_,V2_),a__U83(a__isNatIListKind(V2_),V1_,V2_)> *)
      | DP_R_xml_0_scc_38_large_scc_5_1 :
       forall x12 x1 x13 x3 x11, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   (algebra.Alg.Term algebra.F.id_tt nil) 
          x13) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x1 x12) ->
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     x3 x11) ->
           DP_R_xml_0_scc_38_large_scc_5 (algebra.Alg.Term 
                                          algebra.F.id_a__U83 
                                          ((algebra.Alg.Term 
                                          algebra.F.id_a__isNatIListKind 
                                          (x3::nil))::x1::x3::nil)) 
            (algebra.Alg.Term algebra.F.id_a__U82 (x13::x12::x11::nil))
      
       (* <a__U83(tt,V1_,V2_),a__U84(a__isNatIListKind(V2_),V1_,V2_)> *)
      | DP_R_xml_0_scc_38_large_scc_5_2 :
       forall x12 x1 x13 x3 x11, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   (algebra.Alg.Term algebra.F.id_tt nil) 
          x13) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x1 x12) ->
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     x3 x11) ->
           DP_R_xml_0_scc_38_large_scc_5 (algebra.Alg.Term 
                                          algebra.F.id_a__U84 
                                          ((algebra.Alg.Term 
                                          algebra.F.id_a__isNatIListKind 
                                          (x3::nil))::x1::x3::nil)) 
            (algebra.Alg.Term algebra.F.id_a__U83 (x13::x12::x11::nil))
      
       (* <a__U84(tt,V1_,V2_),a__U85(a__isNat(V1_),V2_)> *)
      | DP_R_xml_0_scc_38_large_scc_5_3 :
       forall x12 x1 x13 x3 x11, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   (algebra.Alg.Term algebra.F.id_tt nil) 
          x13) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x1 x12) ->
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     x3 x11) ->
           DP_R_xml_0_scc_38_large_scc_5 (algebra.Alg.Term 
                                          algebra.F.id_a__U85 
                                          ((algebra.Alg.Term 
                                          algebra.F.id_a__isNat (x1::nil))::
                                          x3::nil)) 
            (algebra.Alg.Term algebra.F.id_a__U84 (x13::x12::x11::nil))
      
       (* <a__U85(tt,V2_),a__isNatList(V2_)> *)
      | DP_R_xml_0_scc_38_large_scc_5_4 :
       forall x12 x3 x11, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   (algebra.Alg.Term algebra.F.id_tt nil) 
          x12) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x3 x11) ->
          DP_R_xml_0_scc_38_large_scc_5 (algebra.Alg.Term 
                                         algebra.F.id_a__isNatList (x3::nil)) 
           (algebra.Alg.Term algebra.F.id_a__U85 (x12::x11::nil))
      
       (* <a__isNatList(cons(V1_,V2_)),a__U81(a__isNatKind(V1_),V1_,V2_)> *)
      | DP_R_xml_0_scc_38_large_scc_5_5 :
       forall x1 x3 x11, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_cons (x1::x3::nil)) x11) ->
         DP_R_xml_0_scc_38_large_scc_5 (algebra.Alg.Term algebra.F.id_a__U81 
                                        ((algebra.Alg.Term 
                                        algebra.F.id_a__isNatKind 
                                        (x1::nil))::x1::x3::nil)) 
          (algebra.Alg.Term algebra.F.id_a__isNatList (x11::nil))
    .
    
    
    Module WF_DP_R_xml_0_scc_38_large_scc_5.
     Inductive DP_R_xml_0_scc_38_large_scc_5_large  :
      algebra.Alg.term ->algebra.Alg.term ->Prop := 
        (* <a__U81(tt,V1_,V2_),a__U82(a__isNatKind(V1_),V1_,V2_)> *)
       | DP_R_xml_0_scc_38_large_scc_5_large_0 :
        forall x12 x1 x13 x3 x11, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    (algebra.Alg.Term algebra.F.id_tt nil) 
           x13) ->
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     x1 x12) ->
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                      x3 x11) ->
            DP_R_xml_0_scc_38_large_scc_5_large (algebra.Alg.Term 
                                                 algebra.F.id_a__U82 
                                                 ((algebra.Alg.Term 
                                                 algebra.F.id_a__isNatKind 
                                                 (x1::nil))::x1::x3::nil)) 
             (algebra.Alg.Term algebra.F.id_a__U81 (x13::x12::x11::nil))
       
        (* <a__U82(tt,V1_,V2_),a__U83(a__isNatIListKind(V2_),V1_,V2_)> *)
       | DP_R_xml_0_scc_38_large_scc_5_large_1 :
        forall x12 x1 x13 x3 x11, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    (algebra.Alg.Term algebra.F.id_tt nil) 
           x13) ->
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     x1 x12) ->
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                      x3 x11) ->
            DP_R_xml_0_scc_38_large_scc_5_large (algebra.Alg.Term 
                                                 algebra.F.id_a__U83 
                                                 ((algebra.Alg.Term 
                                                 algebra.F.id_a__isNatIListKind 
                                                 (x3::nil))::x1::x3::nil)) 
             (algebra.Alg.Term algebra.F.id_a__U82 (x13::x12::x11::nil))
       
        (* <a__U83(tt,V1_,V2_),a__U84(a__isNatIListKind(V2_),V1_,V2_)> *)
       | DP_R_xml_0_scc_38_large_scc_5_large_2 :
        forall x12 x1 x13 x3 x11, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    (algebra.Alg.Term algebra.F.id_tt nil) 
           x13) ->
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     x1 x12) ->
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                      x3 x11) ->
            DP_R_xml_0_scc_38_large_scc_5_large (algebra.Alg.Term 
                                                 algebra.F.id_a__U84 
                                                 ((algebra.Alg.Term 
                                                 algebra.F.id_a__isNatIListKind 
                                                 (x3::nil))::x1::x3::nil)) 
             (algebra.Alg.Term algebra.F.id_a__U83 (x13::x12::x11::nil))
       
        (* <a__U84(tt,V1_,V2_),a__U85(a__isNat(V1_),V2_)> *)
       | DP_R_xml_0_scc_38_large_scc_5_large_3 :
        forall x12 x1 x13 x3 x11, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    (algebra.Alg.Term algebra.F.id_tt nil) 
           x13) ->
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     x1 x12) ->
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                      x3 x11) ->
            DP_R_xml_0_scc_38_large_scc_5_large (algebra.Alg.Term 
                                                 algebra.F.id_a__U85 
                                                 ((algebra.Alg.Term 
                                                 algebra.F.id_a__isNat 
                                                 (x1::nil))::x3::nil)) 
             (algebra.Alg.Term algebra.F.id_a__U84 (x13::x12::x11::nil))
       
        (* <a__U85(tt,V2_),a__isNatList(V2_)> *)
       | DP_R_xml_0_scc_38_large_scc_5_large_4 :
        forall x12 x3 x11, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    (algebra.Alg.Term algebra.F.id_tt nil) 
           x12) ->
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     x3 x11) ->
           DP_R_xml_0_scc_38_large_scc_5_large (algebra.Alg.Term 
                                                algebra.F.id_a__isNatList 
                                                (x3::nil)) 
            (algebra.Alg.Term algebra.F.id_a__U85 (x12::x11::nil))
     .
     
     
     Inductive DP_R_xml_0_scc_38_large_scc_5_strict  :
      algebra.Alg.term ->algebra.Alg.term ->Prop := 
        (* <a__isNatList(cons(V1_,V2_)),a__U81(a__isNatKind(V1_),V1_,V2_)> *)
       | DP_R_xml_0_scc_38_large_scc_5_strict_0 :
        forall x1 x3 x11, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    
           (algebra.Alg.Term algebra.F.id_cons (x1::x3::nil)) x11) ->
          DP_R_xml_0_scc_38_large_scc_5_strict (algebra.Alg.Term 
                                                algebra.F.id_a__U81 
                                                ((algebra.Alg.Term 
                                                algebra.F.id_a__isNatKind 
                                                (x1::nil))::x1::x3::nil)) 
           (algebra.Alg.Term algebra.F.id_a__isNatList (x11::nil))
     .
     
     
     Module WF_DP_R_xml_0_scc_38_large_scc_5_large.
      Inductive DP_R_xml_0_scc_38_large_scc_5_large_non_scc_1  :
       algebra.Alg.term ->algebra.Alg.term ->Prop := 
         (* <a__U85(tt,V2_),a__isNatList(V2_)> *)
        | DP_R_xml_0_scc_38_large_scc_5_large_non_scc_1_0 :
         forall x12 x3 x11, 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     (algebra.Alg.Term algebra.F.id_tt nil) 
            x12) ->
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                      x3 x11) ->
            DP_R_xml_0_scc_38_large_scc_5_large_non_scc_1 (algebra.Alg.Term 
                                                           algebra.F.id_a__isNatList 
                                                           (x3::nil)) 
             (algebra.Alg.Term algebra.F.id_a__U85 (x12::x11::nil))
      .
      
      
      Lemma acc_DP_R_xml_0_scc_38_large_scc_5_large_non_scc_1 :
       forall x y, 
        (DP_R_xml_0_scc_38_large_scc_5_large_non_scc_1 x y) ->
         Acc WF_DP_R_xml_0_scc_38_large_scc_5.DP_R_xml_0_scc_38_large_scc_5_large 
          x.
      Proof.
        intros x y h.
        
        inversion h;clear h;subst;
         constructor;intros _y _h;inversion _h;clear _h;subst;
          (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
          (eapply Hrec;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' )).
      Qed.
      
      
      Inductive DP_R_xml_0_scc_38_large_scc_5_large_non_scc_2  :
       algebra.Alg.term ->algebra.Alg.term ->Prop := 
         (* <a__U84(tt,V1_,V2_),a__U85(a__isNat(V1_),V2_)> *)
        | DP_R_xml_0_scc_38_large_scc_5_large_non_scc_2_0 :
         forall x12 x1 x13 x3 x11, 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     (algebra.Alg.Term algebra.F.id_tt nil) 
            x13) ->
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                      x1 x12) ->
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                       x3 x11) ->
             DP_R_xml_0_scc_38_large_scc_5_large_non_scc_2 (algebra.Alg.Term 
                                                            algebra.F.id_a__U85 
                                                            ((algebra.Alg.Term 
                                                            algebra.F.id_a__isNat 
                                                            (x1::nil))::
                                                            x3::nil)) 
              (algebra.Alg.Term algebra.F.id_a__U84 (x13::x12::x11::nil))
      .
      
      
      Lemma acc_DP_R_xml_0_scc_38_large_scc_5_large_non_scc_2 :
       forall x y, 
        (DP_R_xml_0_scc_38_large_scc_5_large_non_scc_2 x y) ->
         Acc WF_DP_R_xml_0_scc_38_large_scc_5.DP_R_xml_0_scc_38_large_scc_5_large 
          x.
      Proof.
        intros x y h.
        
        inversion h;clear h;subst;
         constructor;intros _y _h;inversion _h;clear _h;subst;
          (eapply acc_DP_R_xml_0_scc_38_large_scc_5_large_non_scc_1;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
          ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
           (eapply Hrec;
             econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))).
      Qed.
      
      
      Inductive DP_R_xml_0_scc_38_large_scc_5_large_non_scc_3  :
       algebra.Alg.term ->algebra.Alg.term ->Prop := 
         (* <a__U83(tt,V1_,V2_),a__U84(a__isNatIListKind(V2_),V1_,V2_)> *)
        | DP_R_xml_0_scc_38_large_scc_5_large_non_scc_3_0 :
         forall x12 x1 x13 x3 x11, 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     (algebra.Alg.Term algebra.F.id_tt nil) 
            x13) ->
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                      x1 x12) ->
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                       x3 x11) ->
             DP_R_xml_0_scc_38_large_scc_5_large_non_scc_3 (algebra.Alg.Term 
                                                            algebra.F.id_a__U84 
                                                            ((algebra.Alg.Term 
                                                            algebra.F.id_a__isNatIListKind 
                                                            (x3::nil))::x1::
                                                            x3::nil)) 
              (algebra.Alg.Term algebra.F.id_a__U83 (x13::x12::x11::nil))
      .
      
      
      Lemma acc_DP_R_xml_0_scc_38_large_scc_5_large_non_scc_3 :
       forall x y, 
        (DP_R_xml_0_scc_38_large_scc_5_large_non_scc_3 x y) ->
         Acc WF_DP_R_xml_0_scc_38_large_scc_5.DP_R_xml_0_scc_38_large_scc_5_large 
          x.
      Proof.
        intros x y h.
        
        inversion h;clear h;subst;
         constructor;intros _y _h;inversion _h;clear _h;subst;
          (eapply acc_DP_R_xml_0_scc_38_large_scc_5_large_non_scc_2;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
          ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
           (eapply Hrec;
             econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))).
      Qed.
      
      
      Inductive DP_R_xml_0_scc_38_large_scc_5_large_non_scc_4  :
       algebra.Alg.term ->algebra.Alg.term ->Prop := 
         (* <a__U82(tt,V1_,V2_),a__U83(a__isNatIListKind(V2_),V1_,V2_)> *)
        | DP_R_xml_0_scc_38_large_scc_5_large_non_scc_4_0 :
         forall x12 x1 x13 x3 x11, 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     (algebra.Alg.Term algebra.F.id_tt nil) 
            x13) ->
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                      x1 x12) ->
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                       x3 x11) ->
             DP_R_xml_0_scc_38_large_scc_5_large_non_scc_4 (algebra.Alg.Term 
                                                            algebra.F.id_a__U83 
                                                            ((algebra.Alg.Term 
                                                            algebra.F.id_a__isNatIListKind 
                                                            (x3::nil))::x1::
                                                            x3::nil)) 
              (algebra.Alg.Term algebra.F.id_a__U82 (x13::x12::x11::nil))
      .
      
      
      Lemma acc_DP_R_xml_0_scc_38_large_scc_5_large_non_scc_4 :
       forall x y, 
        (DP_R_xml_0_scc_38_large_scc_5_large_non_scc_4 x y) ->
         Acc WF_DP_R_xml_0_scc_38_large_scc_5.DP_R_xml_0_scc_38_large_scc_5_large 
          x.
      Proof.
        intros x y h.
        
        inversion h;clear h;subst;
         constructor;intros _y _h;inversion _h;clear _h;subst;
          (eapply acc_DP_R_xml_0_scc_38_large_scc_5_large_non_scc_3;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
          ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
           (eapply Hrec;
             econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))).
      Qed.
      
      
      Inductive DP_R_xml_0_scc_38_large_scc_5_large_non_scc_5  :
       algebra.Alg.term ->algebra.Alg.term ->Prop := 
         (* <a__U81(tt,V1_,V2_),a__U82(a__isNatKind(V1_),V1_,V2_)> *)
        | DP_R_xml_0_scc_38_large_scc_5_large_non_scc_5_0 :
         forall x12 x1 x13 x3 x11, 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     (algebra.Alg.Term algebra.F.id_tt nil) 
            x13) ->
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                      x1 x12) ->
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                       x3 x11) ->
             DP_R_xml_0_scc_38_large_scc_5_large_non_scc_5 (algebra.Alg.Term 
                                                            algebra.F.id_a__U82 
                                                            ((algebra.Alg.Term 
                                                            algebra.F.id_a__isNatKind 
                                                            (x1::nil))::x1::
                                                            x3::nil)) 
              (algebra.Alg.Term algebra.F.id_a__U81 (x13::x12::x11::nil))
      .
      
      
      Lemma acc_DP_R_xml_0_scc_38_large_scc_5_large_non_scc_5 :
       forall x y, 
        (DP_R_xml_0_scc_38_large_scc_5_large_non_scc_5 x y) ->
         Acc WF_DP_R_xml_0_scc_38_large_scc_5.DP_R_xml_0_scc_38_large_scc_5_large 
          x.
      Proof.
        intros x y h.
        
        inversion h;clear h;subst;
         constructor;intros _y _h;inversion _h;clear _h;subst;
          (eapply acc_DP_R_xml_0_scc_38_large_scc_5_large_non_scc_4;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
          ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
           (eapply Hrec;
             econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))).
      Qed.
      
      
      Lemma wf :
       well_founded WF_DP_R_xml_0_scc_38_large_scc_5.DP_R_xml_0_scc_38_large_scc_5_large
        .
      Proof.
        constructor;intros _y _h;inversion _h;clear _h;subst;
         (eapply acc_DP_R_xml_0_scc_38_large_scc_5_large_non_scc_5;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
         ((eapply acc_DP_R_xml_0_scc_38_large_scc_5_large_non_scc_4;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
          ((eapply acc_DP_R_xml_0_scc_38_large_scc_5_large_non_scc_3;
             econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
           ((eapply acc_DP_R_xml_0_scc_38_large_scc_5_large_non_scc_2;
              econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
            ((eapply acc_DP_R_xml_0_scc_38_large_scc_5_large_non_scc_1;
               econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
             ((eapply acc_DP_R_xml_0_scc_38_large_scc_5_large_non_scc_0;
                econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
              ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||(fail))))))).
      Qed.
     End WF_DP_R_xml_0_scc_38_large_scc_5_large.
     
     Open Scope Z_scope.
     
     Import ring_extention.
     
     Notation Local "a <= b" := (Zle a b).
     
     Notation Local "a < b" := (Zlt a b).
     
     Definition P_id_a__zeros  := 2.
     
     Definition P_id_U51 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_a__U84 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_a__U32 (x11:Z) (x12:Z) := 1 + 3* x12.
     
     Definition P_id_isNatList (x11:Z) := 0.
     
     Definition P_id_a__U13 (x11:Z) := 0.
     
     Definition P_id_U85 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_a__length (x11:Z) := 0.
     
     Definition P_id_a__isNatIList (x11:Z) := 1 + 3* x11.
     
     Definition P_id_U33 (x11:Z) := 1.
     
     Definition P_id_a__U11 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_U81 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_a__U92 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_a__U43 (x11:Z) (x12:Z) (x13:Z) := 2* x13.
     
     Definition P_id_U23 (x11:Z) := 0.
     
     Definition P_id_a__isNatKind (x11:Z) := 2.
     
     Definition P_id_U93 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_U11 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_a__U71 (x11:Z) := 2.
     
     Definition P_id_U44 (x11:Z) (x12:Z) (x13:Z) := 1* x13.
     
     Definition P_id_0  := 0.
     
     Definition P_id_U61 (x11:Z) := 0.
     
     Definition P_id_a__U86 (x11:Z) := 0.
     
     Definition P_id_a__U41 (x11:Z) (x12:Z) (x13:Z) := 3 + 3* x13.
     
     Definition P_id_U22 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_a__U21 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_U91 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_length (x11:Z) := 0.
     
     Definition P_id_a__U52 (x11:Z) := 0.
     
     Definition P_id_U42 (x11:Z) (x12:Z) (x13:Z) := 3* x13.
     
     Definition P_id_a__U12 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_U83 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_a__U94 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_a__U45 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_U31 (x11:Z) (x12:Z) := 1 + 2* x12.
     
     Definition P_id_a__isNat (x11:Z) := 2.
     
     Definition P_id_isNatIListKind (x11:Z) := 0.
     
     Definition P_id_a__U82 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_U46 (x11:Z) := 0.
     
     Definition P_id_cons (x11:Z) (x12:Z) := 1 + 1* x12.
     
     Definition P_id_U52 (x11:Z) := 0.
     
     Definition P_id_a__U85 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_a__U33 (x11:Z) := 1.
     
     Definition P_id_U21 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_a__isNatList (x11:Z) := 0.
     
     Definition P_id_U86 (x11:Z) := 0.
     
     Definition P_id_mark (x11:Z) := 2* x11.
     
     Definition P_id_a__U51 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_U41 (x11:Z) (x12:Z) (x13:Z) := 2 + 2* x13.
     
     Definition P_id_tt  := 0.
     
     Definition P_id_U82 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_a__U93 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_a__U44 (x11:Z) (x12:Z) (x13:Z) := 1* x13.
     
     Definition P_id_isNat (x11:Z) := 1.
     
     Definition P_id_a__U23 (x11:Z) := 0.
     
     Definition P_id_U94 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_U12 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_a__U81 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_U45 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_zeros  := 1.
     
     Definition P_id_U71 (x11:Z) := 2.
     
     Definition P_id_a__U91 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_a__U42 (x11:Z) (x12:Z) (x13:Z) := 3* x13.
     
     Definition P_id_isNatKind (x11:Z) := 2.
     
     Definition P_id_a__U22 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_U92 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_nil  := 2.
     
     Definition P_id_a__U61 (x11:Z) := 0.
     
     Definition P_id_U43 (x11:Z) (x12:Z) (x13:Z) := 1* x13.
     
     Definition P_id_a__isNatIListKind (x11:Z) := 0.
     
     Definition P_id_U84 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_s (x11:Z) := 0.
     
     Definition P_id_a__U46 (x11:Z) := 0.
     
     Definition P_id_U32 (x11:Z) (x12:Z) := 1 + 2* x12.
     
     Definition P_id_a__U31 (x11:Z) (x12:Z) := 1 + 3* x12.
     
     Definition P_id_U13 (x11:Z) := 0.
     
     Definition P_id_a__U83 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_isNatIList (x11:Z) := 1 + 2* x11.
     
     Lemma P_id_U51_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U51 x12 x14 <= P_id_U51 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U84_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U84 x12 x14 x16 <= P_id_a__U84 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U32_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U32 x12 x14 <= P_id_a__U32 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNatList_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_isNatList x12 <= P_id_isNatList x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U13_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U13 x12 <= P_id_a__U13 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U85_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U85 x12 x14 <= P_id_U85 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__length_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__length x12 <= P_id_a__length x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNatIList_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__isNatIList x12 <= P_id_a__isNatIList x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U33_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U33 x12 <= P_id_U33 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U11_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U11 x12 x14 <= P_id_a__U11 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U81_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U81 x12 x14 x16 <= P_id_U81 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U92_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U92 x12 x14 x16 <= P_id_a__U92 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U43_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U43 x12 x14 x16 <= P_id_a__U43 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U23_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U23 x12 <= P_id_U23 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNatKind_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__isNatKind x12 <= P_id_a__isNatKind x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U93_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U93 x12 x14 x16 <= P_id_U93 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U11_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U11 x12 x14 <= P_id_U11 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U71_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U71 x12 <= P_id_a__U71 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U44_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U44 x12 x14 x16 <= P_id_U44 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U61_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U61 x12 <= P_id_U61 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U86_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U86 x12 <= P_id_a__U86 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U41_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U41 x12 x14 x16 <= P_id_a__U41 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U22_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U22 x12 x14 <= P_id_U22 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U21_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U21 x12 x14 <= P_id_a__U21 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U91_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U91 x12 x14 x16 <= P_id_U91 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_length_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_length x12 <= P_id_length x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U52_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U52 x12 <= P_id_a__U52 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U42_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U42 x12 x14 x16 <= P_id_U42 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U12_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U12 x12 x14 <= P_id_a__U12 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U83_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U83 x12 x14 x16 <= P_id_U83 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U94_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U94 x12 x14 <= P_id_a__U94 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U45_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U45 x12 x14 <= P_id_a__U45 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U31_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U31 x12 x14 <= P_id_U31 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNat_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__isNat x12 <= P_id_a__isNat x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNatIListKind_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_isNatIListKind x12 <= P_id_isNatIListKind x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U82_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U82 x12 x14 x16 <= P_id_a__U82 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U46_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U46 x12 <= P_id_U46 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_cons_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_cons x12 x14 <= P_id_cons x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U52_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U52 x12 <= P_id_U52 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U85_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U85 x12 x14 <= P_id_a__U85 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U33_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U33 x12 <= P_id_a__U33 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U21_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U21 x12 x14 <= P_id_U21 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNatList_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__isNatList x12 <= P_id_a__isNatList x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U86_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U86 x12 <= P_id_U86 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_mark_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_mark x12 <= P_id_mark x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U51_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U51 x12 x14 <= P_id_a__U51 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U41_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U41 x12 x14 x16 <= P_id_U41 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U82_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U82 x12 x14 x16 <= P_id_U82 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U93_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U93 x12 x14 x16 <= P_id_a__U93 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U44_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U44 x12 x14 x16 <= P_id_a__U44 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNat_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_isNat x12 <= P_id_isNat x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U23_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U23 x12 <= P_id_a__U23 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U94_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U94 x12 x14 <= P_id_U94 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U12_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U12 x12 x14 <= P_id_U12 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U81_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U81 x12 x14 x16 <= P_id_a__U81 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U45_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U45 x12 x14 <= P_id_U45 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U71_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U71 x12 <= P_id_U71 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U91_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U91 x12 x14 x16 <= P_id_a__U91 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U42_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U42 x12 x14 x16 <= P_id_a__U42 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNatKind_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_isNatKind x12 <= P_id_isNatKind x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U22_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U22 x12 x14 <= P_id_a__U22 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U92_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U92 x12 x14 x16 <= P_id_U92 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U61_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U61 x12 <= P_id_a__U61 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U43_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U43 x12 x14 x16 <= P_id_U43 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNatIListKind_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__isNatIListKind x12 <= P_id_a__isNatIListKind x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U84_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U84 x12 x14 x16 <= P_id_U84 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_s_monotonic :
      forall x12 x11, (0 <= x12)/\ (x12 <= x11) ->P_id_s x12 <= P_id_s x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U46_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U46 x12 <= P_id_a__U46 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U32_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U32 x12 x14 <= P_id_U32 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U31_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U31 x12 x14 <= P_id_a__U31 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U13_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U13 x12 <= P_id_U13 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U83_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U83 x12 x14 x16 <= P_id_a__U83 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNatIList_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_isNatIList x12 <= P_id_isNatIList x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__zeros_bounded : 0 <= P_id_a__zeros .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U51_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U51 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U84_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U84 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U32_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U32 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNatList_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_isNatList x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U13_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U13 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U85_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U85 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__length_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__length x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNatIList_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__isNatIList x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U33_bounded : forall x11, (0 <= x11) ->0 <= P_id_U33 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U11_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U11 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U81_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U81 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U92_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U92 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U43_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U43 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U23_bounded : forall x11, (0 <= x11) ->0 <= P_id_U23 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNatKind_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__isNatKind x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U93_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U93 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U11_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U11 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U71_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U71 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U44_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U44 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_0_bounded : 0 <= P_id_0 .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U61_bounded : forall x11, (0 <= x11) ->0 <= P_id_U61 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U86_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U86 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U41_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U41 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U22_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U22 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U21_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U21 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U91_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U91 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_length_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_length x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U52_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U52 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U42_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U42 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U12_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U12 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U83_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U83 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U94_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U94 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U45_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U45 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U31_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U31 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNat_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__isNat x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNatIListKind_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_isNatIListKind x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U82_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U82 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U46_bounded : forall x11, (0 <= x11) ->0 <= P_id_U46 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_cons_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_cons x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U52_bounded : forall x11, (0 <= x11) ->0 <= P_id_U52 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U85_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U85 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U33_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U33 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U21_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U21 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNatList_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__isNatList x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U86_bounded : forall x11, (0 <= x11) ->0 <= P_id_U86 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_mark_bounded : forall x11, (0 <= x11) ->0 <= P_id_mark x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U51_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U51 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U41_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U41 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_tt_bounded : 0 <= P_id_tt .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U82_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U82 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U93_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U93 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U44_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U44 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNat_bounded : forall x11, (0 <= x11) ->0 <= P_id_isNat x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U23_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U23 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U94_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U94 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U12_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U12 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U81_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U81 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U45_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U45 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_zeros_bounded : 0 <= P_id_zeros .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U71_bounded : forall x11, (0 <= x11) ->0 <= P_id_U71 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U91_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U91 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U42_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U42 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNatKind_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_isNatKind x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U22_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U22 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U92_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U92 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_nil_bounded : 0 <= P_id_nil .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U61_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U61 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U43_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U43 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNatIListKind_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__isNatIListKind x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U84_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U84 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_s_bounded : forall x11, (0 <= x11) ->0 <= P_id_s x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U46_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U46 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U32_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U32 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U31_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U31 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U13_bounded : forall x11, (0 <= x11) ->0 <= P_id_U13 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U83_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U83 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNatIList_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_isNatIList x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Definition measure  := 
       InterpZ.measure 0 P_id_a__zeros P_id_U51 P_id_a__U84 P_id_a__U32 
        P_id_isNatList P_id_a__U13 P_id_U85 P_id_a__length 
        P_id_a__isNatIList P_id_U33 P_id_a__U11 P_id_U81 P_id_a__U92 
        P_id_a__U43 P_id_U23 P_id_a__isNatKind P_id_U93 P_id_U11 P_id_a__U71 
        P_id_U44 P_id_0 P_id_U61 P_id_a__U86 P_id_a__U41 P_id_U22 
        P_id_a__U21 P_id_U91 P_id_length P_id_a__U52 P_id_U42 P_id_a__U12 
        P_id_U83 P_id_a__U94 P_id_a__U45 P_id_U31 P_id_a__isNat 
        P_id_isNatIListKind P_id_a__U82 P_id_U46 P_id_cons P_id_U52 
        P_id_a__U85 P_id_a__U33 P_id_U21 P_id_a__isNatList P_id_U86 
        P_id_mark P_id_a__U51 P_id_U41 P_id_tt P_id_U82 P_id_a__U93 
        P_id_a__U44 P_id_isNat P_id_a__U23 P_id_U94 P_id_U12 P_id_a__U81 
        P_id_U45 P_id_zeros P_id_U71 P_id_a__U91 P_id_a__U42 P_id_isNatKind 
        P_id_a__U22 P_id_U92 P_id_nil P_id_a__U61 P_id_U43 
        P_id_a__isNatIListKind P_id_U84 P_id_s P_id_a__U46 P_id_U32 
        P_id_a__U31 P_id_U13 P_id_a__U83 P_id_isNatIList.
     
     Lemma measure_equation :
      forall t, 
       measure t = match t with
                     | (algebra.Alg.Term algebra.F.id_a__zeros nil) =>
                      P_id_a__zeros 
                     | (algebra.Alg.Term algebra.F.id_U51 (x12::x11::nil)) =>
                      P_id_U51 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U84 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U84 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U32 (x12::x11::nil)) =>
                      P_id_a__U32 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_isNatList (x11::nil)) =>
                      P_id_isNatList (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U13 (x11::nil)) =>
                      P_id_a__U13 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U85 (x12::x11::nil)) =>
                      P_id_U85 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__length (x11::nil)) =>
                      P_id_a__length (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__isNatIList 
                        (x11::nil)) =>
                      P_id_a__isNatIList (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U33 (x11::nil)) =>
                      P_id_U33 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U11 (x12::x11::nil)) =>
                      P_id_a__U11 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U81 (x13::x12::
                        x11::nil)) =>
                      P_id_U81 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U92 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U92 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U43 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U43 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U23 (x11::nil)) =>
                      P_id_U23 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__isNatKind 
                        (x11::nil)) =>
                      P_id_a__isNatKind (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U93 (x13::x12::
                        x11::nil)) =>
                      P_id_U93 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U11 (x12::x11::nil)) =>
                      P_id_U11 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U71 (x11::nil)) =>
                      P_id_a__U71 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U44 (x13::x12::
                        x11::nil)) =>
                      P_id_U44 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_0 nil) => P_id_0 
                     | (algebra.Alg.Term algebra.F.id_U61 (x11::nil)) =>
                      P_id_U61 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U86 (x11::nil)) =>
                      P_id_a__U86 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U41 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U41 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U22 (x12::x11::nil)) =>
                      P_id_U22 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U21 (x12::x11::nil)) =>
                      P_id_a__U21 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U91 (x13::x12::
                        x11::nil)) =>
                      P_id_U91 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_length (x11::nil)) =>
                      P_id_length (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U52 (x11::nil)) =>
                      P_id_a__U52 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U42 (x13::x12::
                        x11::nil)) =>
                      P_id_U42 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U12 (x12::x11::nil)) =>
                      P_id_a__U12 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U83 (x13::x12::
                        x11::nil)) =>
                      P_id_U83 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U94 (x12::x11::nil)) =>
                      P_id_a__U94 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U45 (x12::x11::nil)) =>
                      P_id_a__U45 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U31 (x12::x11::nil)) =>
                      P_id_U31 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__isNat (x11::nil)) =>
                      P_id_a__isNat (measure x11)
                     | (algebra.Alg.Term algebra.F.id_isNatIListKind 
                        (x11::nil)) =>
                      P_id_isNatIListKind (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U82 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U82 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U46 (x11::nil)) =>
                      P_id_U46 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_cons (x12::x11::nil)) =>
                      P_id_cons (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U52 (x11::nil)) =>
                      P_id_U52 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U85 (x12::x11::nil)) =>
                      P_id_a__U85 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U33 (x11::nil)) =>
                      P_id_a__U33 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U21 (x12::x11::nil)) =>
                      P_id_U21 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__isNatList 
                        (x11::nil)) =>
                      P_id_a__isNatList (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U86 (x11::nil)) =>
                      P_id_U86 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_mark (x11::nil)) =>
                      P_id_mark (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U51 (x12::x11::nil)) =>
                      P_id_a__U51 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U41 (x13::x12::
                        x11::nil)) =>
                      P_id_U41 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt 
                     | (algebra.Alg.Term algebra.F.id_U82 (x13::x12::
                        x11::nil)) =>
                      P_id_U82 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U93 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U93 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U44 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U44 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_isNat (x11::nil)) =>
                      P_id_isNat (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U23 (x11::nil)) =>
                      P_id_a__U23 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U94 (x12::x11::nil)) =>
                      P_id_U94 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U12 (x12::x11::nil)) =>
                      P_id_U12 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U81 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U81 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U45 (x12::x11::nil)) =>
                      P_id_U45 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_zeros nil) =>
                      P_id_zeros 
                     | (algebra.Alg.Term algebra.F.id_U71 (x11::nil)) =>
                      P_id_U71 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U91 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U91 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U42 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U42 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_isNatKind (x11::nil)) =>
                      P_id_isNatKind (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U22 (x12::x11::nil)) =>
                      P_id_a__U22 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U92 (x13::x12::
                        x11::nil)) =>
                      P_id_U92 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil 
                     | (algebra.Alg.Term algebra.F.id_a__U61 (x11::nil)) =>
                      P_id_a__U61 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U43 (x13::x12::
                        x11::nil)) =>
                      P_id_U43 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
                        (x11::nil)) =>
                      P_id_a__isNatIListKind (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U84 (x13::x12::
                        x11::nil)) =>
                      P_id_U84 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_s (x11::nil)) =>
                      P_id_s (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U46 (x11::nil)) =>
                      P_id_a__U46 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U32 (x12::x11::nil)) =>
                      P_id_U32 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U31 (x12::x11::nil)) =>
                      P_id_a__U31 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U13 (x11::nil)) =>
                      P_id_U13 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U83 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U83 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_isNatIList (x11::nil)) =>
                      P_id_isNatIList (measure x11)
                     | _ => 0
                     end.
     Proof.
       intros t;case t;intros ;apply refl_equal.
     Qed.
     
     Lemma measure_bounded : forall t, 0 <= measure t.
     Proof.
       unfold measure in |-*.
       
       apply InterpZ.measure_bounded;
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Ltac generate_pos_hyp  :=
      match goal with
        | H:context [measure ?x] |- _ =>
         let v := fresh "v" in 
          (let H := fresh "h" in 
            (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
              clearbody H;clearbody v))
        |  |- context [measure ?x] =>
         let v := fresh "v" in 
          (let H := fresh "h" in 
            (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
              clearbody H;clearbody v))
        end
      .
     
     Lemma rules_monotonic :
      forall l r, 
       (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) ->
        measure r <= measure l.
     Proof.
       intros l r H.
       fold measure in |-*.
       
       inversion H;clear H;subst;inversion H0;clear H0;subst;
        simpl algebra.EQT.T.apply_subst in |-*;
        repeat (
        match goal with
          |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
           rewrite (measure_equation (algebra.Alg.Term f t))
          end
        );repeat (generate_pos_hyp );
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma measure_star_monotonic :
      forall l r, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  r l) ->measure r <= measure l.
     Proof.
       unfold measure in *.
       apply InterpZ.measure_star_monotonic.
       intros ;apply P_id_U51_monotonic;assumption.
       intros ;apply P_id_a__U84_monotonic;assumption.
       intros ;apply P_id_a__U32_monotonic;assumption.
       intros ;apply P_id_isNatList_monotonic;assumption.
       intros ;apply P_id_a__U13_monotonic;assumption.
       intros ;apply P_id_U85_monotonic;assumption.
       intros ;apply P_id_a__length_monotonic;assumption.
       intros ;apply P_id_a__isNatIList_monotonic;assumption.
       intros ;apply P_id_U33_monotonic;assumption.
       intros ;apply P_id_a__U11_monotonic;assumption.
       intros ;apply P_id_U81_monotonic;assumption.
       intros ;apply P_id_a__U92_monotonic;assumption.
       intros ;apply P_id_a__U43_monotonic;assumption.
       intros ;apply P_id_U23_monotonic;assumption.
       intros ;apply P_id_a__isNatKind_monotonic;assumption.
       intros ;apply P_id_U93_monotonic;assumption.
       intros ;apply P_id_U11_monotonic;assumption.
       intros ;apply P_id_a__U71_monotonic;assumption.
       intros ;apply P_id_U44_monotonic;assumption.
       intros ;apply P_id_U61_monotonic;assumption.
       intros ;apply P_id_a__U86_monotonic;assumption.
       intros ;apply P_id_a__U41_monotonic;assumption.
       intros ;apply P_id_U22_monotonic;assumption.
       intros ;apply P_id_a__U21_monotonic;assumption.
       intros ;apply P_id_U91_monotonic;assumption.
       intros ;apply P_id_length_monotonic;assumption.
       intros ;apply P_id_a__U52_monotonic;assumption.
       intros ;apply P_id_U42_monotonic;assumption.
       intros ;apply P_id_a__U12_monotonic;assumption.
       intros ;apply P_id_U83_monotonic;assumption.
       intros ;apply P_id_a__U94_monotonic;assumption.
       intros ;apply P_id_a__U45_monotonic;assumption.
       intros ;apply P_id_U31_monotonic;assumption.
       intros ;apply P_id_a__isNat_monotonic;assumption.
       intros ;apply P_id_isNatIListKind_monotonic;assumption.
       intros ;apply P_id_a__U82_monotonic;assumption.
       intros ;apply P_id_U46_monotonic;assumption.
       intros ;apply P_id_cons_monotonic;assumption.
       intros ;apply P_id_U52_monotonic;assumption.
       intros ;apply P_id_a__U85_monotonic;assumption.
       intros ;apply P_id_a__U33_monotonic;assumption.
       intros ;apply P_id_U21_monotonic;assumption.
       intros ;apply P_id_a__isNatList_monotonic;assumption.
       intros ;apply P_id_U86_monotonic;assumption.
       intros ;apply P_id_mark_monotonic;assumption.
       intros ;apply P_id_a__U51_monotonic;assumption.
       intros ;apply P_id_U41_monotonic;assumption.
       intros ;apply P_id_U82_monotonic;assumption.
       intros ;apply P_id_a__U93_monotonic;assumption.
       intros ;apply P_id_a__U44_monotonic;assumption.
       intros ;apply P_id_isNat_monotonic;assumption.
       intros ;apply P_id_a__U23_monotonic;assumption.
       intros ;apply P_id_U94_monotonic;assumption.
       intros ;apply P_id_U12_monotonic;assumption.
       intros ;apply P_id_a__U81_monotonic;assumption.
       intros ;apply P_id_U45_monotonic;assumption.
       intros ;apply P_id_U71_monotonic;assumption.
       intros ;apply P_id_a__U91_monotonic;assumption.
       intros ;apply P_id_a__U42_monotonic;assumption.
       intros ;apply P_id_isNatKind_monotonic;assumption.
       intros ;apply P_id_a__U22_monotonic;assumption.
       intros ;apply P_id_U92_monotonic;assumption.
       intros ;apply P_id_a__U61_monotonic;assumption.
       intros ;apply P_id_U43_monotonic;assumption.
       intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
       intros ;apply P_id_U84_monotonic;assumption.
       intros ;apply P_id_s_monotonic;assumption.
       intros ;apply P_id_a__U46_monotonic;assumption.
       intros ;apply P_id_U32_monotonic;assumption.
       intros ;apply P_id_a__U31_monotonic;assumption.
       intros ;apply P_id_U13_monotonic;assumption.
       intros ;apply P_id_a__U83_monotonic;assumption.
       intros ;apply P_id_isNatIList_monotonic;assumption.
       intros ;apply P_id_a__zeros_bounded;assumption.
       intros ;apply P_id_U51_bounded;assumption.
       intros ;apply P_id_a__U84_bounded;assumption.
       intros ;apply P_id_a__U32_bounded;assumption.
       intros ;apply P_id_isNatList_bounded;assumption.
       intros ;apply P_id_a__U13_bounded;assumption.
       intros ;apply P_id_U85_bounded;assumption.
       intros ;apply P_id_a__length_bounded;assumption.
       intros ;apply P_id_a__isNatIList_bounded;assumption.
       intros ;apply P_id_U33_bounded;assumption.
       intros ;apply P_id_a__U11_bounded;assumption.
       intros ;apply P_id_U81_bounded;assumption.
       intros ;apply P_id_a__U92_bounded;assumption.
       intros ;apply P_id_a__U43_bounded;assumption.
       intros ;apply P_id_U23_bounded;assumption.
       intros ;apply P_id_a__isNatKind_bounded;assumption.
       intros ;apply P_id_U93_bounded;assumption.
       intros ;apply P_id_U11_bounded;assumption.
       intros ;apply P_id_a__U71_bounded;assumption.
       intros ;apply P_id_U44_bounded;assumption.
       intros ;apply P_id_0_bounded;assumption.
       intros ;apply P_id_U61_bounded;assumption.
       intros ;apply P_id_a__U86_bounded;assumption.
       intros ;apply P_id_a__U41_bounded;assumption.
       intros ;apply P_id_U22_bounded;assumption.
       intros ;apply P_id_a__U21_bounded;assumption.
       intros ;apply P_id_U91_bounded;assumption.
       intros ;apply P_id_length_bounded;assumption.
       intros ;apply P_id_a__U52_bounded;assumption.
       intros ;apply P_id_U42_bounded;assumption.
       intros ;apply P_id_a__U12_bounded;assumption.
       intros ;apply P_id_U83_bounded;assumption.
       intros ;apply P_id_a__U94_bounded;assumption.
       intros ;apply P_id_a__U45_bounded;assumption.
       intros ;apply P_id_U31_bounded;assumption.
       intros ;apply P_id_a__isNat_bounded;assumption.
       intros ;apply P_id_isNatIListKind_bounded;assumption.
       intros ;apply P_id_a__U82_bounded;assumption.
       intros ;apply P_id_U46_bounded;assumption.
       intros ;apply P_id_cons_bounded;assumption.
       intros ;apply P_id_U52_bounded;assumption.
       intros ;apply P_id_a__U85_bounded;assumption.
       intros ;apply P_id_a__U33_bounded;assumption.
       intros ;apply P_id_U21_bounded;assumption.
       intros ;apply P_id_a__isNatList_bounded;assumption.
       intros ;apply P_id_U86_bounded;assumption.
       intros ;apply P_id_mark_bounded;assumption.
       intros ;apply P_id_a__U51_bounded;assumption.
       intros ;apply P_id_U41_bounded;assumption.
       intros ;apply P_id_tt_bounded;assumption.
       intros ;apply P_id_U82_bounded;assumption.
       intros ;apply P_id_a__U93_bounded;assumption.
       intros ;apply P_id_a__U44_bounded;assumption.
       intros ;apply P_id_isNat_bounded;assumption.
       intros ;apply P_id_a__U23_bounded;assumption.
       intros ;apply P_id_U94_bounded;assumption.
       intros ;apply P_id_U12_bounded;assumption.
       intros ;apply P_id_a__U81_bounded;assumption.
       intros ;apply P_id_U45_bounded;assumption.
       intros ;apply P_id_zeros_bounded;assumption.
       intros ;apply P_id_U71_bounded;assumption.
       intros ;apply P_id_a__U91_bounded;assumption.
       intros ;apply P_id_a__U42_bounded;assumption.
       intros ;apply P_id_isNatKind_bounded;assumption.
       intros ;apply P_id_a__U22_bounded;assumption.
       intros ;apply P_id_U92_bounded;assumption.
       intros ;apply P_id_nil_bounded;assumption.
       intros ;apply P_id_a__U61_bounded;assumption.
       intros ;apply P_id_U43_bounded;assumption.
       intros ;apply P_id_a__isNatIListKind_bounded;assumption.
       intros ;apply P_id_U84_bounded;assumption.
       intros ;apply P_id_s_bounded;assumption.
       intros ;apply P_id_a__U46_bounded;assumption.
       intros ;apply P_id_U32_bounded;assumption.
       intros ;apply P_id_a__U31_bounded;assumption.
       intros ;apply P_id_U13_bounded;assumption.
       intros ;apply P_id_a__U83_bounded;assumption.
       intros ;apply P_id_isNatIList_bounded;assumption.
       apply rules_monotonic.
     Qed.
     
     Definition P_id_A__U46 (x11:Z) := 0.
     
     Definition P_id_A__ISNATILISTKIND (x11:Z) := 0.
     
     Definition P_id_A__U61 (x11:Z) := 0.
     
     Definition P_id_A__U85 (x11:Z) (x12:Z) := 1* x12.
     
     Definition P_id_A__U31 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__U81 (x11:Z) (x12:Z) (x13:Z) := 1* x13.
     
     Definition P_id_A__U22 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__U93 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__U42 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__U51 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__ISNATLIST (x11:Z) := 1* x11.
     
     Definition P_id_A__ZEROS  := 0.
     
     Definition P_id_A__U91 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__U33 (x11:Z) := 0.
     
     Definition P_id_A__U83 (x11:Z) (x12:Z) (x13:Z) := 1* x13.
     
     Definition P_id_A__U23 (x11:Z) := 0.
     
     Definition P_id_A__U11 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__LENGTH (x11:Z) := 0.
     
     Definition P_id_A__U44 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__ISNATILIST (x11:Z) := 0.
     
     Definition P_id_A__U13 (x11:Z) := 0.
     
     Definition P_id_A__U71 (x11:Z) := 0.
     
     Definition P_id_A__U86 (x11:Z) := 0.
     
     Definition P_id_A__U32 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__U82 (x11:Z) (x12:Z) (x13:Z) := 1* x13.
     
     Definition P_id_A__ISNATKIND (x11:Z) := 0.
     
     Definition P_id_A__U94 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__U43 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__U52 (x11:Z) := 0.
     
     Definition P_id_A__U21 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__U92 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__U41 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__U84 (x11:Z) (x12:Z) (x13:Z) := 1* x13.
     
     Definition P_id_A__ISNAT (x11:Z) := 0.
     
     Definition P_id_A__U12 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_MARK (x11:Z) := 0.
     
     Definition P_id_A__U45 (x11:Z) (x12:Z) := 0.
     
     Lemma P_id_A__U46_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U46 x12 <= P_id_A__U46 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__ISNATILISTKIND_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__ISNATILISTKIND x12 <= P_id_A__ISNATILISTKIND x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U61_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U61 x12 <= P_id_A__U61 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U85_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U85 x12 x14 <= P_id_A__U85 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U31_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U31 x12 x14 <= P_id_A__U31 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U81_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U81 x12 x14 x16 <= P_id_A__U81 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U22_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U22 x12 x14 <= P_id_A__U22 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U93_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U93 x12 x14 x16 <= P_id_A__U93 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U42_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U42 x12 x14 x16 <= P_id_A__U42 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U51_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U51 x12 x14 <= P_id_A__U51 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__ISNATLIST_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__ISNATLIST x12 <= P_id_A__ISNATLIST x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U91_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U91 x12 x14 x16 <= P_id_A__U91 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U33_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U33 x12 <= P_id_A__U33 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U83_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U83 x12 x14 x16 <= P_id_A__U83 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U23_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U23 x12 <= P_id_A__U23 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U11_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U11 x12 x14 <= P_id_A__U11 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__LENGTH_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__LENGTH x12 <= P_id_A__LENGTH x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U44_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U44 x12 x14 x16 <= P_id_A__U44 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__ISNATILIST_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__ISNATILIST x12 <= P_id_A__ISNATILIST x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U13_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U13 x12 <= P_id_A__U13 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U71_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U71 x12 <= P_id_A__U71 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U86_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U86 x12 <= P_id_A__U86 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U32_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U32 x12 x14 <= P_id_A__U32 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U82_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U82 x12 x14 x16 <= P_id_A__U82 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__ISNATKIND_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__ISNATKIND x12 <= P_id_A__ISNATKIND x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U94_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U94 x12 x14 <= P_id_A__U94 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U43_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U43 x12 x14 x16 <= P_id_A__U43 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U52_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U52 x12 <= P_id_A__U52 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U21_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U21 x12 x14 <= P_id_A__U21 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U92_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U92 x12 x14 x16 <= P_id_A__U92 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U41_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U41 x12 x14 x16 <= P_id_A__U41 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U84_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U84 x12 x14 x16 <= P_id_A__U84 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__ISNAT_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__ISNAT x12 <= P_id_A__ISNAT x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U12_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U12 x12 x14 <= P_id_A__U12 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_MARK_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_MARK x12 <= P_id_MARK x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U45_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U45 x12 x14 <= P_id_A__U45 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Definition marked_measure  := 
       InterpZ.marked_measure 0 P_id_a__zeros P_id_U51 P_id_a__U84 
        P_id_a__U32 P_id_isNatList P_id_a__U13 P_id_U85 P_id_a__length 
        P_id_a__isNatIList P_id_U33 P_id_a__U11 P_id_U81 P_id_a__U92 
        P_id_a__U43 P_id_U23 P_id_a__isNatKind P_id_U93 P_id_U11 P_id_a__U71 
        P_id_U44 P_id_0 P_id_U61 P_id_a__U86 P_id_a__U41 P_id_U22 
        P_id_a__U21 P_id_U91 P_id_length P_id_a__U52 P_id_U42 P_id_a__U12 
        P_id_U83 P_id_a__U94 P_id_a__U45 P_id_U31 P_id_a__isNat 
        P_id_isNatIListKind P_id_a__U82 P_id_U46 P_id_cons P_id_U52 
        P_id_a__U85 P_id_a__U33 P_id_U21 P_id_a__isNatList P_id_U86 
        P_id_mark P_id_a__U51 P_id_U41 P_id_tt P_id_U82 P_id_a__U93 
        P_id_a__U44 P_id_isNat P_id_a__U23 P_id_U94 P_id_U12 P_id_a__U81 
        P_id_U45 P_id_zeros P_id_U71 P_id_a__U91 P_id_a__U42 P_id_isNatKind 
        P_id_a__U22 P_id_U92 P_id_nil P_id_a__U61 P_id_U43 
        P_id_a__isNatIListKind P_id_U84 P_id_s P_id_a__U46 P_id_U32 
        P_id_a__U31 P_id_U13 P_id_a__U83 P_id_isNatIList P_id_A__U46 
        P_id_A__ISNATILISTKIND P_id_A__U61 P_id_A__U85 P_id_A__U31 
        P_id_A__U81 P_id_A__U22 P_id_A__U93 P_id_A__U42 P_id_A__U51 
        P_id_A__ISNATLIST P_id_A__ZEROS P_id_A__U91 P_id_A__U33 P_id_A__U83 
        P_id_A__U23 P_id_A__U11 P_id_A__LENGTH P_id_A__U44 
        P_id_A__ISNATILIST P_id_A__U13 P_id_A__U71 P_id_A__U86 P_id_A__U32 
        P_id_A__U82 P_id_A__ISNATKIND P_id_A__U94 P_id_A__U43 P_id_A__U52 
        P_id_A__U21 P_id_A__U92 P_id_A__U41 P_id_A__U84 P_id_A__ISNAT 
        P_id_A__U12 P_id_MARK P_id_A__U45.
     
     Lemma marked_measure_equation :
      forall t, 
       marked_measure t = match t with
                            | (algebra.Alg.Term algebra.F.id_a__U46 
                               (x11::nil)) =>
                             P_id_A__U46 (measure x11)
                            | (algebra.Alg.Term 
                               algebra.F.id_a__isNatIListKind (x11::nil)) =>
                             P_id_A__ISNATILISTKIND (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U61 
                               (x11::nil)) =>
                             P_id_A__U61 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U85 (x12::
                               x11::nil)) =>
                             P_id_A__U85 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U31 (x12::
                               x11::nil)) =>
                             P_id_A__U31 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U81 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U81 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U22 (x12::
                               x11::nil)) =>
                             P_id_A__U22 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U93 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U93 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U42 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U42 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U51 (x12::
                               x11::nil)) =>
                             P_id_A__U51 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__isNatList 
                               (x11::nil)) =>
                             P_id_A__ISNATLIST (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__zeros nil) =>
                             P_id_A__ZEROS 
                            | (algebra.Alg.Term algebra.F.id_a__U91 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U91 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U33 
                               (x11::nil)) =>
                             P_id_A__U33 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U83 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U83 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U23 
                               (x11::nil)) =>
                             P_id_A__U23 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U11 (x12::
                               x11::nil)) =>
                             P_id_A__U11 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__length 
                               (x11::nil)) =>
                             P_id_A__LENGTH (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U44 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U44 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__isNatIList 
                               (x11::nil)) =>
                             P_id_A__ISNATILIST (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U13 
                               (x11::nil)) =>
                             P_id_A__U13 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U71 
                               (x11::nil)) =>
                             P_id_A__U71 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U86 
                               (x11::nil)) =>
                             P_id_A__U86 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U32 (x12::
                               x11::nil)) =>
                             P_id_A__U32 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U82 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U82 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__isNatKind 
                               (x11::nil)) =>
                             P_id_A__ISNATKIND (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U94 (x12::
                               x11::nil)) =>
                             P_id_A__U94 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U43 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U43 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U52 
                               (x11::nil)) =>
                             P_id_A__U52 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U21 (x12::
                               x11::nil)) =>
                             P_id_A__U21 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U92 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U92 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U41 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U41 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U84 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U84 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__isNat 
                               (x11::nil)) =>
                             P_id_A__ISNAT (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U12 (x12::
                               x11::nil)) =>
                             P_id_A__U12 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_mark (x11::nil)) =>
                             P_id_MARK (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U45 (x12::
                               x11::nil)) =>
                             P_id_A__U45 (measure x12) (measure x11)
                            | _ => measure t
                            end.
     Proof.
       reflexivity .
     Qed.
     
     Lemma marked_measure_star_monotonic :
      forall f l1 l2, 
       (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step 
                                                         R_xml_0_deep_rew.R_xml_0_rules)
                                                        ) l1 l2) ->
        marked_measure (algebra.Alg.Term f l1) <= marked_measure (algebra.Alg.Term 
                                                                   f 
                                                                   l2).
     Proof.
       unfold marked_measure in *.
       apply InterpZ.marked_measure_star_monotonic.
       intros ;apply P_id_U51_monotonic;assumption.
       intros ;apply P_id_a__U84_monotonic;assumption.
       intros ;apply P_id_a__U32_monotonic;assumption.
       intros ;apply P_id_isNatList_monotonic;assumption.
       intros ;apply P_id_a__U13_monotonic;assumption.
       intros ;apply P_id_U85_monotonic;assumption.
       intros ;apply P_id_a__length_monotonic;assumption.
       intros ;apply P_id_a__isNatIList_monotonic;assumption.
       intros ;apply P_id_U33_monotonic;assumption.
       intros ;apply P_id_a__U11_monotonic;assumption.
       intros ;apply P_id_U81_monotonic;assumption.
       intros ;apply P_id_a__U92_monotonic;assumption.
       intros ;apply P_id_a__U43_monotonic;assumption.
       intros ;apply P_id_U23_monotonic;assumption.
       intros ;apply P_id_a__isNatKind_monotonic;assumption.
       intros ;apply P_id_U93_monotonic;assumption.
       intros ;apply P_id_U11_monotonic;assumption.
       intros ;apply P_id_a__U71_monotonic;assumption.
       intros ;apply P_id_U44_monotonic;assumption.
       intros ;apply P_id_U61_monotonic;assumption.
       intros ;apply P_id_a__U86_monotonic;assumption.
       intros ;apply P_id_a__U41_monotonic;assumption.
       intros ;apply P_id_U22_monotonic;assumption.
       intros ;apply P_id_a__U21_monotonic;assumption.
       intros ;apply P_id_U91_monotonic;assumption.
       intros ;apply P_id_length_monotonic;assumption.
       intros ;apply P_id_a__U52_monotonic;assumption.
       intros ;apply P_id_U42_monotonic;assumption.
       intros ;apply P_id_a__U12_monotonic;assumption.
       intros ;apply P_id_U83_monotonic;assumption.
       intros ;apply P_id_a__U94_monotonic;assumption.
       intros ;apply P_id_a__U45_monotonic;assumption.
       intros ;apply P_id_U31_monotonic;assumption.
       intros ;apply P_id_a__isNat_monotonic;assumption.
       intros ;apply P_id_isNatIListKind_monotonic;assumption.
       intros ;apply P_id_a__U82_monotonic;assumption.
       intros ;apply P_id_U46_monotonic;assumption.
       intros ;apply P_id_cons_monotonic;assumption.
       intros ;apply P_id_U52_monotonic;assumption.
       intros ;apply P_id_a__U85_monotonic;assumption.
       intros ;apply P_id_a__U33_monotonic;assumption.
       intros ;apply P_id_U21_monotonic;assumption.
       intros ;apply P_id_a__isNatList_monotonic;assumption.
       intros ;apply P_id_U86_monotonic;assumption.
       intros ;apply P_id_mark_monotonic;assumption.
       intros ;apply P_id_a__U51_monotonic;assumption.
       intros ;apply P_id_U41_monotonic;assumption.
       intros ;apply P_id_U82_monotonic;assumption.
       intros ;apply P_id_a__U93_monotonic;assumption.
       intros ;apply P_id_a__U44_monotonic;assumption.
       intros ;apply P_id_isNat_monotonic;assumption.
       intros ;apply P_id_a__U23_monotonic;assumption.
       intros ;apply P_id_U94_monotonic;assumption.
       intros ;apply P_id_U12_monotonic;assumption.
       intros ;apply P_id_a__U81_monotonic;assumption.
       intros ;apply P_id_U45_monotonic;assumption.
       intros ;apply P_id_U71_monotonic;assumption.
       intros ;apply P_id_a__U91_monotonic;assumption.
       intros ;apply P_id_a__U42_monotonic;assumption.
       intros ;apply P_id_isNatKind_monotonic;assumption.
       intros ;apply P_id_a__U22_monotonic;assumption.
       intros ;apply P_id_U92_monotonic;assumption.
       intros ;apply P_id_a__U61_monotonic;assumption.
       intros ;apply P_id_U43_monotonic;assumption.
       intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
       intros ;apply P_id_U84_monotonic;assumption.
       intros ;apply P_id_s_monotonic;assumption.
       intros ;apply P_id_a__U46_monotonic;assumption.
       intros ;apply P_id_U32_monotonic;assumption.
       intros ;apply P_id_a__U31_monotonic;assumption.
       intros ;apply P_id_U13_monotonic;assumption.
       intros ;apply P_id_a__U83_monotonic;assumption.
       intros ;apply P_id_isNatIList_monotonic;assumption.
       intros ;apply P_id_a__zeros_bounded;assumption.
       intros ;apply P_id_U51_bounded;assumption.
       intros ;apply P_id_a__U84_bounded;assumption.
       intros ;apply P_id_a__U32_bounded;assumption.
       intros ;apply P_id_isNatList_bounded;assumption.
       intros ;apply P_id_a__U13_bounded;assumption.
       intros ;apply P_id_U85_bounded;assumption.
       intros ;apply P_id_a__length_bounded;assumption.
       intros ;apply P_id_a__isNatIList_bounded;assumption.
       intros ;apply P_id_U33_bounded;assumption.
       intros ;apply P_id_a__U11_bounded;assumption.
       intros ;apply P_id_U81_bounded;assumption.
       intros ;apply P_id_a__U92_bounded;assumption.
       intros ;apply P_id_a__U43_bounded;assumption.
       intros ;apply P_id_U23_bounded;assumption.
       intros ;apply P_id_a__isNatKind_bounded;assumption.
       intros ;apply P_id_U93_bounded;assumption.
       intros ;apply P_id_U11_bounded;assumption.
       intros ;apply P_id_a__U71_bounded;assumption.
       intros ;apply P_id_U44_bounded;assumption.
       intros ;apply P_id_0_bounded;assumption.
       intros ;apply P_id_U61_bounded;assumption.
       intros ;apply P_id_a__U86_bounded;assumption.
       intros ;apply P_id_a__U41_bounded;assumption.
       intros ;apply P_id_U22_bounded;assumption.
       intros ;apply P_id_a__U21_bounded;assumption.
       intros ;apply P_id_U91_bounded;assumption.
       intros ;apply P_id_length_bounded;assumption.
       intros ;apply P_id_a__U52_bounded;assumption.
       intros ;apply P_id_U42_bounded;assumption.
       intros ;apply P_id_a__U12_bounded;assumption.
       intros ;apply P_id_U83_bounded;assumption.
       intros ;apply P_id_a__U94_bounded;assumption.
       intros ;apply P_id_a__U45_bounded;assumption.
       intros ;apply P_id_U31_bounded;assumption.
       intros ;apply P_id_a__isNat_bounded;assumption.
       intros ;apply P_id_isNatIListKind_bounded;assumption.
       intros ;apply P_id_a__U82_bounded;assumption.
       intros ;apply P_id_U46_bounded;assumption.
       intros ;apply P_id_cons_bounded;assumption.
       intros ;apply P_id_U52_bounded;assumption.
       intros ;apply P_id_a__U85_bounded;assumption.
       intros ;apply P_id_a__U33_bounded;assumption.
       intros ;apply P_id_U21_bounded;assumption.
       intros ;apply P_id_a__isNatList_bounded;assumption.
       intros ;apply P_id_U86_bounded;assumption.
       intros ;apply P_id_mark_bounded;assumption.
       intros ;apply P_id_a__U51_bounded;assumption.
       intros ;apply P_id_U41_bounded;assumption.
       intros ;apply P_id_tt_bounded;assumption.
       intros ;apply P_id_U82_bounded;assumption.
       intros ;apply P_id_a__U93_bounded;assumption.
       intros ;apply P_id_a__U44_bounded;assumption.
       intros ;apply P_id_isNat_bounded;assumption.
       intros ;apply P_id_a__U23_bounded;assumption.
       intros ;apply P_id_U94_bounded;assumption.
       intros ;apply P_id_U12_bounded;assumption.
       intros ;apply P_id_a__U81_bounded;assumption.
       intros ;apply P_id_U45_bounded;assumption.
       intros ;apply P_id_zeros_bounded;assumption.
       intros ;apply P_id_U71_bounded;assumption.
       intros ;apply P_id_a__U91_bounded;assumption.
       intros ;apply P_id_a__U42_bounded;assumption.
       intros ;apply P_id_isNatKind_bounded;assumption.
       intros ;apply P_id_a__U22_bounded;assumption.
       intros ;apply P_id_U92_bounded;assumption.
       intros ;apply P_id_nil_bounded;assumption.
       intros ;apply P_id_a__U61_bounded;assumption.
       intros ;apply P_id_U43_bounded;assumption.
       intros ;apply P_id_a__isNatIListKind_bounded;assumption.
       intros ;apply P_id_U84_bounded;assumption.
       intros ;apply P_id_s_bounded;assumption.
       intros ;apply P_id_a__U46_bounded;assumption.
       intros ;apply P_id_U32_bounded;assumption.
       intros ;apply P_id_a__U31_bounded;assumption.
       intros ;apply P_id_U13_bounded;assumption.
       intros ;apply P_id_a__U83_bounded;assumption.
       intros ;apply P_id_isNatIList_bounded;assumption.
       apply rules_monotonic.
       intros ;apply P_id_A__U46_monotonic;assumption.
       intros ;apply P_id_A__ISNATILISTKIND_monotonic;assumption.
       intros ;apply P_id_A__U61_monotonic;assumption.
       intros ;apply P_id_A__U85_monotonic;assumption.
       intros ;apply P_id_A__U31_monotonic;assumption.
       intros ;apply P_id_A__U81_monotonic;assumption.
       intros ;apply P_id_A__U22_monotonic;assumption.
       intros ;apply P_id_A__U93_monotonic;assumption.
       intros ;apply P_id_A__U42_monotonic;assumption.
       intros ;apply P_id_A__U51_monotonic;assumption.
       intros ;apply P_id_A__ISNATLIST_monotonic;assumption.
       intros ;apply P_id_A__U91_monotonic;assumption.
       intros ;apply P_id_A__U33_monotonic;assumption.
       intros ;apply P_id_A__U83_monotonic;assumption.
       intros ;apply P_id_A__U23_monotonic;assumption.
       intros ;apply P_id_A__U11_monotonic;assumption.
       intros ;apply P_id_A__LENGTH_monotonic;assumption.
       intros ;apply P_id_A__U44_monotonic;assumption.
       intros ;apply P_id_A__ISNATILIST_monotonic;assumption.
       intros ;apply P_id_A__U13_monotonic;assumption.
       intros ;apply P_id_A__U71_monotonic;assumption.
       intros ;apply P_id_A__U86_monotonic;assumption.
       intros ;apply P_id_A__U32_monotonic;assumption.
       intros ;apply P_id_A__U82_monotonic;assumption.
       intros ;apply P_id_A__ISNATKIND_monotonic;assumption.
       intros ;apply P_id_A__U94_monotonic;assumption.
       intros ;apply P_id_A__U43_monotonic;assumption.
       intros ;apply P_id_A__U52_monotonic;assumption.
       intros ;apply P_id_A__U21_monotonic;assumption.
       intros ;apply P_id_A__U92_monotonic;assumption.
       intros ;apply P_id_A__U41_monotonic;assumption.
       intros ;apply P_id_A__U84_monotonic;assumption.
       intros ;apply P_id_A__ISNAT_monotonic;assumption.
       intros ;apply P_id_A__U12_monotonic;assumption.
       intros ;apply P_id_MARK_monotonic;assumption.
       intros ;apply P_id_A__U45_monotonic;assumption.
     Qed.
     
     Ltac rewrite_and_unfold  :=
      do 2 (rewrite marked_measure_equation);
       repeat (
       match goal with
         |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
          rewrite (measure_equation (algebra.Alg.Term f t))
         | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ =>
          rewrite (measure_equation (algebra.Alg.Term f t)) in H|-
         end
       ).
     
     Definition lt a b := (Zwf.Zwf 0) (marked_measure a) (marked_measure b).
     
     Definition le a b := marked_measure a <= marked_measure b.
     
     Lemma lt_le_compat : forall a b c, (lt a b) ->(le b c) ->lt a c.
     Proof.
       unfold lt, le in *.
       intros a b c.
       apply (interp.le_lt_compat_right (interp.o_Z 0)).
     Qed.
     
     Lemma wf_lt : well_founded lt.
     Proof.
       unfold lt in *.
       apply Inverse_Image.wf_inverse_image with  (B:=Z).
       apply Zwf.Zwf_well_founded.
     Qed.
     
     Lemma DP_R_xml_0_scc_38_large_scc_5_strict_in_lt :
      Relation_Definitions.inclusion _ DP_R_xml_0_scc_38_large_scc_5_strict 
       lt.
     Proof.
       unfold Relation_Definitions.inclusion, lt in *.
       
       intros a b H;destruct H;
        match goal with
          |  |- (Zwf.Zwf 0) _ (marked_measure (algebra.Alg.Term ?f ?l)) =>
           let l'' := algebra.Alg_ext.find_replacement l  in 
            ((apply (interp.le_lt_compat_right (interp.o_Z 0)) with
               (marked_measure (algebra.Alg.Term f l''));[idtac|
              apply marked_measure_star_monotonic;
               repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos);
               (assumption)||(constructor 1)]))
          end
        ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma DP_R_xml_0_scc_38_large_scc_5_large_in_le :
      Relation_Definitions.inclusion _ DP_R_xml_0_scc_38_large_scc_5_large le.
     Proof.
       unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *.
       
       intros a b H;destruct H;
        match goal with
          |  |- _ <= marked_measure (algebra.Alg.Term ?f ?l) =>
           let l'' := algebra.Alg_ext.find_replacement l  in 
            ((apply (interp.le_trans (interp.o_Z 0)) with
               (marked_measure (algebra.Alg.Term f l''));[idtac|
              apply marked_measure_star_monotonic;
               repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos);
               (assumption)||(constructor 1)]))
          end
        ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Definition wf_DP_R_xml_0_scc_38_large_scc_5_large  := 
       WF_DP_R_xml_0_scc_38_large_scc_5_large.wf.
     
     
     Lemma wf :
      well_founded WF_DP_R_xml_0_scc_38_large.DP_R_xml_0_scc_38_large_scc_5.
     Proof.
       intros x.
       apply (well_founded_ind wf_lt).
       clear x.
       intros x.
       pattern x.
       apply (@Acc_ind _ DP_R_xml_0_scc_38_large_scc_5_large).
       clear x.
       intros x _ IHx IHx'.
       constructor.
       intros y H.
       
       destruct H;
        (apply IHx';apply DP_R_xml_0_scc_38_large_scc_5_strict_in_lt;
          econstructor eassumption)||
        ((apply IHx;[econstructor eassumption|
          intros y' Hlt;apply IHx';apply lt_le_compat with (1:=Hlt) ;
           apply DP_R_xml_0_scc_38_large_scc_5_large_in_le;
           econstructor eassumption])).
       apply wf_DP_R_xml_0_scc_38_large_scc_5_large.
     Qed.
    End WF_DP_R_xml_0_scc_38_large_scc_5.
    
    Definition wf_DP_R_xml_0_scc_38_large_scc_5  := 
      WF_DP_R_xml_0_scc_38_large_scc_5.wf.
    
    
    Lemma acc_DP_R_xml_0_scc_38_large_scc_5 :
     forall x y, 
      (DP_R_xml_0_scc_38_large_scc_5 x y) ->
       Acc WF_DP_R_xml_0_scc_38.DP_R_xml_0_scc_38_large x.
    Proof.
      intros x.
      pattern x.
      apply (@Acc_ind _ DP_R_xml_0_scc_38_large_scc_5).
      intros x' _ Hrec y h.
      
      inversion h;clear h;subst;
       constructor;intros _y _h;inversion _h;clear _h;subst;
        (eapply Hrec;econstructor eassumption)||
        ((eapply acc_DP_R_xml_0_scc_38_large_non_scc_4;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
         ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
          (eapply Hrec;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))).
      apply wf_DP_R_xml_0_scc_38_large_scc_5.
    Qed.
    
    
    Lemma wf : well_founded WF_DP_R_xml_0_scc_38.DP_R_xml_0_scc_38_large.
    Proof.
      constructor;intros _y _h;inversion _h;clear _h;subst;
       (eapply acc_DP_R_xml_0_scc_38_large_non_scc_4;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_scc_38_large_non_scc_3;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((eapply acc_DP_R_xml_0_scc_38_large_non_scc_2;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
         ((eapply acc_DP_R_xml_0_scc_38_large_non_scc_1;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
          ((eapply acc_DP_R_xml_0_scc_38_large_non_scc_0;
             econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
           ((eapply acc_DP_R_xml_0_scc_38_large_scc_5;
              econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
            ((eapply acc_DP_R_xml_0_scc_38_large_scc_4;
               econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
             ((eapply acc_DP_R_xml_0_scc_38_large_scc_3;
                econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
              ((eapply acc_DP_R_xml_0_scc_38_large_scc_2;
                 econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
               ((eapply acc_DP_R_xml_0_scc_38_large_scc_1;
                  econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
                ((eapply acc_DP_R_xml_0_scc_38_large_scc_0;
                   econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
                 ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
                  (fail)))))))))))).
    Qed.
   End WF_DP_R_xml_0_scc_38_large.
   
   Open Scope Z_scope.
   
   Import ring_extention.
   
   Notation Local "a <= b" := (Zle a b).
   
   Notation Local "a < b" := (Zlt a b).
   
   Definition P_id_a__zeros  := 0.
   
   Definition P_id_U51 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_a__U84 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_a__U32 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_isNatList (x11:Z) := 0.
   
   Definition P_id_a__U13 (x11:Z) := 0.
   
   Definition P_id_U85 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_a__length (x11:Z) := 1 + 1* x11.
   
   Definition P_id_a__isNatIList (x11:Z) := 0.
   
   Definition P_id_U33 (x11:Z) := 0.
   
   Definition P_id_a__U11 (x11:Z) (x12:Z) := 2* x11.
   
   Definition P_id_U81 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_a__U92 (x11:Z) (x12:Z) (x13:Z) := 1 + 2* x12.
   
   Definition P_id_a__U43 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_U23 (x11:Z) := 0.
   
   Definition P_id_a__isNatKind (x11:Z) := 0.
   
   Definition P_id_U93 (x11:Z) (x12:Z) (x13:Z) := 1 + 1* x12.
   
   Definition P_id_U11 (x11:Z) (x12:Z) := 2* x11.
   
   Definition P_id_a__U71 (x11:Z) := 2* x11.
   
   Definition P_id_U44 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_0  := 0.
   
   Definition P_id_U61 (x11:Z) := 0.
   
   Definition P_id_a__U86 (x11:Z) := 0.
   
   Definition P_id_a__U41 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_U22 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_a__U21 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_U91 (x11:Z) (x12:Z) (x13:Z) := 1 + 2* x12.
   
   Definition P_id_length (x11:Z) := 1 + 1* x11.
   
   Definition P_id_a__U52 (x11:Z) := 0.
   
   Definition P_id_U42 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_a__U12 (x11:Z) (x12:Z) := 2* x11.
   
   Definition P_id_U83 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_a__U94 (x11:Z) (x12:Z) := 1 + 1* x12.
   
   Definition P_id_a__U45 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_U31 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_a__isNat (x11:Z) := 0.
   
   Definition P_id_isNatIListKind (x11:Z) := 0.
   
   Definition P_id_a__U82 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_U46 (x11:Z) := 0.
   
   Definition P_id_cons (x11:Z) (x12:Z) := 2* x11 + 2* x12.
   
   Definition P_id_U52 (x11:Z) := 0.
   
   Definition P_id_a__U85 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_a__U33 (x11:Z) := 0.
   
   Definition P_id_U21 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_a__isNatList (x11:Z) := 0.
   
   Definition P_id_U86 (x11:Z) := 0.
   
   Definition P_id_mark (x11:Z) := 1* x11.
   
   Definition P_id_a__U51 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_U41 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_tt  := 0.
   
   Definition P_id_U82 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_a__U93 (x11:Z) (x12:Z) (x13:Z) := 1 + 1* x12.
   
   Definition P_id_a__U44 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_isNat (x11:Z) := 0.
   
   Definition P_id_a__U23 (x11:Z) := 0.
   
   Definition P_id_U94 (x11:Z) (x12:Z) := 1 + 1* x12.
   
   Definition P_id_U12 (x11:Z) (x12:Z) := 2* x11.
   
   Definition P_id_a__U81 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_U45 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_zeros  := 0.
   
   Definition P_id_U71 (x11:Z) := 2* x11.
   
   Definition P_id_a__U91 (x11:Z) (x12:Z) (x13:Z) := 1 + 2* x12.
   
   Definition P_id_a__U42 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_isNatKind (x11:Z) := 0.
   
   Definition P_id_a__U22 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_U92 (x11:Z) (x12:Z) (x13:Z) := 1 + 2* x12.
   
   Definition P_id_nil  := 0.
   
   Definition P_id_a__U61 (x11:Z) := 0.
   
   Definition P_id_U43 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_a__isNatIListKind (x11:Z) := 0.
   
   Definition P_id_U84 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_s (x11:Z) := 1* x11.
   
   Definition P_id_a__U46 (x11:Z) := 0.
   
   Definition P_id_U32 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_a__U31 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_U13 (x11:Z) := 0.
   
   Definition P_id_a__U83 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_isNatIList (x11:Z) := 0.
   
   Lemma P_id_U51_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_U51 x12 x14 <= P_id_U51 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U84_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U84 x12 x14 x16 <= P_id_a__U84 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U32_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_a__U32 x12 x14 <= P_id_a__U32 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_isNatList_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_isNatList x12 <= P_id_isNatList x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U13_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_a__U13 x12 <= P_id_a__U13 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U85_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_U85 x12 x14 <= P_id_U85 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__length_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_a__length x12 <= P_id_a__length x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__isNatIList_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->
      P_id_a__isNatIList x12 <= P_id_a__isNatIList x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U33_monotonic :
    forall x12 x11, (0 <= x12)/\ (x12 <= x11) ->P_id_U33 x12 <= P_id_U33 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U11_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_a__U11 x12 x14 <= P_id_a__U11 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U81_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_U81 x12 x14 x16 <= P_id_U81 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U92_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U92 x12 x14 x16 <= P_id_a__U92 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U43_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U43 x12 x14 x16 <= P_id_a__U43 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U23_monotonic :
    forall x12 x11, (0 <= x12)/\ (x12 <= x11) ->P_id_U23 x12 <= P_id_U23 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__isNatKind_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->
      P_id_a__isNatKind x12 <= P_id_a__isNatKind x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U93_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_U93 x12 x14 x16 <= P_id_U93 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U11_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_U11 x12 x14 <= P_id_U11 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U71_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_a__U71 x12 <= P_id_a__U71 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U44_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_U44 x12 x14 x16 <= P_id_U44 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U61_monotonic :
    forall x12 x11, (0 <= x12)/\ (x12 <= x11) ->P_id_U61 x12 <= P_id_U61 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U86_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_a__U86 x12 <= P_id_a__U86 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U41_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U41 x12 x14 x16 <= P_id_a__U41 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U22_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_U22 x12 x14 <= P_id_U22 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U21_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_a__U21 x12 x14 <= P_id_a__U21 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U91_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_U91 x12 x14 x16 <= P_id_U91 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_length_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_length x12 <= P_id_length x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U52_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_a__U52 x12 <= P_id_a__U52 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U42_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_U42 x12 x14 x16 <= P_id_U42 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U12_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_a__U12 x12 x14 <= P_id_a__U12 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U83_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_U83 x12 x14 x16 <= P_id_U83 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U94_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_a__U94 x12 x14 <= P_id_a__U94 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U45_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_a__U45 x12 x14 <= P_id_a__U45 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U31_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_U31 x12 x14 <= P_id_U31 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__isNat_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_a__isNat x12 <= P_id_a__isNat x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_isNatIListKind_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->
      P_id_isNatIListKind x12 <= P_id_isNatIListKind x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U82_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U82 x12 x14 x16 <= P_id_a__U82 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U46_monotonic :
    forall x12 x11, (0 <= x12)/\ (x12 <= x11) ->P_id_U46 x12 <= P_id_U46 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_cons_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_cons x12 x14 <= P_id_cons x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U52_monotonic :
    forall x12 x11, (0 <= x12)/\ (x12 <= x11) ->P_id_U52 x12 <= P_id_U52 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U85_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_a__U85 x12 x14 <= P_id_a__U85 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U33_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_a__U33 x12 <= P_id_a__U33 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U21_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_U21 x12 x14 <= P_id_U21 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__isNatList_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->
      P_id_a__isNatList x12 <= P_id_a__isNatList x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U86_monotonic :
    forall x12 x11, (0 <= x12)/\ (x12 <= x11) ->P_id_U86 x12 <= P_id_U86 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_mark_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_mark x12 <= P_id_mark x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U51_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_a__U51 x12 x14 <= P_id_a__U51 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U41_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_U41 x12 x14 x16 <= P_id_U41 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U82_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_U82 x12 x14 x16 <= P_id_U82 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U93_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U93 x12 x14 x16 <= P_id_a__U93 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U44_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U44 x12 x14 x16 <= P_id_a__U44 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_isNat_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_isNat x12 <= P_id_isNat x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U23_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_a__U23 x12 <= P_id_a__U23 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U94_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_U94 x12 x14 <= P_id_U94 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U12_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_U12 x12 x14 <= P_id_U12 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U81_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U81 x12 x14 x16 <= P_id_a__U81 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U45_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_U45 x12 x14 <= P_id_U45 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U71_monotonic :
    forall x12 x11, (0 <= x12)/\ (x12 <= x11) ->P_id_U71 x12 <= P_id_U71 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U91_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U91 x12 x14 x16 <= P_id_a__U91 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U42_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U42 x12 x14 x16 <= P_id_a__U42 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_isNatKind_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_isNatKind x12 <= P_id_isNatKind x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U22_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_a__U22 x12 x14 <= P_id_a__U22 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U92_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_U92 x12 x14 x16 <= P_id_U92 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U61_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_a__U61 x12 <= P_id_a__U61 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U43_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_U43 x12 x14 x16 <= P_id_U43 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__isNatIListKind_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->
      P_id_a__isNatIListKind x12 <= P_id_a__isNatIListKind x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U84_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_U84 x12 x14 x16 <= P_id_U84 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_s_monotonic :
    forall x12 x11, (0 <= x12)/\ (x12 <= x11) ->P_id_s x12 <= P_id_s x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U46_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_a__U46 x12 <= P_id_a__U46 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U32_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_U32 x12 x14 <= P_id_U32 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U31_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_a__U31 x12 x14 <= P_id_a__U31 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U13_monotonic :
    forall x12 x11, (0 <= x12)/\ (x12 <= x11) ->P_id_U13 x12 <= P_id_U13 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U83_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U83 x12 x14 x16 <= P_id_a__U83 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_isNatIList_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_isNatIList x12 <= P_id_isNatIList x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__zeros_bounded : 0 <= P_id_a__zeros .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U51_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U51 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U84_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U84 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U32_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U32 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_isNatList_bounded :
    forall x11, (0 <= x11) ->0 <= P_id_isNatList x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U13_bounded : forall x11, (0 <= x11) ->0 <= P_id_a__U13 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U85_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U85 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__length_bounded :
    forall x11, (0 <= x11) ->0 <= P_id_a__length x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__isNatIList_bounded :
    forall x11, (0 <= x11) ->0 <= P_id_a__isNatIList x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U33_bounded : forall x11, (0 <= x11) ->0 <= P_id_U33 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U11_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U11 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U81_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U81 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U92_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U92 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U43_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U43 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U23_bounded : forall x11, (0 <= x11) ->0 <= P_id_U23 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__isNatKind_bounded :
    forall x11, (0 <= x11) ->0 <= P_id_a__isNatKind x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U93_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U93 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U11_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U11 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U71_bounded : forall x11, (0 <= x11) ->0 <= P_id_a__U71 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U44_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U44 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_0_bounded : 0 <= P_id_0 .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U61_bounded : forall x11, (0 <= x11) ->0 <= P_id_U61 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U86_bounded : forall x11, (0 <= x11) ->0 <= P_id_a__U86 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U41_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U41 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U22_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U22 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U21_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U21 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U91_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U91 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_length_bounded : forall x11, (0 <= x11) ->0 <= P_id_length x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U52_bounded : forall x11, (0 <= x11) ->0 <= P_id_a__U52 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U42_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U42 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U12_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U12 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U83_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U83 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U94_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U94 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U45_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U45 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U31_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U31 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__isNat_bounded :
    forall x11, (0 <= x11) ->0 <= P_id_a__isNat x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_isNatIListKind_bounded :
    forall x11, (0 <= x11) ->0 <= P_id_isNatIListKind x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U82_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U82 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U46_bounded : forall x11, (0 <= x11) ->0 <= P_id_U46 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_cons_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_cons x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U52_bounded : forall x11, (0 <= x11) ->0 <= P_id_U52 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U85_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U85 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U33_bounded : forall x11, (0 <= x11) ->0 <= P_id_a__U33 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U21_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U21 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__isNatList_bounded :
    forall x11, (0 <= x11) ->0 <= P_id_a__isNatList x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U86_bounded : forall x11, (0 <= x11) ->0 <= P_id_U86 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_mark_bounded : forall x11, (0 <= x11) ->0 <= P_id_mark x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U51_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U51 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U41_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U41 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_tt_bounded : 0 <= P_id_tt .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U82_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U82 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U93_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U93 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U44_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U44 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_isNat_bounded : forall x11, (0 <= x11) ->0 <= P_id_isNat x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U23_bounded : forall x11, (0 <= x11) ->0 <= P_id_a__U23 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U94_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U94 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U12_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U12 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U81_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U81 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U45_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U45 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_zeros_bounded : 0 <= P_id_zeros .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U71_bounded : forall x11, (0 <= x11) ->0 <= P_id_U71 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U91_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U91 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U42_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U42 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_isNatKind_bounded :
    forall x11, (0 <= x11) ->0 <= P_id_isNatKind x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U22_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U22 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U92_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U92 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_nil_bounded : 0 <= P_id_nil .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U61_bounded : forall x11, (0 <= x11) ->0 <= P_id_a__U61 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U43_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U43 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__isNatIListKind_bounded :
    forall x11, (0 <= x11) ->0 <= P_id_a__isNatIListKind x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U84_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U84 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_s_bounded : forall x11, (0 <= x11) ->0 <= P_id_s x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U46_bounded : forall x11, (0 <= x11) ->0 <= P_id_a__U46 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U32_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U32 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U31_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U31 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U13_bounded : forall x11, (0 <= x11) ->0 <= P_id_U13 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U83_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U83 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_isNatIList_bounded :
    forall x11, (0 <= x11) ->0 <= P_id_isNatIList x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Definition measure  := 
     InterpZ.measure 0 P_id_a__zeros P_id_U51 P_id_a__U84 P_id_a__U32 
      P_id_isNatList P_id_a__U13 P_id_U85 P_id_a__length P_id_a__isNatIList 
      P_id_U33 P_id_a__U11 P_id_U81 P_id_a__U92 P_id_a__U43 P_id_U23 
      P_id_a__isNatKind P_id_U93 P_id_U11 P_id_a__U71 P_id_U44 P_id_0 
      P_id_U61 P_id_a__U86 P_id_a__U41 P_id_U22 P_id_a__U21 P_id_U91 
      P_id_length P_id_a__U52 P_id_U42 P_id_a__U12 P_id_U83 P_id_a__U94 
      P_id_a__U45 P_id_U31 P_id_a__isNat P_id_isNatIListKind P_id_a__U82 
      P_id_U46 P_id_cons P_id_U52 P_id_a__U85 P_id_a__U33 P_id_U21 
      P_id_a__isNatList P_id_U86 P_id_mark P_id_a__U51 P_id_U41 P_id_tt 
      P_id_U82 P_id_a__U93 P_id_a__U44 P_id_isNat P_id_a__U23 P_id_U94 
      P_id_U12 P_id_a__U81 P_id_U45 P_id_zeros P_id_U71 P_id_a__U91 
      P_id_a__U42 P_id_isNatKind P_id_a__U22 P_id_U92 P_id_nil P_id_a__U61 
      P_id_U43 P_id_a__isNatIListKind P_id_U84 P_id_s P_id_a__U46 P_id_U32 
      P_id_a__U31 P_id_U13 P_id_a__U83 P_id_isNatIList.
   
   Lemma measure_equation :
    forall t, 
     measure t = match t with
                   | (algebra.Alg.Term algebra.F.id_a__zeros nil) =>
                    P_id_a__zeros 
                   | (algebra.Alg.Term algebra.F.id_U51 (x12::x11::nil)) =>
                    P_id_U51 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U84 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U84 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U32 (x12::x11::nil)) =>
                    P_id_a__U32 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_isNatList (x11::nil)) =>
                    P_id_isNatList (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U13 (x11::nil)) =>
                    P_id_a__U13 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U85 (x12::x11::nil)) =>
                    P_id_U85 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__length (x11::nil)) =>
                    P_id_a__length (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__isNatIList (x11::nil)) =>
                    P_id_a__isNatIList (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U33 (x11::nil)) =>
                    P_id_U33 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U11 (x12::x11::nil)) =>
                    P_id_a__U11 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U81 (x13::x12::x11::nil)) =>
                    P_id_U81 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U92 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U92 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U43 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U43 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U23 (x11::nil)) =>
                    P_id_U23 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__isNatKind (x11::nil)) =>
                    P_id_a__isNatKind (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U93 (x13::x12::x11::nil)) =>
                    P_id_U93 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U11 (x12::x11::nil)) =>
                    P_id_U11 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U71 (x11::nil)) =>
                    P_id_a__U71 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U44 (x13::x12::x11::nil)) =>
                    P_id_U44 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_0 nil) => P_id_0 
                   | (algebra.Alg.Term algebra.F.id_U61 (x11::nil)) =>
                    P_id_U61 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U86 (x11::nil)) =>
                    P_id_a__U86 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U41 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U41 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U22 (x12::x11::nil)) =>
                    P_id_U22 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U21 (x12::x11::nil)) =>
                    P_id_a__U21 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U91 (x13::x12::x11::nil)) =>
                    P_id_U91 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_length (x11::nil)) =>
                    P_id_length (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U52 (x11::nil)) =>
                    P_id_a__U52 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U42 (x13::x12::x11::nil)) =>
                    P_id_U42 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U12 (x12::x11::nil)) =>
                    P_id_a__U12 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U83 (x13::x12::x11::nil)) =>
                    P_id_U83 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U94 (x12::x11::nil)) =>
                    P_id_a__U94 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U45 (x12::x11::nil)) =>
                    P_id_a__U45 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U31 (x12::x11::nil)) =>
                    P_id_U31 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__isNat (x11::nil)) =>
                    P_id_a__isNat (measure x11)
                   | (algebra.Alg.Term algebra.F.id_isNatIListKind 
                      (x11::nil)) =>
                    P_id_isNatIListKind (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U82 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U82 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U46 (x11::nil)) =>
                    P_id_U46 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_cons (x12::x11::nil)) =>
                    P_id_cons (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U52 (x11::nil)) =>
                    P_id_U52 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U85 (x12::x11::nil)) =>
                    P_id_a__U85 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U33 (x11::nil)) =>
                    P_id_a__U33 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U21 (x12::x11::nil)) =>
                    P_id_U21 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__isNatList (x11::nil)) =>
                    P_id_a__isNatList (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U86 (x11::nil)) =>
                    P_id_U86 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_mark (x11::nil)) =>
                    P_id_mark (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U51 (x12::x11::nil)) =>
                    P_id_a__U51 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U41 (x13::x12::x11::nil)) =>
                    P_id_U41 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt 
                   | (algebra.Alg.Term algebra.F.id_U82 (x13::x12::x11::nil)) =>
                    P_id_U82 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U93 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U93 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U44 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U44 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_isNat (x11::nil)) =>
                    P_id_isNat (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U23 (x11::nil)) =>
                    P_id_a__U23 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U94 (x12::x11::nil)) =>
                    P_id_U94 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U12 (x12::x11::nil)) =>
                    P_id_U12 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U81 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U81 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U45 (x12::x11::nil)) =>
                    P_id_U45 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_zeros nil) => P_id_zeros 
                   | (algebra.Alg.Term algebra.F.id_U71 (x11::nil)) =>
                    P_id_U71 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U91 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U91 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U42 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U42 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_isNatKind (x11::nil)) =>
                    P_id_isNatKind (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U22 (x12::x11::nil)) =>
                    P_id_a__U22 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U92 (x13::x12::x11::nil)) =>
                    P_id_U92 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil 
                   | (algebra.Alg.Term algebra.F.id_a__U61 (x11::nil)) =>
                    P_id_a__U61 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U43 (x13::x12::x11::nil)) =>
                    P_id_U43 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
                      (x11::nil)) =>
                    P_id_a__isNatIListKind (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U84 (x13::x12::x11::nil)) =>
                    P_id_U84 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_s (x11::nil)) =>
                    P_id_s (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U46 (x11::nil)) =>
                    P_id_a__U46 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U32 (x12::x11::nil)) =>
                    P_id_U32 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U31 (x12::x11::nil)) =>
                    P_id_a__U31 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U13 (x11::nil)) =>
                    P_id_U13 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U83 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U83 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_isNatIList (x11::nil)) =>
                    P_id_isNatIList (measure x11)
                   | _ => 0
                   end.
   Proof.
     intros t;case t;intros ;apply refl_equal.
   Qed.
   
   Lemma measure_bounded : forall t, 0 <= measure t.
   Proof.
     unfold measure in |-*.
     
     apply InterpZ.measure_bounded;
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Ltac generate_pos_hyp  :=
    match goal with
      | H:context [measure ?x] |- _ =>
       let v := fresh "v" in 
        (let H := fresh "h" in 
          (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
            clearbody H;clearbody v))
      |  |- context [measure ?x] =>
       let v := fresh "v" in 
        (let H := fresh "h" in 
          (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
            clearbody H;clearbody v))
      end
    .
   
   Lemma rules_monotonic :
    forall l r, 
     (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) ->
      measure r <= measure l.
   Proof.
     intros l r H.
     fold measure in |-*.
     
     inversion H;clear H;subst;inversion H0;clear H0;subst;
      simpl algebra.EQT.T.apply_subst in |-*;
      repeat (
      match goal with
        |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
         rewrite (measure_equation (algebra.Alg.Term f t))
        end
      );repeat (generate_pos_hyp );
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma measure_star_monotonic :
    forall l r, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                r l) ->measure r <= measure l.
   Proof.
     unfold measure in *.
     apply InterpZ.measure_star_monotonic.
     intros ;apply P_id_U51_monotonic;assumption.
     intros ;apply P_id_a__U84_monotonic;assumption.
     intros ;apply P_id_a__U32_monotonic;assumption.
     intros ;apply P_id_isNatList_monotonic;assumption.
     intros ;apply P_id_a__U13_monotonic;assumption.
     intros ;apply P_id_U85_monotonic;assumption.
     intros ;apply P_id_a__length_monotonic;assumption.
     intros ;apply P_id_a__isNatIList_monotonic;assumption.
     intros ;apply P_id_U33_monotonic;assumption.
     intros ;apply P_id_a__U11_monotonic;assumption.
     intros ;apply P_id_U81_monotonic;assumption.
     intros ;apply P_id_a__U92_monotonic;assumption.
     intros ;apply P_id_a__U43_monotonic;assumption.
     intros ;apply P_id_U23_monotonic;assumption.
     intros ;apply P_id_a__isNatKind_monotonic;assumption.
     intros ;apply P_id_U93_monotonic;assumption.
     intros ;apply P_id_U11_monotonic;assumption.
     intros ;apply P_id_a__U71_monotonic;assumption.
     intros ;apply P_id_U44_monotonic;assumption.
     intros ;apply P_id_U61_monotonic;assumption.
     intros ;apply P_id_a__U86_monotonic;assumption.
     intros ;apply P_id_a__U41_monotonic;assumption.
     intros ;apply P_id_U22_monotonic;assumption.
     intros ;apply P_id_a__U21_monotonic;assumption.
     intros ;apply P_id_U91_monotonic;assumption.
     intros ;apply P_id_length_monotonic;assumption.
     intros ;apply P_id_a__U52_monotonic;assumption.
     intros ;apply P_id_U42_monotonic;assumption.
     intros ;apply P_id_a__U12_monotonic;assumption.
     intros ;apply P_id_U83_monotonic;assumption.
     intros ;apply P_id_a__U94_monotonic;assumption.
     intros ;apply P_id_a__U45_monotonic;assumption.
     intros ;apply P_id_U31_monotonic;assumption.
     intros ;apply P_id_a__isNat_monotonic;assumption.
     intros ;apply P_id_isNatIListKind_monotonic;assumption.
     intros ;apply P_id_a__U82_monotonic;assumption.
     intros ;apply P_id_U46_monotonic;assumption.
     intros ;apply P_id_cons_monotonic;assumption.
     intros ;apply P_id_U52_monotonic;assumption.
     intros ;apply P_id_a__U85_monotonic;assumption.
     intros ;apply P_id_a__U33_monotonic;assumption.
     intros ;apply P_id_U21_monotonic;assumption.
     intros ;apply P_id_a__isNatList_monotonic;assumption.
     intros ;apply P_id_U86_monotonic;assumption.
     intros ;apply P_id_mark_monotonic;assumption.
     intros ;apply P_id_a__U51_monotonic;assumption.
     intros ;apply P_id_U41_monotonic;assumption.
     intros ;apply P_id_U82_monotonic;assumption.
     intros ;apply P_id_a__U93_monotonic;assumption.
     intros ;apply P_id_a__U44_monotonic;assumption.
     intros ;apply P_id_isNat_monotonic;assumption.
     intros ;apply P_id_a__U23_monotonic;assumption.
     intros ;apply P_id_U94_monotonic;assumption.
     intros ;apply P_id_U12_monotonic;assumption.
     intros ;apply P_id_a__U81_monotonic;assumption.
     intros ;apply P_id_U45_monotonic;assumption.
     intros ;apply P_id_U71_monotonic;assumption.
     intros ;apply P_id_a__U91_monotonic;assumption.
     intros ;apply P_id_a__U42_monotonic;assumption.
     intros ;apply P_id_isNatKind_monotonic;assumption.
     intros ;apply P_id_a__U22_monotonic;assumption.
     intros ;apply P_id_U92_monotonic;assumption.
     intros ;apply P_id_a__U61_monotonic;assumption.
     intros ;apply P_id_U43_monotonic;assumption.
     intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
     intros ;apply P_id_U84_monotonic;assumption.
     intros ;apply P_id_s_monotonic;assumption.
     intros ;apply P_id_a__U46_monotonic;assumption.
     intros ;apply P_id_U32_monotonic;assumption.
     intros ;apply P_id_a__U31_monotonic;assumption.
     intros ;apply P_id_U13_monotonic;assumption.
     intros ;apply P_id_a__U83_monotonic;assumption.
     intros ;apply P_id_isNatIList_monotonic;assumption.
     intros ;apply P_id_a__zeros_bounded;assumption.
     intros ;apply P_id_U51_bounded;assumption.
     intros ;apply P_id_a__U84_bounded;assumption.
     intros ;apply P_id_a__U32_bounded;assumption.
     intros ;apply P_id_isNatList_bounded;assumption.
     intros ;apply P_id_a__U13_bounded;assumption.
     intros ;apply P_id_U85_bounded;assumption.
     intros ;apply P_id_a__length_bounded;assumption.
     intros ;apply P_id_a__isNatIList_bounded;assumption.
     intros ;apply P_id_U33_bounded;assumption.
     intros ;apply P_id_a__U11_bounded;assumption.
     intros ;apply P_id_U81_bounded;assumption.
     intros ;apply P_id_a__U92_bounded;assumption.
     intros ;apply P_id_a__U43_bounded;assumption.
     intros ;apply P_id_U23_bounded;assumption.
     intros ;apply P_id_a__isNatKind_bounded;assumption.
     intros ;apply P_id_U93_bounded;assumption.
     intros ;apply P_id_U11_bounded;assumption.
     intros ;apply P_id_a__U71_bounded;assumption.
     intros ;apply P_id_U44_bounded;assumption.
     intros ;apply P_id_0_bounded;assumption.
     intros ;apply P_id_U61_bounded;assumption.
     intros ;apply P_id_a__U86_bounded;assumption.
     intros ;apply P_id_a__U41_bounded;assumption.
     intros ;apply P_id_U22_bounded;assumption.
     intros ;apply P_id_a__U21_bounded;assumption.
     intros ;apply P_id_U91_bounded;assumption.
     intros ;apply P_id_length_bounded;assumption.
     intros ;apply P_id_a__U52_bounded;assumption.
     intros ;apply P_id_U42_bounded;assumption.
     intros ;apply P_id_a__U12_bounded;assumption.
     intros ;apply P_id_U83_bounded;assumption.
     intros ;apply P_id_a__U94_bounded;assumption.
     intros ;apply P_id_a__U45_bounded;assumption.
     intros ;apply P_id_U31_bounded;assumption.
     intros ;apply P_id_a__isNat_bounded;assumption.
     intros ;apply P_id_isNatIListKind_bounded;assumption.
     intros ;apply P_id_a__U82_bounded;assumption.
     intros ;apply P_id_U46_bounded;assumption.
     intros ;apply P_id_cons_bounded;assumption.
     intros ;apply P_id_U52_bounded;assumption.
     intros ;apply P_id_a__U85_bounded;assumption.
     intros ;apply P_id_a__U33_bounded;assumption.
     intros ;apply P_id_U21_bounded;assumption.
     intros ;apply P_id_a__isNatList_bounded;assumption.
     intros ;apply P_id_U86_bounded;assumption.
     intros ;apply P_id_mark_bounded;assumption.
     intros ;apply P_id_a__U51_bounded;assumption.
     intros ;apply P_id_U41_bounded;assumption.
     intros ;apply P_id_tt_bounded;assumption.
     intros ;apply P_id_U82_bounded;assumption.
     intros ;apply P_id_a__U93_bounded;assumption.
     intros ;apply P_id_a__U44_bounded;assumption.
     intros ;apply P_id_isNat_bounded;assumption.
     intros ;apply P_id_a__U23_bounded;assumption.
     intros ;apply P_id_U94_bounded;assumption.
     intros ;apply P_id_U12_bounded;assumption.
     intros ;apply P_id_a__U81_bounded;assumption.
     intros ;apply P_id_U45_bounded;assumption.
     intros ;apply P_id_zeros_bounded;assumption.
     intros ;apply P_id_U71_bounded;assumption.
     intros ;apply P_id_a__U91_bounded;assumption.
     intros ;apply P_id_a__U42_bounded;assumption.
     intros ;apply P_id_isNatKind_bounded;assumption.
     intros ;apply P_id_a__U22_bounded;assumption.
     intros ;apply P_id_U92_bounded;assumption.
     intros ;apply P_id_nil_bounded;assumption.
     intros ;apply P_id_a__U61_bounded;assumption.
     intros ;apply P_id_U43_bounded;assumption.
     intros ;apply P_id_a__isNatIListKind_bounded;assumption.
     intros ;apply P_id_U84_bounded;assumption.
     intros ;apply P_id_s_bounded;assumption.
     intros ;apply P_id_a__U46_bounded;assumption.
     intros ;apply P_id_U32_bounded;assumption.
     intros ;apply P_id_a__U31_bounded;assumption.
     intros ;apply P_id_U13_bounded;assumption.
     intros ;apply P_id_a__U83_bounded;assumption.
     intros ;apply P_id_isNatIList_bounded;assumption.
     apply rules_monotonic.
   Qed.
   
   Definition P_id_A__U46 (x11:Z) := 0.
   
   Definition P_id_A__ISNATILISTKIND (x11:Z) := 0.
   
   Definition P_id_A__U61 (x11:Z) := 0.
   
   Definition P_id_A__U85 (x11:Z) (x12:Z) := 2* x12.
   
   Definition P_id_A__U31 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_A__U81 (x11:Z) (x12:Z) (x13:Z) := 2* x12 + 2* x13.
   
   Definition P_id_A__U22 (x11:Z) (x12:Z) := 2* x12.
   
   Definition P_id_A__U93 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_A__U42 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_A__U51 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_A__ISNATLIST (x11:Z) := 2* x11.
   
   Definition P_id_A__ZEROS  := 0.
   
   Definition P_id_A__U91 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_A__U33 (x11:Z) := 0.
   
   Definition P_id_A__U83 (x11:Z) (x12:Z) (x13:Z) := 2* x12 + 2* x13.
   
   Definition P_id_A__U23 (x11:Z) := 0.
   
   Definition P_id_A__U11 (x11:Z) (x12:Z) := 2 + 2* x12.
   
   Definition P_id_A__LENGTH (x11:Z) := 0.
   
   Definition P_id_A__U44 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_A__ISNATILIST (x11:Z) := 0.
   
   Definition P_id_A__U13 (x11:Z) := 0.
   
   Definition P_id_A__U71 (x11:Z) := 0.
   
   Definition P_id_A__U86 (x11:Z) := 0.
   
   Definition P_id_A__U32 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_A__U82 (x11:Z) (x12:Z) (x13:Z) := 2* x12 + 2* x13.
   
   Definition P_id_A__ISNATKIND (x11:Z) := 0.
   
   Definition P_id_A__U94 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_A__U43 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_A__U52 (x11:Z) := 0.
   
   Definition P_id_A__U21 (x11:Z) (x12:Z) := 2* x12.
   
   Definition P_id_A__U92 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_A__U41 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_A__U84 (x11:Z) (x12:Z) (x13:Z) := 2* x12 + 2* x13.
   
   Definition P_id_A__ISNAT (x11:Z) := 2* x11.
   
   Definition P_id_A__U12 (x11:Z) (x12:Z) := 2 + 2* x12.
   
   Definition P_id_MARK (x11:Z) := 0.
   
   Definition P_id_A__U45 (x11:Z) (x12:Z) := 0.
   
   Lemma P_id_A__U46_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_A__U46 x12 <= P_id_A__U46 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__ISNATILISTKIND_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->
      P_id_A__ISNATILISTKIND x12 <= P_id_A__ISNATILISTKIND x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U61_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_A__U61 x12 <= P_id_A__U61 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U85_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_A__U85 x12 x14 <= P_id_A__U85 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U31_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_A__U31 x12 x14 <= P_id_A__U31 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U81_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U81 x12 x14 x16 <= P_id_A__U81 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U22_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_A__U22 x12 x14 <= P_id_A__U22 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U93_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U93 x12 x14 x16 <= P_id_A__U93 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U42_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U42 x12 x14 x16 <= P_id_A__U42 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U51_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_A__U51 x12 x14 <= P_id_A__U51 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__ISNATLIST_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->
      P_id_A__ISNATLIST x12 <= P_id_A__ISNATLIST x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U91_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U91 x12 x14 x16 <= P_id_A__U91 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U33_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_A__U33 x12 <= P_id_A__U33 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U83_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U83 x12 x14 x16 <= P_id_A__U83 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U23_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_A__U23 x12 <= P_id_A__U23 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U11_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_A__U11 x12 x14 <= P_id_A__U11 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__LENGTH_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_A__LENGTH x12 <= P_id_A__LENGTH x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U44_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U44 x12 x14 x16 <= P_id_A__U44 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__ISNATILIST_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->
      P_id_A__ISNATILIST x12 <= P_id_A__ISNATILIST x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U13_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_A__U13 x12 <= P_id_A__U13 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U71_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_A__U71 x12 <= P_id_A__U71 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U86_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_A__U86 x12 <= P_id_A__U86 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U32_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_A__U32 x12 x14 <= P_id_A__U32 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U82_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U82 x12 x14 x16 <= P_id_A__U82 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__ISNATKIND_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->
      P_id_A__ISNATKIND x12 <= P_id_A__ISNATKIND x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U94_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_A__U94 x12 x14 <= P_id_A__U94 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U43_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U43 x12 x14 x16 <= P_id_A__U43 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U52_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_A__U52 x12 <= P_id_A__U52 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U21_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_A__U21 x12 x14 <= P_id_A__U21 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U92_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U92 x12 x14 x16 <= P_id_A__U92 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U41_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U41 x12 x14 x16 <= P_id_A__U41 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U84_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U84 x12 x14 x16 <= P_id_A__U84 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__ISNAT_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_A__ISNAT x12 <= P_id_A__ISNAT x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U12_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_A__U12 x12 x14 <= P_id_A__U12 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_MARK_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_MARK x12 <= P_id_MARK x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U45_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_A__U45 x12 x14 <= P_id_A__U45 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Definition marked_measure  := 
     InterpZ.marked_measure 0 P_id_a__zeros P_id_U51 P_id_a__U84 P_id_a__U32 
      P_id_isNatList P_id_a__U13 P_id_U85 P_id_a__length P_id_a__isNatIList 
      P_id_U33 P_id_a__U11 P_id_U81 P_id_a__U92 P_id_a__U43 P_id_U23 
      P_id_a__isNatKind P_id_U93 P_id_U11 P_id_a__U71 P_id_U44 P_id_0 
      P_id_U61 P_id_a__U86 P_id_a__U41 P_id_U22 P_id_a__U21 P_id_U91 
      P_id_length P_id_a__U52 P_id_U42 P_id_a__U12 P_id_U83 P_id_a__U94 
      P_id_a__U45 P_id_U31 P_id_a__isNat P_id_isNatIListKind P_id_a__U82 
      P_id_U46 P_id_cons P_id_U52 P_id_a__U85 P_id_a__U33 P_id_U21 
      P_id_a__isNatList P_id_U86 P_id_mark P_id_a__U51 P_id_U41 P_id_tt 
      P_id_U82 P_id_a__U93 P_id_a__U44 P_id_isNat P_id_a__U23 P_id_U94 
      P_id_U12 P_id_a__U81 P_id_U45 P_id_zeros P_id_U71 P_id_a__U91 
      P_id_a__U42 P_id_isNatKind P_id_a__U22 P_id_U92 P_id_nil P_id_a__U61 
      P_id_U43 P_id_a__isNatIListKind P_id_U84 P_id_s P_id_a__U46 P_id_U32 
      P_id_a__U31 P_id_U13 P_id_a__U83 P_id_isNatIList P_id_A__U46 
      P_id_A__ISNATILISTKIND P_id_A__U61 P_id_A__U85 P_id_A__U31 P_id_A__U81 
      P_id_A__U22 P_id_A__U93 P_id_A__U42 P_id_A__U51 P_id_A__ISNATLIST 
      P_id_A__ZEROS P_id_A__U91 P_id_A__U33 P_id_A__U83 P_id_A__U23 
      P_id_A__U11 P_id_A__LENGTH P_id_A__U44 P_id_A__ISNATILIST P_id_A__U13 
      P_id_A__U71 P_id_A__U86 P_id_A__U32 P_id_A__U82 P_id_A__ISNATKIND 
      P_id_A__U94 P_id_A__U43 P_id_A__U52 P_id_A__U21 P_id_A__U92 
      P_id_A__U41 P_id_A__U84 P_id_A__ISNAT P_id_A__U12 P_id_MARK P_id_A__U45
     .
   
   Lemma marked_measure_equation :
    forall t, 
     marked_measure t = match t with
                          | (algebra.Alg.Term algebra.F.id_a__U46 (x11::nil)) =>
                           P_id_A__U46 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
                             (x11::nil)) =>
                           P_id_A__ISNATILISTKIND (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U61 (x11::nil)) =>
                           P_id_A__U61 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U85 (x12::
                             x11::nil)) =>
                           P_id_A__U85 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U31 (x12::
                             x11::nil)) =>
                           P_id_A__U31 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U81 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U81 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U22 (x12::
                             x11::nil)) =>
                           P_id_A__U22 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U93 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U93 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U42 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U42 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U51 (x12::
                             x11::nil)) =>
                           P_id_A__U51 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__isNatList 
                             (x11::nil)) =>
                           P_id_A__ISNATLIST (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__zeros nil) =>
                           P_id_A__ZEROS 
                          | (algebra.Alg.Term algebra.F.id_a__U91 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U91 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U33 (x11::nil)) =>
                           P_id_A__U33 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U83 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U83 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U23 (x11::nil)) =>
                           P_id_A__U23 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U11 (x12::
                             x11::nil)) =>
                           P_id_A__U11 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__length 
                             (x11::nil)) =>
                           P_id_A__LENGTH (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U44 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U44 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__isNatIList 
                             (x11::nil)) =>
                           P_id_A__ISNATILIST (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U13 (x11::nil)) =>
                           P_id_A__U13 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U71 (x11::nil)) =>
                           P_id_A__U71 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U86 (x11::nil)) =>
                           P_id_A__U86 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U32 (x12::
                             x11::nil)) =>
                           P_id_A__U32 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U82 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U82 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__isNatKind 
                             (x11::nil)) =>
                           P_id_A__ISNATKIND (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U94 (x12::
                             x11::nil)) =>
                           P_id_A__U94 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U43 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U43 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U52 (x11::nil)) =>
                           P_id_A__U52 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U21 (x12::
                             x11::nil)) =>
                           P_id_A__U21 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U92 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U92 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U41 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U41 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U84 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U84 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__isNat 
                             (x11::nil)) =>
                           P_id_A__ISNAT (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U12 (x12::
                             x11::nil)) =>
                           P_id_A__U12 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_mark (x11::nil)) =>
                           P_id_MARK (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U45 (x12::
                             x11::nil)) =>
                           P_id_A__U45 (measure x12) (measure x11)
                          | _ => measure t
                          end.
   Proof.
     reflexivity .
   Qed.
   
   Lemma marked_measure_star_monotonic :
    forall f l1 l2, 
     (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step 
                                                       R_xml_0_deep_rew.R_xml_0_rules)
                                                      ) l1 l2) ->
      marked_measure (algebra.Alg.Term f l1) <= marked_measure (algebra.Alg.Term 
                                                                 f l2).
   Proof.
     unfold marked_measure in *.
     apply InterpZ.marked_measure_star_monotonic.
     intros ;apply P_id_U51_monotonic;assumption.
     intros ;apply P_id_a__U84_monotonic;assumption.
     intros ;apply P_id_a__U32_monotonic;assumption.
     intros ;apply P_id_isNatList_monotonic;assumption.
     intros ;apply P_id_a__U13_monotonic;assumption.
     intros ;apply P_id_U85_monotonic;assumption.
     intros ;apply P_id_a__length_monotonic;assumption.
     intros ;apply P_id_a__isNatIList_monotonic;assumption.
     intros ;apply P_id_U33_monotonic;assumption.
     intros ;apply P_id_a__U11_monotonic;assumption.
     intros ;apply P_id_U81_monotonic;assumption.
     intros ;apply P_id_a__U92_monotonic;assumption.
     intros ;apply P_id_a__U43_monotonic;assumption.
     intros ;apply P_id_U23_monotonic;assumption.
     intros ;apply P_id_a__isNatKind_monotonic;assumption.
     intros ;apply P_id_U93_monotonic;assumption.
     intros ;apply P_id_U11_monotonic;assumption.
     intros ;apply P_id_a__U71_monotonic;assumption.
     intros ;apply P_id_U44_monotonic;assumption.
     intros ;apply P_id_U61_monotonic;assumption.
     intros ;apply P_id_a__U86_monotonic;assumption.
     intros ;apply P_id_a__U41_monotonic;assumption.
     intros ;apply P_id_U22_monotonic;assumption.
     intros ;apply P_id_a__U21_monotonic;assumption.
     intros ;apply P_id_U91_monotonic;assumption.
     intros ;apply P_id_length_monotonic;assumption.
     intros ;apply P_id_a__U52_monotonic;assumption.
     intros ;apply P_id_U42_monotonic;assumption.
     intros ;apply P_id_a__U12_monotonic;assumption.
     intros ;apply P_id_U83_monotonic;assumption.
     intros ;apply P_id_a__U94_monotonic;assumption.
     intros ;apply P_id_a__U45_monotonic;assumption.
     intros ;apply P_id_U31_monotonic;assumption.
     intros ;apply P_id_a__isNat_monotonic;assumption.
     intros ;apply P_id_isNatIListKind_monotonic;assumption.
     intros ;apply P_id_a__U82_monotonic;assumption.
     intros ;apply P_id_U46_monotonic;assumption.
     intros ;apply P_id_cons_monotonic;assumption.
     intros ;apply P_id_U52_monotonic;assumption.
     intros ;apply P_id_a__U85_monotonic;assumption.
     intros ;apply P_id_a__U33_monotonic;assumption.
     intros ;apply P_id_U21_monotonic;assumption.
     intros ;apply P_id_a__isNatList_monotonic;assumption.
     intros ;apply P_id_U86_monotonic;assumption.
     intros ;apply P_id_mark_monotonic;assumption.
     intros ;apply P_id_a__U51_monotonic;assumption.
     intros ;apply P_id_U41_monotonic;assumption.
     intros ;apply P_id_U82_monotonic;assumption.
     intros ;apply P_id_a__U93_monotonic;assumption.
     intros ;apply P_id_a__U44_monotonic;assumption.
     intros ;apply P_id_isNat_monotonic;assumption.
     intros ;apply P_id_a__U23_monotonic;assumption.
     intros ;apply P_id_U94_monotonic;assumption.
     intros ;apply P_id_U12_monotonic;assumption.
     intros ;apply P_id_a__U81_monotonic;assumption.
     intros ;apply P_id_U45_monotonic;assumption.
     intros ;apply P_id_U71_monotonic;assumption.
     intros ;apply P_id_a__U91_monotonic;assumption.
     intros ;apply P_id_a__U42_monotonic;assumption.
     intros ;apply P_id_isNatKind_monotonic;assumption.
     intros ;apply P_id_a__U22_monotonic;assumption.
     intros ;apply P_id_U92_monotonic;assumption.
     intros ;apply P_id_a__U61_monotonic;assumption.
     intros ;apply P_id_U43_monotonic;assumption.
     intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
     intros ;apply P_id_U84_monotonic;assumption.
     intros ;apply P_id_s_monotonic;assumption.
     intros ;apply P_id_a__U46_monotonic;assumption.
     intros ;apply P_id_U32_monotonic;assumption.
     intros ;apply P_id_a__U31_monotonic;assumption.
     intros ;apply P_id_U13_monotonic;assumption.
     intros ;apply P_id_a__U83_monotonic;assumption.
     intros ;apply P_id_isNatIList_monotonic;assumption.
     intros ;apply P_id_a__zeros_bounded;assumption.
     intros ;apply P_id_U51_bounded;assumption.
     intros ;apply P_id_a__U84_bounded;assumption.
     intros ;apply P_id_a__U32_bounded;assumption.
     intros ;apply P_id_isNatList_bounded;assumption.
     intros ;apply P_id_a__U13_bounded;assumption.
     intros ;apply P_id_U85_bounded;assumption.
     intros ;apply P_id_a__length_bounded;assumption.
     intros ;apply P_id_a__isNatIList_bounded;assumption.
     intros ;apply P_id_U33_bounded;assumption.
     intros ;apply P_id_a__U11_bounded;assumption.
     intros ;apply P_id_U81_bounded;assumption.
     intros ;apply P_id_a__U92_bounded;assumption.
     intros ;apply P_id_a__U43_bounded;assumption.
     intros ;apply P_id_U23_bounded;assumption.
     intros ;apply P_id_a__isNatKind_bounded;assumption.
     intros ;apply P_id_U93_bounded;assumption.
     intros ;apply P_id_U11_bounded;assumption.
     intros ;apply P_id_a__U71_bounded;assumption.
     intros ;apply P_id_U44_bounded;assumption.
     intros ;apply P_id_0_bounded;assumption.
     intros ;apply P_id_U61_bounded;assumption.
     intros ;apply P_id_a__U86_bounded;assumption.
     intros ;apply P_id_a__U41_bounded;assumption.
     intros ;apply P_id_U22_bounded;assumption.
     intros ;apply P_id_a__U21_bounded;assumption.
     intros ;apply P_id_U91_bounded;assumption.
     intros ;apply P_id_length_bounded;assumption.
     intros ;apply P_id_a__U52_bounded;assumption.
     intros ;apply P_id_U42_bounded;assumption.
     intros ;apply P_id_a__U12_bounded;assumption.
     intros ;apply P_id_U83_bounded;assumption.
     intros ;apply P_id_a__U94_bounded;assumption.
     intros ;apply P_id_a__U45_bounded;assumption.
     intros ;apply P_id_U31_bounded;assumption.
     intros ;apply P_id_a__isNat_bounded;assumption.
     intros ;apply P_id_isNatIListKind_bounded;assumption.
     intros ;apply P_id_a__U82_bounded;assumption.
     intros ;apply P_id_U46_bounded;assumption.
     intros ;apply P_id_cons_bounded;assumption.
     intros ;apply P_id_U52_bounded;assumption.
     intros ;apply P_id_a__U85_bounded;assumption.
     intros ;apply P_id_a__U33_bounded;assumption.
     intros ;apply P_id_U21_bounded;assumption.
     intros ;apply P_id_a__isNatList_bounded;assumption.
     intros ;apply P_id_U86_bounded;assumption.
     intros ;apply P_id_mark_bounded;assumption.
     intros ;apply P_id_a__U51_bounded;assumption.
     intros ;apply P_id_U41_bounded;assumption.
     intros ;apply P_id_tt_bounded;assumption.
     intros ;apply P_id_U82_bounded;assumption.
     intros ;apply P_id_a__U93_bounded;assumption.
     intros ;apply P_id_a__U44_bounded;assumption.
     intros ;apply P_id_isNat_bounded;assumption.
     intros ;apply P_id_a__U23_bounded;assumption.
     intros ;apply P_id_U94_bounded;assumption.
     intros ;apply P_id_U12_bounded;assumption.
     intros ;apply P_id_a__U81_bounded;assumption.
     intros ;apply P_id_U45_bounded;assumption.
     intros ;apply P_id_zeros_bounded;assumption.
     intros ;apply P_id_U71_bounded;assumption.
     intros ;apply P_id_a__U91_bounded;assumption.
     intros ;apply P_id_a__U42_bounded;assumption.
     intros ;apply P_id_isNatKind_bounded;assumption.
     intros ;apply P_id_a__U22_bounded;assumption.
     intros ;apply P_id_U92_bounded;assumption.
     intros ;apply P_id_nil_bounded;assumption.
     intros ;apply P_id_a__U61_bounded;assumption.
     intros ;apply P_id_U43_bounded;assumption.
     intros ;apply P_id_a__isNatIListKind_bounded;assumption.
     intros ;apply P_id_U84_bounded;assumption.
     intros ;apply P_id_s_bounded;assumption.
     intros ;apply P_id_a__U46_bounded;assumption.
     intros ;apply P_id_U32_bounded;assumption.
     intros ;apply P_id_a__U31_bounded;assumption.
     intros ;apply P_id_U13_bounded;assumption.
     intros ;apply P_id_a__U83_bounded;assumption.
     intros ;apply P_id_isNatIList_bounded;assumption.
     apply rules_monotonic.
     intros ;apply P_id_A__U46_monotonic;assumption.
     intros ;apply P_id_A__ISNATILISTKIND_monotonic;assumption.
     intros ;apply P_id_A__U61_monotonic;assumption.
     intros ;apply P_id_A__U85_monotonic;assumption.
     intros ;apply P_id_A__U31_monotonic;assumption.
     intros ;apply P_id_A__U81_monotonic;assumption.
     intros ;apply P_id_A__U22_monotonic;assumption.
     intros ;apply P_id_A__U93_monotonic;assumption.
     intros ;apply P_id_A__U42_monotonic;assumption.
     intros ;apply P_id_A__U51_monotonic;assumption.
     intros ;apply P_id_A__ISNATLIST_monotonic;assumption.
     intros ;apply P_id_A__U91_monotonic;assumption.
     intros ;apply P_id_A__U33_monotonic;assumption.
     intros ;apply P_id_A__U83_monotonic;assumption.
     intros ;apply P_id_A__U23_monotonic;assumption.
     intros ;apply P_id_A__U11_monotonic;assumption.
     intros ;apply P_id_A__LENGTH_monotonic;assumption.
     intros ;apply P_id_A__U44_monotonic;assumption.
     intros ;apply P_id_A__ISNATILIST_monotonic;assumption.
     intros ;apply P_id_A__U13_monotonic;assumption.
     intros ;apply P_id_A__U71_monotonic;assumption.
     intros ;apply P_id_A__U86_monotonic;assumption.
     intros ;apply P_id_A__U32_monotonic;assumption.
     intros ;apply P_id_A__U82_monotonic;assumption.
     intros ;apply P_id_A__ISNATKIND_monotonic;assumption.
     intros ;apply P_id_A__U94_monotonic;assumption.
     intros ;apply P_id_A__U43_monotonic;assumption.
     intros ;apply P_id_A__U52_monotonic;assumption.
     intros ;apply P_id_A__U21_monotonic;assumption.
     intros ;apply P_id_A__U92_monotonic;assumption.
     intros ;apply P_id_A__U41_monotonic;assumption.
     intros ;apply P_id_A__U84_monotonic;assumption.
     intros ;apply P_id_A__ISNAT_monotonic;assumption.
     intros ;apply P_id_A__U12_monotonic;assumption.
     intros ;apply P_id_MARK_monotonic;assumption.
     intros ;apply P_id_A__U45_monotonic;assumption.
   Qed.
   
   Ltac rewrite_and_unfold  :=
    do 2 (rewrite marked_measure_equation);
     repeat (
     match goal with
       |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
        rewrite (measure_equation (algebra.Alg.Term f t))
       | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ =>
        rewrite (measure_equation (algebra.Alg.Term f t)) in H|-
       end
     ).
   
   Definition lt a b := (Zwf.Zwf 0) (marked_measure a) (marked_measure b).
   
   Definition le a b := marked_measure a <= marked_measure b.
   
   Lemma lt_le_compat : forall a b c, (lt a b) ->(le b c) ->lt a c.
   Proof.
     unfold lt, le in *.
     intros a b c.
     apply (interp.le_lt_compat_right (interp.o_Z 0)).
   Qed.
   
   Lemma wf_lt : well_founded lt.
   Proof.
     unfold lt in *.
     apply Inverse_Image.wf_inverse_image with  (B:=Z).
     apply Zwf.Zwf_well_founded.
   Qed.
   
   Lemma DP_R_xml_0_scc_38_strict_in_lt :
    Relation_Definitions.inclusion _ DP_R_xml_0_scc_38_strict lt.
   Proof.
     unfold Relation_Definitions.inclusion, lt in *.
     
     intros a b H;destruct H;
      match goal with
        |  |- (Zwf.Zwf 0) _ (marked_measure (algebra.Alg.Term ?f ?l)) =>
         let l'' := algebra.Alg_ext.find_replacement l  in 
          ((apply (interp.le_lt_compat_right (interp.o_Z 0)) with
             (marked_measure (algebra.Alg.Term f l''));[idtac|
            apply marked_measure_star_monotonic;
             repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos);
             (assumption)||(constructor 1)]))
        end
      ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma DP_R_xml_0_scc_38_large_in_le :
    Relation_Definitions.inclusion _ DP_R_xml_0_scc_38_large le.
   Proof.
     unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *.
     
     intros a b H;destruct H;
      match goal with
        |  |- _ <= marked_measure (algebra.Alg.Term ?f ?l) =>
         let l'' := algebra.Alg_ext.find_replacement l  in 
          ((apply (interp.le_trans (interp.o_Z 0)) with
             (marked_measure (algebra.Alg.Term f l''));[idtac|
            apply marked_measure_star_monotonic;
             repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos);
             (assumption)||(constructor 1)]))
        end
      ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Definition wf_DP_R_xml_0_scc_38_large  := WF_DP_R_xml_0_scc_38_large.wf.
   
   
   Lemma wf : well_founded WF_DP_R_xml_0.DP_R_xml_0_scc_38.
   Proof.
     intros x.
     apply (well_founded_ind wf_lt).
     clear x.
     intros x.
     pattern x.
     apply (@Acc_ind _ DP_R_xml_0_scc_38_large).
     clear x.
     intros x _ IHx IHx'.
     constructor.
     intros y H.
     
     destruct H;
      (apply IHx';apply DP_R_xml_0_scc_38_strict_in_lt;
        econstructor eassumption)||
      ((apply IHx;[econstructor eassumption|
        intros y' Hlt;apply IHx';apply lt_le_compat with (1:=Hlt) ;
         apply DP_R_xml_0_scc_38_large_in_le;econstructor eassumption])).
     apply wf_DP_R_xml_0_scc_38_large.
   Qed.
  End WF_DP_R_xml_0_scc_38.
  
  Definition wf_DP_R_xml_0_scc_38  := WF_DP_R_xml_0_scc_38.wf.
  
  
  Lemma acc_DP_R_xml_0_scc_38 :
   forall x y, (DP_R_xml_0_scc_38 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x.
    pattern x.
    apply (@Acc_ind _ DP_R_xml_0_scc_38).
    intros x' _ Hrec y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply Hrec;econstructor eassumption)||
      ((eapply acc_DP_R_xml_0_non_scc_37;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_36;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((eapply acc_DP_R_xml_0_non_scc_35;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
         ((eapply acc_DP_R_xml_0_non_scc_34;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
          ((eapply acc_DP_R_xml_0_non_scc_26;
             econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
           ((eapply acc_DP_R_xml_0_non_scc_25;
              econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
            ((eapply acc_DP_R_xml_0_non_scc_23;
               econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
             ((eapply acc_DP_R_xml_0_non_scc_21;
                econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
              ((eapply acc_DP_R_xml_0_non_scc_19;
                 econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
               ((eapply acc_DP_R_xml_0_non_scc_16;
                  econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
                ((eapply acc_DP_R_xml_0_non_scc_12;
                   econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
                 ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
                  (eapply Hrec;
                    econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))))))))))))).
    apply wf_DP_R_xml_0_scc_38.
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_39  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <mark(U11(X1_,X2_)),a__U11(mark(X1_),X2_)> *)
    | DP_R_xml_0_non_scc_39_0 :
     forall x6 x11 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U11 (x6::x7::nil)) x11) ->
       DP_R_xml_0_non_scc_39 (algebra.Alg.Term algebra.F.id_a__U11 
                              ((algebra.Alg.Term algebra.F.id_mark 
                              (x6::nil))::x7::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_39 :
   forall x y, 
    (DP_R_xml_0_non_scc_39 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_38;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_37;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
        (eapply Hrec;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_40  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <mark(U22(X1_,X2_)),a__U22(mark(X1_),X2_)> *)
    | DP_R_xml_0_non_scc_40_0 :
     forall x6 x11 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U22 (x6::x7::nil)) x11) ->
       DP_R_xml_0_non_scc_40 (algebra.Alg.Term algebra.F.id_a__U22 
                              ((algebra.Alg.Term algebra.F.id_mark 
                              (x6::nil))::x7::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_40 :
   forall x y, 
    (DP_R_xml_0_non_scc_40 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_38;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_34;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
        (eapply Hrec;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_41  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <mark(U21(X1_,X2_)),a__U21(mark(X1_),X2_)> *)
    | DP_R_xml_0_non_scc_41_0 :
     forall x6 x11 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U21 (x6::x7::nil)) x11) ->
       DP_R_xml_0_non_scc_41 (algebra.Alg.Term algebra.F.id_a__U21 
                              ((algebra.Alg.Term algebra.F.id_mark 
                              (x6::nil))::x7::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_41 :
   forall x y, 
    (DP_R_xml_0_non_scc_41 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_38;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_35;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
        (eapply Hrec;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_42  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <mark(isNat(X_)),a__isNat(X_)> *)
    | DP_R_xml_0_non_scc_42_0 :
     forall x8 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_isNat (x8::nil)) x11) ->
       DP_R_xml_0_non_scc_42 (algebra.Alg.Term algebra.F.id_a__isNat 
                              (x8::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_42 :
   forall x y, 
    (DP_R_xml_0_non_scc_42 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_38;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_23;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_19;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
         (eapply Hrec;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))))).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_43  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <a__U92(tt,L_,N_),a__isNat(N_)> *)
    | DP_R_xml_0_non_scc_43_0 :
     forall x4 x12 x5 x13 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_tt nil) 
        x13) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x4 x12) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x5 x11) ->
         DP_R_xml_0_non_scc_43 (algebra.Alg.Term algebra.F.id_a__isNat 
                                (x5::nil)) 
          (algebra.Alg.Term algebra.F.id_a__U92 (x13::x12::x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_43 :
   forall x y, 
    (DP_R_xml_0_non_scc_43 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_38;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_23;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_19;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
         (eapply Hrec;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))))).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_44  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <mark(U83(X1_,X2_,X3_)),a__U83(mark(X1_),X2_,X3_)> *)
    | DP_R_xml_0_non_scc_44_0 :
     forall x6 x9 x11 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U83 (x6::x7::x9::nil)) x11) ->
       DP_R_xml_0_non_scc_44 (algebra.Alg.Term algebra.F.id_a__U83 
                              ((algebra.Alg.Term algebra.F.id_mark 
                              (x6::nil))::x7::x9::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_44 :
   forall x y, 
    (DP_R_xml_0_non_scc_44 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_38;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_25;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
        (eapply Hrec;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_45  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <mark(U82(X1_,X2_,X3_)),a__U82(mark(X1_),X2_,X3_)> *)
    | DP_R_xml_0_non_scc_45_0 :
     forall x6 x9 x11 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U82 (x6::x7::x9::nil)) x11) ->
       DP_R_xml_0_non_scc_45 (algebra.Alg.Term algebra.F.id_a__U82 
                              ((algebra.Alg.Term algebra.F.id_mark 
                              (x6::nil))::x7::x9::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_45 :
   forall x y, 
    (DP_R_xml_0_non_scc_45 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_38;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_26;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
        (eapply Hrec;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_46  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <mark(U81(X1_,X2_,X3_)),a__U81(mark(X1_),X2_,X3_)> *)
    | DP_R_xml_0_non_scc_46_0 :
     forall x6 x9 x11 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U81 (x6::x7::x9::nil)) x11) ->
       DP_R_xml_0_non_scc_46 (algebra.Alg.Term algebra.F.id_a__U81 
                              ((algebra.Alg.Term algebra.F.id_mark 
                              (x6::nil))::x7::x9::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_46 :
   forall x y, 
    (DP_R_xml_0_non_scc_46 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_38;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_21;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
        (eapply Hrec;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_47  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <mark(isNatList(X_)),a__isNatList(X_)> *)
    | DP_R_xml_0_non_scc_47_0 :
     forall x8 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_isNatList (x8::nil)) x11) ->
       DP_R_xml_0_non_scc_47 (algebra.Alg.Term algebra.F.id_a__isNatList 
                              (x8::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_47 :
   forall x y, 
    (DP_R_xml_0_non_scc_47 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_38;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_16;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
        (eapply Hrec;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_48  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <a__length(cons(N_,L_)),a__isNatList(L_)> *)
    | DP_R_xml_0_non_scc_48_0 :
     forall x4 x5 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_cons (x5::x4::nil)) x11) ->
       DP_R_xml_0_non_scc_48 (algebra.Alg.Term algebra.F.id_a__isNatList 
                              (x4::nil)) 
        (algebra.Alg.Term algebra.F.id_a__length (x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_48 :
   forall x y, 
    (DP_R_xml_0_non_scc_48 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_38;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_16;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
        (eapply Hrec;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_49  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <mark(U85(X1_,X2_)),a__U85(mark(X1_),X2_)> *)
    | DP_R_xml_0_non_scc_49_0 :
     forall x6 x11 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U85 (x6::x7::nil)) x11) ->
       DP_R_xml_0_non_scc_49 (algebra.Alg.Term algebra.F.id_a__U85 
                              ((algebra.Alg.Term algebra.F.id_mark 
                              (x6::nil))::x7::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_49 :
   forall x y, 
    (DP_R_xml_0_non_scc_49 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_38;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_12;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
        (eapply Hrec;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_50  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <mark(U84(X1_,X2_,X3_)),a__U84(mark(X1_),X2_,X3_)> *)
    | DP_R_xml_0_non_scc_50_0 :
     forall x6 x9 x11 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U84 (x6::x7::x9::nil)) x11) ->
       DP_R_xml_0_non_scc_50 (algebra.Alg.Term algebra.F.id_a__U84 
                              ((algebra.Alg.Term algebra.F.id_mark 
                              (x6::nil))::x7::x9::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_50 :
   forall x y, 
    (DP_R_xml_0_non_scc_50 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_38;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
       (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_51  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <a__U32(tt,V_),a__isNatList(V_)> *)
    | DP_R_xml_0_non_scc_51_0 :
     forall x12 x2 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_tt nil) 
        x12) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x2 x11) ->
        DP_R_xml_0_non_scc_51 (algebra.Alg.Term algebra.F.id_a__isNatList 
                               (x2::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U32 (x12::x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_51 :
   forall x y, 
    (DP_R_xml_0_non_scc_51 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_38;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_16;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
        (eapply Hrec;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_52  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <mark(U32(X1_,X2_)),a__U32(mark(X1_),X2_)> *)
    | DP_R_xml_0_non_scc_52_0 :
     forall x6 x11 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U32 (x6::x7::nil)) x11) ->
       DP_R_xml_0_non_scc_52 (algebra.Alg.Term algebra.F.id_a__U32 
                              ((algebra.Alg.Term algebra.F.id_mark 
                              (x6::nil))::x7::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_52 :
   forall x y, 
    (DP_R_xml_0_non_scc_52 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_non_scc_51;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_32;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
        (eapply Hrec;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_53  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <a__U31(tt,V_),a__U32(a__isNatIListKind(V_),V_)> *)
    | DP_R_xml_0_non_scc_53_0 :
     forall x12 x2 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_tt nil) 
        x12) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x2 x11) ->
        DP_R_xml_0_non_scc_53 (algebra.Alg.Term algebra.F.id_a__U32 
                               ((algebra.Alg.Term 
                               algebra.F.id_a__isNatIListKind (x2::nil))::
                               x2::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U31 (x12::x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_53 :
   forall x y, 
    (DP_R_xml_0_non_scc_53 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_non_scc_51;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_32;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
        (eapply Hrec;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_54  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <mark(U31(X1_,X2_)),a__U31(mark(X1_),X2_)> *)
    | DP_R_xml_0_non_scc_54_0 :
     forall x6 x11 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U31 (x6::x7::nil)) x11) ->
       DP_R_xml_0_non_scc_54 (algebra.Alg.Term algebra.F.id_a__U31 
                              ((algebra.Alg.Term algebra.F.id_mark 
                              (x6::nil))::x7::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_54 :
   forall x y, 
    (DP_R_xml_0_non_scc_54 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_non_scc_53;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_33;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
        (eapply Hrec;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_55  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <a__isNatIList(V_),a__U31(a__isNatIListKind(V_),V_)> *)
    | DP_R_xml_0_non_scc_55_0 :
     forall x2 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x2 x11) ->
       DP_R_xml_0_non_scc_55 (algebra.Alg.Term algebra.F.id_a__U31 
                              ((algebra.Alg.Term 
                              algebra.F.id_a__isNatIListKind (x2::nil))::
                              x2::nil)) 
        (algebra.Alg.Term algebra.F.id_a__isNatIList (x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_55 :
   forall x y, 
    (DP_R_xml_0_non_scc_55 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_non_scc_53;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_33;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
        (eapply Hrec;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_56  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <mark(U12(X1_,X2_)),a__U12(mark(X1_),X2_)> *)
    | DP_R_xml_0_non_scc_56_0 :
     forall x6 x11 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U12 (x6::x7::nil)) x11) ->
       DP_R_xml_0_non_scc_56 (algebra.Alg.Term algebra.F.id_a__U12 
                              ((algebra.Alg.Term algebra.F.id_mark 
                              (x6::nil))::x7::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_56 :
   forall x y, 
    (DP_R_xml_0_non_scc_56 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_38;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_36;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
        (eapply Hrec;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_57  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <a__U44(tt,V1_,V2_),a__isNat(V1_)> *)
    | DP_R_xml_0_non_scc_57_0 :
     forall x12 x1 x13 x3 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_tt nil) 
        x13) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x1 x12) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x3 x11) ->
         DP_R_xml_0_non_scc_57 (algebra.Alg.Term algebra.F.id_a__isNat 
                                (x1::nil)) 
          (algebra.Alg.Term algebra.F.id_a__U44 (x13::x12::x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_57 :
   forall x y, 
    (DP_R_xml_0_non_scc_57 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_38;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_23;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_19;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
         (eapply Hrec;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))))).
  Qed.
  
  
  Inductive DP_R_xml_0_scc_58  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <a__U44(tt,V1_,V2_),a__U45(a__isNat(V1_),V2_)> *)
    | DP_R_xml_0_scc_58_0 :
     forall x12 x1 x13 x3 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_tt nil) 
        x13) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x1 x12) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x3 x11) ->
         DP_R_xml_0_scc_58 (algebra.Alg.Term algebra.F.id_a__U45 
                            ((algebra.Alg.Term algebra.F.id_a__isNat 
                            (x1::nil))::x3::nil)) 
          (algebra.Alg.Term algebra.F.id_a__U44 (x13::x12::x11::nil))
     (* <a__U45(tt,V2_),a__isNatIList(V2_)> *)
    | DP_R_xml_0_scc_58_1 :
     forall x12 x3 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_tt nil) 
        x12) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x3 x11) ->
        DP_R_xml_0_scc_58 (algebra.Alg.Term algebra.F.id_a__isNatIList 
                           (x3::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U45 (x12::x11::nil))
    
     (* <a__isNatIList(cons(V1_,V2_)),a__U41(a__isNatKind(V1_),V1_,V2_)> *)
    | DP_R_xml_0_scc_58_2 :
     forall x1 x3 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_cons (x1::x3::nil)) x11) ->
       DP_R_xml_0_scc_58 (algebra.Alg.Term algebra.F.id_a__U41 
                          ((algebra.Alg.Term algebra.F.id_a__isNatKind 
                          (x1::nil))::x1::x3::nil)) 
        (algebra.Alg.Term algebra.F.id_a__isNatIList (x11::nil))
    
     (* <a__U41(tt,V1_,V2_),a__U42(a__isNatKind(V1_),V1_,V2_)> *)
    | DP_R_xml_0_scc_58_3 :
     forall x12 x1 x13 x3 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_tt nil) 
        x13) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x1 x12) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x3 x11) ->
         DP_R_xml_0_scc_58 (algebra.Alg.Term algebra.F.id_a__U42 
                            ((algebra.Alg.Term algebra.F.id_a__isNatKind 
                            (x1::nil))::x1::x3::nil)) 
          (algebra.Alg.Term algebra.F.id_a__U41 (x13::x12::x11::nil))
    
     (* <a__U42(tt,V1_,V2_),a__U43(a__isNatIListKind(V2_),V1_,V2_)> *)
    | DP_R_xml_0_scc_58_4 :
     forall x12 x1 x13 x3 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_tt nil) 
        x13) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x1 x12) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x3 x11) ->
         DP_R_xml_0_scc_58 (algebra.Alg.Term algebra.F.id_a__U43 
                            ((algebra.Alg.Term 
                            algebra.F.id_a__isNatIListKind (x3::nil))::x1::
                            x3::nil)) 
          (algebra.Alg.Term algebra.F.id_a__U42 (x13::x12::x11::nil))
    
     (* <a__U43(tt,V1_,V2_),a__U44(a__isNatIListKind(V2_),V1_,V2_)> *)
    | DP_R_xml_0_scc_58_5 :
     forall x12 x1 x13 x3 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_tt nil) 
        x13) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x1 x12) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x3 x11) ->
         DP_R_xml_0_scc_58 (algebra.Alg.Term algebra.F.id_a__U44 
                            ((algebra.Alg.Term 
                            algebra.F.id_a__isNatIListKind (x3::nil))::x1::
                            x3::nil)) 
          (algebra.Alg.Term algebra.F.id_a__U43 (x13::x12::x11::nil))
  .
  
  
  Module WF_DP_R_xml_0_scc_58.
   Inductive DP_R_xml_0_scc_58_large  :
    algebra.Alg.term ->algebra.Alg.term ->Prop := 
      (* <a__U44(tt,V1_,V2_),a__U45(a__isNat(V1_),V2_)> *)
     | DP_R_xml_0_scc_58_large_0 :
      forall x12 x1 x13 x3 x11, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  (algebra.Alg.Term algebra.F.id_tt nil) 
         x13) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x1 x12) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x3 x11) ->
          DP_R_xml_0_scc_58_large (algebra.Alg.Term algebra.F.id_a__U45 
                                   ((algebra.Alg.Term algebra.F.id_a__isNat 
                                   (x1::nil))::x3::nil)) 
           (algebra.Alg.Term algebra.F.id_a__U44 (x13::x12::x11::nil))
      (* <a__U45(tt,V2_),a__isNatIList(V2_)> *)
     | DP_R_xml_0_scc_58_large_1 :
      forall x12 x3 x11, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  (algebra.Alg.Term algebra.F.id_tt nil) 
         x12) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x3 x11) ->
         DP_R_xml_0_scc_58_large (algebra.Alg.Term 
                                  algebra.F.id_a__isNatIList (x3::nil)) 
          (algebra.Alg.Term algebra.F.id_a__U45 (x12::x11::nil))
     
      (* <a__isNatIList(cons(V1_,V2_)),a__U41(a__isNatKind(V1_),V1_,V2_)> *)
     | DP_R_xml_0_scc_58_large_2 :
      forall x1 x3 x11, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_cons (x1::x3::nil)) x11) ->
        DP_R_xml_0_scc_58_large (algebra.Alg.Term algebra.F.id_a__U41 
                                 ((algebra.Alg.Term 
                                 algebra.F.id_a__isNatKind (x1::nil))::x1::
                                 x3::nil)) 
         (algebra.Alg.Term algebra.F.id_a__isNatIList (x11::nil))
     
      (* <a__U42(tt,V1_,V2_),a__U43(a__isNatIListKind(V2_),V1_,V2_)> *)
     | DP_R_xml_0_scc_58_large_3 :
      forall x12 x1 x13 x3 x11, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  (algebra.Alg.Term algebra.F.id_tt nil) 
         x13) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x1 x12) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x3 x11) ->
          DP_R_xml_0_scc_58_large (algebra.Alg.Term algebra.F.id_a__U43 
                                   ((algebra.Alg.Term 
                                   algebra.F.id_a__isNatIListKind 
                                   (x3::nil))::x1::x3::nil)) 
           (algebra.Alg.Term algebra.F.id_a__U42 (x13::x12::x11::nil))
     
      (* <a__U43(tt,V1_,V2_),a__U44(a__isNatIListKind(V2_),V1_,V2_)> *)
     | DP_R_xml_0_scc_58_large_4 :
      forall x12 x1 x13 x3 x11, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  (algebra.Alg.Term algebra.F.id_tt nil) 
         x13) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x1 x12) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x3 x11) ->
          DP_R_xml_0_scc_58_large (algebra.Alg.Term algebra.F.id_a__U44 
                                   ((algebra.Alg.Term 
                                   algebra.F.id_a__isNatIListKind 
                                   (x3::nil))::x1::x3::nil)) 
           (algebra.Alg.Term algebra.F.id_a__U43 (x13::x12::x11::nil))
   .
   
   
   Inductive DP_R_xml_0_scc_58_strict  :
    algebra.Alg.term ->algebra.Alg.term ->Prop := 
      (* <a__U41(tt,V1_,V2_),a__U42(a__isNatKind(V1_),V1_,V2_)> *)
     | DP_R_xml_0_scc_58_strict_0 :
      forall x12 x1 x13 x3 x11, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  (algebra.Alg.Term algebra.F.id_tt nil) 
         x13) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x1 x12) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x3 x11) ->
          DP_R_xml_0_scc_58_strict (algebra.Alg.Term algebra.F.id_a__U42 
                                    ((algebra.Alg.Term 
                                    algebra.F.id_a__isNatKind (x1::nil))::
                                    x1::x3::nil)) 
           (algebra.Alg.Term algebra.F.id_a__U41 (x13::x12::x11::nil))
   .
   
   
   Module WF_DP_R_xml_0_scc_58_large.
    Inductive DP_R_xml_0_scc_58_large_non_scc_1  :
     algebra.Alg.term ->algebra.Alg.term ->Prop := 
       (* <a__isNatIList(cons(V1_,V2_)),a__U41(a__isNatKind(V1_),V1_,V2_)> *)
      | DP_R_xml_0_scc_58_large_non_scc_1_0 :
       forall x1 x3 x11, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_cons (x1::x3::nil)) x11) ->
         DP_R_xml_0_scc_58_large_non_scc_1 (algebra.Alg.Term 
                                            algebra.F.id_a__U41 
                                            ((algebra.Alg.Term 
                                            algebra.F.id_a__isNatKind 
                                            (x1::nil))::x1::x3::nil)) 
          (algebra.Alg.Term algebra.F.id_a__isNatIList (x11::nil))
    .
    
    
    Lemma acc_DP_R_xml_0_scc_58_large_non_scc_1 :
     forall x y, 
      (DP_R_xml_0_scc_58_large_non_scc_1 x y) ->
       Acc WF_DP_R_xml_0_scc_58.DP_R_xml_0_scc_58_large x.
    Proof.
      intros x y h.
      
      inversion h;clear h;subst;
       constructor;intros _y _h;inversion _h;clear _h;subst;
        (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
        (eapply Hrec;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' )).
    Qed.
    
    
    Inductive DP_R_xml_0_scc_58_large_non_scc_2  :
     algebra.Alg.term ->algebra.Alg.term ->Prop := 
       (* <a__U45(tt,V2_),a__isNatIList(V2_)> *)
      | DP_R_xml_0_scc_58_large_non_scc_2_0 :
       forall x12 x3 x11, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   (algebra.Alg.Term algebra.F.id_tt nil) 
          x12) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x3 x11) ->
          DP_R_xml_0_scc_58_large_non_scc_2 (algebra.Alg.Term 
                                             algebra.F.id_a__isNatIList 
                                             (x3::nil)) 
           (algebra.Alg.Term algebra.F.id_a__U45 (x12::x11::nil))
    .
    
    
    Lemma acc_DP_R_xml_0_scc_58_large_non_scc_2 :
     forall x y, 
      (DP_R_xml_0_scc_58_large_non_scc_2 x y) ->
       Acc WF_DP_R_xml_0_scc_58.DP_R_xml_0_scc_58_large x.
    Proof.
      intros x y h.
      
      inversion h;clear h;subst;
       constructor;intros _y _h;inversion _h;clear _h;subst;
        (eapply acc_DP_R_xml_0_scc_58_large_non_scc_1;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
         (eapply Hrec;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))).
    Qed.
    
    
    Inductive DP_R_xml_0_scc_58_large_non_scc_3  :
     algebra.Alg.term ->algebra.Alg.term ->Prop := 
       (* <a__U44(tt,V1_,V2_),a__U45(a__isNat(V1_),V2_)> *)
      | DP_R_xml_0_scc_58_large_non_scc_3_0 :
       forall x12 x1 x13 x3 x11, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   (algebra.Alg.Term algebra.F.id_tt nil) 
          x13) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x1 x12) ->
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     x3 x11) ->
           DP_R_xml_0_scc_58_large_non_scc_3 (algebra.Alg.Term 
                                              algebra.F.id_a__U45 
                                              ((algebra.Alg.Term 
                                              algebra.F.id_a__isNat 
                                              (x1::nil))::x3::nil)) 
            (algebra.Alg.Term algebra.F.id_a__U44 (x13::x12::x11::nil))
    .
    
    
    Lemma acc_DP_R_xml_0_scc_58_large_non_scc_3 :
     forall x y, 
      (DP_R_xml_0_scc_58_large_non_scc_3 x y) ->
       Acc WF_DP_R_xml_0_scc_58.DP_R_xml_0_scc_58_large x.
    Proof.
      intros x y h.
      
      inversion h;clear h;subst;
       constructor;intros _y _h;inversion _h;clear _h;subst;
        (eapply acc_DP_R_xml_0_scc_58_large_non_scc_2;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
         (eapply Hrec;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))).
    Qed.
    
    
    Inductive DP_R_xml_0_scc_58_large_non_scc_4  :
     algebra.Alg.term ->algebra.Alg.term ->Prop := 
       (* <a__U43(tt,V1_,V2_),a__U44(a__isNatIListKind(V2_),V1_,V2_)> *)
      | DP_R_xml_0_scc_58_large_non_scc_4_0 :
       forall x12 x1 x13 x3 x11, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   (algebra.Alg.Term algebra.F.id_tt nil) 
          x13) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x1 x12) ->
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     x3 x11) ->
           DP_R_xml_0_scc_58_large_non_scc_4 (algebra.Alg.Term 
                                              algebra.F.id_a__U44 
                                              ((algebra.Alg.Term 
                                              algebra.F.id_a__isNatIListKind 
                                              (x3::nil))::x1::x3::nil)) 
            (algebra.Alg.Term algebra.F.id_a__U43 (x13::x12::x11::nil))
    .
    
    
    Lemma acc_DP_R_xml_0_scc_58_large_non_scc_4 :
     forall x y, 
      (DP_R_xml_0_scc_58_large_non_scc_4 x y) ->
       Acc WF_DP_R_xml_0_scc_58.DP_R_xml_0_scc_58_large x.
    Proof.
      intros x y h.
      
      inversion h;clear h;subst;
       constructor;intros _y _h;inversion _h;clear _h;subst;
        (eapply acc_DP_R_xml_0_scc_58_large_non_scc_3;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
         (eapply Hrec;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))).
    Qed.
    
    
    Inductive DP_R_xml_0_scc_58_large_non_scc_5  :
     algebra.Alg.term ->algebra.Alg.term ->Prop := 
       (* <a__U42(tt,V1_,V2_),a__U43(a__isNatIListKind(V2_),V1_,V2_)> *)
      | DP_R_xml_0_scc_58_large_non_scc_5_0 :
       forall x12 x1 x13 x3 x11, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   (algebra.Alg.Term algebra.F.id_tt nil) 
          x13) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x1 x12) ->
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     x3 x11) ->
           DP_R_xml_0_scc_58_large_non_scc_5 (algebra.Alg.Term 
                                              algebra.F.id_a__U43 
                                              ((algebra.Alg.Term 
                                              algebra.F.id_a__isNatIListKind 
                                              (x3::nil))::x1::x3::nil)) 
            (algebra.Alg.Term algebra.F.id_a__U42 (x13::x12::x11::nil))
    .
    
    
    Lemma acc_DP_R_xml_0_scc_58_large_non_scc_5 :
     forall x y, 
      (DP_R_xml_0_scc_58_large_non_scc_5 x y) ->
       Acc WF_DP_R_xml_0_scc_58.DP_R_xml_0_scc_58_large x.
    Proof.
      intros x y h.
      
      inversion h;clear h;subst;
       constructor;intros _y _h;inversion _h;clear _h;subst;
        (eapply acc_DP_R_xml_0_scc_58_large_non_scc_4;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
         (eapply Hrec;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))).
    Qed.
    
    
    Lemma wf : well_founded WF_DP_R_xml_0_scc_58.DP_R_xml_0_scc_58_large.
    Proof.
      constructor;intros _y _h;inversion _h;clear _h;subst;
       (eapply acc_DP_R_xml_0_scc_58_large_non_scc_5;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_scc_58_large_non_scc_4;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((eapply acc_DP_R_xml_0_scc_58_large_non_scc_3;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
         ((eapply acc_DP_R_xml_0_scc_58_large_non_scc_2;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
          ((eapply acc_DP_R_xml_0_scc_58_large_non_scc_1;
             econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
           ((eapply acc_DP_R_xml_0_scc_58_large_non_scc_0;
              econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
            ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||(fail))))))).
    Qed.
   End WF_DP_R_xml_0_scc_58_large.
   
   Open Scope Z_scope.
   
   Import ring_extention.
   
   Notation Local "a <= b" := (Zle a b).
   
   Notation Local "a < b" := (Zlt a b).
   
   Definition P_id_a__zeros  := 3.
   
   Definition P_id_U51 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_a__U84 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_a__U32 (x11:Z) (x12:Z) := 2* x12.
   
   Definition P_id_isNatList (x11:Z) := 0.
   
   Definition P_id_a__U13 (x11:Z) := 0.
   
   Definition P_id_U85 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_a__length (x11:Z) := 0.
   
   Definition P_id_a__isNatIList (x11:Z) := 3* x11.
   
   Definition P_id_U33 (x11:Z) := 0.
   
   Definition P_id_a__U11 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_U81 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_a__U92 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_a__U43 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_U23 (x11:Z) := 0.
   
   Definition P_id_a__isNatKind (x11:Z) := 1.
   
   Definition P_id_U93 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_U11 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_a__U71 (x11:Z) := 0.
   
   Definition P_id_U44 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_0  := 0.
   
   Definition P_id_U61 (x11:Z) := 1.
   
   Definition P_id_a__U86 (x11:Z) := 0.
   
   Definition P_id_a__U41 (x11:Z) (x12:Z) (x13:Z) := 2 + 1* x11.
   
   Definition P_id_U22 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_a__U21 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_U91 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_length (x11:Z) := 0.
   
   Definition P_id_a__U52 (x11:Z) := 0.
   
   Definition P_id_U42 (x11:Z) (x12:Z) (x13:Z) := 2* x11.
   
   Definition P_id_a__U12 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_U83 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_a__U94 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_a__U45 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_U31 (x11:Z) (x12:Z) := 1* x12.
   
   Definition P_id_a__isNat (x11:Z) := 0.
   
   Definition P_id_isNatIListKind (x11:Z) := 0.
   
   Definition P_id_a__U82 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_U46 (x11:Z) := 0.
   
   Definition P_id_cons (x11:Z) (x12:Z) := 1 + 1* x12.
   
   Definition P_id_U52 (x11:Z) := 0.
   
   Definition P_id_a__U85 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_a__U33 (x11:Z) := 0.
   
   Definition P_id_U21 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_a__isNatList (x11:Z) := 0.
   
   Definition P_id_U86 (x11:Z) := 0.
   
   Definition P_id_mark (x11:Z) := 2* x11.
   
   Definition P_id_a__U51 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_U41 (x11:Z) (x12:Z) (x13:Z) := 1 + 1* x11.
   
   Definition P_id_tt  := 0.
   
   Definition P_id_U82 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_a__U93 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_a__U44 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_isNat (x11:Z) := 0.
   
   Definition P_id_a__U23 (x11:Z) := 0.
   
   Definition P_id_U94 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_U12 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_a__U81 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_U45 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_zeros  := 2.
   
   Definition P_id_U71 (x11:Z) := 0.
   
   Definition P_id_a__U91 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_a__U42 (x11:Z) (x12:Z) (x13:Z) := 2* x11.
   
   Definition P_id_isNatKind (x11:Z) := 1.
   
   Definition P_id_a__U22 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_U92 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_nil  := 0.
   
   Definition P_id_a__U61 (x11:Z) := 1.
   
   Definition P_id_U43 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_a__isNatIListKind (x11:Z) := 0.
   
   Definition P_id_U84 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_s (x11:Z) := 0.
   
   Definition P_id_a__U46 (x11:Z) := 0.
   
   Definition P_id_U32 (x11:Z) (x12:Z) := 1* x12.
   
   Definition P_id_a__U31 (x11:Z) (x12:Z) := 2* x12.
   
   Definition P_id_U13 (x11:Z) := 0.
   
   Definition P_id_a__U83 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_isNatIList (x11:Z) := 2* x11.
   
   Lemma P_id_U51_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_U51 x12 x14 <= P_id_U51 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U84_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U84 x12 x14 x16 <= P_id_a__U84 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U32_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_a__U32 x12 x14 <= P_id_a__U32 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_isNatList_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_isNatList x12 <= P_id_isNatList x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U13_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_a__U13 x12 <= P_id_a__U13 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U85_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_U85 x12 x14 <= P_id_U85 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__length_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_a__length x12 <= P_id_a__length x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__isNatIList_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->
      P_id_a__isNatIList x12 <= P_id_a__isNatIList x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U33_monotonic :
    forall x12 x11, (0 <= x12)/\ (x12 <= x11) ->P_id_U33 x12 <= P_id_U33 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U11_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_a__U11 x12 x14 <= P_id_a__U11 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U81_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_U81 x12 x14 x16 <= P_id_U81 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U92_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U92 x12 x14 x16 <= P_id_a__U92 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U43_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U43 x12 x14 x16 <= P_id_a__U43 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U23_monotonic :
    forall x12 x11, (0 <= x12)/\ (x12 <= x11) ->P_id_U23 x12 <= P_id_U23 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__isNatKind_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->
      P_id_a__isNatKind x12 <= P_id_a__isNatKind x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U93_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_U93 x12 x14 x16 <= P_id_U93 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U11_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_U11 x12 x14 <= P_id_U11 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U71_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_a__U71 x12 <= P_id_a__U71 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U44_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_U44 x12 x14 x16 <= P_id_U44 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U61_monotonic :
    forall x12 x11, (0 <= x12)/\ (x12 <= x11) ->P_id_U61 x12 <= P_id_U61 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U86_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_a__U86 x12 <= P_id_a__U86 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U41_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U41 x12 x14 x16 <= P_id_a__U41 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U22_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_U22 x12 x14 <= P_id_U22 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U21_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_a__U21 x12 x14 <= P_id_a__U21 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U91_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_U91 x12 x14 x16 <= P_id_U91 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_length_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_length x12 <= P_id_length x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U52_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_a__U52 x12 <= P_id_a__U52 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U42_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_U42 x12 x14 x16 <= P_id_U42 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U12_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_a__U12 x12 x14 <= P_id_a__U12 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U83_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_U83 x12 x14 x16 <= P_id_U83 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U94_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_a__U94 x12 x14 <= P_id_a__U94 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U45_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_a__U45 x12 x14 <= P_id_a__U45 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U31_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_U31 x12 x14 <= P_id_U31 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__isNat_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_a__isNat x12 <= P_id_a__isNat x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_isNatIListKind_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->
      P_id_isNatIListKind x12 <= P_id_isNatIListKind x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U82_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U82 x12 x14 x16 <= P_id_a__U82 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U46_monotonic :
    forall x12 x11, (0 <= x12)/\ (x12 <= x11) ->P_id_U46 x12 <= P_id_U46 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_cons_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_cons x12 x14 <= P_id_cons x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U52_monotonic :
    forall x12 x11, (0 <= x12)/\ (x12 <= x11) ->P_id_U52 x12 <= P_id_U52 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U85_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_a__U85 x12 x14 <= P_id_a__U85 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U33_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_a__U33 x12 <= P_id_a__U33 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U21_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_U21 x12 x14 <= P_id_U21 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__isNatList_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->
      P_id_a__isNatList x12 <= P_id_a__isNatList x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U86_monotonic :
    forall x12 x11, (0 <= x12)/\ (x12 <= x11) ->P_id_U86 x12 <= P_id_U86 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_mark_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_mark x12 <= P_id_mark x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U51_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_a__U51 x12 x14 <= P_id_a__U51 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U41_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_U41 x12 x14 x16 <= P_id_U41 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U82_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_U82 x12 x14 x16 <= P_id_U82 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U93_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U93 x12 x14 x16 <= P_id_a__U93 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U44_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U44 x12 x14 x16 <= P_id_a__U44 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_isNat_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_isNat x12 <= P_id_isNat x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U23_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_a__U23 x12 <= P_id_a__U23 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U94_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_U94 x12 x14 <= P_id_U94 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U12_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_U12 x12 x14 <= P_id_U12 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U81_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U81 x12 x14 x16 <= P_id_a__U81 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U45_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_U45 x12 x14 <= P_id_U45 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U71_monotonic :
    forall x12 x11, (0 <= x12)/\ (x12 <= x11) ->P_id_U71 x12 <= P_id_U71 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U91_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U91 x12 x14 x16 <= P_id_a__U91 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U42_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U42 x12 x14 x16 <= P_id_a__U42 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_isNatKind_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_isNatKind x12 <= P_id_isNatKind x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U22_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_a__U22 x12 x14 <= P_id_a__U22 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U92_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_U92 x12 x14 x16 <= P_id_U92 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U61_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_a__U61 x12 <= P_id_a__U61 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U43_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_U43 x12 x14 x16 <= P_id_U43 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__isNatIListKind_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->
      P_id_a__isNatIListKind x12 <= P_id_a__isNatIListKind x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U84_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_U84 x12 x14 x16 <= P_id_U84 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_s_monotonic :
    forall x12 x11, (0 <= x12)/\ (x12 <= x11) ->P_id_s x12 <= P_id_s x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U46_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_a__U46 x12 <= P_id_a__U46 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U32_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_U32 x12 x14 <= P_id_U32 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U31_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_a__U31 x12 x14 <= P_id_a__U31 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U13_monotonic :
    forall x12 x11, (0 <= x12)/\ (x12 <= x11) ->P_id_U13 x12 <= P_id_U13 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U83_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U83 x12 x14 x16 <= P_id_a__U83 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_isNatIList_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_isNatIList x12 <= P_id_isNatIList x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__zeros_bounded : 0 <= P_id_a__zeros .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U51_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U51 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U84_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U84 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U32_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U32 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_isNatList_bounded :
    forall x11, (0 <= x11) ->0 <= P_id_isNatList x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U13_bounded : forall x11, (0 <= x11) ->0 <= P_id_a__U13 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U85_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U85 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__length_bounded :
    forall x11, (0 <= x11) ->0 <= P_id_a__length x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__isNatIList_bounded :
    forall x11, (0 <= x11) ->0 <= P_id_a__isNatIList x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U33_bounded : forall x11, (0 <= x11) ->0 <= P_id_U33 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U11_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U11 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U81_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U81 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U92_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U92 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U43_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U43 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U23_bounded : forall x11, (0 <= x11) ->0 <= P_id_U23 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__isNatKind_bounded :
    forall x11, (0 <= x11) ->0 <= P_id_a__isNatKind x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U93_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U93 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U11_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U11 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U71_bounded : forall x11, (0 <= x11) ->0 <= P_id_a__U71 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U44_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U44 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_0_bounded : 0 <= P_id_0 .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U61_bounded : forall x11, (0 <= x11) ->0 <= P_id_U61 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U86_bounded : forall x11, (0 <= x11) ->0 <= P_id_a__U86 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U41_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U41 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U22_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U22 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U21_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U21 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U91_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U91 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_length_bounded : forall x11, (0 <= x11) ->0 <= P_id_length x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U52_bounded : forall x11, (0 <= x11) ->0 <= P_id_a__U52 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U42_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U42 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U12_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U12 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U83_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U83 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U94_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U94 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U45_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U45 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U31_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U31 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__isNat_bounded :
    forall x11, (0 <= x11) ->0 <= P_id_a__isNat x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_isNatIListKind_bounded :
    forall x11, (0 <= x11) ->0 <= P_id_isNatIListKind x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U82_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U82 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U46_bounded : forall x11, (0 <= x11) ->0 <= P_id_U46 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_cons_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_cons x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U52_bounded : forall x11, (0 <= x11) ->0 <= P_id_U52 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U85_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U85 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U33_bounded : forall x11, (0 <= x11) ->0 <= P_id_a__U33 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U21_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U21 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__isNatList_bounded :
    forall x11, (0 <= x11) ->0 <= P_id_a__isNatList x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U86_bounded : forall x11, (0 <= x11) ->0 <= P_id_U86 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_mark_bounded : forall x11, (0 <= x11) ->0 <= P_id_mark x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U51_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U51 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U41_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U41 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_tt_bounded : 0 <= P_id_tt .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U82_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U82 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U93_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U93 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U44_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U44 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_isNat_bounded : forall x11, (0 <= x11) ->0 <= P_id_isNat x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U23_bounded : forall x11, (0 <= x11) ->0 <= P_id_a__U23 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U94_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U94 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U12_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U12 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U81_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U81 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U45_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U45 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_zeros_bounded : 0 <= P_id_zeros .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U71_bounded : forall x11, (0 <= x11) ->0 <= P_id_U71 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U91_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U91 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U42_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U42 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_isNatKind_bounded :
    forall x11, (0 <= x11) ->0 <= P_id_isNatKind x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U22_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U22 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U92_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U92 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_nil_bounded : 0 <= P_id_nil .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U61_bounded : forall x11, (0 <= x11) ->0 <= P_id_a__U61 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U43_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U43 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__isNatIListKind_bounded :
    forall x11, (0 <= x11) ->0 <= P_id_a__isNatIListKind x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U84_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U84 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_s_bounded : forall x11, (0 <= x11) ->0 <= P_id_s x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U46_bounded : forall x11, (0 <= x11) ->0 <= P_id_a__U46 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U32_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U32 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U31_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U31 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U13_bounded : forall x11, (0 <= x11) ->0 <= P_id_U13 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U83_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U83 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_isNatIList_bounded :
    forall x11, (0 <= x11) ->0 <= P_id_isNatIList x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Definition measure  := 
     InterpZ.measure 0 P_id_a__zeros P_id_U51 P_id_a__U84 P_id_a__U32 
      P_id_isNatList P_id_a__U13 P_id_U85 P_id_a__length P_id_a__isNatIList 
      P_id_U33 P_id_a__U11 P_id_U81 P_id_a__U92 P_id_a__U43 P_id_U23 
      P_id_a__isNatKind P_id_U93 P_id_U11 P_id_a__U71 P_id_U44 P_id_0 
      P_id_U61 P_id_a__U86 P_id_a__U41 P_id_U22 P_id_a__U21 P_id_U91 
      P_id_length P_id_a__U52 P_id_U42 P_id_a__U12 P_id_U83 P_id_a__U94 
      P_id_a__U45 P_id_U31 P_id_a__isNat P_id_isNatIListKind P_id_a__U82 
      P_id_U46 P_id_cons P_id_U52 P_id_a__U85 P_id_a__U33 P_id_U21 
      P_id_a__isNatList P_id_U86 P_id_mark P_id_a__U51 P_id_U41 P_id_tt 
      P_id_U82 P_id_a__U93 P_id_a__U44 P_id_isNat P_id_a__U23 P_id_U94 
      P_id_U12 P_id_a__U81 P_id_U45 P_id_zeros P_id_U71 P_id_a__U91 
      P_id_a__U42 P_id_isNatKind P_id_a__U22 P_id_U92 P_id_nil P_id_a__U61 
      P_id_U43 P_id_a__isNatIListKind P_id_U84 P_id_s P_id_a__U46 P_id_U32 
      P_id_a__U31 P_id_U13 P_id_a__U83 P_id_isNatIList.
   
   Lemma measure_equation :
    forall t, 
     measure t = match t with
                   | (algebra.Alg.Term algebra.F.id_a__zeros nil) =>
                    P_id_a__zeros 
                   | (algebra.Alg.Term algebra.F.id_U51 (x12::x11::nil)) =>
                    P_id_U51 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U84 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U84 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U32 (x12::x11::nil)) =>
                    P_id_a__U32 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_isNatList (x11::nil)) =>
                    P_id_isNatList (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U13 (x11::nil)) =>
                    P_id_a__U13 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U85 (x12::x11::nil)) =>
                    P_id_U85 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__length (x11::nil)) =>
                    P_id_a__length (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__isNatIList (x11::nil)) =>
                    P_id_a__isNatIList (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U33 (x11::nil)) =>
                    P_id_U33 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U11 (x12::x11::nil)) =>
                    P_id_a__U11 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U81 (x13::x12::x11::nil)) =>
                    P_id_U81 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U92 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U92 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U43 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U43 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U23 (x11::nil)) =>
                    P_id_U23 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__isNatKind (x11::nil)) =>
                    P_id_a__isNatKind (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U93 (x13::x12::x11::nil)) =>
                    P_id_U93 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U11 (x12::x11::nil)) =>
                    P_id_U11 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U71 (x11::nil)) =>
                    P_id_a__U71 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U44 (x13::x12::x11::nil)) =>
                    P_id_U44 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_0 nil) => P_id_0 
                   | (algebra.Alg.Term algebra.F.id_U61 (x11::nil)) =>
                    P_id_U61 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U86 (x11::nil)) =>
                    P_id_a__U86 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U41 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U41 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U22 (x12::x11::nil)) =>
                    P_id_U22 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U21 (x12::x11::nil)) =>
                    P_id_a__U21 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U91 (x13::x12::x11::nil)) =>
                    P_id_U91 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_length (x11::nil)) =>
                    P_id_length (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U52 (x11::nil)) =>
                    P_id_a__U52 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U42 (x13::x12::x11::nil)) =>
                    P_id_U42 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U12 (x12::x11::nil)) =>
                    P_id_a__U12 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U83 (x13::x12::x11::nil)) =>
                    P_id_U83 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U94 (x12::x11::nil)) =>
                    P_id_a__U94 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U45 (x12::x11::nil)) =>
                    P_id_a__U45 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U31 (x12::x11::nil)) =>
                    P_id_U31 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__isNat (x11::nil)) =>
                    P_id_a__isNat (measure x11)
                   | (algebra.Alg.Term algebra.F.id_isNatIListKind 
                      (x11::nil)) =>
                    P_id_isNatIListKind (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U82 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U82 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U46 (x11::nil)) =>
                    P_id_U46 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_cons (x12::x11::nil)) =>
                    P_id_cons (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U52 (x11::nil)) =>
                    P_id_U52 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U85 (x12::x11::nil)) =>
                    P_id_a__U85 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U33 (x11::nil)) =>
                    P_id_a__U33 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U21 (x12::x11::nil)) =>
                    P_id_U21 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__isNatList (x11::nil)) =>
                    P_id_a__isNatList (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U86 (x11::nil)) =>
                    P_id_U86 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_mark (x11::nil)) =>
                    P_id_mark (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U51 (x12::x11::nil)) =>
                    P_id_a__U51 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U41 (x13::x12::x11::nil)) =>
                    P_id_U41 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt 
                   | (algebra.Alg.Term algebra.F.id_U82 (x13::x12::x11::nil)) =>
                    P_id_U82 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U93 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U93 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U44 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U44 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_isNat (x11::nil)) =>
                    P_id_isNat (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U23 (x11::nil)) =>
                    P_id_a__U23 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U94 (x12::x11::nil)) =>
                    P_id_U94 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U12 (x12::x11::nil)) =>
                    P_id_U12 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U81 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U81 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U45 (x12::x11::nil)) =>
                    P_id_U45 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_zeros nil) => P_id_zeros 
                   | (algebra.Alg.Term algebra.F.id_U71 (x11::nil)) =>
                    P_id_U71 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U91 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U91 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U42 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U42 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_isNatKind (x11::nil)) =>
                    P_id_isNatKind (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U22 (x12::x11::nil)) =>
                    P_id_a__U22 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U92 (x13::x12::x11::nil)) =>
                    P_id_U92 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil 
                   | (algebra.Alg.Term algebra.F.id_a__U61 (x11::nil)) =>
                    P_id_a__U61 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U43 (x13::x12::x11::nil)) =>
                    P_id_U43 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
                      (x11::nil)) =>
                    P_id_a__isNatIListKind (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U84 (x13::x12::x11::nil)) =>
                    P_id_U84 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_s (x11::nil)) =>
                    P_id_s (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U46 (x11::nil)) =>
                    P_id_a__U46 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U32 (x12::x11::nil)) =>
                    P_id_U32 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U31 (x12::x11::nil)) =>
                    P_id_a__U31 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U13 (x11::nil)) =>
                    P_id_U13 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U83 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U83 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_isNatIList (x11::nil)) =>
                    P_id_isNatIList (measure x11)
                   | _ => 0
                   end.
   Proof.
     intros t;case t;intros ;apply refl_equal.
   Qed.
   
   Lemma measure_bounded : forall t, 0 <= measure t.
   Proof.
     unfold measure in |-*.
     
     apply InterpZ.measure_bounded;
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Ltac generate_pos_hyp  :=
    match goal with
      | H:context [measure ?x] |- _ =>
       let v := fresh "v" in 
        (let H := fresh "h" in 
          (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
            clearbody H;clearbody v))
      |  |- context [measure ?x] =>
       let v := fresh "v" in 
        (let H := fresh "h" in 
          (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
            clearbody H;clearbody v))
      end
    .
   
   Lemma rules_monotonic :
    forall l r, 
     (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) ->
      measure r <= measure l.
   Proof.
     intros l r H.
     fold measure in |-*.
     
     inversion H;clear H;subst;inversion H0;clear H0;subst;
      simpl algebra.EQT.T.apply_subst in |-*;
      repeat (
      match goal with
        |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
         rewrite (measure_equation (algebra.Alg.Term f t))
        end
      );repeat (generate_pos_hyp );
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma measure_star_monotonic :
    forall l r, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                r l) ->measure r <= measure l.
   Proof.
     unfold measure in *.
     apply InterpZ.measure_star_monotonic.
     intros ;apply P_id_U51_monotonic;assumption.
     intros ;apply P_id_a__U84_monotonic;assumption.
     intros ;apply P_id_a__U32_monotonic;assumption.
     intros ;apply P_id_isNatList_monotonic;assumption.
     intros ;apply P_id_a__U13_monotonic;assumption.
     intros ;apply P_id_U85_monotonic;assumption.
     intros ;apply P_id_a__length_monotonic;assumption.
     intros ;apply P_id_a__isNatIList_monotonic;assumption.
     intros ;apply P_id_U33_monotonic;assumption.
     intros ;apply P_id_a__U11_monotonic;assumption.
     intros ;apply P_id_U81_monotonic;assumption.
     intros ;apply P_id_a__U92_monotonic;assumption.
     intros ;apply P_id_a__U43_monotonic;assumption.
     intros ;apply P_id_U23_monotonic;assumption.
     intros ;apply P_id_a__isNatKind_monotonic;assumption.
     intros ;apply P_id_U93_monotonic;assumption.
     intros ;apply P_id_U11_monotonic;assumption.
     intros ;apply P_id_a__U71_monotonic;assumption.
     intros ;apply P_id_U44_monotonic;assumption.
     intros ;apply P_id_U61_monotonic;assumption.
     intros ;apply P_id_a__U86_monotonic;assumption.
     intros ;apply P_id_a__U41_monotonic;assumption.
     intros ;apply P_id_U22_monotonic;assumption.
     intros ;apply P_id_a__U21_monotonic;assumption.
     intros ;apply P_id_U91_monotonic;assumption.
     intros ;apply P_id_length_monotonic;assumption.
     intros ;apply P_id_a__U52_monotonic;assumption.
     intros ;apply P_id_U42_monotonic;assumption.
     intros ;apply P_id_a__U12_monotonic;assumption.
     intros ;apply P_id_U83_monotonic;assumption.
     intros ;apply P_id_a__U94_monotonic;assumption.
     intros ;apply P_id_a__U45_monotonic;assumption.
     intros ;apply P_id_U31_monotonic;assumption.
     intros ;apply P_id_a__isNat_monotonic;assumption.
     intros ;apply P_id_isNatIListKind_monotonic;assumption.
     intros ;apply P_id_a__U82_monotonic;assumption.
     intros ;apply P_id_U46_monotonic;assumption.
     intros ;apply P_id_cons_monotonic;assumption.
     intros ;apply P_id_U52_monotonic;assumption.
     intros ;apply P_id_a__U85_monotonic;assumption.
     intros ;apply P_id_a__U33_monotonic;assumption.
     intros ;apply P_id_U21_monotonic;assumption.
     intros ;apply P_id_a__isNatList_monotonic;assumption.
     intros ;apply P_id_U86_monotonic;assumption.
     intros ;apply P_id_mark_monotonic;assumption.
     intros ;apply P_id_a__U51_monotonic;assumption.
     intros ;apply P_id_U41_monotonic;assumption.
     intros ;apply P_id_U82_monotonic;assumption.
     intros ;apply P_id_a__U93_monotonic;assumption.
     intros ;apply P_id_a__U44_monotonic;assumption.
     intros ;apply P_id_isNat_monotonic;assumption.
     intros ;apply P_id_a__U23_monotonic;assumption.
     intros ;apply P_id_U94_monotonic;assumption.
     intros ;apply P_id_U12_monotonic;assumption.
     intros ;apply P_id_a__U81_monotonic;assumption.
     intros ;apply P_id_U45_monotonic;assumption.
     intros ;apply P_id_U71_monotonic;assumption.
     intros ;apply P_id_a__U91_monotonic;assumption.
     intros ;apply P_id_a__U42_monotonic;assumption.
     intros ;apply P_id_isNatKind_monotonic;assumption.
     intros ;apply P_id_a__U22_monotonic;assumption.
     intros ;apply P_id_U92_monotonic;assumption.
     intros ;apply P_id_a__U61_monotonic;assumption.
     intros ;apply P_id_U43_monotonic;assumption.
     intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
     intros ;apply P_id_U84_monotonic;assumption.
     intros ;apply P_id_s_monotonic;assumption.
     intros ;apply P_id_a__U46_monotonic;assumption.
     intros ;apply P_id_U32_monotonic;assumption.
     intros ;apply P_id_a__U31_monotonic;assumption.
     intros ;apply P_id_U13_monotonic;assumption.
     intros ;apply P_id_a__U83_monotonic;assumption.
     intros ;apply P_id_isNatIList_monotonic;assumption.
     intros ;apply P_id_a__zeros_bounded;assumption.
     intros ;apply P_id_U51_bounded;assumption.
     intros ;apply P_id_a__U84_bounded;assumption.
     intros ;apply P_id_a__U32_bounded;assumption.
     intros ;apply P_id_isNatList_bounded;assumption.
     intros ;apply P_id_a__U13_bounded;assumption.
     intros ;apply P_id_U85_bounded;assumption.
     intros ;apply P_id_a__length_bounded;assumption.
     intros ;apply P_id_a__isNatIList_bounded;assumption.
     intros ;apply P_id_U33_bounded;assumption.
     intros ;apply P_id_a__U11_bounded;assumption.
     intros ;apply P_id_U81_bounded;assumption.
     intros ;apply P_id_a__U92_bounded;assumption.
     intros ;apply P_id_a__U43_bounded;assumption.
     intros ;apply P_id_U23_bounded;assumption.
     intros ;apply P_id_a__isNatKind_bounded;assumption.
     intros ;apply P_id_U93_bounded;assumption.
     intros ;apply P_id_U11_bounded;assumption.
     intros ;apply P_id_a__U71_bounded;assumption.
     intros ;apply P_id_U44_bounded;assumption.
     intros ;apply P_id_0_bounded;assumption.
     intros ;apply P_id_U61_bounded;assumption.
     intros ;apply P_id_a__U86_bounded;assumption.
     intros ;apply P_id_a__U41_bounded;assumption.
     intros ;apply P_id_U22_bounded;assumption.
     intros ;apply P_id_a__U21_bounded;assumption.
     intros ;apply P_id_U91_bounded;assumption.
     intros ;apply P_id_length_bounded;assumption.
     intros ;apply P_id_a__U52_bounded;assumption.
     intros ;apply P_id_U42_bounded;assumption.
     intros ;apply P_id_a__U12_bounded;assumption.
     intros ;apply P_id_U83_bounded;assumption.
     intros ;apply P_id_a__U94_bounded;assumption.
     intros ;apply P_id_a__U45_bounded;assumption.
     intros ;apply P_id_U31_bounded;assumption.
     intros ;apply P_id_a__isNat_bounded;assumption.
     intros ;apply P_id_isNatIListKind_bounded;assumption.
     intros ;apply P_id_a__U82_bounded;assumption.
     intros ;apply P_id_U46_bounded;assumption.
     intros ;apply P_id_cons_bounded;assumption.
     intros ;apply P_id_U52_bounded;assumption.
     intros ;apply P_id_a__U85_bounded;assumption.
     intros ;apply P_id_a__U33_bounded;assumption.
     intros ;apply P_id_U21_bounded;assumption.
     intros ;apply P_id_a__isNatList_bounded;assumption.
     intros ;apply P_id_U86_bounded;assumption.
     intros ;apply P_id_mark_bounded;assumption.
     intros ;apply P_id_a__U51_bounded;assumption.
     intros ;apply P_id_U41_bounded;assumption.
     intros ;apply P_id_tt_bounded;assumption.
     intros ;apply P_id_U82_bounded;assumption.
     intros ;apply P_id_a__U93_bounded;assumption.
     intros ;apply P_id_a__U44_bounded;assumption.
     intros ;apply P_id_isNat_bounded;assumption.
     intros ;apply P_id_a__U23_bounded;assumption.
     intros ;apply P_id_U94_bounded;assumption.
     intros ;apply P_id_U12_bounded;assumption.
     intros ;apply P_id_a__U81_bounded;assumption.
     intros ;apply P_id_U45_bounded;assumption.
     intros ;apply P_id_zeros_bounded;assumption.
     intros ;apply P_id_U71_bounded;assumption.
     intros ;apply P_id_a__U91_bounded;assumption.
     intros ;apply P_id_a__U42_bounded;assumption.
     intros ;apply P_id_isNatKind_bounded;assumption.
     intros ;apply P_id_a__U22_bounded;assumption.
     intros ;apply P_id_U92_bounded;assumption.
     intros ;apply P_id_nil_bounded;assumption.
     intros ;apply P_id_a__U61_bounded;assumption.
     intros ;apply P_id_U43_bounded;assumption.
     intros ;apply P_id_a__isNatIListKind_bounded;assumption.
     intros ;apply P_id_U84_bounded;assumption.
     intros ;apply P_id_s_bounded;assumption.
     intros ;apply P_id_a__U46_bounded;assumption.
     intros ;apply P_id_U32_bounded;assumption.
     intros ;apply P_id_a__U31_bounded;assumption.
     intros ;apply P_id_U13_bounded;assumption.
     intros ;apply P_id_a__U83_bounded;assumption.
     intros ;apply P_id_isNatIList_bounded;assumption.
     apply rules_monotonic.
   Qed.
   
   Definition P_id_A__U46 (x11:Z) := 0.
   
   Definition P_id_A__ISNATILISTKIND (x11:Z) := 0.
   
   Definition P_id_A__U61 (x11:Z) := 0.
   
   Definition P_id_A__U85 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_A__U31 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_A__U81 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_A__U22 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_A__U93 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_A__U42 (x11:Z) (x12:Z) (x13:Z) := 1* x13.
   
   Definition P_id_A__U51 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_A__ISNATLIST (x11:Z) := 0.
   
   Definition P_id_A__ZEROS  := 0.
   
   Definition P_id_A__U91 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_A__U33 (x11:Z) := 0.
   
   Definition P_id_A__U83 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_A__U23 (x11:Z) := 0.
   
   Definition P_id_A__U11 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_A__LENGTH (x11:Z) := 0.
   
   Definition P_id_A__U44 (x11:Z) (x12:Z) (x13:Z) := 1* x13.
   
   Definition P_id_A__ISNATILIST (x11:Z) := 1* x11.
   
   Definition P_id_A__U13 (x11:Z) := 0.
   
   Definition P_id_A__U71 (x11:Z) := 0.
   
   Definition P_id_A__U86 (x11:Z) := 0.
   
   Definition P_id_A__U32 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_A__U82 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_A__ISNATKIND (x11:Z) := 0.
   
   Definition P_id_A__U94 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_A__U43 (x11:Z) (x12:Z) (x13:Z) := 1* x13.
   
   Definition P_id_A__U52 (x11:Z) := 0.
   
   Definition P_id_A__U21 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_A__U92 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_A__U41 (x11:Z) (x12:Z) (x13:Z) := 1 + 1* x13.
   
   Definition P_id_A__U84 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_A__ISNAT (x11:Z) := 0.
   
   Definition P_id_A__U12 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_MARK (x11:Z) := 0.
   
   Definition P_id_A__U45 (x11:Z) (x12:Z) := 1* x12.
   
   Lemma P_id_A__U46_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_A__U46 x12 <= P_id_A__U46 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__ISNATILISTKIND_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->
      P_id_A__ISNATILISTKIND x12 <= P_id_A__ISNATILISTKIND x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U61_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_A__U61 x12 <= P_id_A__U61 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U85_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_A__U85 x12 x14 <= P_id_A__U85 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U31_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_A__U31 x12 x14 <= P_id_A__U31 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U81_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U81 x12 x14 x16 <= P_id_A__U81 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U22_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_A__U22 x12 x14 <= P_id_A__U22 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U93_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U93 x12 x14 x16 <= P_id_A__U93 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U42_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U42 x12 x14 x16 <= P_id_A__U42 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U51_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_A__U51 x12 x14 <= P_id_A__U51 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__ISNATLIST_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->
      P_id_A__ISNATLIST x12 <= P_id_A__ISNATLIST x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U91_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U91 x12 x14 x16 <= P_id_A__U91 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U33_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_A__U33 x12 <= P_id_A__U33 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U83_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U83 x12 x14 x16 <= P_id_A__U83 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U23_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_A__U23 x12 <= P_id_A__U23 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U11_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_A__U11 x12 x14 <= P_id_A__U11 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__LENGTH_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_A__LENGTH x12 <= P_id_A__LENGTH x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U44_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U44 x12 x14 x16 <= P_id_A__U44 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__ISNATILIST_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->
      P_id_A__ISNATILIST x12 <= P_id_A__ISNATILIST x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U13_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_A__U13 x12 <= P_id_A__U13 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U71_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_A__U71 x12 <= P_id_A__U71 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U86_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_A__U86 x12 <= P_id_A__U86 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U32_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_A__U32 x12 x14 <= P_id_A__U32 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U82_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U82 x12 x14 x16 <= P_id_A__U82 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__ISNATKIND_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->
      P_id_A__ISNATKIND x12 <= P_id_A__ISNATKIND x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U94_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_A__U94 x12 x14 <= P_id_A__U94 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U43_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U43 x12 x14 x16 <= P_id_A__U43 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U52_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_A__U52 x12 <= P_id_A__U52 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U21_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_A__U21 x12 x14 <= P_id_A__U21 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U92_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U92 x12 x14 x16 <= P_id_A__U92 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U41_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U41 x12 x14 x16 <= P_id_A__U41 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U84_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U84 x12 x14 x16 <= P_id_A__U84 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__ISNAT_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_A__ISNAT x12 <= P_id_A__ISNAT x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U12_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_A__U12 x12 x14 <= P_id_A__U12 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_MARK_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_MARK x12 <= P_id_MARK x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U45_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_A__U45 x12 x14 <= P_id_A__U45 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Definition marked_measure  := 
     InterpZ.marked_measure 0 P_id_a__zeros P_id_U51 P_id_a__U84 P_id_a__U32 
      P_id_isNatList P_id_a__U13 P_id_U85 P_id_a__length P_id_a__isNatIList 
      P_id_U33 P_id_a__U11 P_id_U81 P_id_a__U92 P_id_a__U43 P_id_U23 
      P_id_a__isNatKind P_id_U93 P_id_U11 P_id_a__U71 P_id_U44 P_id_0 
      P_id_U61 P_id_a__U86 P_id_a__U41 P_id_U22 P_id_a__U21 P_id_U91 
      P_id_length P_id_a__U52 P_id_U42 P_id_a__U12 P_id_U83 P_id_a__U94 
      P_id_a__U45 P_id_U31 P_id_a__isNat P_id_isNatIListKind P_id_a__U82 
      P_id_U46 P_id_cons P_id_U52 P_id_a__U85 P_id_a__U33 P_id_U21 
      P_id_a__isNatList P_id_U86 P_id_mark P_id_a__U51 P_id_U41 P_id_tt 
      P_id_U82 P_id_a__U93 P_id_a__U44 P_id_isNat P_id_a__U23 P_id_U94 
      P_id_U12 P_id_a__U81 P_id_U45 P_id_zeros P_id_U71 P_id_a__U91 
      P_id_a__U42 P_id_isNatKind P_id_a__U22 P_id_U92 P_id_nil P_id_a__U61 
      P_id_U43 P_id_a__isNatIListKind P_id_U84 P_id_s P_id_a__U46 P_id_U32 
      P_id_a__U31 P_id_U13 P_id_a__U83 P_id_isNatIList P_id_A__U46 
      P_id_A__ISNATILISTKIND P_id_A__U61 P_id_A__U85 P_id_A__U31 P_id_A__U81 
      P_id_A__U22 P_id_A__U93 P_id_A__U42 P_id_A__U51 P_id_A__ISNATLIST 
      P_id_A__ZEROS P_id_A__U91 P_id_A__U33 P_id_A__U83 P_id_A__U23 
      P_id_A__U11 P_id_A__LENGTH P_id_A__U44 P_id_A__ISNATILIST P_id_A__U13 
      P_id_A__U71 P_id_A__U86 P_id_A__U32 P_id_A__U82 P_id_A__ISNATKIND 
      P_id_A__U94 P_id_A__U43 P_id_A__U52 P_id_A__U21 P_id_A__U92 
      P_id_A__U41 P_id_A__U84 P_id_A__ISNAT P_id_A__U12 P_id_MARK P_id_A__U45
     .
   
   Lemma marked_measure_equation :
    forall t, 
     marked_measure t = match t with
                          | (algebra.Alg.Term algebra.F.id_a__U46 (x11::nil)) =>
                           P_id_A__U46 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
                             (x11::nil)) =>
                           P_id_A__ISNATILISTKIND (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U61 (x11::nil)) =>
                           P_id_A__U61 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U85 (x12::
                             x11::nil)) =>
                           P_id_A__U85 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U31 (x12::
                             x11::nil)) =>
                           P_id_A__U31 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U81 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U81 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U22 (x12::
                             x11::nil)) =>
                           P_id_A__U22 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U93 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U93 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U42 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U42 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U51 (x12::
                             x11::nil)) =>
                           P_id_A__U51 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__isNatList 
                             (x11::nil)) =>
                           P_id_A__ISNATLIST (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__zeros nil) =>
                           P_id_A__ZEROS 
                          | (algebra.Alg.Term algebra.F.id_a__U91 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U91 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U33 (x11::nil)) =>
                           P_id_A__U33 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U83 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U83 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U23 (x11::nil)) =>
                           P_id_A__U23 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U11 (x12::
                             x11::nil)) =>
                           P_id_A__U11 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__length 
                             (x11::nil)) =>
                           P_id_A__LENGTH (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U44 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U44 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__isNatIList 
                             (x11::nil)) =>
                           P_id_A__ISNATILIST (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U13 (x11::nil)) =>
                           P_id_A__U13 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U71 (x11::nil)) =>
                           P_id_A__U71 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U86 (x11::nil)) =>
                           P_id_A__U86 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U32 (x12::
                             x11::nil)) =>
                           P_id_A__U32 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U82 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U82 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__isNatKind 
                             (x11::nil)) =>
                           P_id_A__ISNATKIND (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U94 (x12::
                             x11::nil)) =>
                           P_id_A__U94 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U43 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U43 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U52 (x11::nil)) =>
                           P_id_A__U52 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U21 (x12::
                             x11::nil)) =>
                           P_id_A__U21 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U92 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U92 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U41 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U41 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U84 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U84 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__isNat 
                             (x11::nil)) =>
                           P_id_A__ISNAT (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U12 (x12::
                             x11::nil)) =>
                           P_id_A__U12 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_mark (x11::nil)) =>
                           P_id_MARK (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U45 (x12::
                             x11::nil)) =>
                           P_id_A__U45 (measure x12) (measure x11)
                          | _ => measure t
                          end.
   Proof.
     reflexivity .
   Qed.
   
   Lemma marked_measure_star_monotonic :
    forall f l1 l2, 
     (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step 
                                                       R_xml_0_deep_rew.R_xml_0_rules)
                                                      ) l1 l2) ->
      marked_measure (algebra.Alg.Term f l1) <= marked_measure (algebra.Alg.Term 
                                                                 f l2).
   Proof.
     unfold marked_measure in *.
     apply InterpZ.marked_measure_star_monotonic.
     intros ;apply P_id_U51_monotonic;assumption.
     intros ;apply P_id_a__U84_monotonic;assumption.
     intros ;apply P_id_a__U32_monotonic;assumption.
     intros ;apply P_id_isNatList_monotonic;assumption.
     intros ;apply P_id_a__U13_monotonic;assumption.
     intros ;apply P_id_U85_monotonic;assumption.
     intros ;apply P_id_a__length_monotonic;assumption.
     intros ;apply P_id_a__isNatIList_monotonic;assumption.
     intros ;apply P_id_U33_monotonic;assumption.
     intros ;apply P_id_a__U11_monotonic;assumption.
     intros ;apply P_id_U81_monotonic;assumption.
     intros ;apply P_id_a__U92_monotonic;assumption.
     intros ;apply P_id_a__U43_monotonic;assumption.
     intros ;apply P_id_U23_monotonic;assumption.
     intros ;apply P_id_a__isNatKind_monotonic;assumption.
     intros ;apply P_id_U93_monotonic;assumption.
     intros ;apply P_id_U11_monotonic;assumption.
     intros ;apply P_id_a__U71_monotonic;assumption.
     intros ;apply P_id_U44_monotonic;assumption.
     intros ;apply P_id_U61_monotonic;assumption.
     intros ;apply P_id_a__U86_monotonic;assumption.
     intros ;apply P_id_a__U41_monotonic;assumption.
     intros ;apply P_id_U22_monotonic;assumption.
     intros ;apply P_id_a__U21_monotonic;assumption.
     intros ;apply P_id_U91_monotonic;assumption.
     intros ;apply P_id_length_monotonic;assumption.
     intros ;apply P_id_a__U52_monotonic;assumption.
     intros ;apply P_id_U42_monotonic;assumption.
     intros ;apply P_id_a__U12_monotonic;assumption.
     intros ;apply P_id_U83_monotonic;assumption.
     intros ;apply P_id_a__U94_monotonic;assumption.
     intros ;apply P_id_a__U45_monotonic;assumption.
     intros ;apply P_id_U31_monotonic;assumption.
     intros ;apply P_id_a__isNat_monotonic;assumption.
     intros ;apply P_id_isNatIListKind_monotonic;assumption.
     intros ;apply P_id_a__U82_monotonic;assumption.
     intros ;apply P_id_U46_monotonic;assumption.
     intros ;apply P_id_cons_monotonic;assumption.
     intros ;apply P_id_U52_monotonic;assumption.
     intros ;apply P_id_a__U85_monotonic;assumption.
     intros ;apply P_id_a__U33_monotonic;assumption.
     intros ;apply P_id_U21_monotonic;assumption.
     intros ;apply P_id_a__isNatList_monotonic;assumption.
     intros ;apply P_id_U86_monotonic;assumption.
     intros ;apply P_id_mark_monotonic;assumption.
     intros ;apply P_id_a__U51_monotonic;assumption.
     intros ;apply P_id_U41_monotonic;assumption.
     intros ;apply P_id_U82_monotonic;assumption.
     intros ;apply P_id_a__U93_monotonic;assumption.
     intros ;apply P_id_a__U44_monotonic;assumption.
     intros ;apply P_id_isNat_monotonic;assumption.
     intros ;apply P_id_a__U23_monotonic;assumption.
     intros ;apply P_id_U94_monotonic;assumption.
     intros ;apply P_id_U12_monotonic;assumption.
     intros ;apply P_id_a__U81_monotonic;assumption.
     intros ;apply P_id_U45_monotonic;assumption.
     intros ;apply P_id_U71_monotonic;assumption.
     intros ;apply P_id_a__U91_monotonic;assumption.
     intros ;apply P_id_a__U42_monotonic;assumption.
     intros ;apply P_id_isNatKind_monotonic;assumption.
     intros ;apply P_id_a__U22_monotonic;assumption.
     intros ;apply P_id_U92_monotonic;assumption.
     intros ;apply P_id_a__U61_monotonic;assumption.
     intros ;apply P_id_U43_monotonic;assumption.
     intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
     intros ;apply P_id_U84_monotonic;assumption.
     intros ;apply P_id_s_monotonic;assumption.
     intros ;apply P_id_a__U46_monotonic;assumption.
     intros ;apply P_id_U32_monotonic;assumption.
     intros ;apply P_id_a__U31_monotonic;assumption.
     intros ;apply P_id_U13_monotonic;assumption.
     intros ;apply P_id_a__U83_monotonic;assumption.
     intros ;apply P_id_isNatIList_monotonic;assumption.
     intros ;apply P_id_a__zeros_bounded;assumption.
     intros ;apply P_id_U51_bounded;assumption.
     intros ;apply P_id_a__U84_bounded;assumption.
     intros ;apply P_id_a__U32_bounded;assumption.
     intros ;apply P_id_isNatList_bounded;assumption.
     intros ;apply P_id_a__U13_bounded;assumption.
     intros ;apply P_id_U85_bounded;assumption.
     intros ;apply P_id_a__length_bounded;assumption.
     intros ;apply P_id_a__isNatIList_bounded;assumption.
     intros ;apply P_id_U33_bounded;assumption.
     intros ;apply P_id_a__U11_bounded;assumption.
     intros ;apply P_id_U81_bounded;assumption.
     intros ;apply P_id_a__U92_bounded;assumption.
     intros ;apply P_id_a__U43_bounded;assumption.
     intros ;apply P_id_U23_bounded;assumption.
     intros ;apply P_id_a__isNatKind_bounded;assumption.
     intros ;apply P_id_U93_bounded;assumption.
     intros ;apply P_id_U11_bounded;assumption.
     intros ;apply P_id_a__U71_bounded;assumption.
     intros ;apply P_id_U44_bounded;assumption.
     intros ;apply P_id_0_bounded;assumption.
     intros ;apply P_id_U61_bounded;assumption.
     intros ;apply P_id_a__U86_bounded;assumption.
     intros ;apply P_id_a__U41_bounded;assumption.
     intros ;apply P_id_U22_bounded;assumption.
     intros ;apply P_id_a__U21_bounded;assumption.
     intros ;apply P_id_U91_bounded;assumption.
     intros ;apply P_id_length_bounded;assumption.
     intros ;apply P_id_a__U52_bounded;assumption.
     intros ;apply P_id_U42_bounded;assumption.
     intros ;apply P_id_a__U12_bounded;assumption.
     intros ;apply P_id_U83_bounded;assumption.
     intros ;apply P_id_a__U94_bounded;assumption.
     intros ;apply P_id_a__U45_bounded;assumption.
     intros ;apply P_id_U31_bounded;assumption.
     intros ;apply P_id_a__isNat_bounded;assumption.
     intros ;apply P_id_isNatIListKind_bounded;assumption.
     intros ;apply P_id_a__U82_bounded;assumption.
     intros ;apply P_id_U46_bounded;assumption.
     intros ;apply P_id_cons_bounded;assumption.
     intros ;apply P_id_U52_bounded;assumption.
     intros ;apply P_id_a__U85_bounded;assumption.
     intros ;apply P_id_a__U33_bounded;assumption.
     intros ;apply P_id_U21_bounded;assumption.
     intros ;apply P_id_a__isNatList_bounded;assumption.
     intros ;apply P_id_U86_bounded;assumption.
     intros ;apply P_id_mark_bounded;assumption.
     intros ;apply P_id_a__U51_bounded;assumption.
     intros ;apply P_id_U41_bounded;assumption.
     intros ;apply P_id_tt_bounded;assumption.
     intros ;apply P_id_U82_bounded;assumption.
     intros ;apply P_id_a__U93_bounded;assumption.
     intros ;apply P_id_a__U44_bounded;assumption.
     intros ;apply P_id_isNat_bounded;assumption.
     intros ;apply P_id_a__U23_bounded;assumption.
     intros ;apply P_id_U94_bounded;assumption.
     intros ;apply P_id_U12_bounded;assumption.
     intros ;apply P_id_a__U81_bounded;assumption.
     intros ;apply P_id_U45_bounded;assumption.
     intros ;apply P_id_zeros_bounded;assumption.
     intros ;apply P_id_U71_bounded;assumption.
     intros ;apply P_id_a__U91_bounded;assumption.
     intros ;apply P_id_a__U42_bounded;assumption.
     intros ;apply P_id_isNatKind_bounded;assumption.
     intros ;apply P_id_a__U22_bounded;assumption.
     intros ;apply P_id_U92_bounded;assumption.
     intros ;apply P_id_nil_bounded;assumption.
     intros ;apply P_id_a__U61_bounded;assumption.
     intros ;apply P_id_U43_bounded;assumption.
     intros ;apply P_id_a__isNatIListKind_bounded;assumption.
     intros ;apply P_id_U84_bounded;assumption.
     intros ;apply P_id_s_bounded;assumption.
     intros ;apply P_id_a__U46_bounded;assumption.
     intros ;apply P_id_U32_bounded;assumption.
     intros ;apply P_id_a__U31_bounded;assumption.
     intros ;apply P_id_U13_bounded;assumption.
     intros ;apply P_id_a__U83_bounded;assumption.
     intros ;apply P_id_isNatIList_bounded;assumption.
     apply rules_monotonic.
     intros ;apply P_id_A__U46_monotonic;assumption.
     intros ;apply P_id_A__ISNATILISTKIND_monotonic;assumption.
     intros ;apply P_id_A__U61_monotonic;assumption.
     intros ;apply P_id_A__U85_monotonic;assumption.
     intros ;apply P_id_A__U31_monotonic;assumption.
     intros ;apply P_id_A__U81_monotonic;assumption.
     intros ;apply P_id_A__U22_monotonic;assumption.
     intros ;apply P_id_A__U93_monotonic;assumption.
     intros ;apply P_id_A__U42_monotonic;assumption.
     intros ;apply P_id_A__U51_monotonic;assumption.
     intros ;apply P_id_A__ISNATLIST_monotonic;assumption.
     intros ;apply P_id_A__U91_monotonic;assumption.
     intros ;apply P_id_A__U33_monotonic;assumption.
     intros ;apply P_id_A__U83_monotonic;assumption.
     intros ;apply P_id_A__U23_monotonic;assumption.
     intros ;apply P_id_A__U11_monotonic;assumption.
     intros ;apply P_id_A__LENGTH_monotonic;assumption.
     intros ;apply P_id_A__U44_monotonic;assumption.
     intros ;apply P_id_A__ISNATILIST_monotonic;assumption.
     intros ;apply P_id_A__U13_monotonic;assumption.
     intros ;apply P_id_A__U71_monotonic;assumption.
     intros ;apply P_id_A__U86_monotonic;assumption.
     intros ;apply P_id_A__U32_monotonic;assumption.
     intros ;apply P_id_A__U82_monotonic;assumption.
     intros ;apply P_id_A__ISNATKIND_monotonic;assumption.
     intros ;apply P_id_A__U94_monotonic;assumption.
     intros ;apply P_id_A__U43_monotonic;assumption.
     intros ;apply P_id_A__U52_monotonic;assumption.
     intros ;apply P_id_A__U21_monotonic;assumption.
     intros ;apply P_id_A__U92_monotonic;assumption.
     intros ;apply P_id_A__U41_monotonic;assumption.
     intros ;apply P_id_A__U84_monotonic;assumption.
     intros ;apply P_id_A__ISNAT_monotonic;assumption.
     intros ;apply P_id_A__U12_monotonic;assumption.
     intros ;apply P_id_MARK_monotonic;assumption.
     intros ;apply P_id_A__U45_monotonic;assumption.
   Qed.
   
   Ltac rewrite_and_unfold  :=
    do 2 (rewrite marked_measure_equation);
     repeat (
     match goal with
       |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
        rewrite (measure_equation (algebra.Alg.Term f t))
       | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ =>
        rewrite (measure_equation (algebra.Alg.Term f t)) in H|-
       end
     ).
   
   Definition lt a b := (Zwf.Zwf 0) (marked_measure a) (marked_measure b).
   
   Definition le a b := marked_measure a <= marked_measure b.
   
   Lemma lt_le_compat : forall a b c, (lt a b) ->(le b c) ->lt a c.
   Proof.
     unfold lt, le in *.
     intros a b c.
     apply (interp.le_lt_compat_right (interp.o_Z 0)).
   Qed.
   
   Lemma wf_lt : well_founded lt.
   Proof.
     unfold lt in *.
     apply Inverse_Image.wf_inverse_image with  (B:=Z).
     apply Zwf.Zwf_well_founded.
   Qed.
   
   Lemma DP_R_xml_0_scc_58_strict_in_lt :
    Relation_Definitions.inclusion _ DP_R_xml_0_scc_58_strict lt.
   Proof.
     unfold Relation_Definitions.inclusion, lt in *.
     
     intros a b H;destruct H;
      match goal with
        |  |- (Zwf.Zwf 0) _ (marked_measure (algebra.Alg.Term ?f ?l)) =>
         let l'' := algebra.Alg_ext.find_replacement l  in 
          ((apply (interp.le_lt_compat_right (interp.o_Z 0)) with
             (marked_measure (algebra.Alg.Term f l''));[idtac|
            apply marked_measure_star_monotonic;
             repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos);
             (assumption)||(constructor 1)]))
        end
      ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma DP_R_xml_0_scc_58_large_in_le :
    Relation_Definitions.inclusion _ DP_R_xml_0_scc_58_large le.
   Proof.
     unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *.
     
     intros a b H;destruct H;
      match goal with
        |  |- _ <= marked_measure (algebra.Alg.Term ?f ?l) =>
         let l'' := algebra.Alg_ext.find_replacement l  in 
          ((apply (interp.le_trans (interp.o_Z 0)) with
             (marked_measure (algebra.Alg.Term f l''));[idtac|
            apply marked_measure_star_monotonic;
             repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos);
             (assumption)||(constructor 1)]))
        end
      ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Definition wf_DP_R_xml_0_scc_58_large  := WF_DP_R_xml_0_scc_58_large.wf.
   
   
   Lemma wf : well_founded WF_DP_R_xml_0.DP_R_xml_0_scc_58.
   Proof.
     intros x.
     apply (well_founded_ind wf_lt).
     clear x.
     intros x.
     pattern x.
     apply (@Acc_ind _ DP_R_xml_0_scc_58_large).
     clear x.
     intros x _ IHx IHx'.
     constructor.
     intros y H.
     
     destruct H;
      (apply IHx';apply DP_R_xml_0_scc_58_strict_in_lt;
        econstructor eassumption)||
      ((apply IHx;[econstructor eassumption|
        intros y' Hlt;apply IHx';apply lt_le_compat with (1:=Hlt) ;
         apply DP_R_xml_0_scc_58_large_in_le;econstructor eassumption])).
     apply wf_DP_R_xml_0_scc_58_large.
   Qed.
  End WF_DP_R_xml_0_scc_58.
  
  Definition wf_DP_R_xml_0_scc_58  := WF_DP_R_xml_0_scc_58.wf.
  
  
  Lemma acc_DP_R_xml_0_scc_58 :
   forall x y, (DP_R_xml_0_scc_58 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x.
    pattern x.
    apply (@Acc_ind _ DP_R_xml_0_scc_58).
    intros x' _ Hrec y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply Hrec;econstructor eassumption)||
      ((eapply acc_DP_R_xml_0_non_scc_57;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_55;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((eapply acc_DP_R_xml_0_non_scc_31;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
         ((eapply acc_DP_R_xml_0_non_scc_30;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
          ((eapply acc_DP_R_xml_0_non_scc_29;
             econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
           ((eapply acc_DP_R_xml_0_non_scc_28;
              econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
            ((eapply acc_DP_R_xml_0_non_scc_22;
               econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
             ((eapply acc_DP_R_xml_0_non_scc_18;
                econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
              ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
               (eapply Hrec;
                 econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))))))))))).
    apply wf_DP_R_xml_0_scc_58.
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_59  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <mark(U43(X1_,X2_,X3_)),a__U43(mark(X1_),X2_,X3_)> *)
    | DP_R_xml_0_non_scc_59_0 :
     forall x6 x9 x11 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U43 (x6::x7::x9::nil)) x11) ->
       DP_R_xml_0_non_scc_59 (algebra.Alg.Term algebra.F.id_a__U43 
                              ((algebra.Alg.Term algebra.F.id_mark 
                              (x6::nil))::x7::x9::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_59 :
   forall x y, 
    (DP_R_xml_0_non_scc_59 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_58;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_29;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
        (eapply Hrec;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_60  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <mark(U42(X1_,X2_,X3_)),a__U42(mark(X1_),X2_,X3_)> *)
    | DP_R_xml_0_non_scc_60_0 :
     forall x6 x9 x11 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U42 (x6::x7::x9::nil)) x11) ->
       DP_R_xml_0_non_scc_60 (algebra.Alg.Term algebra.F.id_a__U42 
                              ((algebra.Alg.Term algebra.F.id_mark 
                              (x6::nil))::x7::x9::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_60 :
   forall x y, 
    (DP_R_xml_0_non_scc_60 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_58;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_30;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
        (eapply Hrec;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_61  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <mark(U41(X1_,X2_,X3_)),a__U41(mark(X1_),X2_,X3_)> *)
    | DP_R_xml_0_non_scc_61_0 :
     forall x6 x9 x11 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U41 (x6::x7::x9::nil)) x11) ->
       DP_R_xml_0_non_scc_61 (algebra.Alg.Term algebra.F.id_a__U41 
                              ((algebra.Alg.Term algebra.F.id_mark 
                              (x6::nil))::x7::x9::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_61 :
   forall x y, 
    (DP_R_xml_0_non_scc_61 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_58;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_31;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
        (eapply Hrec;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_62  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <mark(isNatIList(X_)),a__isNatIList(X_)> *)
    | DP_R_xml_0_non_scc_62_0 :
     forall x8 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_isNatIList (x8::nil)) x11) ->
       DP_R_xml_0_non_scc_62 (algebra.Alg.Term algebra.F.id_a__isNatIList 
                              (x8::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_62 :
   forall x y, 
    (DP_R_xml_0_non_scc_62 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_58;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_55;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_22;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((eapply acc_DP_R_xml_0_non_scc_18;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
         ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
          (eapply Hrec;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))))).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_63  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <mark(U45(X1_,X2_)),a__U45(mark(X1_),X2_)> *)
    | DP_R_xml_0_non_scc_63_0 :
     forall x6 x11 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U45 (x6::x7::nil)) x11) ->
       DP_R_xml_0_non_scc_63 (algebra.Alg.Term algebra.F.id_a__U45 
                              ((algebra.Alg.Term algebra.F.id_mark 
                              (x6::nil))::x7::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_63 :
   forall x y, 
    (DP_R_xml_0_non_scc_63 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_58;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_28;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
        (eapply Hrec;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_64  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <mark(U44(X1_,X2_,X3_)),a__U44(mark(X1_),X2_,X3_)> *)
    | DP_R_xml_0_non_scc_64_0 :
     forall x6 x9 x11 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U44 (x6::x7::x9::nil)) x11) ->
       DP_R_xml_0_non_scc_64 (algebra.Alg.Term algebra.F.id_a__U44 
                              ((algebra.Alg.Term algebra.F.id_mark 
                              (x6::nil))::x7::x9::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_64 :
   forall x y, 
    (DP_R_xml_0_non_scc_64 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_58;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_57;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
        (eapply Hrec;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))).
  Qed.
  
  
  Inductive DP_R_xml_0_scc_65  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <mark(U11(X1_,X2_)),mark(X1_)> *)
    | DP_R_xml_0_scc_65_0 :
     forall x6 x11 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U11 (x6::x7::nil)) x11) ->
       DP_R_xml_0_scc_65 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <mark(U12(X1_,X2_)),mark(X1_)> *)
    | DP_R_xml_0_scc_65_1 :
     forall x6 x11 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U12 (x6::x7::nil)) x11) ->
       DP_R_xml_0_scc_65 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <mark(U13(X_)),mark(X_)> *)
    | DP_R_xml_0_scc_65_2 :
     forall x8 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U13 (x8::nil)) x11) ->
       DP_R_xml_0_scc_65 (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <mark(U21(X1_,X2_)),mark(X1_)> *)
    | DP_R_xml_0_scc_65_3 :
     forall x6 x11 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U21 (x6::x7::nil)) x11) ->
       DP_R_xml_0_scc_65 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <mark(U22(X1_,X2_)),mark(X1_)> *)
    | DP_R_xml_0_scc_65_4 :
     forall x6 x11 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U22 (x6::x7::nil)) x11) ->
       DP_R_xml_0_scc_65 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <mark(U23(X_)),mark(X_)> *)
    | DP_R_xml_0_scc_65_5 :
     forall x8 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U23 (x8::nil)) x11) ->
       DP_R_xml_0_scc_65 (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <mark(U31(X1_,X2_)),mark(X1_)> *)
    | DP_R_xml_0_scc_65_6 :
     forall x6 x11 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U31 (x6::x7::nil)) x11) ->
       DP_R_xml_0_scc_65 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <mark(U32(X1_,X2_)),mark(X1_)> *)
    | DP_R_xml_0_scc_65_7 :
     forall x6 x11 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U32 (x6::x7::nil)) x11) ->
       DP_R_xml_0_scc_65 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <mark(U33(X_)),mark(X_)> *)
    | DP_R_xml_0_scc_65_8 :
     forall x8 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U33 (x8::nil)) x11) ->
       DP_R_xml_0_scc_65 (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <mark(U41(X1_,X2_,X3_)),mark(X1_)> *)
    | DP_R_xml_0_scc_65_9 :
     forall x6 x9 x11 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U41 (x6::x7::x9::nil)) x11) ->
       DP_R_xml_0_scc_65 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <mark(U42(X1_,X2_,X3_)),mark(X1_)> *)
    | DP_R_xml_0_scc_65_10 :
     forall x6 x9 x11 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U42 (x6::x7::x9::nil)) x11) ->
       DP_R_xml_0_scc_65 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <mark(U43(X1_,X2_,X3_)),mark(X1_)> *)
    | DP_R_xml_0_scc_65_11 :
     forall x6 x9 x11 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U43 (x6::x7::x9::nil)) x11) ->
       DP_R_xml_0_scc_65 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <mark(U44(X1_,X2_,X3_)),mark(X1_)> *)
    | DP_R_xml_0_scc_65_12 :
     forall x6 x9 x11 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U44 (x6::x7::x9::nil)) x11) ->
       DP_R_xml_0_scc_65 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <mark(U45(X1_,X2_)),mark(X1_)> *)
    | DP_R_xml_0_scc_65_13 :
     forall x6 x11 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U45 (x6::x7::nil)) x11) ->
       DP_R_xml_0_scc_65 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <mark(U46(X_)),mark(X_)> *)
    | DP_R_xml_0_scc_65_14 :
     forall x8 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U46 (x8::nil)) x11) ->
       DP_R_xml_0_scc_65 (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <mark(U51(X1_,X2_)),mark(X1_)> *)
    | DP_R_xml_0_scc_65_15 :
     forall x6 x11 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U51 (x6::x7::nil)) x11) ->
       DP_R_xml_0_scc_65 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <mark(U52(X_)),mark(X_)> *)
    | DP_R_xml_0_scc_65_16 :
     forall x8 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U52 (x8::nil)) x11) ->
       DP_R_xml_0_scc_65 (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <mark(U61(X_)),mark(X_)> *)
    | DP_R_xml_0_scc_65_17 :
     forall x8 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U61 (x8::nil)) x11) ->
       DP_R_xml_0_scc_65 (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <mark(U71(X_)),mark(X_)> *)
    | DP_R_xml_0_scc_65_18 :
     forall x8 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U71 (x8::nil)) x11) ->
       DP_R_xml_0_scc_65 (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <mark(U81(X1_,X2_,X3_)),mark(X1_)> *)
    | DP_R_xml_0_scc_65_19 :
     forall x6 x9 x11 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U81 (x6::x7::x9::nil)) x11) ->
       DP_R_xml_0_scc_65 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <mark(U82(X1_,X2_,X3_)),mark(X1_)> *)
    | DP_R_xml_0_scc_65_20 :
     forall x6 x9 x11 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U82 (x6::x7::x9::nil)) x11) ->
       DP_R_xml_0_scc_65 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <mark(U83(X1_,X2_,X3_)),mark(X1_)> *)
    | DP_R_xml_0_scc_65_21 :
     forall x6 x9 x11 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U83 (x6::x7::x9::nil)) x11) ->
       DP_R_xml_0_scc_65 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <mark(U84(X1_,X2_,X3_)),mark(X1_)> *)
    | DP_R_xml_0_scc_65_22 :
     forall x6 x9 x11 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U84 (x6::x7::x9::nil)) x11) ->
       DP_R_xml_0_scc_65 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <mark(U85(X1_,X2_)),mark(X1_)> *)
    | DP_R_xml_0_scc_65_23 :
     forall x6 x11 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U85 (x6::x7::nil)) x11) ->
       DP_R_xml_0_scc_65 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <mark(U86(X_)),mark(X_)> *)
    | DP_R_xml_0_scc_65_24 :
     forall x8 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U86 (x8::nil)) x11) ->
       DP_R_xml_0_scc_65 (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    
     (* <mark(U91(X1_,X2_,X3_)),a__U91(mark(X1_),X2_,X3_)> *)
    | DP_R_xml_0_scc_65_25 :
     forall x6 x9 x11 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U91 (x6::x7::x9::nil)) x11) ->
       DP_R_xml_0_scc_65 (algebra.Alg.Term algebra.F.id_a__U91 
                          ((algebra.Alg.Term algebra.F.id_mark (x6::nil))::
                          x7::x9::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    
     (* <a__U91(tt,L_,N_),a__U92(a__isNatIListKind(L_),L_,N_)> *)
    | DP_R_xml_0_scc_65_26 :
     forall x4 x12 x5 x13 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_tt nil) 
        x13) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x4 x12) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x5 x11) ->
         DP_R_xml_0_scc_65 (algebra.Alg.Term algebra.F.id_a__U92 
                            ((algebra.Alg.Term 
                            algebra.F.id_a__isNatIListKind (x4::nil))::x4::
                            x5::nil)) 
          (algebra.Alg.Term algebra.F.id_a__U91 (x13::x12::x11::nil))
    
     (* <a__U92(tt,L_,N_),a__U93(a__isNat(N_),L_,N_)> *)
    | DP_R_xml_0_scc_65_27 :
     forall x4 x12 x5 x13 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_tt nil) 
        x13) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x4 x12) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x5 x11) ->
         DP_R_xml_0_scc_65 (algebra.Alg.Term algebra.F.id_a__U93 
                            ((algebra.Alg.Term algebra.F.id_a__isNat 
                            (x5::nil))::x4::x5::nil)) 
          (algebra.Alg.Term algebra.F.id_a__U92 (x13::x12::x11::nil))
    
     (* <a__U93(tt,L_,N_),a__U94(a__isNatKind(N_),L_)> *)
    | DP_R_xml_0_scc_65_28 :
     forall x4 x12 x5 x13 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_tt nil) 
        x13) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x4 x12) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x5 x11) ->
         DP_R_xml_0_scc_65 (algebra.Alg.Term algebra.F.id_a__U94 
                            ((algebra.Alg.Term algebra.F.id_a__isNatKind 
                            (x5::nil))::x4::nil)) 
          (algebra.Alg.Term algebra.F.id_a__U93 (x13::x12::x11::nil))
     (* <a__U94(tt,L_),a__length(mark(L_))> *)
    | DP_R_xml_0_scc_65_29 :
     forall x4 x12 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_tt nil) 
        x12) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x4 x11) ->
        DP_R_xml_0_scc_65 (algebra.Alg.Term algebra.F.id_a__length 
                           ((algebra.Alg.Term algebra.F.id_mark 
                           (x4::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U94 (x12::x11::nil))
    
     (* <a__length(cons(N_,L_)),a__U91(a__isNatList(L_),L_,N_)> *)
    | DP_R_xml_0_scc_65_30 :
     forall x4 x5 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_cons (x5::x4::nil)) x11) ->
       DP_R_xml_0_scc_65 (algebra.Alg.Term algebra.F.id_a__U91 
                          ((algebra.Alg.Term algebra.F.id_a__isNatList 
                          (x4::nil))::x4::x5::nil)) 
        (algebra.Alg.Term algebra.F.id_a__length (x11::nil))
     (* <a__U94(tt,L_),mark(L_)> *)
    | DP_R_xml_0_scc_65_31 :
     forall x4 x12 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_tt nil) 
        x12) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x4 x11) ->
        DP_R_xml_0_scc_65 (algebra.Alg.Term algebra.F.id_mark (x4::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U94 (x12::x11::nil))
     (* <mark(U91(X1_,X2_,X3_)),mark(X1_)> *)
    | DP_R_xml_0_scc_65_32 :
     forall x6 x9 x11 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U91 (x6::x7::x9::nil)) x11) ->
       DP_R_xml_0_scc_65 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    
     (* <mark(U92(X1_,X2_,X3_)),a__U92(mark(X1_),X2_,X3_)> *)
    | DP_R_xml_0_scc_65_33 :
     forall x6 x9 x11 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U92 (x6::x7::x9::nil)) x11) ->
       DP_R_xml_0_scc_65 (algebra.Alg.Term algebra.F.id_a__U92 
                          ((algebra.Alg.Term algebra.F.id_mark (x6::nil))::
                          x7::x9::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <mark(U92(X1_,X2_,X3_)),mark(X1_)> *)
    | DP_R_xml_0_scc_65_34 :
     forall x6 x9 x11 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U92 (x6::x7::x9::nil)) x11) ->
       DP_R_xml_0_scc_65 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    
     (* <mark(U93(X1_,X2_,X3_)),a__U93(mark(X1_),X2_,X3_)> *)
    | DP_R_xml_0_scc_65_35 :
     forall x6 x9 x11 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U93 (x6::x7::x9::nil)) x11) ->
       DP_R_xml_0_scc_65 (algebra.Alg.Term algebra.F.id_a__U93 
                          ((algebra.Alg.Term algebra.F.id_mark (x6::nil))::
                          x7::x9::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <mark(U93(X1_,X2_,X3_)),mark(X1_)> *)
    | DP_R_xml_0_scc_65_36 :
     forall x6 x9 x11 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U93 (x6::x7::x9::nil)) x11) ->
       DP_R_xml_0_scc_65 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <mark(U94(X1_,X2_)),a__U94(mark(X1_),X2_)> *)
    | DP_R_xml_0_scc_65_37 :
     forall x6 x11 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U94 (x6::x7::nil)) x11) ->
       DP_R_xml_0_scc_65 (algebra.Alg.Term algebra.F.id_a__U94 
                          ((algebra.Alg.Term algebra.F.id_mark (x6::nil))::
                          x7::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <mark(U94(X1_,X2_)),mark(X1_)> *)
    | DP_R_xml_0_scc_65_38 :
     forall x6 x11 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_U94 (x6::x7::nil)) x11) ->
       DP_R_xml_0_scc_65 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <mark(length(X_)),a__length(mark(X_))> *)
    | DP_R_xml_0_scc_65_39 :
     forall x8 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_length (x8::nil)) x11) ->
       DP_R_xml_0_scc_65 (algebra.Alg.Term algebra.F.id_a__length 
                          ((algebra.Alg.Term algebra.F.id_mark 
                          (x8::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <mark(length(X_)),mark(X_)> *)
    | DP_R_xml_0_scc_65_40 :
     forall x8 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_length (x8::nil)) x11) ->
       DP_R_xml_0_scc_65 (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <mark(cons(X1_,X2_)),mark(X1_)> *)
    | DP_R_xml_0_scc_65_41 :
     forall x6 x11 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_cons (x6::x7::nil)) x11) ->
       DP_R_xml_0_scc_65 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <mark(s(X_)),mark(X_)> *)
    | DP_R_xml_0_scc_65_42 :
     forall x8 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_s (x8::nil)) 
        x11) ->
       DP_R_xml_0_scc_65 (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
  .
  
  
  Module WF_DP_R_xml_0_scc_65.
   Inductive DP_R_xml_0_scc_65_large  :
    algebra.Alg.term ->algebra.Alg.term ->Prop := 
      (* <mark(U11(X1_,X2_)),mark(X1_)> *)
     | DP_R_xml_0_scc_65_large_0 :
      forall x6 x11 x7, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_U11 (x6::x7::nil)) x11) ->
        DP_R_xml_0_scc_65_large (algebra.Alg.Term algebra.F.id_mark 
                                 (x6::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      (* <mark(U12(X1_,X2_)),mark(X1_)> *)
     | DP_R_xml_0_scc_65_large_1 :
      forall x6 x11 x7, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_U12 (x6::x7::nil)) x11) ->
        DP_R_xml_0_scc_65_large (algebra.Alg.Term algebra.F.id_mark 
                                 (x6::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      (* <mark(U13(X_)),mark(X_)> *)
     | DP_R_xml_0_scc_65_large_2 :
      forall x8 x11, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_U13 (x8::nil)) x11) ->
        DP_R_xml_0_scc_65_large (algebra.Alg.Term algebra.F.id_mark 
                                 (x8::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      (* <mark(U21(X1_,X2_)),mark(X1_)> *)
     | DP_R_xml_0_scc_65_large_3 :
      forall x6 x11 x7, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_U21 (x6::x7::nil)) x11) ->
        DP_R_xml_0_scc_65_large (algebra.Alg.Term algebra.F.id_mark 
                                 (x6::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      (* <mark(U22(X1_,X2_)),mark(X1_)> *)
     | DP_R_xml_0_scc_65_large_4 :
      forall x6 x11 x7, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_U22 (x6::x7::nil)) x11) ->
        DP_R_xml_0_scc_65_large (algebra.Alg.Term algebra.F.id_mark 
                                 (x6::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      (* <mark(U23(X_)),mark(X_)> *)
     | DP_R_xml_0_scc_65_large_5 :
      forall x8 x11, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_U23 (x8::nil)) x11) ->
        DP_R_xml_0_scc_65_large (algebra.Alg.Term algebra.F.id_mark 
                                 (x8::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      (* <mark(U31(X1_,X2_)),mark(X1_)> *)
     | DP_R_xml_0_scc_65_large_6 :
      forall x6 x11 x7, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_U31 (x6::x7::nil)) x11) ->
        DP_R_xml_0_scc_65_large (algebra.Alg.Term algebra.F.id_mark 
                                 (x6::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      (* <mark(U32(X1_,X2_)),mark(X1_)> *)
     | DP_R_xml_0_scc_65_large_7 :
      forall x6 x11 x7, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_U32 (x6::x7::nil)) x11) ->
        DP_R_xml_0_scc_65_large (algebra.Alg.Term algebra.F.id_mark 
                                 (x6::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      (* <mark(U33(X_)),mark(X_)> *)
     | DP_R_xml_0_scc_65_large_8 :
      forall x8 x11, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_U33 (x8::nil)) x11) ->
        DP_R_xml_0_scc_65_large (algebra.Alg.Term algebra.F.id_mark 
                                 (x8::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      (* <mark(U41(X1_,X2_,X3_)),mark(X1_)> *)
     | DP_R_xml_0_scc_65_large_9 :
      forall x6 x9 x11 x7, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_U41 (x6::x7::x9::nil)) x11) ->
        DP_R_xml_0_scc_65_large (algebra.Alg.Term algebra.F.id_mark 
                                 (x6::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      (* <mark(U42(X1_,X2_,X3_)),mark(X1_)> *)
     | DP_R_xml_0_scc_65_large_10 :
      forall x6 x9 x11 x7, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_U42 (x6::x7::x9::nil)) x11) ->
        DP_R_xml_0_scc_65_large (algebra.Alg.Term algebra.F.id_mark 
                                 (x6::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      (* <mark(U43(X1_,X2_,X3_)),mark(X1_)> *)
     | DP_R_xml_0_scc_65_large_11 :
      forall x6 x9 x11 x7, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_U43 (x6::x7::x9::nil)) x11) ->
        DP_R_xml_0_scc_65_large (algebra.Alg.Term algebra.F.id_mark 
                                 (x6::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      (* <mark(U44(X1_,X2_,X3_)),mark(X1_)> *)
     | DP_R_xml_0_scc_65_large_12 :
      forall x6 x9 x11 x7, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_U44 (x6::x7::x9::nil)) x11) ->
        DP_R_xml_0_scc_65_large (algebra.Alg.Term algebra.F.id_mark 
                                 (x6::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      (* <mark(U45(X1_,X2_)),mark(X1_)> *)
     | DP_R_xml_0_scc_65_large_13 :
      forall x6 x11 x7, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_U45 (x6::x7::nil)) x11) ->
        DP_R_xml_0_scc_65_large (algebra.Alg.Term algebra.F.id_mark 
                                 (x6::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      (* <mark(U46(X_)),mark(X_)> *)
     | DP_R_xml_0_scc_65_large_14 :
      forall x8 x11, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_U46 (x8::nil)) x11) ->
        DP_R_xml_0_scc_65_large (algebra.Alg.Term algebra.F.id_mark 
                                 (x8::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      (* <mark(U51(X1_,X2_)),mark(X1_)> *)
     | DP_R_xml_0_scc_65_large_15 :
      forall x6 x11 x7, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_U51 (x6::x7::nil)) x11) ->
        DP_R_xml_0_scc_65_large (algebra.Alg.Term algebra.F.id_mark 
                                 (x6::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      (* <mark(U52(X_)),mark(X_)> *)
     | DP_R_xml_0_scc_65_large_16 :
      forall x8 x11, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_U52 (x8::nil)) x11) ->
        DP_R_xml_0_scc_65_large (algebra.Alg.Term algebra.F.id_mark 
                                 (x8::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      (* <mark(U61(X_)),mark(X_)> *)
     | DP_R_xml_0_scc_65_large_17 :
      forall x8 x11, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_U61 (x8::nil)) x11) ->
        DP_R_xml_0_scc_65_large (algebra.Alg.Term algebra.F.id_mark 
                                 (x8::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      (* <mark(U71(X_)),mark(X_)> *)
     | DP_R_xml_0_scc_65_large_18 :
      forall x8 x11, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_U71 (x8::nil)) x11) ->
        DP_R_xml_0_scc_65_large (algebra.Alg.Term algebra.F.id_mark 
                                 (x8::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      (* <mark(U81(X1_,X2_,X3_)),mark(X1_)> *)
     | DP_R_xml_0_scc_65_large_19 :
      forall x6 x9 x11 x7, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_U81 (x6::x7::x9::nil)) x11) ->
        DP_R_xml_0_scc_65_large (algebra.Alg.Term algebra.F.id_mark 
                                 (x6::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      (* <mark(U82(X1_,X2_,X3_)),mark(X1_)> *)
     | DP_R_xml_0_scc_65_large_20 :
      forall x6 x9 x11 x7, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_U82 (x6::x7::x9::nil)) x11) ->
        DP_R_xml_0_scc_65_large (algebra.Alg.Term algebra.F.id_mark 
                                 (x6::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      (* <mark(U83(X1_,X2_,X3_)),mark(X1_)> *)
     | DP_R_xml_0_scc_65_large_21 :
      forall x6 x9 x11 x7, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_U83 (x6::x7::x9::nil)) x11) ->
        DP_R_xml_0_scc_65_large (algebra.Alg.Term algebra.F.id_mark 
                                 (x6::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      (* <mark(U84(X1_,X2_,X3_)),mark(X1_)> *)
     | DP_R_xml_0_scc_65_large_22 :
      forall x6 x9 x11 x7, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_U84 (x6::x7::x9::nil)) x11) ->
        DP_R_xml_0_scc_65_large (algebra.Alg.Term algebra.F.id_mark 
                                 (x6::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      (* <mark(U85(X1_,X2_)),mark(X1_)> *)
     | DP_R_xml_0_scc_65_large_23 :
      forall x6 x11 x7, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_U85 (x6::x7::nil)) x11) ->
        DP_R_xml_0_scc_65_large (algebra.Alg.Term algebra.F.id_mark 
                                 (x6::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      (* <mark(U86(X_)),mark(X_)> *)
     | DP_R_xml_0_scc_65_large_24 :
      forall x8 x11, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_U86 (x8::nil)) x11) ->
        DP_R_xml_0_scc_65_large (algebra.Alg.Term algebra.F.id_mark 
                                 (x8::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     
      (* <a__U91(tt,L_,N_),a__U92(a__isNatIListKind(L_),L_,N_)> *)
     | DP_R_xml_0_scc_65_large_25 :
      forall x4 x12 x5 x13 x11, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  (algebra.Alg.Term algebra.F.id_tt nil) 
         x13) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x4 x12) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x5 x11) ->
          DP_R_xml_0_scc_65_large (algebra.Alg.Term algebra.F.id_a__U92 
                                   ((algebra.Alg.Term 
                                   algebra.F.id_a__isNatIListKind 
                                   (x4::nil))::x4::x5::nil)) 
           (algebra.Alg.Term algebra.F.id_a__U91 (x13::x12::x11::nil))
     
      (* <a__U92(tt,L_,N_),a__U93(a__isNat(N_),L_,N_)> *)
     | DP_R_xml_0_scc_65_large_26 :
      forall x4 x12 x5 x13 x11, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  (algebra.Alg.Term algebra.F.id_tt nil) 
         x13) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x4 x12) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x5 x11) ->
          DP_R_xml_0_scc_65_large (algebra.Alg.Term algebra.F.id_a__U93 
                                   ((algebra.Alg.Term algebra.F.id_a__isNat 
                                   (x5::nil))::x4::x5::nil)) 
           (algebra.Alg.Term algebra.F.id_a__U92 (x13::x12::x11::nil))
     
      (* <a__U93(tt,L_,N_),a__U94(a__isNatKind(N_),L_)> *)
     | DP_R_xml_0_scc_65_large_27 :
      forall x4 x12 x5 x13 x11, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  (algebra.Alg.Term algebra.F.id_tt nil) 
         x13) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x4 x12) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x5 x11) ->
          DP_R_xml_0_scc_65_large (algebra.Alg.Term algebra.F.id_a__U94 
                                   ((algebra.Alg.Term 
                                   algebra.F.id_a__isNatKind (x5::nil))::
                                   x4::nil)) 
           (algebra.Alg.Term algebra.F.id_a__U93 (x13::x12::x11::nil))
      (* <a__U94(tt,L_),a__length(mark(L_))> *)
     | DP_R_xml_0_scc_65_large_28 :
      forall x4 x12 x11, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  (algebra.Alg.Term algebra.F.id_tt nil) 
         x12) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x4 x11) ->
         DP_R_xml_0_scc_65_large (algebra.Alg.Term algebra.F.id_a__length 
                                  ((algebra.Alg.Term algebra.F.id_mark 
                                  (x4::nil))::nil)) 
          (algebra.Alg.Term algebra.F.id_a__U94 (x12::x11::nil))
     
      (* <a__length(cons(N_,L_)),a__U91(a__isNatList(L_),L_,N_)> *)
     | DP_R_xml_0_scc_65_large_29 :
      forall x4 x5 x11, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_cons (x5::x4::nil)) x11) ->
        DP_R_xml_0_scc_65_large (algebra.Alg.Term algebra.F.id_a__U91 
                                 ((algebra.Alg.Term 
                                 algebra.F.id_a__isNatList (x4::nil))::x4::
                                 x5::nil)) 
         (algebra.Alg.Term algebra.F.id_a__length (x11::nil))
      (* <a__U94(tt,L_),mark(L_)> *)
     | DP_R_xml_0_scc_65_large_30 :
      forall x4 x12 x11, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  (algebra.Alg.Term algebra.F.id_tt nil) 
         x12) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x4 x11) ->
         DP_R_xml_0_scc_65_large (algebra.Alg.Term algebra.F.id_mark 
                                  (x4::nil)) 
          (algebra.Alg.Term algebra.F.id_a__U94 (x12::x11::nil))
      (* <mark(cons(X1_,X2_)),mark(X1_)> *)
     | DP_R_xml_0_scc_65_large_31 :
      forall x6 x11 x7, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_cons (x6::x7::nil)) x11) ->
        DP_R_xml_0_scc_65_large (algebra.Alg.Term algebra.F.id_mark 
                                 (x6::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      (* <mark(s(X_)),mark(X_)> *)
     | DP_R_xml_0_scc_65_large_32 :
      forall x8 x11, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_s (x8::nil)) x11) ->
        DP_R_xml_0_scc_65_large (algebra.Alg.Term algebra.F.id_mark 
                                 (x8::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
   .
   
   
   Inductive DP_R_xml_0_scc_65_strict  :
    algebra.Alg.term ->algebra.Alg.term ->Prop := 
      (* <mark(U91(X1_,X2_,X3_)),a__U91(mark(X1_),X2_,X3_)> *)
     | DP_R_xml_0_scc_65_strict_0 :
      forall x6 x9 x11 x7, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_U91 (x6::x7::x9::nil)) x11) ->
        DP_R_xml_0_scc_65_strict (algebra.Alg.Term algebra.F.id_a__U91 
                                  ((algebra.Alg.Term algebra.F.id_mark 
                                  (x6::nil))::x7::x9::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      (* <mark(U91(X1_,X2_,X3_)),mark(X1_)> *)
     | DP_R_xml_0_scc_65_strict_1 :
      forall x6 x9 x11 x7, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_U91 (x6::x7::x9::nil)) x11) ->
        DP_R_xml_0_scc_65_strict (algebra.Alg.Term algebra.F.id_mark 
                                  (x6::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     
      (* <mark(U92(X1_,X2_,X3_)),a__U92(mark(X1_),X2_,X3_)> *)
     | DP_R_xml_0_scc_65_strict_2 :
      forall x6 x9 x11 x7, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_U92 (x6::x7::x9::nil)) x11) ->
        DP_R_xml_0_scc_65_strict (algebra.Alg.Term algebra.F.id_a__U92 
                                  ((algebra.Alg.Term algebra.F.id_mark 
                                  (x6::nil))::x7::x9::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      (* <mark(U92(X1_,X2_,X3_)),mark(X1_)> *)
     | DP_R_xml_0_scc_65_strict_3 :
      forall x6 x9 x11 x7, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_U92 (x6::x7::x9::nil)) x11) ->
        DP_R_xml_0_scc_65_strict (algebra.Alg.Term algebra.F.id_mark 
                                  (x6::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     
      (* <mark(U93(X1_,X2_,X3_)),a__U93(mark(X1_),X2_,X3_)> *)
     | DP_R_xml_0_scc_65_strict_4 :
      forall x6 x9 x11 x7, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_U93 (x6::x7::x9::nil)) x11) ->
        DP_R_xml_0_scc_65_strict (algebra.Alg.Term algebra.F.id_a__U93 
                                  ((algebra.Alg.Term algebra.F.id_mark 
                                  (x6::nil))::x7::x9::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      (* <mark(U93(X1_,X2_,X3_)),mark(X1_)> *)
     | DP_R_xml_0_scc_65_strict_5 :
      forall x6 x9 x11 x7, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_U93 (x6::x7::x9::nil)) x11) ->
        DP_R_xml_0_scc_65_strict (algebra.Alg.Term algebra.F.id_mark 
                                  (x6::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     
      (* <mark(U94(X1_,X2_)),a__U94(mark(X1_),X2_)> *)
     | DP_R_xml_0_scc_65_strict_6 :
      forall x6 x11 x7, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_U94 (x6::x7::nil)) x11) ->
        DP_R_xml_0_scc_65_strict (algebra.Alg.Term algebra.F.id_a__U94 
                                  ((algebra.Alg.Term algebra.F.id_mark 
                                  (x6::nil))::x7::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      (* <mark(U94(X1_,X2_)),mark(X1_)> *)
     | DP_R_xml_0_scc_65_strict_7 :
      forall x6 x11 x7, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_U94 (x6::x7::nil)) x11) ->
        DP_R_xml_0_scc_65_strict (algebra.Alg.Term algebra.F.id_mark 
                                  (x6::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     
      (* <mark(length(X_)),a__length(mark(X_))> *)
     | DP_R_xml_0_scc_65_strict_8 :
      forall x8 x11, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_length (x8::nil)) x11) ->
        DP_R_xml_0_scc_65_strict (algebra.Alg.Term algebra.F.id_a__length 
                                  ((algebra.Alg.Term algebra.F.id_mark 
                                  (x8::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      (* <mark(length(X_)),mark(X_)> *)
     | DP_R_xml_0_scc_65_strict_9 :
      forall x8 x11, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_length (x8::nil)) x11) ->
        DP_R_xml_0_scc_65_strict (algebra.Alg.Term algebra.F.id_mark 
                                  (x8::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
   .
   
   
   Module WF_DP_R_xml_0_scc_65_large.
    Inductive DP_R_xml_0_scc_65_large_scc_1  :
     algebra.Alg.term ->algebra.Alg.term ->Prop := 
       (* <mark(U12(X1_,X2_)),mark(X1_)> *)
      | DP_R_xml_0_scc_65_large_scc_1_0 :
       forall x6 x11 x7, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_U12 (x6::x7::nil)) x11) ->
         DP_R_xml_0_scc_65_large_scc_1 (algebra.Alg.Term algebra.F.id_mark 
                                        (x6::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      
       (* <mark(U11(X1_,X2_)),mark(X1_)> *)
      | DP_R_xml_0_scc_65_large_scc_1_1 :
       forall x6 x11 x7, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_U11 (x6::x7::nil)) x11) ->
         DP_R_xml_0_scc_65_large_scc_1 (algebra.Alg.Term algebra.F.id_mark 
                                        (x6::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       (* <mark(U13(X_)),mark(X_)> *)
      | DP_R_xml_0_scc_65_large_scc_1_2 :
       forall x8 x11, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_U13 (x8::nil)) x11) ->
         DP_R_xml_0_scc_65_large_scc_1 (algebra.Alg.Term algebra.F.id_mark 
                                        (x8::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      
       (* <mark(U21(X1_,X2_)),mark(X1_)> *)
      | DP_R_xml_0_scc_65_large_scc_1_3 :
       forall x6 x11 x7, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_U21 (x6::x7::nil)) x11) ->
         DP_R_xml_0_scc_65_large_scc_1 (algebra.Alg.Term algebra.F.id_mark 
                                        (x6::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      
       (* <mark(U22(X1_,X2_)),mark(X1_)> *)
      | DP_R_xml_0_scc_65_large_scc_1_4 :
       forall x6 x11 x7, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_U22 (x6::x7::nil)) x11) ->
         DP_R_xml_0_scc_65_large_scc_1 (algebra.Alg.Term algebra.F.id_mark 
                                        (x6::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       (* <mark(U23(X_)),mark(X_)> *)
      | DP_R_xml_0_scc_65_large_scc_1_5 :
       forall x8 x11, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_U23 (x8::nil)) x11) ->
         DP_R_xml_0_scc_65_large_scc_1 (algebra.Alg.Term algebra.F.id_mark 
                                        (x8::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      
       (* <mark(U31(X1_,X2_)),mark(X1_)> *)
      | DP_R_xml_0_scc_65_large_scc_1_6 :
       forall x6 x11 x7, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_U31 (x6::x7::nil)) x11) ->
         DP_R_xml_0_scc_65_large_scc_1 (algebra.Alg.Term algebra.F.id_mark 
                                        (x6::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      
       (* <mark(U32(X1_,X2_)),mark(X1_)> *)
      | DP_R_xml_0_scc_65_large_scc_1_7 :
       forall x6 x11 x7, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_U32 (x6::x7::nil)) x11) ->
         DP_R_xml_0_scc_65_large_scc_1 (algebra.Alg.Term algebra.F.id_mark 
                                        (x6::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       (* <mark(U33(X_)),mark(X_)> *)
      | DP_R_xml_0_scc_65_large_scc_1_8 :
       forall x8 x11, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_U33 (x8::nil)) x11) ->
         DP_R_xml_0_scc_65_large_scc_1 (algebra.Alg.Term algebra.F.id_mark 
                                        (x8::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      
       (* <mark(U41(X1_,X2_,X3_)),mark(X1_)> *)
      | DP_R_xml_0_scc_65_large_scc_1_9 :
       forall x6 x9 x11 x7, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_U41 (x6::x7::x9::nil)) x11) ->
         DP_R_xml_0_scc_65_large_scc_1 (algebra.Alg.Term algebra.F.id_mark 
                                        (x6::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      
       (* <mark(U42(X1_,X2_,X3_)),mark(X1_)> *)
      | DP_R_xml_0_scc_65_large_scc_1_10 :
       forall x6 x9 x11 x7, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_U42 (x6::x7::x9::nil)) x11) ->
         DP_R_xml_0_scc_65_large_scc_1 (algebra.Alg.Term algebra.F.id_mark 
                                        (x6::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      
       (* <mark(U43(X1_,X2_,X3_)),mark(X1_)> *)
      | DP_R_xml_0_scc_65_large_scc_1_11 :
       forall x6 x9 x11 x7, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_U43 (x6::x7::x9::nil)) x11) ->
         DP_R_xml_0_scc_65_large_scc_1 (algebra.Alg.Term algebra.F.id_mark 
                                        (x6::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      
       (* <mark(U44(X1_,X2_,X3_)),mark(X1_)> *)
      | DP_R_xml_0_scc_65_large_scc_1_12 :
       forall x6 x9 x11 x7, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_U44 (x6::x7::x9::nil)) x11) ->
         DP_R_xml_0_scc_65_large_scc_1 (algebra.Alg.Term algebra.F.id_mark 
                                        (x6::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      
       (* <mark(U45(X1_,X2_)),mark(X1_)> *)
      | DP_R_xml_0_scc_65_large_scc_1_13 :
       forall x6 x11 x7, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_U45 (x6::x7::nil)) x11) ->
         DP_R_xml_0_scc_65_large_scc_1 (algebra.Alg.Term algebra.F.id_mark 
                                        (x6::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       (* <mark(U46(X_)),mark(X_)> *)
      | DP_R_xml_0_scc_65_large_scc_1_14 :
       forall x8 x11, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_U46 (x8::nil)) x11) ->
         DP_R_xml_0_scc_65_large_scc_1 (algebra.Alg.Term algebra.F.id_mark 
                                        (x8::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      
       (* <mark(U51(X1_,X2_)),mark(X1_)> *)
      | DP_R_xml_0_scc_65_large_scc_1_15 :
       forall x6 x11 x7, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_U51 (x6::x7::nil)) x11) ->
         DP_R_xml_0_scc_65_large_scc_1 (algebra.Alg.Term algebra.F.id_mark 
                                        (x6::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       (* <mark(U52(X_)),mark(X_)> *)
      | DP_R_xml_0_scc_65_large_scc_1_16 :
       forall x8 x11, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_U52 (x8::nil)) x11) ->
         DP_R_xml_0_scc_65_large_scc_1 (algebra.Alg.Term algebra.F.id_mark 
                                        (x8::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       (* <mark(U61(X_)),mark(X_)> *)
      | DP_R_xml_0_scc_65_large_scc_1_17 :
       forall x8 x11, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_U61 (x8::nil)) x11) ->
         DP_R_xml_0_scc_65_large_scc_1 (algebra.Alg.Term algebra.F.id_mark 
                                        (x8::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       (* <mark(U71(X_)),mark(X_)> *)
      | DP_R_xml_0_scc_65_large_scc_1_18 :
       forall x8 x11, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_U71 (x8::nil)) x11) ->
         DP_R_xml_0_scc_65_large_scc_1 (algebra.Alg.Term algebra.F.id_mark 
                                        (x8::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      
       (* <mark(U81(X1_,X2_,X3_)),mark(X1_)> *)
      | DP_R_xml_0_scc_65_large_scc_1_19 :
       forall x6 x9 x11 x7, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_U81 (x6::x7::x9::nil)) x11) ->
         DP_R_xml_0_scc_65_large_scc_1 (algebra.Alg.Term algebra.F.id_mark 
                                        (x6::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      
       (* <mark(U82(X1_,X2_,X3_)),mark(X1_)> *)
      | DP_R_xml_0_scc_65_large_scc_1_20 :
       forall x6 x9 x11 x7, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_U82 (x6::x7::x9::nil)) x11) ->
         DP_R_xml_0_scc_65_large_scc_1 (algebra.Alg.Term algebra.F.id_mark 
                                        (x6::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      
       (* <mark(U83(X1_,X2_,X3_)),mark(X1_)> *)
      | DP_R_xml_0_scc_65_large_scc_1_21 :
       forall x6 x9 x11 x7, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_U83 (x6::x7::x9::nil)) x11) ->
         DP_R_xml_0_scc_65_large_scc_1 (algebra.Alg.Term algebra.F.id_mark 
                                        (x6::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      
       (* <mark(U84(X1_,X2_,X3_)),mark(X1_)> *)
      | DP_R_xml_0_scc_65_large_scc_1_22 :
       forall x6 x9 x11 x7, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_U84 (x6::x7::x9::nil)) x11) ->
         DP_R_xml_0_scc_65_large_scc_1 (algebra.Alg.Term algebra.F.id_mark 
                                        (x6::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      
       (* <mark(U85(X1_,X2_)),mark(X1_)> *)
      | DP_R_xml_0_scc_65_large_scc_1_23 :
       forall x6 x11 x7, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_U85 (x6::x7::nil)) x11) ->
         DP_R_xml_0_scc_65_large_scc_1 (algebra.Alg.Term algebra.F.id_mark 
                                        (x6::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       (* <mark(U86(X_)),mark(X_)> *)
      | DP_R_xml_0_scc_65_large_scc_1_24 :
       forall x8 x11, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_U86 (x8::nil)) x11) ->
         DP_R_xml_0_scc_65_large_scc_1 (algebra.Alg.Term algebra.F.id_mark 
                                        (x8::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      
       (* <mark(cons(X1_,X2_)),mark(X1_)> *)
      | DP_R_xml_0_scc_65_large_scc_1_25 :
       forall x6 x11 x7, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_cons (x6::x7::nil)) x11) ->
         DP_R_xml_0_scc_65_large_scc_1 (algebra.Alg.Term algebra.F.id_mark 
                                        (x6::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       (* <mark(s(X_)),mark(X_)> *)
      | DP_R_xml_0_scc_65_large_scc_1_26 :
       forall x8 x11, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_s (x8::nil)) x11) ->
         DP_R_xml_0_scc_65_large_scc_1 (algebra.Alg.Term algebra.F.id_mark 
                                        (x8::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    .
    
    
    Module WF_DP_R_xml_0_scc_65_large_scc_1.
     Inductive DP_R_xml_0_scc_65_large_scc_1_large  :
      algebra.Alg.term ->algebra.Alg.term ->Prop := 
        (* <mark(U12(X1_,X2_)),mark(X1_)> *)
       | DP_R_xml_0_scc_65_large_scc_1_large_0 :
        forall x6 x11 x7, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    
           (algebra.Alg.Term algebra.F.id_U12 (x6::x7::nil)) x11) ->
          DP_R_xml_0_scc_65_large_scc_1_large (algebra.Alg.Term 
                                               algebra.F.id_mark (x6::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       
        (* <mark(U11(X1_,X2_)),mark(X1_)> *)
       | DP_R_xml_0_scc_65_large_scc_1_large_1 :
        forall x6 x11 x7, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    
           (algebra.Alg.Term algebra.F.id_U11 (x6::x7::nil)) x11) ->
          DP_R_xml_0_scc_65_large_scc_1_large (algebra.Alg.Term 
                                               algebra.F.id_mark (x6::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       
        (* <mark(U13(X_)),mark(X_)> *)
       | DP_R_xml_0_scc_65_large_scc_1_large_2 :
        forall x8 x11, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    
           (algebra.Alg.Term algebra.F.id_U13 (x8::nil)) x11) ->
          DP_R_xml_0_scc_65_large_scc_1_large (algebra.Alg.Term 
                                               algebra.F.id_mark (x8::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       
        (* <mark(U21(X1_,X2_)),mark(X1_)> *)
       | DP_R_xml_0_scc_65_large_scc_1_large_3 :
        forall x6 x11 x7, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    
           (algebra.Alg.Term algebra.F.id_U21 (x6::x7::nil)) x11) ->
          DP_R_xml_0_scc_65_large_scc_1_large (algebra.Alg.Term 
                                               algebra.F.id_mark (x6::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       
        (* <mark(U22(X1_,X2_)),mark(X1_)> *)
       | DP_R_xml_0_scc_65_large_scc_1_large_4 :
        forall x6 x11 x7, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    
           (algebra.Alg.Term algebra.F.id_U22 (x6::x7::nil)) x11) ->
          DP_R_xml_0_scc_65_large_scc_1_large (algebra.Alg.Term 
                                               algebra.F.id_mark (x6::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       
        (* <mark(U23(X_)),mark(X_)> *)
       | DP_R_xml_0_scc_65_large_scc_1_large_5 :
        forall x8 x11, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    
           (algebra.Alg.Term algebra.F.id_U23 (x8::nil)) x11) ->
          DP_R_xml_0_scc_65_large_scc_1_large (algebra.Alg.Term 
                                               algebra.F.id_mark (x8::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       
        (* <mark(U31(X1_,X2_)),mark(X1_)> *)
       | DP_R_xml_0_scc_65_large_scc_1_large_6 :
        forall x6 x11 x7, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    
           (algebra.Alg.Term algebra.F.id_U31 (x6::x7::nil)) x11) ->
          DP_R_xml_0_scc_65_large_scc_1_large (algebra.Alg.Term 
                                               algebra.F.id_mark (x6::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       
        (* <mark(U32(X1_,X2_)),mark(X1_)> *)
       | DP_R_xml_0_scc_65_large_scc_1_large_7 :
        forall x6 x11 x7, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    
           (algebra.Alg.Term algebra.F.id_U32 (x6::x7::nil)) x11) ->
          DP_R_xml_0_scc_65_large_scc_1_large (algebra.Alg.Term 
                                               algebra.F.id_mark (x6::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       
        (* <mark(U33(X_)),mark(X_)> *)
       | DP_R_xml_0_scc_65_large_scc_1_large_8 :
        forall x8 x11, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    
           (algebra.Alg.Term algebra.F.id_U33 (x8::nil)) x11) ->
          DP_R_xml_0_scc_65_large_scc_1_large (algebra.Alg.Term 
                                               algebra.F.id_mark (x8::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       
        (* <mark(U41(X1_,X2_,X3_)),mark(X1_)> *)
       | DP_R_xml_0_scc_65_large_scc_1_large_9 :
        forall x6 x9 x11 x7, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    
           (algebra.Alg.Term algebra.F.id_U41 (x6::x7::x9::nil)) x11) ->
          DP_R_xml_0_scc_65_large_scc_1_large (algebra.Alg.Term 
                                               algebra.F.id_mark (x6::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       
        (* <mark(U42(X1_,X2_,X3_)),mark(X1_)> *)
       | DP_R_xml_0_scc_65_large_scc_1_large_10 :
        forall x6 x9 x11 x7, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    
           (algebra.Alg.Term algebra.F.id_U42 (x6::x7::x9::nil)) x11) ->
          DP_R_xml_0_scc_65_large_scc_1_large (algebra.Alg.Term 
                                               algebra.F.id_mark (x6::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       
        (* <mark(U43(X1_,X2_,X3_)),mark(X1_)> *)
       | DP_R_xml_0_scc_65_large_scc_1_large_11 :
        forall x6 x9 x11 x7, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    
           (algebra.Alg.Term algebra.F.id_U43 (x6::x7::x9::nil)) x11) ->
          DP_R_xml_0_scc_65_large_scc_1_large (algebra.Alg.Term 
                                               algebra.F.id_mark (x6::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       
        (* <mark(U44(X1_,X2_,X3_)),mark(X1_)> *)
       | DP_R_xml_0_scc_65_large_scc_1_large_12 :
        forall x6 x9 x11 x7, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    
           (algebra.Alg.Term algebra.F.id_U44 (x6::x7::x9::nil)) x11) ->
          DP_R_xml_0_scc_65_large_scc_1_large (algebra.Alg.Term 
                                               algebra.F.id_mark (x6::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       
        (* <mark(U45(X1_,X2_)),mark(X1_)> *)
       | DP_R_xml_0_scc_65_large_scc_1_large_13 :
        forall x6 x11 x7, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    
           (algebra.Alg.Term algebra.F.id_U45 (x6::x7::nil)) x11) ->
          DP_R_xml_0_scc_65_large_scc_1_large (algebra.Alg.Term 
                                               algebra.F.id_mark (x6::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       
        (* <mark(U46(X_)),mark(X_)> *)
       | DP_R_xml_0_scc_65_large_scc_1_large_14 :
        forall x8 x11, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    
           (algebra.Alg.Term algebra.F.id_U46 (x8::nil)) x11) ->
          DP_R_xml_0_scc_65_large_scc_1_large (algebra.Alg.Term 
                                               algebra.F.id_mark (x8::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       
        (* <mark(U51(X1_,X2_)),mark(X1_)> *)
       | DP_R_xml_0_scc_65_large_scc_1_large_15 :
        forall x6 x11 x7, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    
           (algebra.Alg.Term algebra.F.id_U51 (x6::x7::nil)) x11) ->
          DP_R_xml_0_scc_65_large_scc_1_large (algebra.Alg.Term 
                                               algebra.F.id_mark (x6::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       
        (* <mark(U52(X_)),mark(X_)> *)
       | DP_R_xml_0_scc_65_large_scc_1_large_16 :
        forall x8 x11, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    
           (algebra.Alg.Term algebra.F.id_U52 (x8::nil)) x11) ->
          DP_R_xml_0_scc_65_large_scc_1_large (algebra.Alg.Term 
                                               algebra.F.id_mark (x8::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       
        (* <mark(U61(X_)),mark(X_)> *)
       | DP_R_xml_0_scc_65_large_scc_1_large_17 :
        forall x8 x11, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    
           (algebra.Alg.Term algebra.F.id_U61 (x8::nil)) x11) ->
          DP_R_xml_0_scc_65_large_scc_1_large (algebra.Alg.Term 
                                               algebra.F.id_mark (x8::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       
        (* <mark(U71(X_)),mark(X_)> *)
       | DP_R_xml_0_scc_65_large_scc_1_large_18 :
        forall x8 x11, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    
           (algebra.Alg.Term algebra.F.id_U71 (x8::nil)) x11) ->
          DP_R_xml_0_scc_65_large_scc_1_large (algebra.Alg.Term 
                                               algebra.F.id_mark (x8::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       
        (* <mark(U81(X1_,X2_,X3_)),mark(X1_)> *)
       | DP_R_xml_0_scc_65_large_scc_1_large_19 :
        forall x6 x9 x11 x7, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    
           (algebra.Alg.Term algebra.F.id_U81 (x6::x7::x9::nil)) x11) ->
          DP_R_xml_0_scc_65_large_scc_1_large (algebra.Alg.Term 
                                               algebra.F.id_mark (x6::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       
        (* <mark(U82(X1_,X2_,X3_)),mark(X1_)> *)
       | DP_R_xml_0_scc_65_large_scc_1_large_20 :
        forall x6 x9 x11 x7, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    
           (algebra.Alg.Term algebra.F.id_U82 (x6::x7::x9::nil)) x11) ->
          DP_R_xml_0_scc_65_large_scc_1_large (algebra.Alg.Term 
                                               algebra.F.id_mark (x6::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       
        (* <mark(U83(X1_,X2_,X3_)),mark(X1_)> *)
       | DP_R_xml_0_scc_65_large_scc_1_large_21 :
        forall x6 x9 x11 x7, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    
           (algebra.Alg.Term algebra.F.id_U83 (x6::x7::x9::nil)) x11) ->
          DP_R_xml_0_scc_65_large_scc_1_large (algebra.Alg.Term 
                                               algebra.F.id_mark (x6::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       
        (* <mark(U84(X1_,X2_,X3_)),mark(X1_)> *)
       | DP_R_xml_0_scc_65_large_scc_1_large_22 :
        forall x6 x9 x11 x7, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    
           (algebra.Alg.Term algebra.F.id_U84 (x6::x7::x9::nil)) x11) ->
          DP_R_xml_0_scc_65_large_scc_1_large (algebra.Alg.Term 
                                               algebra.F.id_mark (x6::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       
        (* <mark(U85(X1_,X2_)),mark(X1_)> *)
       | DP_R_xml_0_scc_65_large_scc_1_large_23 :
        forall x6 x11 x7, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    
           (algebra.Alg.Term algebra.F.id_U85 (x6::x7::nil)) x11) ->
          DP_R_xml_0_scc_65_large_scc_1_large (algebra.Alg.Term 
                                               algebra.F.id_mark (x6::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       
        (* <mark(U86(X_)),mark(X_)> *)
       | DP_R_xml_0_scc_65_large_scc_1_large_24 :
        forall x8 x11, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    
           (algebra.Alg.Term algebra.F.id_U86 (x8::nil)) x11) ->
          DP_R_xml_0_scc_65_large_scc_1_large (algebra.Alg.Term 
                                               algebra.F.id_mark (x8::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       
        (* <mark(s(X_)),mark(X_)> *)
       | DP_R_xml_0_scc_65_large_scc_1_large_25 :
        forall x8 x11, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    
           (algebra.Alg.Term algebra.F.id_s (x8::nil)) x11) ->
          DP_R_xml_0_scc_65_large_scc_1_large (algebra.Alg.Term 
                                               algebra.F.id_mark (x8::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     .
     
     
     Inductive DP_R_xml_0_scc_65_large_scc_1_strict  :
      algebra.Alg.term ->algebra.Alg.term ->Prop := 
        (* <mark(cons(X1_,X2_)),mark(X1_)> *)
       | DP_R_xml_0_scc_65_large_scc_1_strict_0 :
        forall x6 x11 x7, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    
           (algebra.Alg.Term algebra.F.id_cons (x6::x7::nil)) x11) ->
          DP_R_xml_0_scc_65_large_scc_1_strict (algebra.Alg.Term 
                                                algebra.F.id_mark (x6::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     .
     
     
     Module WF_DP_R_xml_0_scc_65_large_scc_1_large.
      Inductive DP_R_xml_0_scc_65_large_scc_1_large_large  :
       algebra.Alg.term ->algebra.Alg.term ->Prop := 
         (* <mark(U12(X1_,X2_)),mark(X1_)> *)
        | DP_R_xml_0_scc_65_large_scc_1_large_large_0 :
         forall x6 x11 x7, 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     
            (algebra.Alg.Term algebra.F.id_U12 (x6::x7::nil)) x11) ->
           DP_R_xml_0_scc_65_large_scc_1_large_large (algebra.Alg.Term 
                                                      algebra.F.id_mark 
                                                      (x6::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x11::nil))
        
         (* <mark(U11(X1_,X2_)),mark(X1_)> *)
        | DP_R_xml_0_scc_65_large_scc_1_large_large_1 :
         forall x6 x11 x7, 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     
            (algebra.Alg.Term algebra.F.id_U11 (x6::x7::nil)) x11) ->
           DP_R_xml_0_scc_65_large_scc_1_large_large (algebra.Alg.Term 
                                                      algebra.F.id_mark 
                                                      (x6::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x11::nil))
        
         (* <mark(U13(X_)),mark(X_)> *)
        | DP_R_xml_0_scc_65_large_scc_1_large_large_2 :
         forall x8 x11, 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     
            (algebra.Alg.Term algebra.F.id_U13 (x8::nil)) x11) ->
           DP_R_xml_0_scc_65_large_scc_1_large_large (algebra.Alg.Term 
                                                      algebra.F.id_mark 
                                                      (x8::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x11::nil))
        
         (* <mark(U21(X1_,X2_)),mark(X1_)> *)
        | DP_R_xml_0_scc_65_large_scc_1_large_large_3 :
         forall x6 x11 x7, 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     
            (algebra.Alg.Term algebra.F.id_U21 (x6::x7::nil)) x11) ->
           DP_R_xml_0_scc_65_large_scc_1_large_large (algebra.Alg.Term 
                                                      algebra.F.id_mark 
                                                      (x6::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x11::nil))
        
         (* <mark(U22(X1_,X2_)),mark(X1_)> *)
        | DP_R_xml_0_scc_65_large_scc_1_large_large_4 :
         forall x6 x11 x7, 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     
            (algebra.Alg.Term algebra.F.id_U22 (x6::x7::nil)) x11) ->
           DP_R_xml_0_scc_65_large_scc_1_large_large (algebra.Alg.Term 
                                                      algebra.F.id_mark 
                                                      (x6::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x11::nil))
        
         (* <mark(U23(X_)),mark(X_)> *)
        | DP_R_xml_0_scc_65_large_scc_1_large_large_5 :
         forall x8 x11, 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     
            (algebra.Alg.Term algebra.F.id_U23 (x8::nil)) x11) ->
           DP_R_xml_0_scc_65_large_scc_1_large_large (algebra.Alg.Term 
                                                      algebra.F.id_mark 
                                                      (x8::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x11::nil))
        
         (* <mark(U32(X1_,X2_)),mark(X1_)> *)
        | DP_R_xml_0_scc_65_large_scc_1_large_large_6 :
         forall x6 x11 x7, 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     
            (algebra.Alg.Term algebra.F.id_U32 (x6::x7::nil)) x11) ->
           DP_R_xml_0_scc_65_large_scc_1_large_large (algebra.Alg.Term 
                                                      algebra.F.id_mark 
                                                      (x6::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x11::nil))
        
         (* <mark(U33(X_)),mark(X_)> *)
        | DP_R_xml_0_scc_65_large_scc_1_large_large_7 :
         forall x8 x11, 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     
            (algebra.Alg.Term algebra.F.id_U33 (x8::nil)) x11) ->
           DP_R_xml_0_scc_65_large_scc_1_large_large (algebra.Alg.Term 
                                                      algebra.F.id_mark 
                                                      (x8::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x11::nil))
        
         (* <mark(U46(X_)),mark(X_)> *)
        | DP_R_xml_0_scc_65_large_scc_1_large_large_8 :
         forall x8 x11, 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     
            (algebra.Alg.Term algebra.F.id_U46 (x8::nil)) x11) ->
           DP_R_xml_0_scc_65_large_scc_1_large_large (algebra.Alg.Term 
                                                      algebra.F.id_mark 
                                                      (x8::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x11::nil))
        
         (* <mark(U51(X1_,X2_)),mark(X1_)> *)
        | DP_R_xml_0_scc_65_large_scc_1_large_large_9 :
         forall x6 x11 x7, 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     
            (algebra.Alg.Term algebra.F.id_U51 (x6::x7::nil)) x11) ->
           DP_R_xml_0_scc_65_large_scc_1_large_large (algebra.Alg.Term 
                                                      algebra.F.id_mark 
                                                      (x6::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x11::nil))
        
         (* <mark(U52(X_)),mark(X_)> *)
        | DP_R_xml_0_scc_65_large_scc_1_large_large_10 :
         forall x8 x11, 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     
            (algebra.Alg.Term algebra.F.id_U52 (x8::nil)) x11) ->
           DP_R_xml_0_scc_65_large_scc_1_large_large (algebra.Alg.Term 
                                                      algebra.F.id_mark 
                                                      (x8::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x11::nil))
        
         (* <mark(U61(X_)),mark(X_)> *)
        | DP_R_xml_0_scc_65_large_scc_1_large_large_11 :
         forall x8 x11, 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     
            (algebra.Alg.Term algebra.F.id_U61 (x8::nil)) x11) ->
           DP_R_xml_0_scc_65_large_scc_1_large_large (algebra.Alg.Term 
                                                      algebra.F.id_mark 
                                                      (x8::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x11::nil))
        
         (* <mark(U71(X_)),mark(X_)> *)
        | DP_R_xml_0_scc_65_large_scc_1_large_large_12 :
         forall x8 x11, 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     
            (algebra.Alg.Term algebra.F.id_U71 (x8::nil)) x11) ->
           DP_R_xml_0_scc_65_large_scc_1_large_large (algebra.Alg.Term 
                                                      algebra.F.id_mark 
                                                      (x8::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x11::nil))
        
         (* <mark(U81(X1_,X2_,X3_)),mark(X1_)> *)
        | DP_R_xml_0_scc_65_large_scc_1_large_large_13 :
         forall x6 x9 x11 x7, 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     
            (algebra.Alg.Term algebra.F.id_U81 (x6::x7::x9::nil)) x11) ->
           DP_R_xml_0_scc_65_large_scc_1_large_large (algebra.Alg.Term 
                                                      algebra.F.id_mark 
                                                      (x6::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x11::nil))
        
         (* <mark(U82(X1_,X2_,X3_)),mark(X1_)> *)
        | DP_R_xml_0_scc_65_large_scc_1_large_large_14 :
         forall x6 x9 x11 x7, 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     
            (algebra.Alg.Term algebra.F.id_U82 (x6::x7::x9::nil)) x11) ->
           DP_R_xml_0_scc_65_large_scc_1_large_large (algebra.Alg.Term 
                                                      algebra.F.id_mark 
                                                      (x6::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x11::nil))
        
         (* <mark(U83(X1_,X2_,X3_)),mark(X1_)> *)
        | DP_R_xml_0_scc_65_large_scc_1_large_large_15 :
         forall x6 x9 x11 x7, 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     
            (algebra.Alg.Term algebra.F.id_U83 (x6::x7::x9::nil)) x11) ->
           DP_R_xml_0_scc_65_large_scc_1_large_large (algebra.Alg.Term 
                                                      algebra.F.id_mark 
                                                      (x6::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x11::nil))
        
         (* <mark(U84(X1_,X2_,X3_)),mark(X1_)> *)
        | DP_R_xml_0_scc_65_large_scc_1_large_large_16 :
         forall x6 x9 x11 x7, 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     
            (algebra.Alg.Term algebra.F.id_U84 (x6::x7::x9::nil)) x11) ->
           DP_R_xml_0_scc_65_large_scc_1_large_large (algebra.Alg.Term 
                                                      algebra.F.id_mark 
                                                      (x6::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x11::nil))
        
         (* <mark(U85(X1_,X2_)),mark(X1_)> *)
        | DP_R_xml_0_scc_65_large_scc_1_large_large_17 :
         forall x6 x11 x7, 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     
            (algebra.Alg.Term algebra.F.id_U85 (x6::x7::nil)) x11) ->
           DP_R_xml_0_scc_65_large_scc_1_large_large (algebra.Alg.Term 
                                                      algebra.F.id_mark 
                                                      (x6::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x11::nil))
        
         (* <mark(U86(X_)),mark(X_)> *)
        | DP_R_xml_0_scc_65_large_scc_1_large_large_18 :
         forall x8 x11, 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     
            (algebra.Alg.Term algebra.F.id_U86 (x8::nil)) x11) ->
           DP_R_xml_0_scc_65_large_scc_1_large_large (algebra.Alg.Term 
                                                      algebra.F.id_mark 
                                                      (x8::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x11::nil))
        
         (* <mark(s(X_)),mark(X_)> *)
        | DP_R_xml_0_scc_65_large_scc_1_large_large_19 :
         forall x8 x11, 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     
            (algebra.Alg.Term algebra.F.id_s (x8::nil)) x11) ->
           DP_R_xml_0_scc_65_large_scc_1_large_large (algebra.Alg.Term 
                                                      algebra.F.id_mark 
                                                      (x8::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      .
      
      
      Inductive DP_R_xml_0_scc_65_large_scc_1_large_strict  :
       algebra.Alg.term ->algebra.Alg.term ->Prop := 
         (* <mark(U31(X1_,X2_)),mark(X1_)> *)
        | DP_R_xml_0_scc_65_large_scc_1_large_strict_0 :
         forall x6 x11 x7, 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     
            (algebra.Alg.Term algebra.F.id_U31 (x6::x7::nil)) x11) ->
           DP_R_xml_0_scc_65_large_scc_1_large_strict (algebra.Alg.Term 
                                                       algebra.F.id_mark 
                                                       (x6::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x11::nil))
        
         (* <mark(U41(X1_,X2_,X3_)),mark(X1_)> *)
        | DP_R_xml_0_scc_65_large_scc_1_large_strict_1 :
         forall x6 x9 x11 x7, 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     
            (algebra.Alg.Term algebra.F.id_U41 (x6::x7::x9::nil)) x11) ->
           DP_R_xml_0_scc_65_large_scc_1_large_strict (algebra.Alg.Term 
                                                       algebra.F.id_mark 
                                                       (x6::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x11::nil))
        
         (* <mark(U42(X1_,X2_,X3_)),mark(X1_)> *)
        | DP_R_xml_0_scc_65_large_scc_1_large_strict_2 :
         forall x6 x9 x11 x7, 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     
            (algebra.Alg.Term algebra.F.id_U42 (x6::x7::x9::nil)) x11) ->
           DP_R_xml_0_scc_65_large_scc_1_large_strict (algebra.Alg.Term 
                                                       algebra.F.id_mark 
                                                       (x6::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x11::nil))
        
         (* <mark(U43(X1_,X2_,X3_)),mark(X1_)> *)
        | DP_R_xml_0_scc_65_large_scc_1_large_strict_3 :
         forall x6 x9 x11 x7, 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     
            (algebra.Alg.Term algebra.F.id_U43 (x6::x7::x9::nil)) x11) ->
           DP_R_xml_0_scc_65_large_scc_1_large_strict (algebra.Alg.Term 
                                                       algebra.F.id_mark 
                                                       (x6::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x11::nil))
        
         (* <mark(U44(X1_,X2_,X3_)),mark(X1_)> *)
        | DP_R_xml_0_scc_65_large_scc_1_large_strict_4 :
         forall x6 x9 x11 x7, 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     
            (algebra.Alg.Term algebra.F.id_U44 (x6::x7::x9::nil)) x11) ->
           DP_R_xml_0_scc_65_large_scc_1_large_strict (algebra.Alg.Term 
                                                       algebra.F.id_mark 
                                                       (x6::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x11::nil))
        
         (* <mark(U45(X1_,X2_)),mark(X1_)> *)
        | DP_R_xml_0_scc_65_large_scc_1_large_strict_5 :
         forall x6 x11 x7, 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     
            (algebra.Alg.Term algebra.F.id_U45 (x6::x7::nil)) x11) ->
           DP_R_xml_0_scc_65_large_scc_1_large_strict (algebra.Alg.Term 
                                                       algebra.F.id_mark 
                                                       (x6::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      .
      
      
      Module WF_DP_R_xml_0_scc_65_large_scc_1_large_large.
       Inductive DP_R_xml_0_scc_65_large_scc_1_large_large_large  :
        algebra.Alg.term ->algebra.Alg.term ->Prop := 
          (* <mark(U12(X1_,X2_)),mark(X1_)> *)
         | DP_R_xml_0_scc_65_large_scc_1_large_large_large_0 :
          forall x6 x11 x7, 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                      
             (algebra.Alg.Term algebra.F.id_U12 (x6::x7::nil)) x11) ->
            DP_R_xml_0_scc_65_large_scc_1_large_large_large (algebra.Alg.Term 
                                                             algebra.F.id_mark 
                                                             (x6::nil)) 
             (algebra.Alg.Term algebra.F.id_mark (x11::nil))
         
          (* <mark(U11(X1_,X2_)),mark(X1_)> *)
         | DP_R_xml_0_scc_65_large_scc_1_large_large_large_1 :
          forall x6 x11 x7, 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                      
             (algebra.Alg.Term algebra.F.id_U11 (x6::x7::nil)) x11) ->
            DP_R_xml_0_scc_65_large_scc_1_large_large_large (algebra.Alg.Term 
                                                             algebra.F.id_mark 
                                                             (x6::nil)) 
             (algebra.Alg.Term algebra.F.id_mark (x11::nil))
         
          (* <mark(U13(X_)),mark(X_)> *)
         | DP_R_xml_0_scc_65_large_scc_1_large_large_large_2 :
          forall x8 x11, 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                      
             (algebra.Alg.Term algebra.F.id_U13 (x8::nil)) x11) ->
            DP_R_xml_0_scc_65_large_scc_1_large_large_large (algebra.Alg.Term 
                                                             algebra.F.id_mark 
                                                             (x8::nil)) 
             (algebra.Alg.Term algebra.F.id_mark (x11::nil))
         
          (* <mark(U21(X1_,X2_)),mark(X1_)> *)
         | DP_R_xml_0_scc_65_large_scc_1_large_large_large_3 :
          forall x6 x11 x7, 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                      
             (algebra.Alg.Term algebra.F.id_U21 (x6::x7::nil)) x11) ->
            DP_R_xml_0_scc_65_large_scc_1_large_large_large (algebra.Alg.Term 
                                                             algebra.F.id_mark 
                                                             (x6::nil)) 
             (algebra.Alg.Term algebra.F.id_mark (x11::nil))
         
          (* <mark(U22(X1_,X2_)),mark(X1_)> *)
         | DP_R_xml_0_scc_65_large_scc_1_large_large_large_4 :
          forall x6 x11 x7, 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                      
             (algebra.Alg.Term algebra.F.id_U22 (x6::x7::nil)) x11) ->
            DP_R_xml_0_scc_65_large_scc_1_large_large_large (algebra.Alg.Term 
                                                             algebra.F.id_mark 
                                                             (x6::nil)) 
             (algebra.Alg.Term algebra.F.id_mark (x11::nil))
         
          (* <mark(U23(X_)),mark(X_)> *)
         | DP_R_xml_0_scc_65_large_scc_1_large_large_large_5 :
          forall x8 x11, 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                      
             (algebra.Alg.Term algebra.F.id_U23 (x8::nil)) x11) ->
            DP_R_xml_0_scc_65_large_scc_1_large_large_large (algebra.Alg.Term 
                                                             algebra.F.id_mark 
                                                             (x8::nil)) 
             (algebra.Alg.Term algebra.F.id_mark (x11::nil))
         
          (* <mark(U33(X_)),mark(X_)> *)
         | DP_R_xml_0_scc_65_large_scc_1_large_large_large_6 :
          forall x8 x11, 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                      
             (algebra.Alg.Term algebra.F.id_U33 (x8::nil)) x11) ->
            DP_R_xml_0_scc_65_large_scc_1_large_large_large (algebra.Alg.Term 
                                                             algebra.F.id_mark 
                                                             (x8::nil)) 
             (algebra.Alg.Term algebra.F.id_mark (x11::nil))
         
          (* <mark(U46(X_)),mark(X_)> *)
         | DP_R_xml_0_scc_65_large_scc_1_large_large_large_7 :
          forall x8 x11, 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                      
             (algebra.Alg.Term algebra.F.id_U46 (x8::nil)) x11) ->
            DP_R_xml_0_scc_65_large_scc_1_large_large_large (algebra.Alg.Term 
                                                             algebra.F.id_mark 
                                                             (x8::nil)) 
             (algebra.Alg.Term algebra.F.id_mark (x11::nil))
         
          (* <mark(U51(X1_,X2_)),mark(X1_)> *)
         | DP_R_xml_0_scc_65_large_scc_1_large_large_large_8 :
          forall x6 x11 x7, 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                      
             (algebra.Alg.Term algebra.F.id_U51 (x6::x7::nil)) x11) ->
            DP_R_xml_0_scc_65_large_scc_1_large_large_large (algebra.Alg.Term 
                                                             algebra.F.id_mark 
                                                             (x6::nil)) 
             (algebra.Alg.Term algebra.F.id_mark (x11::nil))
         
          (* <mark(U52(X_)),mark(X_)> *)
         | DP_R_xml_0_scc_65_large_scc_1_large_large_large_9 :
          forall x8 x11, 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                      
             (algebra.Alg.Term algebra.F.id_U52 (x8::nil)) x11) ->
            DP_R_xml_0_scc_65_large_scc_1_large_large_large (algebra.Alg.Term 
                                                             algebra.F.id_mark 
                                                             (x8::nil)) 
             (algebra.Alg.Term algebra.F.id_mark (x11::nil))
         
          (* <mark(U61(X_)),mark(X_)> *)
         | DP_R_xml_0_scc_65_large_scc_1_large_large_large_10 :
          forall x8 x11, 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                      
             (algebra.Alg.Term algebra.F.id_U61 (x8::nil)) x11) ->
            DP_R_xml_0_scc_65_large_scc_1_large_large_large (algebra.Alg.Term 
                                                             algebra.F.id_mark 
                                                             (x8::nil)) 
             (algebra.Alg.Term algebra.F.id_mark (x11::nil))
         
          (* <mark(U71(X_)),mark(X_)> *)
         | DP_R_xml_0_scc_65_large_scc_1_large_large_large_11 :
          forall x8 x11, 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                      
             (algebra.Alg.Term algebra.F.id_U71 (x8::nil)) x11) ->
            DP_R_xml_0_scc_65_large_scc_1_large_large_large (algebra.Alg.Term 
                                                             algebra.F.id_mark 
                                                             (x8::nil)) 
             (algebra.Alg.Term algebra.F.id_mark (x11::nil))
         
          (* <mark(U81(X1_,X2_,X3_)),mark(X1_)> *)
         | DP_R_xml_0_scc_65_large_scc_1_large_large_large_12 :
          forall x6 x9 x11 x7, 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                      
             (algebra.Alg.Term algebra.F.id_U81 (x6::x7::x9::nil)) x11) ->
            DP_R_xml_0_scc_65_large_scc_1_large_large_large (algebra.Alg.Term 
                                                             algebra.F.id_mark 
                                                             (x6::nil)) 
             (algebra.Alg.Term algebra.F.id_mark (x11::nil))
         
          (* <mark(U82(X1_,X2_,X3_)),mark(X1_)> *)
         | DP_R_xml_0_scc_65_large_scc_1_large_large_large_13 :
          forall x6 x9 x11 x7, 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                      
             (algebra.Alg.Term algebra.F.id_U82 (x6::x7::x9::nil)) x11) ->
            DP_R_xml_0_scc_65_large_scc_1_large_large_large (algebra.Alg.Term 
                                                             algebra.F.id_mark 
                                                             (x6::nil)) 
             (algebra.Alg.Term algebra.F.id_mark (x11::nil))
         
          (* <mark(U83(X1_,X2_,X3_)),mark(X1_)> *)
         | DP_R_xml_0_scc_65_large_scc_1_large_large_large_14 :
          forall x6 x9 x11 x7, 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                      
             (algebra.Alg.Term algebra.F.id_U83 (x6::x7::x9::nil)) x11) ->
            DP_R_xml_0_scc_65_large_scc_1_large_large_large (algebra.Alg.Term 
                                                             algebra.F.id_mark 
                                                             (x6::nil)) 
             (algebra.Alg.Term algebra.F.id_mark (x11::nil))
         
          (* <mark(U84(X1_,X2_,X3_)),mark(X1_)> *)
         | DP_R_xml_0_scc_65_large_scc_1_large_large_large_15 :
          forall x6 x9 x11 x7, 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                      
             (algebra.Alg.Term algebra.F.id_U84 (x6::x7::x9::nil)) x11) ->
            DP_R_xml_0_scc_65_large_scc_1_large_large_large (algebra.Alg.Term 
                                                             algebra.F.id_mark 
                                                             (x6::nil)) 
             (algebra.Alg.Term algebra.F.id_mark (x11::nil))
         
          (* <mark(U85(X1_,X2_)),mark(X1_)> *)
         | DP_R_xml_0_scc_65_large_scc_1_large_large_large_16 :
          forall x6 x11 x7, 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                      
             (algebra.Alg.Term algebra.F.id_U85 (x6::x7::nil)) x11) ->
            DP_R_xml_0_scc_65_large_scc_1_large_large_large (algebra.Alg.Term 
                                                             algebra.F.id_mark 
                                                             (x6::nil)) 
             (algebra.Alg.Term algebra.F.id_mark (x11::nil))
         
          (* <mark(U86(X_)),mark(X_)> *)
         | DP_R_xml_0_scc_65_large_scc_1_large_large_large_17 :
          forall x8 x11, 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                      
             (algebra.Alg.Term algebra.F.id_U86 (x8::nil)) x11) ->
            DP_R_xml_0_scc_65_large_scc_1_large_large_large (algebra.Alg.Term 
                                                             algebra.F.id_mark 
                                                             (x8::nil)) 
             (algebra.Alg.Term algebra.F.id_mark (x11::nil))
         
          (* <mark(s(X_)),mark(X_)> *)
         | DP_R_xml_0_scc_65_large_scc_1_large_large_large_18 :
          forall x8 x11, 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                      
             (algebra.Alg.Term algebra.F.id_s (x8::nil)) x11) ->
            DP_R_xml_0_scc_65_large_scc_1_large_large_large (algebra.Alg.Term 
                                                             algebra.F.id_mark 
                                                             (x8::nil)) 
             (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       .
       
       
       Inductive DP_R_xml_0_scc_65_large_scc_1_large_large_strict  :
        algebra.Alg.term ->algebra.Alg.term ->Prop := 
          (* <mark(U32(X1_,X2_)),mark(X1_)> *)
         | DP_R_xml_0_scc_65_large_scc_1_large_large_strict_0 :
          forall x6 x11 x7, 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                      
             (algebra.Alg.Term algebra.F.id_U32 (x6::x7::nil)) x11) ->
            DP_R_xml_0_scc_65_large_scc_1_large_large_strict (algebra.Alg.Term 
                                                              algebra.F.id_mark 
                                                              (x6::nil)) 
             (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       .
       
       
       Module WF_DP_R_xml_0_scc_65_large_scc_1_large_large_large.
        Inductive DP_R_xml_0_scc_65_large_scc_1_large_large_large_large  :
         algebra.Alg.term ->algebra.Alg.term ->Prop := 
           (* <mark(U12(X1_,X2_)),mark(X1_)> *)
          | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_0 :
           forall x6 x11 x7, 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                       
              (algebra.Alg.Term algebra.F.id_U12 (x6::x7::nil)) x11) ->
             DP_R_xml_0_scc_65_large_scc_1_large_large_large_large (algebra.Alg.Term 
                                                                    algebra.F.id_mark 
                                                                    (x6::nil)) 
              (algebra.Alg.Term algebra.F.id_mark (x11::nil))
          
           (* <mark(U11(X1_,X2_)),mark(X1_)> *)
          | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_1 :
           forall x6 x11 x7, 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                       
              (algebra.Alg.Term algebra.F.id_U11 (x6::x7::nil)) x11) ->
             DP_R_xml_0_scc_65_large_scc_1_large_large_large_large (algebra.Alg.Term 
                                                                    algebra.F.id_mark 
                                                                    (x6::nil)) 
              (algebra.Alg.Term algebra.F.id_mark (x11::nil))
          
           (* <mark(U13(X_)),mark(X_)> *)
          | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_2 :
           forall x8 x11, 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                       
              (algebra.Alg.Term algebra.F.id_U13 (x8::nil)) x11) ->
             DP_R_xml_0_scc_65_large_scc_1_large_large_large_large (algebra.Alg.Term 
                                                                    algebra.F.id_mark 
                                                                    (x8::nil)) 
              (algebra.Alg.Term algebra.F.id_mark (x11::nil))
          
           (* <mark(U21(X1_,X2_)),mark(X1_)> *)
          | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_3 :
           forall x6 x11 x7, 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                       
              (algebra.Alg.Term algebra.F.id_U21 (x6::x7::nil)) x11) ->
             DP_R_xml_0_scc_65_large_scc_1_large_large_large_large (algebra.Alg.Term 
                                                                    algebra.F.id_mark 
                                                                    (x6::nil)) 
              (algebra.Alg.Term algebra.F.id_mark (x11::nil))
          
           (* <mark(U22(X1_,X2_)),mark(X1_)> *)
          | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_4 :
           forall x6 x11 x7, 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                       
              (algebra.Alg.Term algebra.F.id_U22 (x6::x7::nil)) x11) ->
             DP_R_xml_0_scc_65_large_scc_1_large_large_large_large (algebra.Alg.Term 
                                                                    algebra.F.id_mark 
                                                                    (x6::nil)) 
              (algebra.Alg.Term algebra.F.id_mark (x11::nil))
          
           (* <mark(U23(X_)),mark(X_)> *)
          | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_5 :
           forall x8 x11, 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                       
              (algebra.Alg.Term algebra.F.id_U23 (x8::nil)) x11) ->
             DP_R_xml_0_scc_65_large_scc_1_large_large_large_large (algebra.Alg.Term 
                                                                    algebra.F.id_mark 
                                                                    (x8::nil)) 
              (algebra.Alg.Term algebra.F.id_mark (x11::nil))
          
           (* <mark(U46(X_)),mark(X_)> *)
          | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_6 :
           forall x8 x11, 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                       
              (algebra.Alg.Term algebra.F.id_U46 (x8::nil)) x11) ->
             DP_R_xml_0_scc_65_large_scc_1_large_large_large_large (algebra.Alg.Term 
                                                                    algebra.F.id_mark 
                                                                    (x8::nil)) 
              (algebra.Alg.Term algebra.F.id_mark (x11::nil))
          
           (* <mark(U51(X1_,X2_)),mark(X1_)> *)
          | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_7 :
           forall x6 x11 x7, 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                       
              (algebra.Alg.Term algebra.F.id_U51 (x6::x7::nil)) x11) ->
             DP_R_xml_0_scc_65_large_scc_1_large_large_large_large (algebra.Alg.Term 
                                                                    algebra.F.id_mark 
                                                                    (x6::nil)) 
              (algebra.Alg.Term algebra.F.id_mark (x11::nil))
          
           (* <mark(U52(X_)),mark(X_)> *)
          | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_8 :
           forall x8 x11, 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                       
              (algebra.Alg.Term algebra.F.id_U52 (x8::nil)) x11) ->
             DP_R_xml_0_scc_65_large_scc_1_large_large_large_large (algebra.Alg.Term 
                                                                    algebra.F.id_mark 
                                                                    (x8::nil)) 
              (algebra.Alg.Term algebra.F.id_mark (x11::nil))
          
           (* <mark(U61(X_)),mark(X_)> *)
          | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_9 :
           forall x8 x11, 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                       
              (algebra.Alg.Term algebra.F.id_U61 (x8::nil)) x11) ->
             DP_R_xml_0_scc_65_large_scc_1_large_large_large_large (algebra.Alg.Term 
                                                                    algebra.F.id_mark 
                                                                    (x8::nil)) 
              (algebra.Alg.Term algebra.F.id_mark (x11::nil))
          
           (* <mark(U71(X_)),mark(X_)> *)
          | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_10 :
           forall x8 x11, 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                       
              (algebra.Alg.Term algebra.F.id_U71 (x8::nil)) x11) ->
             DP_R_xml_0_scc_65_large_scc_1_large_large_large_large (algebra.Alg.Term 
                                                                    algebra.F.id_mark 
                                                                    (x8::nil)) 
              (algebra.Alg.Term algebra.F.id_mark (x11::nil))
          
           (* <mark(U81(X1_,X2_,X3_)),mark(X1_)> *)
          | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_11 :
           forall x6 x9 x11 x7, 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                       
              (algebra.Alg.Term algebra.F.id_U81 (x6::x7::x9::nil)) x11) ->
             DP_R_xml_0_scc_65_large_scc_1_large_large_large_large (algebra.Alg.Term 
                                                                    algebra.F.id_mark 
                                                                    (x6::nil)) 
              (algebra.Alg.Term algebra.F.id_mark (x11::nil))
          
           (* <mark(U82(X1_,X2_,X3_)),mark(X1_)> *)
          | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_12 :
           forall x6 x9 x11 x7, 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                       
              (algebra.Alg.Term algebra.F.id_U82 (x6::x7::x9::nil)) x11) ->
             DP_R_xml_0_scc_65_large_scc_1_large_large_large_large (algebra.Alg.Term 
                                                                    algebra.F.id_mark 
                                                                    (x6::nil)) 
              (algebra.Alg.Term algebra.F.id_mark (x11::nil))
          
           (* <mark(U83(X1_,X2_,X3_)),mark(X1_)> *)
          | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_13 :
           forall x6 x9 x11 x7, 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                       
              (algebra.Alg.Term algebra.F.id_U83 (x6::x7::x9::nil)) x11) ->
             DP_R_xml_0_scc_65_large_scc_1_large_large_large_large (algebra.Alg.Term 
                                                                    algebra.F.id_mark 
                                                                    (x6::nil)) 
              (algebra.Alg.Term algebra.F.id_mark (x11::nil))
          
           (* <mark(U84(X1_,X2_,X3_)),mark(X1_)> *)
          | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_14 :
           forall x6 x9 x11 x7, 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                       
              (algebra.Alg.Term algebra.F.id_U84 (x6::x7::x9::nil)) x11) ->
             DP_R_xml_0_scc_65_large_scc_1_large_large_large_large (algebra.Alg.Term 
                                                                    algebra.F.id_mark 
                                                                    (x6::nil)) 
              (algebra.Alg.Term algebra.F.id_mark (x11::nil))
          
           (* <mark(U85(X1_,X2_)),mark(X1_)> *)
          | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_15 :
           forall x6 x11 x7, 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                       
              (algebra.Alg.Term algebra.F.id_U85 (x6::x7::nil)) x11) ->
             DP_R_xml_0_scc_65_large_scc_1_large_large_large_large (algebra.Alg.Term 
                                                                    algebra.F.id_mark 
                                                                    (x6::nil)) 
              (algebra.Alg.Term algebra.F.id_mark (x11::nil))
          
           (* <mark(U86(X_)),mark(X_)> *)
          | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_16 :
           forall x8 x11, 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                       
              (algebra.Alg.Term algebra.F.id_U86 (x8::nil)) x11) ->
             DP_R_xml_0_scc_65_large_scc_1_large_large_large_large (algebra.Alg.Term 
                                                                    algebra.F.id_mark 
                                                                    (x8::nil)) 
              (algebra.Alg.Term algebra.F.id_mark (x11::nil))
          
           (* <mark(s(X_)),mark(X_)> *)
          | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_17 :
           forall x8 x11, 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                       
              (algebra.Alg.Term algebra.F.id_s (x8::nil)) x11) ->
             DP_R_xml_0_scc_65_large_scc_1_large_large_large_large (algebra.Alg.Term 
                                                                    algebra.F.id_mark 
                                                                    (x8::nil)) 
              (algebra.Alg.Term algebra.F.id_mark (x11::nil))
        .
        
        
        Inductive DP_R_xml_0_scc_65_large_scc_1_large_large_large_strict  :
         algebra.Alg.term ->algebra.Alg.term ->Prop := 
           (* <mark(U33(X_)),mark(X_)> *)
          | DP_R_xml_0_scc_65_large_scc_1_large_large_large_strict_0 :
           forall x8 x11, 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                       
              (algebra.Alg.Term algebra.F.id_U33 (x8::nil)) x11) ->
             DP_R_xml_0_scc_65_large_scc_1_large_large_large_strict (algebra.Alg.Term 
                                                                    algebra.F.id_mark 
                                                                    (x8::nil)) 
              (algebra.Alg.Term algebra.F.id_mark (x11::nil))
        .
        
        
        Module WF_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large.
         Inductive DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large  :
          algebra.Alg.term ->algebra.Alg.term ->Prop := 
            (* <mark(U12(X1_,X2_)),mark(X1_)> *)
           | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_0 :
            forall x6 x11 x7, 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                        
               (algebra.Alg.Term algebra.F.id_U12 (x6::x7::nil)) x11) ->
              DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large 
               (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
               (algebra.Alg.Term algebra.F.id_mark (x11::nil))
           
            (* <mark(U11(X1_,X2_)),mark(X1_)> *)
           | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_1 :
            forall x6 x11 x7, 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                        
               (algebra.Alg.Term algebra.F.id_U11 (x6::x7::nil)) x11) ->
              DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large 
               (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
               (algebra.Alg.Term algebra.F.id_mark (x11::nil))
           
            (* <mark(U13(X_)),mark(X_)> *)
           | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_2 :
            forall x8 x11, 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                        
               (algebra.Alg.Term algebra.F.id_U13 (x8::nil)) x11) ->
              DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large 
               (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
               (algebra.Alg.Term algebra.F.id_mark (x11::nil))
           
            (* <mark(U21(X1_,X2_)),mark(X1_)> *)
           | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_3 :
            forall x6 x11 x7, 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                        
               (algebra.Alg.Term algebra.F.id_U21 (x6::x7::nil)) x11) ->
              DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large 
               (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
               (algebra.Alg.Term algebra.F.id_mark (x11::nil))
           
            (* <mark(U22(X1_,X2_)),mark(X1_)> *)
           | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_4 :
            forall x6 x11 x7, 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                        
               (algebra.Alg.Term algebra.F.id_U22 (x6::x7::nil)) x11) ->
              DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large 
               (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
               (algebra.Alg.Term algebra.F.id_mark (x11::nil))
           
            (* <mark(U23(X_)),mark(X_)> *)
           | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_5 :
            forall x8 x11, 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                        
               (algebra.Alg.Term algebra.F.id_U23 (x8::nil)) x11) ->
              DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large 
               (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
               (algebra.Alg.Term algebra.F.id_mark (x11::nil))
           
            (* <mark(U51(X1_,X2_)),mark(X1_)> *)
           | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_6 :
            forall x6 x11 x7, 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                        
               (algebra.Alg.Term algebra.F.id_U51 (x6::x7::nil)) x11) ->
              DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large 
               (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
               (algebra.Alg.Term algebra.F.id_mark (x11::nil))
           
            (* <mark(U52(X_)),mark(X_)> *)
           | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_7 :
            forall x8 x11, 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                        
               (algebra.Alg.Term algebra.F.id_U52 (x8::nil)) x11) ->
              DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large 
               (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
               (algebra.Alg.Term algebra.F.id_mark (x11::nil))
           
            (* <mark(U61(X_)),mark(X_)> *)
           | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_8 :
            forall x8 x11, 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                        
               (algebra.Alg.Term algebra.F.id_U61 (x8::nil)) x11) ->
              DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large 
               (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
               (algebra.Alg.Term algebra.F.id_mark (x11::nil))
           
            (* <mark(U71(X_)),mark(X_)> *)
           | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_9 :
            forall x8 x11, 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                        
               (algebra.Alg.Term algebra.F.id_U71 (x8::nil)) x11) ->
              DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large 
               (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
               (algebra.Alg.Term algebra.F.id_mark (x11::nil))
           
            (* <mark(U81(X1_,X2_,X3_)),mark(X1_)> *)
           | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_10 :
            forall x6 x9 x11 x7, 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                        
               (algebra.Alg.Term algebra.F.id_U81 (x6::x7::x9::nil)) 
               x11) ->
              DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large 
               (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
               (algebra.Alg.Term algebra.F.id_mark (x11::nil))
           
            (* <mark(U82(X1_,X2_,X3_)),mark(X1_)> *)
           | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_11 :
            forall x6 x9 x11 x7, 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                        
               (algebra.Alg.Term algebra.F.id_U82 (x6::x7::x9::nil)) 
               x11) ->
              DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large 
               (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
               (algebra.Alg.Term algebra.F.id_mark (x11::nil))
           
            (* <mark(U83(X1_,X2_,X3_)),mark(X1_)> *)
           | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_12 :
            forall x6 x9 x11 x7, 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                        
               (algebra.Alg.Term algebra.F.id_U83 (x6::x7::x9::nil)) 
               x11) ->
              DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large 
               (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
               (algebra.Alg.Term algebra.F.id_mark (x11::nil))
           
            (* <mark(U84(X1_,X2_,X3_)),mark(X1_)> *)
           | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_13 :
            forall x6 x9 x11 x7, 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                        
               (algebra.Alg.Term algebra.F.id_U84 (x6::x7::x9::nil)) 
               x11) ->
              DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large 
               (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
               (algebra.Alg.Term algebra.F.id_mark (x11::nil))
           
            (* <mark(U85(X1_,X2_)),mark(X1_)> *)
           | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_14 :
            forall x6 x11 x7, 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                        
               (algebra.Alg.Term algebra.F.id_U85 (x6::x7::nil)) x11) ->
              DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large 
               (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
               (algebra.Alg.Term algebra.F.id_mark (x11::nil))
           
            (* <mark(U86(X_)),mark(X_)> *)
           | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_15 :
            forall x8 x11, 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                        
               (algebra.Alg.Term algebra.F.id_U86 (x8::nil)) x11) ->
              DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large 
               (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
               (algebra.Alg.Term algebra.F.id_mark (x11::nil))
           
            (* <mark(s(X_)),mark(X_)> *)
           | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_16 :
            forall x8 x11, 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                        
               (algebra.Alg.Term algebra.F.id_s (x8::nil)) x11) ->
              DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large 
               (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
               (algebra.Alg.Term algebra.F.id_mark (x11::nil))
         .
         
         
         Inductive DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_strict  :
          algebra.Alg.term ->algebra.Alg.term ->Prop := 
            (* <mark(U46(X_)),mark(X_)> *)
           | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_strict_0 :
            forall x8 x11, 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                        
               (algebra.Alg.Term algebra.F.id_U46 (x8::nil)) x11) ->
              DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_strict 
               (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
               (algebra.Alg.Term algebra.F.id_mark (x11::nil))
         .
         
         
         Module WF_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large.
          Inductive DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large  :
           algebra.Alg.term ->algebra.Alg.term ->Prop := 
             (* <mark(U12(X1_,X2_)),mark(X1_)> *)
            | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_0 :
             forall x6 x11 x7, 
              (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                         
                (algebra.Alg.Term algebra.F.id_U12 (x6::x7::nil)) x11) ->
               DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large 
                (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
                (algebra.Alg.Term algebra.F.id_mark (x11::nil))
            
             (* <mark(U13(X_)),mark(X_)> *)
            | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_1 :
             forall x8 x11, 
              (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                         
                (algebra.Alg.Term algebra.F.id_U13 (x8::nil)) x11) ->
               DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large 
                (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
                (algebra.Alg.Term algebra.F.id_mark (x11::nil))
            
             (* <mark(U21(X1_,X2_)),mark(X1_)> *)
            | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_2 :
             forall x6 x11 x7, 
              (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                         
                (algebra.Alg.Term algebra.F.id_U21 (x6::x7::nil)) x11) ->
               DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large 
                (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
                (algebra.Alg.Term algebra.F.id_mark (x11::nil))
            
             (* <mark(U22(X1_,X2_)),mark(X1_)> *)
            | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_3 :
             forall x6 x11 x7, 
              (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                         
                (algebra.Alg.Term algebra.F.id_U22 (x6::x7::nil)) x11) ->
               DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large 
                (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
                (algebra.Alg.Term algebra.F.id_mark (x11::nil))
            
             (* <mark(U23(X_)),mark(X_)> *)
            | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_4 :
             forall x8 x11, 
              (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                         
                (algebra.Alg.Term algebra.F.id_U23 (x8::nil)) x11) ->
               DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large 
                (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
                (algebra.Alg.Term algebra.F.id_mark (x11::nil))
            
             (* <mark(U51(X1_,X2_)),mark(X1_)> *)
            | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_5 :
             forall x6 x11 x7, 
              (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                         
                (algebra.Alg.Term algebra.F.id_U51 (x6::x7::nil)) x11) ->
               DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large 
                (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
                (algebra.Alg.Term algebra.F.id_mark (x11::nil))
            
             (* <mark(U52(X_)),mark(X_)> *)
            | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_6 :
             forall x8 x11, 
              (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                         
                (algebra.Alg.Term algebra.F.id_U52 (x8::nil)) x11) ->
               DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large 
                (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
                (algebra.Alg.Term algebra.F.id_mark (x11::nil))
            
             (* <mark(U61(X_)),mark(X_)> *)
            | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_7 :
             forall x8 x11, 
              (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                         
                (algebra.Alg.Term algebra.F.id_U61 (x8::nil)) x11) ->
               DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large 
                (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
                (algebra.Alg.Term algebra.F.id_mark (x11::nil))
            
             (* <mark(U71(X_)),mark(X_)> *)
            | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_8 :
             forall x8 x11, 
              (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                         
                (algebra.Alg.Term algebra.F.id_U71 (x8::nil)) x11) ->
               DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large 
                (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
                (algebra.Alg.Term algebra.F.id_mark (x11::nil))
            
             (* <mark(U81(X1_,X2_,X3_)),mark(X1_)> *)
            | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_9 :
             forall x6 x9 x11 x7, 
              (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                         
                (algebra.Alg.Term algebra.F.id_U81 (x6::x7::x9::nil)) 
                x11) ->
               DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large 
                (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
                (algebra.Alg.Term algebra.F.id_mark (x11::nil))
            
             (* <mark(U82(X1_,X2_,X3_)),mark(X1_)> *)
            | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_10 :
             forall x6 x9 x11 x7, 
              (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                         
                (algebra.Alg.Term algebra.F.id_U82 (x6::x7::x9::nil)) 
                x11) ->
               DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large 
                (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
                (algebra.Alg.Term algebra.F.id_mark (x11::nil))
            
             (* <mark(U83(X1_,X2_,X3_)),mark(X1_)> *)
            | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_11 :
             forall x6 x9 x11 x7, 
              (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                         
                (algebra.Alg.Term algebra.F.id_U83 (x6::x7::x9::nil)) 
                x11) ->
               DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large 
                (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
                (algebra.Alg.Term algebra.F.id_mark (x11::nil))
            
             (* <mark(U84(X1_,X2_,X3_)),mark(X1_)> *)
            | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_12 :
             forall x6 x9 x11 x7, 
              (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                         
                (algebra.Alg.Term algebra.F.id_U84 (x6::x7::x9::nil)) 
                x11) ->
               DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large 
                (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
                (algebra.Alg.Term algebra.F.id_mark (x11::nil))
            
             (* <mark(U85(X1_,X2_)),mark(X1_)> *)
            | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_13 :
             forall x6 x11 x7, 
              (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                         
                (algebra.Alg.Term algebra.F.id_U85 (x6::x7::nil)) x11) ->
               DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large 
                (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
                (algebra.Alg.Term algebra.F.id_mark (x11::nil))
            
             (* <mark(U86(X_)),mark(X_)> *)
            | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_14 :
             forall x8 x11, 
              (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                         
                (algebra.Alg.Term algebra.F.id_U86 (x8::nil)) x11) ->
               DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large 
                (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
                (algebra.Alg.Term algebra.F.id_mark (x11::nil))
            
             (* <mark(s(X_)),mark(X_)> *)
            | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_15 :
             forall x8 x11, 
              (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                         
                (algebra.Alg.Term algebra.F.id_s (x8::nil)) x11) ->
               DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large 
                (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
                (algebra.Alg.Term algebra.F.id_mark (x11::nil))
          .
          
          
          Inductive DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_strict  :
           algebra.Alg.term ->algebra.Alg.term ->Prop := 
             (* <mark(U11(X1_,X2_)),mark(X1_)> *)
            | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_strict_0 :
             forall x6 x11 x7, 
              (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                         
                (algebra.Alg.Term algebra.F.id_U11 (x6::x7::nil)) x11) ->
               DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_strict 
                (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
                (algebra.Alg.Term algebra.F.id_mark (x11::nil))
          .
          
          
          Module WF_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large.
           Inductive DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large  :
            algebra.Alg.term ->algebra.Alg.term ->Prop := 
              (* <mark(U13(X_)),mark(X_)> *)
             | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_0 :
              forall x8 x11, 
               (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                          
                 (algebra.Alg.Term algebra.F.id_U13 (x8::nil)) x11) ->
                DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large 
                 (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
                 (algebra.Alg.Term algebra.F.id_mark (x11::nil))
             
              (* <mark(U21(X1_,X2_)),mark(X1_)> *)
             | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_1 :
              forall x6 x11 x7, 
               (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                          
                 (algebra.Alg.Term algebra.F.id_U21 (x6::x7::nil)) x11) ->
                DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large 
                 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
                 (algebra.Alg.Term algebra.F.id_mark (x11::nil))
             
              (* <mark(U22(X1_,X2_)),mark(X1_)> *)
             | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_2 :
              forall x6 x11 x7, 
               (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                          
                 (algebra.Alg.Term algebra.F.id_U22 (x6::x7::nil)) x11) ->
                DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large 
                 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
                 (algebra.Alg.Term algebra.F.id_mark (x11::nil))
             
              (* <mark(U23(X_)),mark(X_)> *)
             | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_3 :
              forall x8 x11, 
               (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                          
                 (algebra.Alg.Term algebra.F.id_U23 (x8::nil)) x11) ->
                DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large 
                 (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
                 (algebra.Alg.Term algebra.F.id_mark (x11::nil))
             
              (* <mark(U51(X1_,X2_)),mark(X1_)> *)
             | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_4 :
              forall x6 x11 x7, 
               (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                          
                 (algebra.Alg.Term algebra.F.id_U51 (x6::x7::nil)) x11) ->
                DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large 
                 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
                 (algebra.Alg.Term algebra.F.id_mark (x11::nil))
             
              (* <mark(U52(X_)),mark(X_)> *)
             | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_5 :
              forall x8 x11, 
               (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                          
                 (algebra.Alg.Term algebra.F.id_U52 (x8::nil)) x11) ->
                DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large 
                 (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
                 (algebra.Alg.Term algebra.F.id_mark (x11::nil))
             
              (* <mark(U61(X_)),mark(X_)> *)
             | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_6 :
              forall x8 x11, 
               (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                          
                 (algebra.Alg.Term algebra.F.id_U61 (x8::nil)) x11) ->
                DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large 
                 (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
                 (algebra.Alg.Term algebra.F.id_mark (x11::nil))
             
              (* <mark(U71(X_)),mark(X_)> *)
             | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_7 :
              forall x8 x11, 
               (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                          
                 (algebra.Alg.Term algebra.F.id_U71 (x8::nil)) x11) ->
                DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large 
                 (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
                 (algebra.Alg.Term algebra.F.id_mark (x11::nil))
             
              (* <mark(U81(X1_,X2_,X3_)),mark(X1_)> *)
             | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_8 :
              forall x6 x9 x11 x7, 
               (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                          
                 (algebra.Alg.Term algebra.F.id_U81 (x6::x7::x9::nil)) 
                 x11) ->
                DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large 
                 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
                 (algebra.Alg.Term algebra.F.id_mark (x11::nil))
             
              (* <mark(U82(X1_,X2_,X3_)),mark(X1_)> *)
             | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_9 :
              forall x6 x9 x11 x7, 
               (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                          
                 (algebra.Alg.Term algebra.F.id_U82 (x6::x7::x9::nil)) 
                 x11) ->
                DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large 
                 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
                 (algebra.Alg.Term algebra.F.id_mark (x11::nil))
             
              (* <mark(U83(X1_,X2_,X3_)),mark(X1_)> *)
             | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_10 :
              forall x6 x9 x11 x7, 
               (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                          
                 (algebra.Alg.Term algebra.F.id_U83 (x6::x7::x9::nil)) 
                 x11) ->
                DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large 
                 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
                 (algebra.Alg.Term algebra.F.id_mark (x11::nil))
             
              (* <mark(U84(X1_,X2_,X3_)),mark(X1_)> *)
             | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_11 :
              forall x6 x9 x11 x7, 
               (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                          
                 (algebra.Alg.Term algebra.F.id_U84 (x6::x7::x9::nil)) 
                 x11) ->
                DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large 
                 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
                 (algebra.Alg.Term algebra.F.id_mark (x11::nil))
             
              (* <mark(U85(X1_,X2_)),mark(X1_)> *)
             | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_12 :
              forall x6 x11 x7, 
               (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                          
                 (algebra.Alg.Term algebra.F.id_U85 (x6::x7::nil)) x11) ->
                DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large 
                 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
                 (algebra.Alg.Term algebra.F.id_mark (x11::nil))
             
              (* <mark(U86(X_)),mark(X_)> *)
             | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_13 :
              forall x8 x11, 
               (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                          
                 (algebra.Alg.Term algebra.F.id_U86 (x8::nil)) x11) ->
                DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large 
                 (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
                 (algebra.Alg.Term algebra.F.id_mark (x11::nil))
             
              (* <mark(s(X_)),mark(X_)> *)
             | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_14 :
              forall x8 x11, 
               (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                          
                 (algebra.Alg.Term algebra.F.id_s (x8::nil)) x11) ->
                DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large 
                 (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
                 (algebra.Alg.Term algebra.F.id_mark (x11::nil))
           .
           
           
           Inductive DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_strict  :
            algebra.Alg.term ->algebra.Alg.term ->Prop := 
              (* <mark(U12(X1_,X2_)),mark(X1_)> *)
             | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_strict_0 :
              forall x6 x11 x7, 
               (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                          
                 (algebra.Alg.Term algebra.F.id_U12 (x6::x7::nil)) x11) ->
                DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_strict 
                 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
                 (algebra.Alg.Term algebra.F.id_mark (x11::nil))
           .
           
           
           Module WF_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large.
            Inductive DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large  :
             algebra.Alg.term ->algebra.Alg.term ->Prop := 
               (* <mark(U21(X1_,X2_)),mark(X1_)> *)
              | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_0 :
               forall x6 x11 x7, 
                (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                           
                  (algebra.Alg.Term algebra.F.id_U21 (x6::x7::nil)) x11) ->
                 DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large 
                  (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
                  (algebra.Alg.Term algebra.F.id_mark (x11::nil))
              
               (* <mark(U22(X1_,X2_)),mark(X1_)> *)
              | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_1 :
               forall x6 x11 x7, 
                (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                           
                  (algebra.Alg.Term algebra.F.id_U22 (x6::x7::nil)) x11) ->
                 DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large 
                  (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
                  (algebra.Alg.Term algebra.F.id_mark (x11::nil))
              
               (* <mark(U23(X_)),mark(X_)> *)
              | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_2 :
               forall x8 x11, 
                (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                           
                  (algebra.Alg.Term algebra.F.id_U23 (x8::nil)) x11) ->
                 DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large 
                  (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
                  (algebra.Alg.Term algebra.F.id_mark (x11::nil))
              
               (* <mark(U51(X1_,X2_)),mark(X1_)> *)
              | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_3 :
               forall x6 x11 x7, 
                (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                           
                  (algebra.Alg.Term algebra.F.id_U51 (x6::x7::nil)) x11) ->
                 DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large 
                  (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
                  (algebra.Alg.Term algebra.F.id_mark (x11::nil))
              
               (* <mark(U52(X_)),mark(X_)> *)
              | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_4 :
               forall x8 x11, 
                (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                           
                  (algebra.Alg.Term algebra.F.id_U52 (x8::nil)) x11) ->
                 DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large 
                  (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
                  (algebra.Alg.Term algebra.F.id_mark (x11::nil))
              
               (* <mark(U61(X_)),mark(X_)> *)
              | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_5 :
               forall x8 x11, 
                (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                           
                  (algebra.Alg.Term algebra.F.id_U61 (x8::nil)) x11) ->
                 DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large 
                  (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
                  (algebra.Alg.Term algebra.F.id_mark (x11::nil))
              
               (* <mark(U71(X_)),mark(X_)> *)
              | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_6 :
               forall x8 x11, 
                (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                           
                  (algebra.Alg.Term algebra.F.id_U71 (x8::nil)) x11) ->
                 DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large 
                  (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
                  (algebra.Alg.Term algebra.F.id_mark (x11::nil))
              
               (* <mark(U81(X1_,X2_,X3_)),mark(X1_)> *)
              | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_7 :
               forall x6 x9 x11 x7, 
                (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                           
                  (algebra.Alg.Term algebra.F.id_U81 (x6::x7::x9::nil)) 
                  x11) ->
                 DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large 
                  (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
                  (algebra.Alg.Term algebra.F.id_mark (x11::nil))
              
               (* <mark(U82(X1_,X2_,X3_)),mark(X1_)> *)
              | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_8 :
               forall x6 x9 x11 x7, 
                (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                           
                  (algebra.Alg.Term algebra.F.id_U82 (x6::x7::x9::nil)) 
                  x11) ->
                 DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large 
                  (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
                  (algebra.Alg.Term algebra.F.id_mark (x11::nil))
              
               (* <mark(U83(X1_,X2_,X3_)),mark(X1_)> *)
              | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_9 :
               forall x6 x9 x11 x7, 
                (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                           
                  (algebra.Alg.Term algebra.F.id_U83 (x6::x7::x9::nil)) 
                  x11) ->
                 DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large 
                  (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
                  (algebra.Alg.Term algebra.F.id_mark (x11::nil))
              
               (* <mark(U84(X1_,X2_,X3_)),mark(X1_)> *)
              | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_10 :
               forall x6 x9 x11 x7, 
                (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                           
                  (algebra.Alg.Term algebra.F.id_U84 (x6::x7::x9::nil)) 
                  x11) ->
                 DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large 
                  (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
                  (algebra.Alg.Term algebra.F.id_mark (x11::nil))
              
               (* <mark(U85(X1_,X2_)),mark(X1_)> *)
              | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_11 :
               forall x6 x11 x7, 
                (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                           
                  (algebra.Alg.Term algebra.F.id_U85 (x6::x7::nil)) x11) ->
                 DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large 
                  (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
                  (algebra.Alg.Term algebra.F.id_mark (x11::nil))
              
               (* <mark(U86(X_)),mark(X_)> *)
              | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_12 :
               forall x8 x11, 
                (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                           
                  (algebra.Alg.Term algebra.F.id_U86 (x8::nil)) x11) ->
                 DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large 
                  (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
                  (algebra.Alg.Term algebra.F.id_mark (x11::nil))
              
               (* <mark(s(X_)),mark(X_)> *)
              | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_13 :
               forall x8 x11, 
                (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                           
                  (algebra.Alg.Term algebra.F.id_s (x8::nil)) x11) ->
                 DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large 
                  (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
                  (algebra.Alg.Term algebra.F.id_mark (x11::nil))
            .
            
            
            Inductive DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_strict  :
             algebra.Alg.term ->algebra.Alg.term ->Prop := 
               (* <mark(U13(X_)),mark(X_)> *)
              | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_strict_0 :
               forall x8 x11, 
                (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                           
                  (algebra.Alg.Term algebra.F.id_U13 (x8::nil)) x11) ->
                 DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_strict 
                  (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
                  (algebra.Alg.Term algebra.F.id_mark (x11::nil))
            .
            
            
            Module WF_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large.
             Inductive DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large  :
              algebra.Alg.term ->algebra.Alg.term ->Prop := 
                (* <mark(U21(X1_,X2_)),mark(X1_)> *)
               | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_0 :
                forall x6 x11 x7, 
                 (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                            
                   (algebra.Alg.Term algebra.F.id_U21 (x6::x7::nil)) 
                   x11) ->
                  DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large 
                   (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
                   (algebra.Alg.Term algebra.F.id_mark (x11::nil))
               
                (* <mark(U22(X1_,X2_)),mark(X1_)> *)
               | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_1 :
                forall x6 x11 x7, 
                 (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                            
                   (algebra.Alg.Term algebra.F.id_U22 (x6::x7::nil)) 
                   x11) ->
                  DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large 
                   (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
                   (algebra.Alg.Term algebra.F.id_mark (x11::nil))
               
                (* <mark(U23(X_)),mark(X_)> *)
               | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_2 :
                forall x8 x11, 
                 (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                            
                   (algebra.Alg.Term algebra.F.id_U23 (x8::nil)) x11) ->
                  DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large 
                   (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
                   (algebra.Alg.Term algebra.F.id_mark (x11::nil))
               
                (* <mark(U51(X1_,X2_)),mark(X1_)> *)
               | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_3 :
                forall x6 x11 x7, 
                 (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                            
                   (algebra.Alg.Term algebra.F.id_U51 (x6::x7::nil)) 
                   x11) ->
                  DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large 
                   (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
                   (algebra.Alg.Term algebra.F.id_mark (x11::nil))
               
                (* <mark(U52(X_)),mark(X_)> *)
               | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_4 :
                forall x8 x11, 
                 (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                            
                   (algebra.Alg.Term algebra.F.id_U52 (x8::nil)) x11) ->
                  DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large 
                   (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
                   (algebra.Alg.Term algebra.F.id_mark (x11::nil))
               
                (* <mark(U61(X_)),mark(X_)> *)
               | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_5 :
                forall x8 x11, 
                 (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                            
                   (algebra.Alg.Term algebra.F.id_U61 (x8::nil)) x11) ->
                  DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large 
                   (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
                   (algebra.Alg.Term algebra.F.id_mark (x11::nil))
               
                (* <mark(U71(X_)),mark(X_)> *)
               | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_6 :
                forall x8 x11, 
                 (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                            
                   (algebra.Alg.Term algebra.F.id_U71 (x8::nil)) x11) ->
                  DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large 
                   (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
                   (algebra.Alg.Term algebra.F.id_mark (x11::nil))
               
                (* <mark(U86(X_)),mark(X_)> *)
               | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_7 :
                forall x8 x11, 
                 (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                            
                   (algebra.Alg.Term algebra.F.id_U86 (x8::nil)) x11) ->
                  DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large 
                   (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
                   (algebra.Alg.Term algebra.F.id_mark (x11::nil))
               
                (* <mark(s(X_)),mark(X_)> *)
               | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_8 :
                forall x8 x11, 
                 (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                            
                   (algebra.Alg.Term algebra.F.id_s (x8::nil)) x11) ->
                  DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large 
                   (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
                   (algebra.Alg.Term algebra.F.id_mark (x11::nil))
             .
             
             
             Inductive DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_strict  :
              algebra.Alg.term ->algebra.Alg.term ->Prop := 
                (* <mark(U81(X1_,X2_,X3_)),mark(X1_)> *)
               | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_strict_0 :
                forall x6 x9 x11 x7, 
                 (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                            
                   (algebra.Alg.Term algebra.F.id_U81 (x6::x7::x9::nil)) 
                   x11) ->
                  DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_strict 
                   (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
                   (algebra.Alg.Term algebra.F.id_mark (x11::nil))
               
                (* <mark(U82(X1_,X2_,X3_)),mark(X1_)> *)
               | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_strict_1 :
                forall x6 x9 x11 x7, 
                 (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                            
                   (algebra.Alg.Term algebra.F.id_U82 (x6::x7::x9::nil)) 
                   x11) ->
                  DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_strict 
                   (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
                   (algebra.Alg.Term algebra.F.id_mark (x11::nil))
               
                (* <mark(U83(X1_,X2_,X3_)),mark(X1_)> *)
               | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_strict_2 :
                forall x6 x9 x11 x7, 
                 (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                            
                   (algebra.Alg.Term algebra.F.id_U83 (x6::x7::x9::nil)) 
                   x11) ->
                  DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_strict 
                   (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
                   (algebra.Alg.Term algebra.F.id_mark (x11::nil))
               
                (* <mark(U84(X1_,X2_,X3_)),mark(X1_)> *)
               | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_strict_3 :
                forall x6 x9 x11 x7, 
                 (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                            
                   (algebra.Alg.Term algebra.F.id_U84 (x6::x7::x9::nil)) 
                   x11) ->
                  DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_strict 
                   (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
                   (algebra.Alg.Term algebra.F.id_mark (x11::nil))
               
                (* <mark(U85(X1_,X2_)),mark(X1_)> *)
               | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_strict_4 :
                forall x6 x11 x7, 
                 (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                            
                   (algebra.Alg.Term algebra.F.id_U85 (x6::x7::nil)) 
                   x11) ->
                  DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_strict 
                   (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
                   (algebra.Alg.Term algebra.F.id_mark (x11::nil))
             .
             
             
             Module WF_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large.
              Inductive DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large  :
               algebra.Alg.term ->algebra.Alg.term ->Prop := 
                 (* <mark(U23(X_)),mark(X_)> *)
                | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_0 :
                 forall x8 x11, 
                  (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                             
                    (algebra.Alg.Term algebra.F.id_U23 (x8::nil)) x11) ->
                   DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large 
                    (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
                    (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                
                 (* <mark(U51(X1_,X2_)),mark(X1_)> *)
                | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_1 :
                 forall x6 x11 x7, 
                  (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                             
                    (algebra.Alg.Term algebra.F.id_U51 (x6::x7::nil)) 
                    x11) ->
                   DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large 
                    (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
                    (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                
                 (* <mark(U52(X_)),mark(X_)> *)
                | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_2 :
                 forall x8 x11, 
                  (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                             
                    (algebra.Alg.Term algebra.F.id_U52 (x8::nil)) x11) ->
                   DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large 
                    (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
                    (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                
                 (* <mark(U61(X_)),mark(X_)> *)
                | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_3 :
                 forall x8 x11, 
                  (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                             
                    (algebra.Alg.Term algebra.F.id_U61 (x8::nil)) x11) ->
                   DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large 
                    (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
                    (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                
                 (* <mark(U71(X_)),mark(X_)> *)
                | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_4 :
                 forall x8 x11, 
                  (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                             
                    (algebra.Alg.Term algebra.F.id_U71 (x8::nil)) x11) ->
                   DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large 
                    (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
                    (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                
                 (* <mark(U86(X_)),mark(X_)> *)
                | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_5 :
                 forall x8 x11, 
                  (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                             
                    (algebra.Alg.Term algebra.F.id_U86 (x8::nil)) x11) ->
                   DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large 
                    (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
                    (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                
                 (* <mark(s(X_)),mark(X_)> *)
                | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_6 :
                 forall x8 x11, 
                  (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                             
                    (algebra.Alg.Term algebra.F.id_s (x8::nil)) x11) ->
                   DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large 
                    (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
                    (algebra.Alg.Term algebra.F.id_mark (x11::nil))
              .
              
              
              Inductive DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_strict  :
               algebra.Alg.term ->algebra.Alg.term ->Prop := 
                 (* <mark(U21(X1_,X2_)),mark(X1_)> *)
                | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_strict_0 :
                 forall x6 x11 x7, 
                  (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                             
                    (algebra.Alg.Term algebra.F.id_U21 (x6::x7::nil)) 
                    x11) ->
                   DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_strict 
                    (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
                    (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                
                 (* <mark(U22(X1_,X2_)),mark(X1_)> *)
                | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_strict_1 :
                 forall x6 x11 x7, 
                  (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                             
                    (algebra.Alg.Term algebra.F.id_U22 (x6::x7::nil)) 
                    x11) ->
                   DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_strict 
                    (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
                    (algebra.Alg.Term algebra.F.id_mark (x11::nil))
              .
              
              
              Module WF_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large.
               Inductive DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large  :
                algebra.Alg.term ->algebra.Alg.term ->Prop := 
                  (* <mark(U23(X_)),mark(X_)> *)
                 | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_0 :
                  forall x8 x11, 
                   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                              
                     (algebra.Alg.Term algebra.F.id_U23 (x8::nil)) x11) ->
                    DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large 
                     (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
                     (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                 
                  (* <mark(U51(X1_,X2_)),mark(X1_)> *)
                 | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_1 :
                  forall x6 x11 x7, 
                   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                              
                     (algebra.Alg.Term algebra.F.id_U51 (x6::x7::nil)) 
                     x11) ->
                    DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large 
                     (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
                     (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                 
                  (* <mark(U52(X_)),mark(X_)> *)
                 | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_2 :
                  forall x8 x11, 
                   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                              
                     (algebra.Alg.Term algebra.F.id_U52 (x8::nil)) x11) ->
                    DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large 
                     (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
                     (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                 
                  (* <mark(U61(X_)),mark(X_)> *)
                 | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_3 :
                  forall x8 x11, 
                   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                              
                     (algebra.Alg.Term algebra.F.id_U61 (x8::nil)) x11) ->
                    DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large 
                     (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
                     (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                 
                  (* <mark(U71(X_)),mark(X_)> *)
                 | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_4 :
                  forall x8 x11, 
                   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                              
                     (algebra.Alg.Term algebra.F.id_U71 (x8::nil)) x11) ->
                    DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large 
                     (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
                     (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                 
                  (* <mark(s(X_)),mark(X_)> *)
                 | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_5 :
                  forall x8 x11, 
                   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                              
                     (algebra.Alg.Term algebra.F.id_s (x8::nil)) x11) ->
                    DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large 
                     (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
                     (algebra.Alg.Term algebra.F.id_mark (x11::nil))
               .
               
               
               Inductive DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_strict  :
                algebra.Alg.term ->algebra.Alg.term ->Prop := 
                  (* <mark(U86(X_)),mark(X_)> *)
                 | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_strict_0 :
                  forall x8 x11, 
                   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                              
                     (algebra.Alg.Term algebra.F.id_U86 (x8::nil)) x11) ->
                    DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_strict 
                     (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
                     (algebra.Alg.Term algebra.F.id_mark (x11::nil))
               .
               
               
               Module WF_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large.
                Inductive DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large  :
                 algebra.Alg.term ->algebra.Alg.term ->Prop := 
                   (* <mark(U23(X_)),mark(X_)> *)
                  | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_0 :
                   forall x8 x11, 
                    (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                               
                      (algebra.Alg.Term algebra.F.id_U23 (x8::nil)) x11) ->
                     DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large 
                      (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
                      (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                  
                   (* <mark(U51(X1_,X2_)),mark(X1_)> *)
                  | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_1 :
                   forall x6 x11 x7, 
                    (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                               
                      (algebra.Alg.Term algebra.F.id_U51 (x6::x7::nil)) 
                      x11) ->
                     DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large 
                      (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
                      (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                  
                   (* <mark(U52(X_)),mark(X_)> *)
                  | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_2 :
                   forall x8 x11, 
                    (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                               
                      (algebra.Alg.Term algebra.F.id_U52 (x8::nil)) x11) ->
                     DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large 
                      (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
                      (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                  
                   (* <mark(U71(X_)),mark(X_)> *)
                  | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_3 :
                   forall x8 x11, 
                    (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                               
                      (algebra.Alg.Term algebra.F.id_U71 (x8::nil)) x11) ->
                     DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large 
                      (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
                      (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                  
                   (* <mark(s(X_)),mark(X_)> *)
                  | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_4 :
                   forall x8 x11, 
                    (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                               
                      (algebra.Alg.Term algebra.F.id_s (x8::nil)) x11) ->
                     DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large 
                      (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
                      (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                .
                
                
                Inductive DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_strict  :
                 algebra.Alg.term ->algebra.Alg.term ->Prop := 
                   (* <mark(U61(X_)),mark(X_)> *)
                  | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_strict_0 :
                   forall x8 x11, 
                    (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                               
                      (algebra.Alg.Term algebra.F.id_U61 (x8::nil)) x11) ->
                     DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_strict 
                      (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
                      (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                .
                
                
                Module WF_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large.
                 Inductive DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large  :
                  algebra.Alg.term ->algebra.Alg.term ->Prop := 
                    (* <mark(U23(X_)),mark(X_)> *)
                   | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_0 :
                    forall x8 x11, 
                     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                                
                       (algebra.Alg.Term algebra.F.id_U23 (x8::nil)) 
                       x11) ->
                      DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large 
                       (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
                       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                   
                    (* <mark(U71(X_)),mark(X_)> *)
                   | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_1 :
                    forall x8 x11, 
                     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                                
                       (algebra.Alg.Term algebra.F.id_U71 (x8::nil)) 
                       x11) ->
                      DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large 
                       (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
                       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                   
                    (* <mark(s(X_)),mark(X_)> *)
                   | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_2 :
                    forall x8 x11, 
                     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                                
                       (algebra.Alg.Term algebra.F.id_s (x8::nil)) x11) ->
                      DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large 
                       (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
                       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                 .
                 
                 
                 Inductive DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_strict  :
                  algebra.Alg.term ->algebra.Alg.term ->Prop := 
                    (* <mark(U51(X1_,X2_)),mark(X1_)> *)
                   | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_strict_0 :
                    forall x6 x11 x7, 
                     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                                
                       (algebra.Alg.Term algebra.F.id_U51 (x6::x7::nil)) 
                       x11) ->
                      DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_strict 
                       (algebra.Alg.Term algebra.F.id_mark (x6::nil)) 
                       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                   
                    (* <mark(U52(X_)),mark(X_)> *)
                   | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_strict_1 :
                    forall x8 x11, 
                     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                                
                       (algebra.Alg.Term algebra.F.id_U52 (x8::nil)) 
                       x11) ->
                      DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_strict 
                       (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
                       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                 .
                 
                 
                 Module WF_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large.
                  Inductive DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_large  :
                   algebra.Alg.term ->algebra.Alg.term ->Prop := 
                     (* <mark(U23(X_)),mark(X_)> *)
                    | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_large_0 :
                     forall x8 x11, 
                      (closure.refl_trans_clos (algebra.EQT.one_step 
                                                 R_xml_0_deep_rew.R_xml_0_rules)
                                                 
                        (algebra.Alg.Term algebra.F.id_U23 (x8::nil)) 
                        x11) ->
                       DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_large 
                        (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
                        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                    
                     (* <mark(U71(X_)),mark(X_)> *)
                    | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_large_1 :
                     forall x8 x11, 
                      (closure.refl_trans_clos (algebra.EQT.one_step 
                                                 R_xml_0_deep_rew.R_xml_0_rules)
                                                 
                        (algebra.Alg.Term algebra.F.id_U71 (x8::nil)) 
                        x11) ->
                       DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_large 
                        (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
                        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                  .
                  
                  
                  Inductive DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_strict  :
                   algebra.Alg.term ->algebra.Alg.term ->Prop := 
                     (* <mark(s(X_)),mark(X_)> *)
                    | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_strict_0 :
                     forall x8 x11, 
                      (closure.refl_trans_clos (algebra.EQT.one_step 
                                                 R_xml_0_deep_rew.R_xml_0_rules)
                                                 
                        (algebra.Alg.Term algebra.F.id_s (x8::nil)) x11) ->
                       DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_strict 
                        (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
                        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                  .
                  
                  
                  Module WF_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_large.
                   Inductive DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_large_large  :
                    algebra.Alg.term ->algebra.Alg.term ->Prop := 
                      (* <mark(U71(X_)),mark(X_)> *)
                     | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_large_large_0 :
                      forall x8 x11, 
                       (closure.refl_trans_clos (algebra.EQT.one_step 
                                                  R_xml_0_deep_rew.R_xml_0_rules)
                                                  
                         (algebra.Alg.Term algebra.F.id_U71 (x8::nil)) 
                         x11) ->
                        DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_large_large 
                         (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
                         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                   .
                   
                   
                   Inductive DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_large_strict  :
                    algebra.Alg.term ->algebra.Alg.term ->Prop := 
                      (* <mark(U23(X_)),mark(X_)> *)
                     | DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_large_strict_0 :
                      forall x8 x11, 
                       (closure.refl_trans_clos (algebra.EQT.one_step 
                                                  R_xml_0_deep_rew.R_xml_0_rules)
                                                  
                         (algebra.Alg.Term algebra.F.id_U23 (x8::nil)) 
                         x11) ->
                        DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_large_strict 
                         (algebra.Alg.Term algebra.F.id_mark (x8::nil)) 
                         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                   .
                   
                   
                   Module WF_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_large_large.
                    Open Scope Z_scope.
                    
                    Import ring_extention.
                    
                    Notation Local "a <= b" := (Zle a b).
                    
                    Notation Local "a < b" := (Zlt a b).
                    
                    Definition P_id_a__zeros  := 0.
                    
                    Definition P_id_U51 (x11:Z) (x12:Z) := 1.
                    
                    Definition P_id_a__U84 (x11:Z) (x12:Z) (x13:Z) := 2* x13.
                    
                    Definition P_id_a__U32 (x11:Z) (x12:Z) := 1* x11.
                    
                    Definition P_id_isNatList (x11:Z) := 1* x11.
                    
                    Definition P_id_a__U13 (x11:Z) := 1.
                    
                    Definition P_id_U85 (x11:Z) (x12:Z) := 2* x12.
                    
                    Definition P_id_a__length (x11:Z) := 1 + 1* x11.
                    
                    Definition P_id_a__isNatIList (x11:Z) := 1.
                    
                    Definition P_id_U33 (x11:Z) := 1.
                    
                    Definition P_id_a__U11 (x11:Z) (x12:Z) := 1 + 1* x11.
                    
                    Definition P_id_U81 (x11:Z) (x12:Z) (x13:Z) := 2* x13.
                    
                    Definition P_id_a__U92 (x11:Z) (x12:Z) (x13:Z) := 
                      2* x11 + 2* x12.
                    
                    Definition P_id_a__U43 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
                    
                    Definition P_id_U23 (x11:Z) := 1.
                    
                    Definition P_id_a__isNatKind (x11:Z) := 1* x11.
                    
                    Definition P_id_U93 (x11:Z) (x12:Z) (x13:Z) := 1 + 1* x12
                      .
                    
                    Definition P_id_U11 (x11:Z) (x12:Z) := 1 + 1* x11.
                    
                    Definition P_id_a__U71 (x11:Z) := 1 + 1* x11.
                    
                    Definition P_id_U44 (x11:Z) (x12:Z) (x13:Z) := 1.
                    
                    Definition P_id_0  := 2.
                    
                    Definition P_id_U61 (x11:Z) := 1* x11.
                    
                    Definition P_id_a__U86 (x11:Z) := 2* x11.
                    
                    Definition P_id_a__U41 (x11:Z) (x12:Z) (x13:Z) := 1.
                    
                    Definition P_id_U22 (x11:Z) (x12:Z) := 1.
                    
                    Definition P_id_a__U21 (x11:Z) (x12:Z) := 1.
                    
                    Definition P_id_U91 (x11:Z) (x12:Z) (x13:Z) := 
                      1 + 1* x11 + 1* x12.
                    
                    Definition P_id_length (x11:Z) := 1 + 1* x11.
                    
                    Definition P_id_a__U52 (x11:Z) := 1.
                    
                    Definition P_id_U42 (x11:Z) (x12:Z) (x13:Z) := 1.
                    
                    Definition P_id_a__U12 (x11:Z) (x12:Z) := 1* x11.
                    
                    Definition P_id_U83 (x11:Z) (x12:Z) (x13:Z) := 1* x13.
                    
                    Definition P_id_a__U94 (x11:Z) (x12:Z) := 2 + 2* x12.
                    
                    Definition P_id_a__U45 (x11:Z) (x12:Z) := 1.
                    
                    Definition P_id_U31 (x11:Z) (x12:Z) := 1.
                    
                    Definition P_id_a__isNat (x11:Z) := 2.
                    
                    Definition P_id_isNatIListKind (x11:Z) := 1.
                    
                    Definition P_id_a__U82 (x11:Z) (x12:Z) (x13:Z) := 2* x13.
                    
                    Definition P_id_U46 (x11:Z) := 1.
                    
                    Definition P_id_cons (x11:Z) (x12:Z) := 3* x12.
                    
                    Definition P_id_U52 (x11:Z) := 1.
                    
                    Definition P_id_a__U85 (x11:Z) (x12:Z) := 2* x12.
                    
                    Definition P_id_a__U33 (x11:Z) := 1.
                    
                    Definition P_id_U21 (x11:Z) (x12:Z) := 1.
                    
                    Definition P_id_a__isNatList (x11:Z) := 1* x11.
                    
                    Definition P_id_U86 (x11:Z) := 2* x11.
                    
                    Definition P_id_mark (x11:Z) := 2* x11.
                    
                    Definition P_id_a__U51 (x11:Z) (x12:Z) := 1.
                    
                    Definition P_id_U41 (x11:Z) (x12:Z) (x13:Z) := 1.
                    
                    Definition P_id_tt  := 1.
                    
                    Definition P_id_U82 (x11:Z) (x12:Z) (x13:Z) := 2* x13.
                    
                    Definition P_id_a__U93 (x11:Z) (x12:Z) (x13:Z) := 
                      2 + 2* x12.
                    
                    Definition P_id_a__U44 (x11:Z) (x12:Z) (x13:Z) := 1.
                    
                    Definition P_id_isNat (x11:Z) := 1.
                    
                    Definition P_id_a__U23 (x11:Z) := 1.
                    
                    Definition P_id_U94 (x11:Z) (x12:Z) := 2 + 2* x12.
                    
                    Definition P_id_U12 (x11:Z) (x12:Z) := 1* x11.
                    
                    Definition P_id_a__U81 (x11:Z) (x12:Z) (x13:Z) := 3* x13.
                    
                    Definition P_id_U45 (x11:Z) (x12:Z) := 1.
                    
                    Definition P_id_zeros  := 0.
                    
                    Definition P_id_U71 (x11:Z) := 1 + 1* x11.
                    
                    Definition P_id_a__U91 (x11:Z) (x12:Z) (x13:Z) := 
                      1 + 1* x11 + 2* x12.
                    
                    Definition P_id_a__U42 (x11:Z) (x12:Z) (x13:Z) := 1.
                    
                    Definition P_id_isNatKind (x11:Z) := 1* x11.
                    
                    Definition P_id_a__U22 (x11:Z) (x12:Z) := 1.
                    
                    Definition P_id_U92 (x11:Z) (x12:Z) (x13:Z) := 
                      2* x11 + 1* x12.
                    
                    Definition P_id_nil  := 1.
                    
                    Definition P_id_a__U61 (x11:Z) := 1* x11.
                    
                    Definition P_id_U43 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
                    
                    Definition P_id_a__isNatIListKind (x11:Z) := 1.
                    
                    Definition P_id_U84 (x11:Z) (x12:Z) (x13:Z) := 1* x13.
                    
                    Definition P_id_s (x11:Z) := 1 + 1* x11.
                    
                    Definition P_id_a__U46 (x11:Z) := 1.
                    
                    Definition P_id_U32 (x11:Z) (x12:Z) := 1* x11.
                    
                    Definition P_id_a__U31 (x11:Z) (x12:Z) := 1.
                    
                    Definition P_id_U13 (x11:Z) := 1.
                    
                    Definition P_id_a__U83 (x11:Z) (x12:Z) (x13:Z) := 2* x13.
                    
                    Definition P_id_isNatIList (x11:Z) := 1.
                    
                    Lemma P_id_U51_monotonic :
                     forall x12 x14 x13 x11, 
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_U51 x12 x14 <= P_id_U51 x11 x13.
                    Proof.
                      intros x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U84_monotonic :
                     forall x16 x12 x14 x13 x11 x15, 
                      (0 <= x16)/\ (x16 <= x15) ->
                       (0 <= x14)/\ (x14 <= x13) ->
                        (0 <= x12)/\ (x12 <= x11) ->
                         P_id_a__U84 x12 x14 x16 <= P_id_a__U84 x11 x13 x15.
                    Proof.
                      intros x16 x15 x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      intros [H_5 H_4].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U32_monotonic :
                     forall x12 x14 x13 x11, 
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_a__U32 x12 x14 <= P_id_a__U32 x11 x13.
                    Proof.
                      intros x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_isNatList_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_isNatList x12 <= P_id_isNatList x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_0].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U13_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_a__U13 x12 <= P_id_a__U13 x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_0].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U85_monotonic :
                     forall x12 x14 x13 x11, 
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_U85 x12 x14 <= P_id_U85 x11 x13.
                    Proof.
                      intros x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__length_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_a__length x12 <= P_id_a__length x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_0].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__isNatIList_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_a__isNatIList x12 <= P_id_a__isNatIList x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_0].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U33_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_U33 x12 <= P_id_U33 x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_0].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U11_monotonic :
                     forall x12 x14 x13 x11, 
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_a__U11 x12 x14 <= P_id_a__U11 x11 x13.
                    Proof.
                      intros x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U81_monotonic :
                     forall x16 x12 x14 x13 x11 x15, 
                      (0 <= x16)/\ (x16 <= x15) ->
                       (0 <= x14)/\ (x14 <= x13) ->
                        (0 <= x12)/\ (x12 <= x11) ->
                         P_id_U81 x12 x14 x16 <= P_id_U81 x11 x13 x15.
                    Proof.
                      intros x16 x15 x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      intros [H_5 H_4].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U92_monotonic :
                     forall x16 x12 x14 x13 x11 x15, 
                      (0 <= x16)/\ (x16 <= x15) ->
                       (0 <= x14)/\ (x14 <= x13) ->
                        (0 <= x12)/\ (x12 <= x11) ->
                         P_id_a__U92 x12 x14 x16 <= P_id_a__U92 x11 x13 x15.
                    Proof.
                      intros x16 x15 x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      intros [H_5 H_4].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U43_monotonic :
                     forall x16 x12 x14 x13 x11 x15, 
                      (0 <= x16)/\ (x16 <= x15) ->
                       (0 <= x14)/\ (x14 <= x13) ->
                        (0 <= x12)/\ (x12 <= x11) ->
                         P_id_a__U43 x12 x14 x16 <= P_id_a__U43 x11 x13 x15.
                    Proof.
                      intros x16 x15 x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      intros [H_5 H_4].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U23_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_U23 x12 <= P_id_U23 x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_0].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__isNatKind_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_a__isNatKind x12 <= P_id_a__isNatKind x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_0].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U93_monotonic :
                     forall x16 x12 x14 x13 x11 x15, 
                      (0 <= x16)/\ (x16 <= x15) ->
                       (0 <= x14)/\ (x14 <= x13) ->
                        (0 <= x12)/\ (x12 <= x11) ->
                         P_id_U93 x12 x14 x16 <= P_id_U93 x11 x13 x15.
                    Proof.
                      intros x16 x15 x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      intros [H_5 H_4].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U11_monotonic :
                     forall x12 x14 x13 x11, 
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_U11 x12 x14 <= P_id_U11 x11 x13.
                    Proof.
                      intros x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U71_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_a__U71 x12 <= P_id_a__U71 x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_0].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U44_monotonic :
                     forall x16 x12 x14 x13 x11 x15, 
                      (0 <= x16)/\ (x16 <= x15) ->
                       (0 <= x14)/\ (x14 <= x13) ->
                        (0 <= x12)/\ (x12 <= x11) ->
                         P_id_U44 x12 x14 x16 <= P_id_U44 x11 x13 x15.
                    Proof.
                      intros x16 x15 x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      intros [H_5 H_4].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U61_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_U61 x12 <= P_id_U61 x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_0].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U86_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_a__U86 x12 <= P_id_a__U86 x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_0].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U41_monotonic :
                     forall x16 x12 x14 x13 x11 x15, 
                      (0 <= x16)/\ (x16 <= x15) ->
                       (0 <= x14)/\ (x14 <= x13) ->
                        (0 <= x12)/\ (x12 <= x11) ->
                         P_id_a__U41 x12 x14 x16 <= P_id_a__U41 x11 x13 x15.
                    Proof.
                      intros x16 x15 x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      intros [H_5 H_4].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U22_monotonic :
                     forall x12 x14 x13 x11, 
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_U22 x12 x14 <= P_id_U22 x11 x13.
                    Proof.
                      intros x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U21_monotonic :
                     forall x12 x14 x13 x11, 
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_a__U21 x12 x14 <= P_id_a__U21 x11 x13.
                    Proof.
                      intros x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U91_monotonic :
                     forall x16 x12 x14 x13 x11 x15, 
                      (0 <= x16)/\ (x16 <= x15) ->
                       (0 <= x14)/\ (x14 <= x13) ->
                        (0 <= x12)/\ (x12 <= x11) ->
                         P_id_U91 x12 x14 x16 <= P_id_U91 x11 x13 x15.
                    Proof.
                      intros x16 x15 x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      intros [H_5 H_4].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_length_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_length x12 <= P_id_length x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_0].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U52_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_a__U52 x12 <= P_id_a__U52 x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_0].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U42_monotonic :
                     forall x16 x12 x14 x13 x11 x15, 
                      (0 <= x16)/\ (x16 <= x15) ->
                       (0 <= x14)/\ (x14 <= x13) ->
                        (0 <= x12)/\ (x12 <= x11) ->
                         P_id_U42 x12 x14 x16 <= P_id_U42 x11 x13 x15.
                    Proof.
                      intros x16 x15 x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      intros [H_5 H_4].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U12_monotonic :
                     forall x12 x14 x13 x11, 
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_a__U12 x12 x14 <= P_id_a__U12 x11 x13.
                    Proof.
                      intros x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U83_monotonic :
                     forall x16 x12 x14 x13 x11 x15, 
                      (0 <= x16)/\ (x16 <= x15) ->
                       (0 <= x14)/\ (x14 <= x13) ->
                        (0 <= x12)/\ (x12 <= x11) ->
                         P_id_U83 x12 x14 x16 <= P_id_U83 x11 x13 x15.
                    Proof.
                      intros x16 x15 x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      intros [H_5 H_4].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U94_monotonic :
                     forall x12 x14 x13 x11, 
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_a__U94 x12 x14 <= P_id_a__U94 x11 x13.
                    Proof.
                      intros x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U45_monotonic :
                     forall x12 x14 x13 x11, 
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_a__U45 x12 x14 <= P_id_a__U45 x11 x13.
                    Proof.
                      intros x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U31_monotonic :
                     forall x12 x14 x13 x11, 
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_U31 x12 x14 <= P_id_U31 x11 x13.
                    Proof.
                      intros x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__isNat_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_a__isNat x12 <= P_id_a__isNat x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_0].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_isNatIListKind_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_isNatIListKind x12 <= P_id_isNatIListKind x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_0].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U82_monotonic :
                     forall x16 x12 x14 x13 x11 x15, 
                      (0 <= x16)/\ (x16 <= x15) ->
                       (0 <= x14)/\ (x14 <= x13) ->
                        (0 <= x12)/\ (x12 <= x11) ->
                         P_id_a__U82 x12 x14 x16 <= P_id_a__U82 x11 x13 x15.
                    Proof.
                      intros x16 x15 x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      intros [H_5 H_4].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U46_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_U46 x12 <= P_id_U46 x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_0].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_cons_monotonic :
                     forall x12 x14 x13 x11, 
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_cons x12 x14 <= P_id_cons x11 x13.
                    Proof.
                      intros x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U52_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_U52 x12 <= P_id_U52 x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_0].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U85_monotonic :
                     forall x12 x14 x13 x11, 
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_a__U85 x12 x14 <= P_id_a__U85 x11 x13.
                    Proof.
                      intros x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U33_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_a__U33 x12 <= P_id_a__U33 x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_0].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U21_monotonic :
                     forall x12 x14 x13 x11, 
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_U21 x12 x14 <= P_id_U21 x11 x13.
                    Proof.
                      intros x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__isNatList_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_a__isNatList x12 <= P_id_a__isNatList x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_0].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U86_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_U86 x12 <= P_id_U86 x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_0].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_mark_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_mark x12 <= P_id_mark x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_0].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U51_monotonic :
                     forall x12 x14 x13 x11, 
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_a__U51 x12 x14 <= P_id_a__U51 x11 x13.
                    Proof.
                      intros x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U41_monotonic :
                     forall x16 x12 x14 x13 x11 x15, 
                      (0 <= x16)/\ (x16 <= x15) ->
                       (0 <= x14)/\ (x14 <= x13) ->
                        (0 <= x12)/\ (x12 <= x11) ->
                         P_id_U41 x12 x14 x16 <= P_id_U41 x11 x13 x15.
                    Proof.
                      intros x16 x15 x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      intros [H_5 H_4].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U82_monotonic :
                     forall x16 x12 x14 x13 x11 x15, 
                      (0 <= x16)/\ (x16 <= x15) ->
                       (0 <= x14)/\ (x14 <= x13) ->
                        (0 <= x12)/\ (x12 <= x11) ->
                         P_id_U82 x12 x14 x16 <= P_id_U82 x11 x13 x15.
                    Proof.
                      intros x16 x15 x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      intros [H_5 H_4].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U93_monotonic :
                     forall x16 x12 x14 x13 x11 x15, 
                      (0 <= x16)/\ (x16 <= x15) ->
                       (0 <= x14)/\ (x14 <= x13) ->
                        (0 <= x12)/\ (x12 <= x11) ->
                         P_id_a__U93 x12 x14 x16 <= P_id_a__U93 x11 x13 x15.
                    Proof.
                      intros x16 x15 x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      intros [H_5 H_4].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U44_monotonic :
                     forall x16 x12 x14 x13 x11 x15, 
                      (0 <= x16)/\ (x16 <= x15) ->
                       (0 <= x14)/\ (x14 <= x13) ->
                        (0 <= x12)/\ (x12 <= x11) ->
                         P_id_a__U44 x12 x14 x16 <= P_id_a__U44 x11 x13 x15.
                    Proof.
                      intros x16 x15 x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      intros [H_5 H_4].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_isNat_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_isNat x12 <= P_id_isNat x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_0].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U23_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_a__U23 x12 <= P_id_a__U23 x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_0].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U94_monotonic :
                     forall x12 x14 x13 x11, 
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_U94 x12 x14 <= P_id_U94 x11 x13.
                    Proof.
                      intros x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U12_monotonic :
                     forall x12 x14 x13 x11, 
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_U12 x12 x14 <= P_id_U12 x11 x13.
                    Proof.
                      intros x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U81_monotonic :
                     forall x16 x12 x14 x13 x11 x15, 
                      (0 <= x16)/\ (x16 <= x15) ->
                       (0 <= x14)/\ (x14 <= x13) ->
                        (0 <= x12)/\ (x12 <= x11) ->
                         P_id_a__U81 x12 x14 x16 <= P_id_a__U81 x11 x13 x15.
                    Proof.
                      intros x16 x15 x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      intros [H_5 H_4].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U45_monotonic :
                     forall x12 x14 x13 x11, 
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_U45 x12 x14 <= P_id_U45 x11 x13.
                    Proof.
                      intros x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U71_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_U71 x12 <= P_id_U71 x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_0].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U91_monotonic :
                     forall x16 x12 x14 x13 x11 x15, 
                      (0 <= x16)/\ (x16 <= x15) ->
                       (0 <= x14)/\ (x14 <= x13) ->
                        (0 <= x12)/\ (x12 <= x11) ->
                         P_id_a__U91 x12 x14 x16 <= P_id_a__U91 x11 x13 x15.
                    Proof.
                      intros x16 x15 x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      intros [H_5 H_4].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U42_monotonic :
                     forall x16 x12 x14 x13 x11 x15, 
                      (0 <= x16)/\ (x16 <= x15) ->
                       (0 <= x14)/\ (x14 <= x13) ->
                        (0 <= x12)/\ (x12 <= x11) ->
                         P_id_a__U42 x12 x14 x16 <= P_id_a__U42 x11 x13 x15.
                    Proof.
                      intros x16 x15 x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      intros [H_5 H_4].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_isNatKind_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_isNatKind x12 <= P_id_isNatKind x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_0].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U22_monotonic :
                     forall x12 x14 x13 x11, 
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_a__U22 x12 x14 <= P_id_a__U22 x11 x13.
                    Proof.
                      intros x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U92_monotonic :
                     forall x16 x12 x14 x13 x11 x15, 
                      (0 <= x16)/\ (x16 <= x15) ->
                       (0 <= x14)/\ (x14 <= x13) ->
                        (0 <= x12)/\ (x12 <= x11) ->
                         P_id_U92 x12 x14 x16 <= P_id_U92 x11 x13 x15.
                    Proof.
                      intros x16 x15 x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      intros [H_5 H_4].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U61_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_a__U61 x12 <= P_id_a__U61 x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_0].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U43_monotonic :
                     forall x16 x12 x14 x13 x11 x15, 
                      (0 <= x16)/\ (x16 <= x15) ->
                       (0 <= x14)/\ (x14 <= x13) ->
                        (0 <= x12)/\ (x12 <= x11) ->
                         P_id_U43 x12 x14 x16 <= P_id_U43 x11 x13 x15.
                    Proof.
                      intros x16 x15 x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      intros [H_5 H_4].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__isNatIListKind_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_a__isNatIListKind x12 <= P_id_a__isNatIListKind 
                                                      x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_0].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U84_monotonic :
                     forall x16 x12 x14 x13 x11 x15, 
                      (0 <= x16)/\ (x16 <= x15) ->
                       (0 <= x14)/\ (x14 <= x13) ->
                        (0 <= x12)/\ (x12 <= x11) ->
                         P_id_U84 x12 x14 x16 <= P_id_U84 x11 x13 x15.
                    Proof.
                      intros x16 x15 x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      intros [H_5 H_4].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_s_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->P_id_s x12 <= P_id_s x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_0].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U46_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_a__U46 x12 <= P_id_a__U46 x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_0].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U32_monotonic :
                     forall x12 x14 x13 x11, 
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_U32 x12 x14 <= P_id_U32 x11 x13.
                    Proof.
                      intros x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U31_monotonic :
                     forall x12 x14 x13 x11, 
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_a__U31 x12 x14 <= P_id_a__U31 x11 x13.
                    Proof.
                      intros x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U13_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_U13 x12 <= P_id_U13 x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_0].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U83_monotonic :
                     forall x16 x12 x14 x13 x11 x15, 
                      (0 <= x16)/\ (x16 <= x15) ->
                       (0 <= x14)/\ (x14 <= x13) ->
                        (0 <= x12)/\ (x12 <= x11) ->
                         P_id_a__U83 x12 x14 x16 <= P_id_a__U83 x11 x13 x15.
                    Proof.
                      intros x16 x15 x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      intros [H_5 H_4].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_isNatIList_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_isNatIList x12 <= P_id_isNatIList x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_0].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__zeros_bounded : 0 <= P_id_a__zeros .
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U51_bounded :
                     forall x12 x11, 
                      (0 <= x11) ->(0 <= x12) ->0 <= P_id_U51 x12 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U84_bounded :
                     forall x12 x13 x11, 
                      (0 <= x11) ->
                       (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U84 x13 x12 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U32_bounded :
                     forall x12 x11, 
                      (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U32 x12 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_isNatList_bounded :
                     forall x11, (0 <= x11) ->0 <= P_id_isNatList x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U13_bounded :
                     forall x11, (0 <= x11) ->0 <= P_id_a__U13 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U85_bounded :
                     forall x12 x11, 
                      (0 <= x11) ->(0 <= x12) ->0 <= P_id_U85 x12 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__length_bounded :
                     forall x11, (0 <= x11) ->0 <= P_id_a__length x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__isNatIList_bounded :
                     forall x11, (0 <= x11) ->0 <= P_id_a__isNatIList x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U33_bounded :
                     forall x11, (0 <= x11) ->0 <= P_id_U33 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U11_bounded :
                     forall x12 x11, 
                      (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U11 x12 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U81_bounded :
                     forall x12 x13 x11, 
                      (0 <= x11) ->
                       (0 <= x12) ->(0 <= x13) ->0 <= P_id_U81 x13 x12 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U92_bounded :
                     forall x12 x13 x11, 
                      (0 <= x11) ->
                       (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U92 x13 x12 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U43_bounded :
                     forall x12 x13 x11, 
                      (0 <= x11) ->
                       (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U43 x13 x12 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U23_bounded :
                     forall x11, (0 <= x11) ->0 <= P_id_U23 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__isNatKind_bounded :
                     forall x11, (0 <= x11) ->0 <= P_id_a__isNatKind x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U93_bounded :
                     forall x12 x13 x11, 
                      (0 <= x11) ->
                       (0 <= x12) ->(0 <= x13) ->0 <= P_id_U93 x13 x12 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U11_bounded :
                     forall x12 x11, 
                      (0 <= x11) ->(0 <= x12) ->0 <= P_id_U11 x12 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U71_bounded :
                     forall x11, (0 <= x11) ->0 <= P_id_a__U71 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U44_bounded :
                     forall x12 x13 x11, 
                      (0 <= x11) ->
                       (0 <= x12) ->(0 <= x13) ->0 <= P_id_U44 x13 x12 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_0_bounded : 0 <= P_id_0 .
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U61_bounded :
                     forall x11, (0 <= x11) ->0 <= P_id_U61 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U86_bounded :
                     forall x11, (0 <= x11) ->0 <= P_id_a__U86 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U41_bounded :
                     forall x12 x13 x11, 
                      (0 <= x11) ->
                       (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U41 x13 x12 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U22_bounded :
                     forall x12 x11, 
                      (0 <= x11) ->(0 <= x12) ->0 <= P_id_U22 x12 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U21_bounded :
                     forall x12 x11, 
                      (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U21 x12 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U91_bounded :
                     forall x12 x13 x11, 
                      (0 <= x11) ->
                       (0 <= x12) ->(0 <= x13) ->0 <= P_id_U91 x13 x12 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_length_bounded :
                     forall x11, (0 <= x11) ->0 <= P_id_length x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U52_bounded :
                     forall x11, (0 <= x11) ->0 <= P_id_a__U52 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U42_bounded :
                     forall x12 x13 x11, 
                      (0 <= x11) ->
                       (0 <= x12) ->(0 <= x13) ->0 <= P_id_U42 x13 x12 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U12_bounded :
                     forall x12 x11, 
                      (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U12 x12 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U83_bounded :
                     forall x12 x13 x11, 
                      (0 <= x11) ->
                       (0 <= x12) ->(0 <= x13) ->0 <= P_id_U83 x13 x12 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U94_bounded :
                     forall x12 x11, 
                      (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U94 x12 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U45_bounded :
                     forall x12 x11, 
                      (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U45 x12 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U31_bounded :
                     forall x12 x11, 
                      (0 <= x11) ->(0 <= x12) ->0 <= P_id_U31 x12 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__isNat_bounded :
                     forall x11, (0 <= x11) ->0 <= P_id_a__isNat x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_isNatIListKind_bounded :
                     forall x11, (0 <= x11) ->0 <= P_id_isNatIListKind x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U82_bounded :
                     forall x12 x13 x11, 
                      (0 <= x11) ->
                       (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U82 x13 x12 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U46_bounded :
                     forall x11, (0 <= x11) ->0 <= P_id_U46 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_cons_bounded :
                     forall x12 x11, 
                      (0 <= x11) ->(0 <= x12) ->0 <= P_id_cons x12 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U52_bounded :
                     forall x11, (0 <= x11) ->0 <= P_id_U52 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U85_bounded :
                     forall x12 x11, 
                      (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U85 x12 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U33_bounded :
                     forall x11, (0 <= x11) ->0 <= P_id_a__U33 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U21_bounded :
                     forall x12 x11, 
                      (0 <= x11) ->(0 <= x12) ->0 <= P_id_U21 x12 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__isNatList_bounded :
                     forall x11, (0 <= x11) ->0 <= P_id_a__isNatList x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U86_bounded :
                     forall x11, (0 <= x11) ->0 <= P_id_U86 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_mark_bounded :
                     forall x11, (0 <= x11) ->0 <= P_id_mark x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U51_bounded :
                     forall x12 x11, 
                      (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U51 x12 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U41_bounded :
                     forall x12 x13 x11, 
                      (0 <= x11) ->
                       (0 <= x12) ->(0 <= x13) ->0 <= P_id_U41 x13 x12 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_tt_bounded : 0 <= P_id_tt .
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U82_bounded :
                     forall x12 x13 x11, 
                      (0 <= x11) ->
                       (0 <= x12) ->(0 <= x13) ->0 <= P_id_U82 x13 x12 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U93_bounded :
                     forall x12 x13 x11, 
                      (0 <= x11) ->
                       (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U93 x13 x12 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U44_bounded :
                     forall x12 x13 x11, 
                      (0 <= x11) ->
                       (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U44 x13 x12 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_isNat_bounded :
                     forall x11, (0 <= x11) ->0 <= P_id_isNat x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U23_bounded :
                     forall x11, (0 <= x11) ->0 <= P_id_a__U23 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U94_bounded :
                     forall x12 x11, 
                      (0 <= x11) ->(0 <= x12) ->0 <= P_id_U94 x12 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U12_bounded :
                     forall x12 x11, 
                      (0 <= x11) ->(0 <= x12) ->0 <= P_id_U12 x12 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U81_bounded :
                     forall x12 x13 x11, 
                      (0 <= x11) ->
                       (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U81 x13 x12 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U45_bounded :
                     forall x12 x11, 
                      (0 <= x11) ->(0 <= x12) ->0 <= P_id_U45 x12 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_zeros_bounded : 0 <= P_id_zeros .
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U71_bounded :
                     forall x11, (0 <= x11) ->0 <= P_id_U71 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U91_bounded :
                     forall x12 x13 x11, 
                      (0 <= x11) ->
                       (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U91 x13 x12 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U42_bounded :
                     forall x12 x13 x11, 
                      (0 <= x11) ->
                       (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U42 x13 x12 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_isNatKind_bounded :
                     forall x11, (0 <= x11) ->0 <= P_id_isNatKind x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U22_bounded :
                     forall x12 x11, 
                      (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U22 x12 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U92_bounded :
                     forall x12 x13 x11, 
                      (0 <= x11) ->
                       (0 <= x12) ->(0 <= x13) ->0 <= P_id_U92 x13 x12 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_nil_bounded : 0 <= P_id_nil .
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U61_bounded :
                     forall x11, (0 <= x11) ->0 <= P_id_a__U61 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U43_bounded :
                     forall x12 x13 x11, 
                      (0 <= x11) ->
                       (0 <= x12) ->(0 <= x13) ->0 <= P_id_U43 x13 x12 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__isNatIListKind_bounded :
                     forall x11, (0 <= x11) ->0 <= P_id_a__isNatIListKind x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U84_bounded :
                     forall x12 x13 x11, 
                      (0 <= x11) ->
                       (0 <= x12) ->(0 <= x13) ->0 <= P_id_U84 x13 x12 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_s_bounded :
                     forall x11, (0 <= x11) ->0 <= P_id_s x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U46_bounded :
                     forall x11, (0 <= x11) ->0 <= P_id_a__U46 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U32_bounded :
                     forall x12 x11, 
                      (0 <= x11) ->(0 <= x12) ->0 <= P_id_U32 x12 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U31_bounded :
                     forall x12 x11, 
                      (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U31 x12 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U13_bounded :
                     forall x11, (0 <= x11) ->0 <= P_id_U13 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_a__U83_bounded :
                     forall x12 x13 x11, 
                      (0 <= x11) ->
                       (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U83 x13 x12 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_isNatIList_bounded :
                     forall x11, (0 <= x11) ->0 <= P_id_isNatIList x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Definition measure  := 
                      InterpZ.measure 0 P_id_a__zeros P_id_U51 P_id_a__U84 
                       P_id_a__U32 P_id_isNatList P_id_a__U13 P_id_U85 
                       P_id_a__length P_id_a__isNatIList P_id_U33 
                       P_id_a__U11 P_id_U81 P_id_a__U92 P_id_a__U43 P_id_U23 
                       P_id_a__isNatKind P_id_U93 P_id_U11 P_id_a__U71 
                       P_id_U44 P_id_0 P_id_U61 P_id_a__U86 P_id_a__U41 
                       P_id_U22 P_id_a__U21 P_id_U91 P_id_length P_id_a__U52 
                       P_id_U42 P_id_a__U12 P_id_U83 P_id_a__U94 P_id_a__U45 
                       P_id_U31 P_id_a__isNat P_id_isNatIListKind 
                       P_id_a__U82 P_id_U46 P_id_cons P_id_U52 P_id_a__U85 
                       P_id_a__U33 P_id_U21 P_id_a__isNatList P_id_U86 
                       P_id_mark P_id_a__U51 P_id_U41 P_id_tt P_id_U82 
                       P_id_a__U93 P_id_a__U44 P_id_isNat P_id_a__U23 
                       P_id_U94 P_id_U12 P_id_a__U81 P_id_U45 P_id_zeros 
                       P_id_U71 P_id_a__U91 P_id_a__U42 P_id_isNatKind 
                       P_id_a__U22 P_id_U92 P_id_nil P_id_a__U61 P_id_U43 
                       P_id_a__isNatIListKind P_id_U84 P_id_s P_id_a__U46 
                       P_id_U32 P_id_a__U31 P_id_U13 P_id_a__U83 
                       P_id_isNatIList.
                    
                    Lemma measure_equation :
                     forall t, 
                      measure t = match t with
                                    | (algebra.Alg.Term 
                                       algebra.F.id_a__zeros nil) =>
                                     P_id_a__zeros 
                                    | (algebra.Alg.Term algebra.F.id_U51 
                                       (x12::x11::nil)) =>
                                     P_id_U51 (measure x12) (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U84 
                                       (x13::x12::x11::nil)) =>
                                     P_id_a__U84 (measure x13) (measure x12) 
                                      (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U32 
                                       (x12::x11::nil)) =>
                                     P_id_a__U32 (measure x12) (measure x11)
                                    | (algebra.Alg.Term 
                                       algebra.F.id_isNatList (x11::nil)) =>
                                     P_id_isNatList (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U13 
                                       (x11::nil)) =>
                                     P_id_a__U13 (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_U85 
                                       (x12::x11::nil)) =>
                                     P_id_U85 (measure x12) (measure x11)
                                    | (algebra.Alg.Term 
                                       algebra.F.id_a__length (x11::nil)) =>
                                     P_id_a__length (measure x11)
                                    | (algebra.Alg.Term 
                                       algebra.F.id_a__isNatIList (x11::nil)) =>
                                     P_id_a__isNatIList (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_U33 
                                       (x11::nil)) =>
                                     P_id_U33 (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U11 
                                       (x12::x11::nil)) =>
                                     P_id_a__U11 (measure x12) (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_U81 
                                       (x13::x12::x11::nil)) =>
                                     P_id_U81 (measure x13) (measure x12) 
                                      (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U92 
                                       (x13::x12::x11::nil)) =>
                                     P_id_a__U92 (measure x13) (measure x12) 
                                      (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U43 
                                       (x13::x12::x11::nil)) =>
                                     P_id_a__U43 (measure x13) (measure x12) 
                                      (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_U23 
                                       (x11::nil)) =>
                                     P_id_U23 (measure x11)
                                    | (algebra.Alg.Term 
                                       algebra.F.id_a__isNatKind (x11::nil)) =>
                                     P_id_a__isNatKind (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_U93 
                                       (x13::x12::x11::nil)) =>
                                     P_id_U93 (measure x13) (measure x12) 
                                      (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_U11 
                                       (x12::x11::nil)) =>
                                     P_id_U11 (measure x12) (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U71 
                                       (x11::nil)) =>
                                     P_id_a__U71 (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_U44 
                                       (x13::x12::x11::nil)) =>
                                     P_id_U44 (measure x13) (measure x12) 
                                      (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_0 nil) =>
                                     P_id_0 
                                    | (algebra.Alg.Term algebra.F.id_U61 
                                       (x11::nil)) =>
                                     P_id_U61 (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U86 
                                       (x11::nil)) =>
                                     P_id_a__U86 (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U41 
                                       (x13::x12::x11::nil)) =>
                                     P_id_a__U41 (measure x13) (measure x12) 
                                      (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_U22 
                                       (x12::x11::nil)) =>
                                     P_id_U22 (measure x12) (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U21 
                                       (x12::x11::nil)) =>
                                     P_id_a__U21 (measure x12) (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_U91 
                                       (x13::x12::x11::nil)) =>
                                     P_id_U91 (measure x13) (measure x12) 
                                      (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_length 
                                       (x11::nil)) =>
                                     P_id_length (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U52 
                                       (x11::nil)) =>
                                     P_id_a__U52 (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_U42 
                                       (x13::x12::x11::nil)) =>
                                     P_id_U42 (measure x13) (measure x12) 
                                      (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U12 
                                       (x12::x11::nil)) =>
                                     P_id_a__U12 (measure x12) (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_U83 
                                       (x13::x12::x11::nil)) =>
                                     P_id_U83 (measure x13) (measure x12) 
                                      (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U94 
                                       (x12::x11::nil)) =>
                                     P_id_a__U94 (measure x12) (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U45 
                                       (x12::x11::nil)) =>
                                     P_id_a__U45 (measure x12) (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_U31 
                                       (x12::x11::nil)) =>
                                     P_id_U31 (measure x12) (measure x11)
                                    | (algebra.Alg.Term 
                                       algebra.F.id_a__isNat (x11::nil)) =>
                                     P_id_a__isNat (measure x11)
                                    | (algebra.Alg.Term 
                                       algebra.F.id_isNatIListKind 
                                       (x11::nil)) =>
                                     P_id_isNatIListKind (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U82 
                                       (x13::x12::x11::nil)) =>
                                     P_id_a__U82 (measure x13) (measure x12) 
                                      (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_U46 
                                       (x11::nil)) =>
                                     P_id_U46 (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_cons 
                                       (x12::x11::nil)) =>
                                     P_id_cons (measure x12) (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_U52 
                                       (x11::nil)) =>
                                     P_id_U52 (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U85 
                                       (x12::x11::nil)) =>
                                     P_id_a__U85 (measure x12) (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U33 
                                       (x11::nil)) =>
                                     P_id_a__U33 (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_U21 
                                       (x12::x11::nil)) =>
                                     P_id_U21 (measure x12) (measure x11)
                                    | (algebra.Alg.Term 
                                       algebra.F.id_a__isNatList (x11::nil)) =>
                                     P_id_a__isNatList (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_U86 
                                       (x11::nil)) =>
                                     P_id_U86 (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_mark 
                                       (x11::nil)) =>
                                     P_id_mark (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U51 
                                       (x12::x11::nil)) =>
                                     P_id_a__U51 (measure x12) (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_U41 
                                       (x13::x12::x11::nil)) =>
                                     P_id_U41 (measure x13) (measure x12) 
                                      (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_tt nil) =>
                                     P_id_tt 
                                    | (algebra.Alg.Term algebra.F.id_U82 
                                       (x13::x12::x11::nil)) =>
                                     P_id_U82 (measure x13) (measure x12) 
                                      (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U93 
                                       (x13::x12::x11::nil)) =>
                                     P_id_a__U93 (measure x13) (measure x12) 
                                      (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U44 
                                       (x13::x12::x11::nil)) =>
                                     P_id_a__U44 (measure x13) (measure x12) 
                                      (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_isNat 
                                       (x11::nil)) =>
                                     P_id_isNat (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U23 
                                       (x11::nil)) =>
                                     P_id_a__U23 (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_U94 
                                       (x12::x11::nil)) =>
                                     P_id_U94 (measure x12) (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_U12 
                                       (x12::x11::nil)) =>
                                     P_id_U12 (measure x12) (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U81 
                                       (x13::x12::x11::nil)) =>
                                     P_id_a__U81 (measure x13) (measure x12) 
                                      (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_U45 
                                       (x12::x11::nil)) =>
                                     P_id_U45 (measure x12) (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_zeros 
                                       nil) =>
                                     P_id_zeros 
                                    | (algebra.Alg.Term algebra.F.id_U71 
                                       (x11::nil)) =>
                                     P_id_U71 (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U91 
                                       (x13::x12::x11::nil)) =>
                                     P_id_a__U91 (measure x13) (measure x12) 
                                      (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U42 
                                       (x13::x12::x11::nil)) =>
                                     P_id_a__U42 (measure x13) (measure x12) 
                                      (measure x11)
                                    | (algebra.Alg.Term 
                                       algebra.F.id_isNatKind (x11::nil)) =>
                                     P_id_isNatKind (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U22 
                                       (x12::x11::nil)) =>
                                     P_id_a__U22 (measure x12) (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_U92 
                                       (x13::x12::x11::nil)) =>
                                     P_id_U92 (measure x13) (measure x12) 
                                      (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_nil nil) =>
                                     P_id_nil 
                                    | (algebra.Alg.Term algebra.F.id_a__U61 
                                       (x11::nil)) =>
                                     P_id_a__U61 (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_U43 
                                       (x13::x12::x11::nil)) =>
                                     P_id_U43 (measure x13) (measure x12) 
                                      (measure x11)
                                    | (algebra.Alg.Term 
                                       algebra.F.id_a__isNatIListKind 
                                       (x11::nil)) =>
                                     P_id_a__isNatIListKind (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_U84 
                                       (x13::x12::x11::nil)) =>
                                     P_id_U84 (measure x13) (measure x12) 
                                      (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_s 
                                       (x11::nil)) =>
                                     P_id_s (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U46 
                                       (x11::nil)) =>
                                     P_id_a__U46 (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_U32 
                                       (x12::x11::nil)) =>
                                     P_id_U32 (measure x12) (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U31 
                                       (x12::x11::nil)) =>
                                     P_id_a__U31 (measure x12) (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_U13 
                                       (x11::nil)) =>
                                     P_id_U13 (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U83 
                                       (x13::x12::x11::nil)) =>
                                     P_id_a__U83 (measure x13) (measure x12) 
                                      (measure x11)
                                    | (algebra.Alg.Term 
                                       algebra.F.id_isNatIList (x11::nil)) =>
                                     P_id_isNatIList (measure x11)
                                    | _ => 0
                                    end.
                    Proof.
                      intros t;case t;intros ;apply refl_equal.
                    Qed.
                    
                    Lemma measure_bounded : forall t, 0 <= measure t.
                    Proof.
                      unfold measure in |-*.
                      
                      apply InterpZ.measure_bounded;
                       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                        intuition;
                        (auto with zarith)||
                        (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Ltac generate_pos_hyp  :=
                     match goal with
                       | H:context [measure ?x] |- _ =>
                        let v := fresh "v" in 
                         (let H := fresh "h" in 
                           (set (H:=measure_bounded x) in *;
                             set (v:=measure x) in *;clearbody H;clearbody v))
                       |  |- context [measure ?x] =>
                        let v := fresh "v" in 
                         (let H := fresh "h" in 
                           (set (H:=measure_bounded x) in *;
                             set (v:=measure x) in *;clearbody H;clearbody v))
                       end
                     .
                    
                    Lemma rules_monotonic :
                     forall l r, 
                      (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) ->
                       measure r <= measure l.
                    Proof.
                      intros l r H.
                      fold measure in |-*.
                      
                      inversion H;clear H;subst;inversion H0;clear H0;
                       subst;simpl algebra.EQT.T.apply_subst in |-*;
                       repeat (
                       match goal with
                         |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
                          rewrite (measure_equation (algebra.Alg.Term f t))
                         end
                       );repeat (generate_pos_hyp );
                       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                        intuition;
                        (auto with zarith)||
                        (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma measure_star_monotonic :
                     forall l r, 
                      (closure.refl_trans_clos (algebra.EQT.one_step 
                                                 R_xml_0_deep_rew.R_xml_0_rules)
                                                 r l) ->
                       measure r <= measure l.
                    Proof.
                      unfold measure in *.
                      apply InterpZ.measure_star_monotonic.
                      intros ;apply P_id_U51_monotonic;assumption.
                      intros ;apply P_id_a__U84_monotonic;assumption.
                      intros ;apply P_id_a__U32_monotonic;assumption.
                      intros ;apply P_id_isNatList_monotonic;assumption.
                      intros ;apply P_id_a__U13_monotonic;assumption.
                      intros ;apply P_id_U85_monotonic;assumption.
                      intros ;apply P_id_a__length_monotonic;assumption.
                      intros ;apply P_id_a__isNatIList_monotonic;assumption.
                      intros ;apply P_id_U33_monotonic;assumption.
                      intros ;apply P_id_a__U11_monotonic;assumption.
                      intros ;apply P_id_U81_monotonic;assumption.
                      intros ;apply P_id_a__U92_monotonic;assumption.
                      intros ;apply P_id_a__U43_monotonic;assumption.
                      intros ;apply P_id_U23_monotonic;assumption.
                      intros ;apply P_id_a__isNatKind_monotonic;assumption.
                      intros ;apply P_id_U93_monotonic;assumption.
                      intros ;apply P_id_U11_monotonic;assumption.
                      intros ;apply P_id_a__U71_monotonic;assumption.
                      intros ;apply P_id_U44_monotonic;assumption.
                      intros ;apply P_id_U61_monotonic;assumption.
                      intros ;apply P_id_a__U86_monotonic;assumption.
                      intros ;apply P_id_a__U41_monotonic;assumption.
                      intros ;apply P_id_U22_monotonic;assumption.
                      intros ;apply P_id_a__U21_monotonic;assumption.
                      intros ;apply P_id_U91_monotonic;assumption.
                      intros ;apply P_id_length_monotonic;assumption.
                      intros ;apply P_id_a__U52_monotonic;assumption.
                      intros ;apply P_id_U42_monotonic;assumption.
                      intros ;apply P_id_a__U12_monotonic;assumption.
                      intros ;apply P_id_U83_monotonic;assumption.
                      intros ;apply P_id_a__U94_monotonic;assumption.
                      intros ;apply P_id_a__U45_monotonic;assumption.
                      intros ;apply P_id_U31_monotonic;assumption.
                      intros ;apply P_id_a__isNat_monotonic;assumption.
                      intros ;apply P_id_isNatIListKind_monotonic;assumption.
                      intros ;apply P_id_a__U82_monotonic;assumption.
                      intros ;apply P_id_U46_monotonic;assumption.
                      intros ;apply P_id_cons_monotonic;assumption.
                      intros ;apply P_id_U52_monotonic;assumption.
                      intros ;apply P_id_a__U85_monotonic;assumption.
                      intros ;apply P_id_a__U33_monotonic;assumption.
                      intros ;apply P_id_U21_monotonic;assumption.
                      intros ;apply P_id_a__isNatList_monotonic;assumption.
                      intros ;apply P_id_U86_monotonic;assumption.
                      intros ;apply P_id_mark_monotonic;assumption.
                      intros ;apply P_id_a__U51_monotonic;assumption.
                      intros ;apply P_id_U41_monotonic;assumption.
                      intros ;apply P_id_U82_monotonic;assumption.
                      intros ;apply P_id_a__U93_monotonic;assumption.
                      intros ;apply P_id_a__U44_monotonic;assumption.
                      intros ;apply P_id_isNat_monotonic;assumption.
                      intros ;apply P_id_a__U23_monotonic;assumption.
                      intros ;apply P_id_U94_monotonic;assumption.
                      intros ;apply P_id_U12_monotonic;assumption.
                      intros ;apply P_id_a__U81_monotonic;assumption.
                      intros ;apply P_id_U45_monotonic;assumption.
                      intros ;apply P_id_U71_monotonic;assumption.
                      intros ;apply P_id_a__U91_monotonic;assumption.
                      intros ;apply P_id_a__U42_monotonic;assumption.
                      intros ;apply P_id_isNatKind_monotonic;assumption.
                      intros ;apply P_id_a__U22_monotonic;assumption.
                      intros ;apply P_id_U92_monotonic;assumption.
                      intros ;apply P_id_a__U61_monotonic;assumption.
                      intros ;apply P_id_U43_monotonic;assumption.
                      
                      intros ;apply P_id_a__isNatIListKind_monotonic;
                       assumption.
                      intros ;apply P_id_U84_monotonic;assumption.
                      intros ;apply P_id_s_monotonic;assumption.
                      intros ;apply P_id_a__U46_monotonic;assumption.
                      intros ;apply P_id_U32_monotonic;assumption.
                      intros ;apply P_id_a__U31_monotonic;assumption.
                      intros ;apply P_id_U13_monotonic;assumption.
                      intros ;apply P_id_a__U83_monotonic;assumption.
                      intros ;apply P_id_isNatIList_monotonic;assumption.
                      intros ;apply P_id_a__zeros_bounded;assumption.
                      intros ;apply P_id_U51_bounded;assumption.
                      intros ;apply P_id_a__U84_bounded;assumption.
                      intros ;apply P_id_a__U32_bounded;assumption.
                      intros ;apply P_id_isNatList_bounded;assumption.
                      intros ;apply P_id_a__U13_bounded;assumption.
                      intros ;apply P_id_U85_bounded;assumption.
                      intros ;apply P_id_a__length_bounded;assumption.
                      intros ;apply P_id_a__isNatIList_bounded;assumption.
                      intros ;apply P_id_U33_bounded;assumption.
                      intros ;apply P_id_a__U11_bounded;assumption.
                      intros ;apply P_id_U81_bounded;assumption.
                      intros ;apply P_id_a__U92_bounded;assumption.
                      intros ;apply P_id_a__U43_bounded;assumption.
                      intros ;apply P_id_U23_bounded;assumption.
                      intros ;apply P_id_a__isNatKind_bounded;assumption.
                      intros ;apply P_id_U93_bounded;assumption.
                      intros ;apply P_id_U11_bounded;assumption.
                      intros ;apply P_id_a__U71_bounded;assumption.
                      intros ;apply P_id_U44_bounded;assumption.
                      intros ;apply P_id_0_bounded;assumption.
                      intros ;apply P_id_U61_bounded;assumption.
                      intros ;apply P_id_a__U86_bounded;assumption.
                      intros ;apply P_id_a__U41_bounded;assumption.
                      intros ;apply P_id_U22_bounded;assumption.
                      intros ;apply P_id_a__U21_bounded;assumption.
                      intros ;apply P_id_U91_bounded;assumption.
                      intros ;apply P_id_length_bounded;assumption.
                      intros ;apply P_id_a__U52_bounded;assumption.
                      intros ;apply P_id_U42_bounded;assumption.
                      intros ;apply P_id_a__U12_bounded;assumption.
                      intros ;apply P_id_U83_bounded;assumption.
                      intros ;apply P_id_a__U94_bounded;assumption.
                      intros ;apply P_id_a__U45_bounded;assumption.
                      intros ;apply P_id_U31_bounded;assumption.
                      intros ;apply P_id_a__isNat_bounded;assumption.
                      intros ;apply P_id_isNatIListKind_bounded;assumption.
                      intros ;apply P_id_a__U82_bounded;assumption.
                      intros ;apply P_id_U46_bounded;assumption.
                      intros ;apply P_id_cons_bounded;assumption.
                      intros ;apply P_id_U52_bounded;assumption.
                      intros ;apply P_id_a__U85_bounded;assumption.
                      intros ;apply P_id_a__U33_bounded;assumption.
                      intros ;apply P_id_U21_bounded;assumption.
                      intros ;apply P_id_a__isNatList_bounded;assumption.
                      intros ;apply P_id_U86_bounded;assumption.
                      intros ;apply P_id_mark_bounded;assumption.
                      intros ;apply P_id_a__U51_bounded;assumption.
                      intros ;apply P_id_U41_bounded;assumption.
                      intros ;apply P_id_tt_bounded;assumption.
                      intros ;apply P_id_U82_bounded;assumption.
                      intros ;apply P_id_a__U93_bounded;assumption.
                      intros ;apply P_id_a__U44_bounded;assumption.
                      intros ;apply P_id_isNat_bounded;assumption.
                      intros ;apply P_id_a__U23_bounded;assumption.
                      intros ;apply P_id_U94_bounded;assumption.
                      intros ;apply P_id_U12_bounded;assumption.
                      intros ;apply P_id_a__U81_bounded;assumption.
                      intros ;apply P_id_U45_bounded;assumption.
                      intros ;apply P_id_zeros_bounded;assumption.
                      intros ;apply P_id_U71_bounded;assumption.
                      intros ;apply P_id_a__U91_bounded;assumption.
                      intros ;apply P_id_a__U42_bounded;assumption.
                      intros ;apply P_id_isNatKind_bounded;assumption.
                      intros ;apply P_id_a__U22_bounded;assumption.
                      intros ;apply P_id_U92_bounded;assumption.
                      intros ;apply P_id_nil_bounded;assumption.
                      intros ;apply P_id_a__U61_bounded;assumption.
                      intros ;apply P_id_U43_bounded;assumption.
                      
                      intros ;apply P_id_a__isNatIListKind_bounded;assumption.
                      intros ;apply P_id_U84_bounded;assumption.
                      intros ;apply P_id_s_bounded;assumption.
                      intros ;apply P_id_a__U46_bounded;assumption.
                      intros ;apply P_id_U32_bounded;assumption.
                      intros ;apply P_id_a__U31_bounded;assumption.
                      intros ;apply P_id_U13_bounded;assumption.
                      intros ;apply P_id_a__U83_bounded;assumption.
                      intros ;apply P_id_isNatIList_bounded;assumption.
                      apply rules_monotonic.
                    Qed.
                    
                    Definition P_id_A__U46 (x11:Z) := 0.
                    
                    Definition P_id_A__ISNATILISTKIND (x11:Z) := 0.
                    
                    Definition P_id_A__U61 (x11:Z) := 0.
                    
                    Definition P_id_A__U85 (x11:Z) (x12:Z) := 0.
                    
                    Definition P_id_A__U31 (x11:Z) (x12:Z) := 0.
                    
                    Definition P_id_A__U81 (x11:Z) (x12:Z) (x13:Z) := 0.
                    
                    Definition P_id_A__U22 (x11:Z) (x12:Z) := 0.
                    
                    Definition P_id_A__U93 (x11:Z) (x12:Z) (x13:Z) := 0.
                    
                    Definition P_id_A__U42 (x11:Z) (x12:Z) (x13:Z) := 0.
                    
                    Definition P_id_A__U51 (x11:Z) (x12:Z) := 0.
                    
                    Definition P_id_A__ISNATLIST (x11:Z) := 0.
                    
                    Definition P_id_A__ZEROS  := 0.
                    
                    Definition P_id_A__U91 (x11:Z) (x12:Z) (x13:Z) := 0.
                    
                    Definition P_id_A__U33 (x11:Z) := 0.
                    
                    Definition P_id_A__U83 (x11:Z) (x12:Z) (x13:Z) := 0.
                    
                    Definition P_id_A__U23 (x11:Z) := 0.
                    
                    Definition P_id_A__U11 (x11:Z) (x12:Z) := 0.
                    
                    Definition P_id_A__LENGTH (x11:Z) := 0.
                    
                    Definition P_id_A__U44 (x11:Z) (x12:Z) (x13:Z) := 0.
                    
                    Definition P_id_A__ISNATILIST (x11:Z) := 0.
                    
                    Definition P_id_A__U13 (x11:Z) := 0.
                    
                    Definition P_id_A__U71 (x11:Z) := 0.
                    
                    Definition P_id_A__U86 (x11:Z) := 0.
                    
                    Definition P_id_A__U32 (x11:Z) (x12:Z) := 0.
                    
                    Definition P_id_A__U82 (x11:Z) (x12:Z) (x13:Z) := 0.
                    
                    Definition P_id_A__ISNATKIND (x11:Z) := 0.
                    
                    Definition P_id_A__U94 (x11:Z) (x12:Z) := 0.
                    
                    Definition P_id_A__U43 (x11:Z) (x12:Z) (x13:Z) := 0.
                    
                    Definition P_id_A__U52 (x11:Z) := 0.
                    
                    Definition P_id_A__U21 (x11:Z) (x12:Z) := 0.
                    
                    Definition P_id_A__U92 (x11:Z) (x12:Z) (x13:Z) := 0.
                    
                    Definition P_id_A__U41 (x11:Z) (x12:Z) (x13:Z) := 0.
                    
                    Definition P_id_A__U84 (x11:Z) (x12:Z) (x13:Z) := 0.
                    
                    Definition P_id_A__ISNAT (x11:Z) := 0.
                    
                    Definition P_id_A__U12 (x11:Z) (x12:Z) := 0.
                    
                    Definition P_id_MARK (x11:Z) := 2* x11.
                    
                    Definition P_id_A__U45 (x11:Z) (x12:Z) := 0.
                    
                    Lemma P_id_A__U46_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_A__U46 x12 <= P_id_A__U46 x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_0].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_A__ISNATILISTKIND_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_A__ISNATILISTKIND x12 <= P_id_A__ISNATILISTKIND 
                                                      x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_0].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_A__U61_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_A__U61 x12 <= P_id_A__U61 x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_0].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_A__U85_monotonic :
                     forall x12 x14 x13 x11, 
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_A__U85 x12 x14 <= P_id_A__U85 x11 x13.
                    Proof.
                      intros x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_A__U31_monotonic :
                     forall x12 x14 x13 x11, 
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_A__U31 x12 x14 <= P_id_A__U31 x11 x13.
                    Proof.
                      intros x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_A__U81_monotonic :
                     forall x16 x12 x14 x13 x11 x15, 
                      (0 <= x16)/\ (x16 <= x15) ->
                       (0 <= x14)/\ (x14 <= x13) ->
                        (0 <= x12)/\ (x12 <= x11) ->
                         P_id_A__U81 x12 x14 x16 <= P_id_A__U81 x11 x13 x15.
                    Proof.
                      intros x16 x15 x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      intros [H_5 H_4].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_A__U22_monotonic :
                     forall x12 x14 x13 x11, 
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_A__U22 x12 x14 <= P_id_A__U22 x11 x13.
                    Proof.
                      intros x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_A__U93_monotonic :
                     forall x16 x12 x14 x13 x11 x15, 
                      (0 <= x16)/\ (x16 <= x15) ->
                       (0 <= x14)/\ (x14 <= x13) ->
                        (0 <= x12)/\ (x12 <= x11) ->
                         P_id_A__U93 x12 x14 x16 <= P_id_A__U93 x11 x13 x15.
                    Proof.
                      intros x16 x15 x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      intros [H_5 H_4].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_A__U42_monotonic :
                     forall x16 x12 x14 x13 x11 x15, 
                      (0 <= x16)/\ (x16 <= x15) ->
                       (0 <= x14)/\ (x14 <= x13) ->
                        (0 <= x12)/\ (x12 <= x11) ->
                         P_id_A__U42 x12 x14 x16 <= P_id_A__U42 x11 x13 x15.
                    Proof.
                      intros x16 x15 x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      intros [H_5 H_4].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_A__U51_monotonic :
                     forall x12 x14 x13 x11, 
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_A__U51 x12 x14 <= P_id_A__U51 x11 x13.
                    Proof.
                      intros x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_A__ISNATLIST_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_A__ISNATLIST x12 <= P_id_A__ISNATLIST x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_0].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_A__U91_monotonic :
                     forall x16 x12 x14 x13 x11 x15, 
                      (0 <= x16)/\ (x16 <= x15) ->
                       (0 <= x14)/\ (x14 <= x13) ->
                        (0 <= x12)/\ (x12 <= x11) ->
                         P_id_A__U91 x12 x14 x16 <= P_id_A__U91 x11 x13 x15.
                    Proof.
                      intros x16 x15 x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      intros [H_5 H_4].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_A__U33_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_A__U33 x12 <= P_id_A__U33 x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_0].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_A__U83_monotonic :
                     forall x16 x12 x14 x13 x11 x15, 
                      (0 <= x16)/\ (x16 <= x15) ->
                       (0 <= x14)/\ (x14 <= x13) ->
                        (0 <= x12)/\ (x12 <= x11) ->
                         P_id_A__U83 x12 x14 x16 <= P_id_A__U83 x11 x13 x15.
                    Proof.
                      intros x16 x15 x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      intros [H_5 H_4].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_A__U23_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_A__U23 x12 <= P_id_A__U23 x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_0].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_A__U11_monotonic :
                     forall x12 x14 x13 x11, 
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_A__U11 x12 x14 <= P_id_A__U11 x11 x13.
                    Proof.
                      intros x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_A__LENGTH_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_A__LENGTH x12 <= P_id_A__LENGTH x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_0].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_A__U44_monotonic :
                     forall x16 x12 x14 x13 x11 x15, 
                      (0 <= x16)/\ (x16 <= x15) ->
                       (0 <= x14)/\ (x14 <= x13) ->
                        (0 <= x12)/\ (x12 <= x11) ->
                         P_id_A__U44 x12 x14 x16 <= P_id_A__U44 x11 x13 x15.
                    Proof.
                      intros x16 x15 x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      intros [H_5 H_4].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_A__ISNATILIST_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_A__ISNATILIST x12 <= P_id_A__ISNATILIST x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_0].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_A__U13_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_A__U13 x12 <= P_id_A__U13 x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_0].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_A__U71_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_A__U71 x12 <= P_id_A__U71 x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_0].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_A__U86_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_A__U86 x12 <= P_id_A__U86 x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_0].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_A__U32_monotonic :
                     forall x12 x14 x13 x11, 
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_A__U32 x12 x14 <= P_id_A__U32 x11 x13.
                    Proof.
                      intros x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_A__U82_monotonic :
                     forall x16 x12 x14 x13 x11 x15, 
                      (0 <= x16)/\ (x16 <= x15) ->
                       (0 <= x14)/\ (x14 <= x13) ->
                        (0 <= x12)/\ (x12 <= x11) ->
                         P_id_A__U82 x12 x14 x16 <= P_id_A__U82 x11 x13 x15.
                    Proof.
                      intros x16 x15 x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      intros [H_5 H_4].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_A__ISNATKIND_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_A__ISNATKIND x12 <= P_id_A__ISNATKIND x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_0].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_A__U94_monotonic :
                     forall x12 x14 x13 x11, 
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_A__U94 x12 x14 <= P_id_A__U94 x11 x13.
                    Proof.
                      intros x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_A__U43_monotonic :
                     forall x16 x12 x14 x13 x11 x15, 
                      (0 <= x16)/\ (x16 <= x15) ->
                       (0 <= x14)/\ (x14 <= x13) ->
                        (0 <= x12)/\ (x12 <= x11) ->
                         P_id_A__U43 x12 x14 x16 <= P_id_A__U43 x11 x13 x15.
                    Proof.
                      intros x16 x15 x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      intros [H_5 H_4].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_A__U52_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_A__U52 x12 <= P_id_A__U52 x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_0].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_A__U21_monotonic :
                     forall x12 x14 x13 x11, 
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_A__U21 x12 x14 <= P_id_A__U21 x11 x13.
                    Proof.
                      intros x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_A__U92_monotonic :
                     forall x16 x12 x14 x13 x11 x15, 
                      (0 <= x16)/\ (x16 <= x15) ->
                       (0 <= x14)/\ (x14 <= x13) ->
                        (0 <= x12)/\ (x12 <= x11) ->
                         P_id_A__U92 x12 x14 x16 <= P_id_A__U92 x11 x13 x15.
                    Proof.
                      intros x16 x15 x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      intros [H_5 H_4].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_A__U41_monotonic :
                     forall x16 x12 x14 x13 x11 x15, 
                      (0 <= x16)/\ (x16 <= x15) ->
                       (0 <= x14)/\ (x14 <= x13) ->
                        (0 <= x12)/\ (x12 <= x11) ->
                         P_id_A__U41 x12 x14 x16 <= P_id_A__U41 x11 x13 x15.
                    Proof.
                      intros x16 x15 x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      intros [H_5 H_4].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_A__U84_monotonic :
                     forall x16 x12 x14 x13 x11 x15, 
                      (0 <= x16)/\ (x16 <= x15) ->
                       (0 <= x14)/\ (x14 <= x13) ->
                        (0 <= x12)/\ (x12 <= x11) ->
                         P_id_A__U84 x12 x14 x16 <= P_id_A__U84 x11 x13 x15.
                    Proof.
                      intros x16 x15 x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      intros [H_5 H_4].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_A__ISNAT_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_A__ISNAT x12 <= P_id_A__ISNAT x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_0].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_A__U12_monotonic :
                     forall x12 x14 x13 x11, 
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_A__U12 x12 x14 <= P_id_A__U12 x11 x13.
                    Proof.
                      intros x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_MARK_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_MARK x12 <= P_id_MARK x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_0].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_A__U45_monotonic :
                     forall x12 x14 x13 x11, 
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_A__U45 x12 x14 <= P_id_A__U45 x11 x13.
                    Proof.
                      intros x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Definition marked_measure  := 
                      InterpZ.marked_measure 0 P_id_a__zeros P_id_U51 
                       P_id_a__U84 P_id_a__U32 P_id_isNatList P_id_a__U13 
                       P_id_U85 P_id_a__length P_id_a__isNatIList P_id_U33 
                       P_id_a__U11 P_id_U81 P_id_a__U92 P_id_a__U43 P_id_U23 
                       P_id_a__isNatKind P_id_U93 P_id_U11 P_id_a__U71 
                       P_id_U44 P_id_0 P_id_U61 P_id_a__U86 P_id_a__U41 
                       P_id_U22 P_id_a__U21 P_id_U91 P_id_length P_id_a__U52 
                       P_id_U42 P_id_a__U12 P_id_U83 P_id_a__U94 P_id_a__U45 
                       P_id_U31 P_id_a__isNat P_id_isNatIListKind 
                       P_id_a__U82 P_id_U46 P_id_cons P_id_U52 P_id_a__U85 
                       P_id_a__U33 P_id_U21 P_id_a__isNatList P_id_U86 
                       P_id_mark P_id_a__U51 P_id_U41 P_id_tt P_id_U82 
                       P_id_a__U93 P_id_a__U44 P_id_isNat P_id_a__U23 
                       P_id_U94 P_id_U12 P_id_a__U81 P_id_U45 P_id_zeros 
                       P_id_U71 P_id_a__U91 P_id_a__U42 P_id_isNatKind 
                       P_id_a__U22 P_id_U92 P_id_nil P_id_a__U61 P_id_U43 
                       P_id_a__isNatIListKind P_id_U84 P_id_s P_id_a__U46 
                       P_id_U32 P_id_a__U31 P_id_U13 P_id_a__U83 
                       P_id_isNatIList P_id_A__U46 P_id_A__ISNATILISTKIND 
                       P_id_A__U61 P_id_A__U85 P_id_A__U31 P_id_A__U81 
                       P_id_A__U22 P_id_A__U93 P_id_A__U42 P_id_A__U51 
                       P_id_A__ISNATLIST P_id_A__ZEROS P_id_A__U91 
                       P_id_A__U33 P_id_A__U83 P_id_A__U23 P_id_A__U11 
                       P_id_A__LENGTH P_id_A__U44 P_id_A__ISNATILIST 
                       P_id_A__U13 P_id_A__U71 P_id_A__U86 P_id_A__U32 
                       P_id_A__U82 P_id_A__ISNATKIND P_id_A__U94 P_id_A__U43 
                       P_id_A__U52 P_id_A__U21 P_id_A__U92 P_id_A__U41 
                       P_id_A__U84 P_id_A__ISNAT P_id_A__U12 P_id_MARK 
                       P_id_A__U45.
                    
                    Lemma marked_measure_equation :
                     forall t, 
                      marked_measure t = match t with
                                           | (algebra.Alg.Term 
                                              algebra.F.id_a__U46 (x11::nil)) =>
                                            P_id_A__U46 (measure x11)
                                           | (algebra.Alg.Term 
                                              algebra.F.id_a__isNatIListKind 
                                              (x11::nil)) =>
                                            P_id_A__ISNATILISTKIND (measure 
                                                                    x11)
                                           | (algebra.Alg.Term 
                                              algebra.F.id_a__U61 (x11::nil)) =>
                                            P_id_A__U61 (measure x11)
                                           | (algebra.Alg.Term 
                                              algebra.F.id_a__U85 (x12::
                                              x11::nil)) =>
                                            P_id_A__U85 (measure x12) 
                                             (measure x11)
                                           | (algebra.Alg.Term 
                                              algebra.F.id_a__U31 (x12::
                                              x11::nil)) =>
                                            P_id_A__U31 (measure x12) 
                                             (measure x11)
                                           | (algebra.Alg.Term 
                                              algebra.F.id_a__U81 (x13::x12::
                                              x11::nil)) =>
                                            P_id_A__U81 (measure x13) 
                                             (measure x12) (measure x11)
                                           | (algebra.Alg.Term 
                                              algebra.F.id_a__U22 (x12::
                                              x11::nil)) =>
                                            P_id_A__U22 (measure x12) 
                                             (measure x11)
                                           | (algebra.Alg.Term 
                                              algebra.F.id_a__U93 (x13::x12::
                                              x11::nil)) =>
                                            P_id_A__U93 (measure x13) 
                                             (measure x12) (measure x11)
                                           | (algebra.Alg.Term 
                                              algebra.F.id_a__U42 (x13::x12::
                                              x11::nil)) =>
                                            P_id_A__U42 (measure x13) 
                                             (measure x12) (measure x11)
                                           | (algebra.Alg.Term 
                                              algebra.F.id_a__U51 (x12::
                                              x11::nil)) =>
                                            P_id_A__U51 (measure x12) 
                                             (measure x11)
                                           | (algebra.Alg.Term 
                                              algebra.F.id_a__isNatList 
                                              (x11::nil)) =>
                                            P_id_A__ISNATLIST (measure x11)
                                           | (algebra.Alg.Term 
                                              algebra.F.id_a__zeros nil) =>
                                            P_id_A__ZEROS 
                                           | (algebra.Alg.Term 
                                              algebra.F.id_a__U91 (x13::x12::
                                              x11::nil)) =>
                                            P_id_A__U91 (measure x13) 
                                             (measure x12) (measure x11)
                                           | (algebra.Alg.Term 
                                              algebra.F.id_a__U33 (x11::nil)) =>
                                            P_id_A__U33 (measure x11)
                                           | (algebra.Alg.Term 
                                              algebra.F.id_a__U83 (x13::x12::
                                              x11::nil)) =>
                                            P_id_A__U83 (measure x13) 
                                             (measure x12) (measure x11)
                                           | (algebra.Alg.Term 
                                              algebra.F.id_a__U23 (x11::nil)) =>
                                            P_id_A__U23 (measure x11)
                                           | (algebra.Alg.Term 
                                              algebra.F.id_a__U11 (x12::
                                              x11::nil)) =>
                                            P_id_A__U11 (measure x12) 
                                             (measure x11)
                                           | (algebra.Alg.Term 
                                              algebra.F.id_a__length 
                                              (x11::nil)) =>
                                            P_id_A__LENGTH (measure x11)
                                           | (algebra.Alg.Term 
                                              algebra.F.id_a__U44 (x13::x12::
                                              x11::nil)) =>
                                            P_id_A__U44 (measure x13) 
                                             (measure x12) (measure x11)
                                           | (algebra.Alg.Term 
                                              algebra.F.id_a__isNatIList 
                                              (x11::nil)) =>
                                            P_id_A__ISNATILIST (measure x11)
                                           | (algebra.Alg.Term 
                                              algebra.F.id_a__U13 (x11::nil)) =>
                                            P_id_A__U13 (measure x11)
                                           | (algebra.Alg.Term 
                                              algebra.F.id_a__U71 (x11::nil)) =>
                                            P_id_A__U71 (measure x11)
                                           | (algebra.Alg.Term 
                                              algebra.F.id_a__U86 (x11::nil)) =>
                                            P_id_A__U86 (measure x11)
                                           | (algebra.Alg.Term 
                                              algebra.F.id_a__U32 (x12::
                                              x11::nil)) =>
                                            P_id_A__U32 (measure x12) 
                                             (measure x11)
                                           | (algebra.Alg.Term 
                                              algebra.F.id_a__U82 (x13::x12::
                                              x11::nil)) =>
                                            P_id_A__U82 (measure x13) 
                                             (measure x12) (measure x11)
                                           | (algebra.Alg.Term 
                                              algebra.F.id_a__isNatKind 
                                              (x11::nil)) =>
                                            P_id_A__ISNATKIND (measure x11)
                                           | (algebra.Alg.Term 
                                              algebra.F.id_a__U94 (x12::
                                              x11::nil)) =>
                                            P_id_A__U94 (measure x12) 
                                             (measure x11)
                                           | (algebra.Alg.Term 
                                              algebra.F.id_a__U43 (x13::x12::
                                              x11::nil)) =>
                                            P_id_A__U43 (measure x13) 
                                             (measure x12) (measure x11)
                                           | (algebra.Alg.Term 
                                              algebra.F.id_a__U52 (x11::nil)) =>
                                            P_id_A__U52 (measure x11)
                                           | (algebra.Alg.Term 
                                              algebra.F.id_a__U21 (x12::
                                              x11::nil)) =>
                                            P_id_A__U21 (measure x12) 
                                             (measure x11)
                                           | (algebra.Alg.Term 
                                              algebra.F.id_a__U92 (x13::x12::
                                              x11::nil)) =>
                                            P_id_A__U92 (measure x13) 
                                             (measure x12) (measure x11)
                                           | (algebra.Alg.Term 
                                              algebra.F.id_a__U41 (x13::x12::
                                              x11::nil)) =>
                                            P_id_A__U41 (measure x13) 
                                             (measure x12) (measure x11)
                                           | (algebra.Alg.Term 
                                              algebra.F.id_a__U84 (x13::x12::
                                              x11::nil)) =>
                                            P_id_A__U84 (measure x13) 
                                             (measure x12) (measure x11)
                                           | (algebra.Alg.Term 
                                              algebra.F.id_a__isNat 
                                              (x11::nil)) =>
                                            P_id_A__ISNAT (measure x11)
                                           | (algebra.Alg.Term 
                                              algebra.F.id_a__U12 (x12::
                                              x11::nil)) =>
                                            P_id_A__U12 (measure x12) 
                                             (measure x11)
                                           | (algebra.Alg.Term 
                                              algebra.F.id_mark (x11::nil)) =>
                                            P_id_MARK (measure x11)
                                           | (algebra.Alg.Term 
                                              algebra.F.id_a__U45 (x12::
                                              x11::nil)) =>
                                            P_id_A__U45 (measure x12) 
                                             (measure x11)
                                           | _ => measure t
                                           end.
                    Proof.
                      reflexivity .
                    Qed.
                    
                    Lemma marked_measure_star_monotonic :
                     forall f l1 l2, 
                      (closure.refl_trans_clos (closure.one_step_list 
                                                 (algebra.EQT.one_step 
                                                   R_xml_0_deep_rew.R_xml_0_rules)
                                                  ) l1 l2) ->
                       marked_measure (algebra.Alg.Term f l1) <= marked_measure 
                                                                  (algebra.Alg.Term 
                                                                    f 
                                                                    l2).
                    Proof.
                      unfold marked_measure in *.
                      apply InterpZ.marked_measure_star_monotonic.
                      intros ;apply P_id_U51_monotonic;assumption.
                      intros ;apply P_id_a__U84_monotonic;assumption.
                      intros ;apply P_id_a__U32_monotonic;assumption.
                      intros ;apply P_id_isNatList_monotonic;assumption.
                      intros ;apply P_id_a__U13_monotonic;assumption.
                      intros ;apply P_id_U85_monotonic;assumption.
                      intros ;apply P_id_a__length_monotonic;assumption.
                      intros ;apply P_id_a__isNatIList_monotonic;assumption.
                      intros ;apply P_id_U33_monotonic;assumption.
                      intros ;apply P_id_a__U11_monotonic;assumption.
                      intros ;apply P_id_U81_monotonic;assumption.
                      intros ;apply P_id_a__U92_monotonic;assumption.
                      intros ;apply P_id_a__U43_monotonic;assumption.
                      intros ;apply P_id_U23_monotonic;assumption.
                      intros ;apply P_id_a__isNatKind_monotonic;assumption.
                      intros ;apply P_id_U93_monotonic;assumption.
                      intros ;apply P_id_U11_monotonic;assumption.
                      intros ;apply P_id_a__U71_monotonic;assumption.
                      intros ;apply P_id_U44_monotonic;assumption.
                      intros ;apply P_id_U61_monotonic;assumption.
                      intros ;apply P_id_a__U86_monotonic;assumption.
                      intros ;apply P_id_a__U41_monotonic;assumption.
                      intros ;apply P_id_U22_monotonic;assumption.
                      intros ;apply P_id_a__U21_monotonic;assumption.
                      intros ;apply P_id_U91_monotonic;assumption.
                      intros ;apply P_id_length_monotonic;assumption.
                      intros ;apply P_id_a__U52_monotonic;assumption.
                      intros ;apply P_id_U42_monotonic;assumption.
                      intros ;apply P_id_a__U12_monotonic;assumption.
                      intros ;apply P_id_U83_monotonic;assumption.
                      intros ;apply P_id_a__U94_monotonic;assumption.
                      intros ;apply P_id_a__U45_monotonic;assumption.
                      intros ;apply P_id_U31_monotonic;assumption.
                      intros ;apply P_id_a__isNat_monotonic;assumption.
                      intros ;apply P_id_isNatIListKind_monotonic;assumption.
                      intros ;apply P_id_a__U82_monotonic;assumption.
                      intros ;apply P_id_U46_monotonic;assumption.
                      intros ;apply P_id_cons_monotonic;assumption.
                      intros ;apply P_id_U52_monotonic;assumption.
                      intros ;apply P_id_a__U85_monotonic;assumption.
                      intros ;apply P_id_a__U33_monotonic;assumption.
                      intros ;apply P_id_U21_monotonic;assumption.
                      intros ;apply P_id_a__isNatList_monotonic;assumption.
                      intros ;apply P_id_U86_monotonic;assumption.
                      intros ;apply P_id_mark_monotonic;assumption.
                      intros ;apply P_id_a__U51_monotonic;assumption.
                      intros ;apply P_id_U41_monotonic;assumption.
                      intros ;apply P_id_U82_monotonic;assumption.
                      intros ;apply P_id_a__U93_monotonic;assumption.
                      intros ;apply P_id_a__U44_monotonic;assumption.
                      intros ;apply P_id_isNat_monotonic;assumption.
                      intros ;apply P_id_a__U23_monotonic;assumption.
                      intros ;apply P_id_U94_monotonic;assumption.
                      intros ;apply P_id_U12_monotonic;assumption.
                      intros ;apply P_id_a__U81_monotonic;assumption.
                      intros ;apply P_id_U45_monotonic;assumption.
                      intros ;apply P_id_U71_monotonic;assumption.
                      intros ;apply P_id_a__U91_monotonic;assumption.
                      intros ;apply P_id_a__U42_monotonic;assumption.
                      intros ;apply P_id_isNatKind_monotonic;assumption.
                      intros ;apply P_id_a__U22_monotonic;assumption.
                      intros ;apply P_id_U92_monotonic;assumption.
                      intros ;apply P_id_a__U61_monotonic;assumption.
                      intros ;apply P_id_U43_monotonic;assumption.
                      
                      intros ;apply P_id_a__isNatIListKind_monotonic;
                       assumption.
                      intros ;apply P_id_U84_monotonic;assumption.
                      intros ;apply P_id_s_monotonic;assumption.
                      intros ;apply P_id_a__U46_monotonic;assumption.
                      intros ;apply P_id_U32_monotonic;assumption.
                      intros ;apply P_id_a__U31_monotonic;assumption.
                      intros ;apply P_id_U13_monotonic;assumption.
                      intros ;apply P_id_a__U83_monotonic;assumption.
                      intros ;apply P_id_isNatIList_monotonic;assumption.
                      intros ;apply P_id_a__zeros_bounded;assumption.
                      intros ;apply P_id_U51_bounded;assumption.
                      intros ;apply P_id_a__U84_bounded;assumption.
                      intros ;apply P_id_a__U32_bounded;assumption.
                      intros ;apply P_id_isNatList_bounded;assumption.
                      intros ;apply P_id_a__U13_bounded;assumption.
                      intros ;apply P_id_U85_bounded;assumption.
                      intros ;apply P_id_a__length_bounded;assumption.
                      intros ;apply P_id_a__isNatIList_bounded;assumption.
                      intros ;apply P_id_U33_bounded;assumption.
                      intros ;apply P_id_a__U11_bounded;assumption.
                      intros ;apply P_id_U81_bounded;assumption.
                      intros ;apply P_id_a__U92_bounded;assumption.
                      intros ;apply P_id_a__U43_bounded;assumption.
                      intros ;apply P_id_U23_bounded;assumption.
                      intros ;apply P_id_a__isNatKind_bounded;assumption.
                      intros ;apply P_id_U93_bounded;assumption.
                      intros ;apply P_id_U11_bounded;assumption.
                      intros ;apply P_id_a__U71_bounded;assumption.
                      intros ;apply P_id_U44_bounded;assumption.
                      intros ;apply P_id_0_bounded;assumption.
                      intros ;apply P_id_U61_bounded;assumption.
                      intros ;apply P_id_a__U86_bounded;assumption.
                      intros ;apply P_id_a__U41_bounded;assumption.
                      intros ;apply P_id_U22_bounded;assumption.
                      intros ;apply P_id_a__U21_bounded;assumption.
                      intros ;apply P_id_U91_bounded;assumption.
                      intros ;apply P_id_length_bounded;assumption.
                      intros ;apply P_id_a__U52_bounded;assumption.
                      intros ;apply P_id_U42_bounded;assumption.
                      intros ;apply P_id_a__U12_bounded;assumption.
                      intros ;apply P_id_U83_bounded;assumption.
                      intros ;apply P_id_a__U94_bounded;assumption.
                      intros ;apply P_id_a__U45_bounded;assumption.
                      intros ;apply P_id_U31_bounded;assumption.
                      intros ;apply P_id_a__isNat_bounded;assumption.
                      intros ;apply P_id_isNatIListKind_bounded;assumption.
                      intros ;apply P_id_a__U82_bounded;assumption.
                      intros ;apply P_id_U46_bounded;assumption.
                      intros ;apply P_id_cons_bounded;assumption.
                      intros ;apply P_id_U52_bounded;assumption.
                      intros ;apply P_id_a__U85_bounded;assumption.
                      intros ;apply P_id_a__U33_bounded;assumption.
                      intros ;apply P_id_U21_bounded;assumption.
                      intros ;apply P_id_a__isNatList_bounded;assumption.
                      intros ;apply P_id_U86_bounded;assumption.
                      intros ;apply P_id_mark_bounded;assumption.
                      intros ;apply P_id_a__U51_bounded;assumption.
                      intros ;apply P_id_U41_bounded;assumption.
                      intros ;apply P_id_tt_bounded;assumption.
                      intros ;apply P_id_U82_bounded;assumption.
                      intros ;apply P_id_a__U93_bounded;assumption.
                      intros ;apply P_id_a__U44_bounded;assumption.
                      intros ;apply P_id_isNat_bounded;assumption.
                      intros ;apply P_id_a__U23_bounded;assumption.
                      intros ;apply P_id_U94_bounded;assumption.
                      intros ;apply P_id_U12_bounded;assumption.
                      intros ;apply P_id_a__U81_bounded;assumption.
                      intros ;apply P_id_U45_bounded;assumption.
                      intros ;apply P_id_zeros_bounded;assumption.
                      intros ;apply P_id_U71_bounded;assumption.
                      intros ;apply P_id_a__U91_bounded;assumption.
                      intros ;apply P_id_a__U42_bounded;assumption.
                      intros ;apply P_id_isNatKind_bounded;assumption.
                      intros ;apply P_id_a__U22_bounded;assumption.
                      intros ;apply P_id_U92_bounded;assumption.
                      intros ;apply P_id_nil_bounded;assumption.
                      intros ;apply P_id_a__U61_bounded;assumption.
                      intros ;apply P_id_U43_bounded;assumption.
                      
                      intros ;apply P_id_a__isNatIListKind_bounded;assumption.
                      intros ;apply P_id_U84_bounded;assumption.
                      intros ;apply P_id_s_bounded;assumption.
                      intros ;apply P_id_a__U46_bounded;assumption.
                      intros ;apply P_id_U32_bounded;assumption.
                      intros ;apply P_id_a__U31_bounded;assumption.
                      intros ;apply P_id_U13_bounded;assumption.
                      intros ;apply P_id_a__U83_bounded;assumption.
                      intros ;apply P_id_isNatIList_bounded;assumption.
                      apply rules_monotonic.
                      intros ;apply P_id_A__U46_monotonic;assumption.
                      
                      intros ;apply P_id_A__ISNATILISTKIND_monotonic;
                       assumption.
                      intros ;apply P_id_A__U61_monotonic;assumption.
                      intros ;apply P_id_A__U85_monotonic;assumption.
                      intros ;apply P_id_A__U31_monotonic;assumption.
                      intros ;apply P_id_A__U81_monotonic;assumption.
                      intros ;apply P_id_A__U22_monotonic;assumption.
                      intros ;apply P_id_A__U93_monotonic;assumption.
                      intros ;apply P_id_A__U42_monotonic;assumption.
                      intros ;apply P_id_A__U51_monotonic;assumption.
                      intros ;apply P_id_A__ISNATLIST_monotonic;assumption.
                      intros ;apply P_id_A__U91_monotonic;assumption.
                      intros ;apply P_id_A__U33_monotonic;assumption.
                      intros ;apply P_id_A__U83_monotonic;assumption.
                      intros ;apply P_id_A__U23_monotonic;assumption.
                      intros ;apply P_id_A__U11_monotonic;assumption.
                      intros ;apply P_id_A__LENGTH_monotonic;assumption.
                      intros ;apply P_id_A__U44_monotonic;assumption.
                      intros ;apply P_id_A__ISNATILIST_monotonic;assumption.
                      intros ;apply P_id_A__U13_monotonic;assumption.
                      intros ;apply P_id_A__U71_monotonic;assumption.
                      intros ;apply P_id_A__U86_monotonic;assumption.
                      intros ;apply P_id_A__U32_monotonic;assumption.
                      intros ;apply P_id_A__U82_monotonic;assumption.
                      intros ;apply P_id_A__ISNATKIND_monotonic;assumption.
                      intros ;apply P_id_A__U94_monotonic;assumption.
                      intros ;apply P_id_A__U43_monotonic;assumption.
                      intros ;apply P_id_A__U52_monotonic;assumption.
                      intros ;apply P_id_A__U21_monotonic;assumption.
                      intros ;apply P_id_A__U92_monotonic;assumption.
                      intros ;apply P_id_A__U41_monotonic;assumption.
                      intros ;apply P_id_A__U84_monotonic;assumption.
                      intros ;apply P_id_A__ISNAT_monotonic;assumption.
                      intros ;apply P_id_A__U12_monotonic;assumption.
                      intros ;apply P_id_MARK_monotonic;assumption.
                      intros ;apply P_id_A__U45_monotonic;assumption.
                    Qed.
                    
                    Ltac rewrite_and_unfold  :=
                     do 2 (rewrite marked_measure_equation);
                      repeat (
                      match goal with
                        |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
                         rewrite (measure_equation (algebra.Alg.Term f t))
                        | H:context [measure (algebra.Alg.Term ?f ?t)] |- 
                        _ =>
                         rewrite (measure_equation (algebra.Alg.Term f t)) in 
                         H|-
                        end
                      ).
                    
                    
                    Lemma wf :
                     well_founded WF_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_large.DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_large_large
                      .
                    Proof.
                      intros x.
                      
                      apply well_founded_ind with
                        (R:=fun x 
                             y => (Zwf.Zwf 0) (marked_measure x) 
                                   (marked_measure y)).
                      apply Inverse_Image.wf_inverse_image with  (B:=Z).
                      apply Zwf.Zwf_well_founded.
                      clear x.
                      intros x IHx.
                      
                      repeat (
                      constructor;inversion 1;subst;
                       full_prove_ineq algebra.Alg.Term 
                       ltac:(algebra.Alg_ext.find_replacement ) 
                       algebra.EQT_ext.one_step_list_refl_trans_clos 
                       marked_measure marked_measure_star_monotonic 
                       (Zwf.Zwf 0) (interp.o_Z 0) 
                       ltac:(fun _ => R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )
                        ltac:(fun _ => rewrite_and_unfold ) 
                       ltac:(fun _ => generate_pos_hyp ) 
                       ltac:(fun _ => cbv beta iota zeta delta - [Zplus Zmult Zle Zlt] in * ;
                                       try (constructor))
                        IHx
                      ).
                    Qed.
                   End WF_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_large_large.
                   
                   Open Scope Z_scope.
                   
                   Import ring_extention.
                   
                   Notation Local "a <= b" := (Zle a b).
                   
                   Notation Local "a < b" := (Zlt a b).
                   
                   Definition P_id_a__zeros  := 0.
                   
                   Definition P_id_U51 (x11:Z) (x12:Z) := 1.
                   
                   Definition P_id_a__U84 (x11:Z) (x12:Z) (x13:Z) := 2* x13.
                   
                   Definition P_id_a__U32 (x11:Z) (x12:Z) := 2* x12.
                   
                   Definition P_id_isNatList (x11:Z) := 1* x11.
                   
                   Definition P_id_a__U13 (x11:Z) := 1.
                   
                   Definition P_id_U85 (x11:Z) (x12:Z) := 2* x12.
                   
                   Definition P_id_a__length (x11:Z) := 1 + 1* x11.
                   
                   Definition P_id_a__isNatIList (x11:Z) := 2 + 2* x11.
                   
                   Definition P_id_U33 (x11:Z) := 1* x11.
                   
                   Definition P_id_a__U11 (x11:Z) (x12:Z) := 1* x11.
                   
                   Definition P_id_U81 (x11:Z) (x12:Z) (x13:Z) := 3* x13.
                   
                   Definition P_id_a__U92 (x11:Z) (x12:Z) (x13:Z) := 
                     2 + 2* x12.
                   
                   Definition P_id_a__U43 (x11:Z) (x12:Z) (x13:Z) := 1.
                   
                   Definition P_id_U23 (x11:Z) := 1 + 1* x11.
                   
                   Definition P_id_a__isNatKind (x11:Z) := 1* x11.
                   
                   Definition P_id_U93 (x11:Z) (x12:Z) (x13:Z) := 2 + 2* x12.
                   
                   Definition P_id_U11 (x11:Z) (x12:Z) := 1* x11.
                   
                   Definition P_id_a__U71 (x11:Z) := 2* x11.
                   
                   Definition P_id_U44 (x11:Z) (x12:Z) (x13:Z) := 1.
                   
                   Definition P_id_0  := 1.
                   
                   Definition P_id_U61 (x11:Z) := 1.
                   
                   Definition P_id_a__U86 (x11:Z) := 2* x11.
                   
                   Definition P_id_a__U41 (x11:Z) (x12:Z) (x13:Z) := 1.
                   
                   Definition P_id_U22 (x11:Z) (x12:Z) := 1 + 2* x12.
                   
                   Definition P_id_a__U21 (x11:Z) (x12:Z) := 1* x11 + 3* x12.
                   
                   Definition P_id_U91 (x11:Z) (x12:Z) (x13:Z) := 
                     1 + 1* x11 + 2* x12.
                   
                   Definition P_id_length (x11:Z) := 1 + 1* x11.
                   
                   Definition P_id_a__U52 (x11:Z) := 1.
                   
                   Definition P_id_U42 (x11:Z) (x12:Z) (x13:Z) := 1.
                   
                   Definition P_id_a__U12 (x11:Z) (x12:Z) := 1.
                   
                   Definition P_id_U83 (x11:Z) (x12:Z) (x13:Z) := 2* x13.
                   
                   Definition P_id_a__U94 (x11:Z) (x12:Z) := 2 + 2* x12.
                   
                   Definition P_id_a__U45 (x11:Z) (x12:Z) := 1.
                   
                   Definition P_id_U31 (x11:Z) (x12:Z) := 2* x12.
                   
                   Definition P_id_a__isNat (x11:Z) := 2* x11.
                   
                   Definition P_id_isNatIListKind (x11:Z) := 2.
                   
                   Definition P_id_a__U82 (x11:Z) (x12:Z) (x13:Z) := 2* x13.
                   
                   Definition P_id_U46 (x11:Z) := 1.
                   
                   Definition P_id_cons (x11:Z) (x12:Z) := 3* x12.
                   
                   Definition P_id_U52 (x11:Z) := 1.
                   
                   Definition P_id_a__U85 (x11:Z) (x12:Z) := 2* x12.
                   
                   Definition P_id_a__U33 (x11:Z) := 1* x11.
                   
                   Definition P_id_U21 (x11:Z) (x12:Z) := 1* x11 + 3* x12.
                   
                   Definition P_id_a__isNatList (x11:Z) := 1* x11.
                   
                   Definition P_id_U86 (x11:Z) := 2* x11.
                   
                   Definition P_id_mark (x11:Z) := 1* x11.
                   
                   Definition P_id_a__U51 (x11:Z) (x12:Z) := 1.
                   
                   Definition P_id_U41 (x11:Z) (x12:Z) (x13:Z) := 1.
                   
                   Definition P_id_tt  := 1.
                   
                   Definition P_id_U82 (x11:Z) (x12:Z) (x13:Z) := 2* x13.
                   
                   Definition P_id_a__U93 (x11:Z) (x12:Z) (x13:Z) := 
                     2 + 2* x12.
                   
                   Definition P_id_a__U44 (x11:Z) (x12:Z) (x13:Z) := 1.
                   
                   Definition P_id_isNat (x11:Z) := 2* x11.
                   
                   Definition P_id_a__U23 (x11:Z) := 1 + 1* x11.
                   
                   Definition P_id_U94 (x11:Z) (x12:Z) := 2 + 2* x12.
                   
                   Definition P_id_U12 (x11:Z) (x12:Z) := 1.
                   
                   Definition P_id_a__U81 (x11:Z) (x12:Z) (x13:Z) := 3* x13.
                   
                   Definition P_id_U45 (x11:Z) (x12:Z) := 1.
                   
                   Definition P_id_zeros  := 0.
                   
                   Definition P_id_U71 (x11:Z) := 2* x11.
                   
                   Definition P_id_a__U91 (x11:Z) (x12:Z) (x13:Z) := 
                     1 + 1* x11 + 2* x12.
                   
                   Definition P_id_a__U42 (x11:Z) (x12:Z) (x13:Z) := 1.
                   
                   Definition P_id_isNatKind (x11:Z) := 1* x11.
                   
                   Definition P_id_a__U22 (x11:Z) (x12:Z) := 1 + 2* x12.
                   
                   Definition P_id_U92 (x11:Z) (x12:Z) (x13:Z) := 2 + 2* x12.
                   
                   Definition P_id_nil  := 1.
                   
                   Definition P_id_a__U61 (x11:Z) := 1.
                   
                   Definition P_id_U43 (x11:Z) (x12:Z) (x13:Z) := 1.
                   
                   Definition P_id_a__isNatIListKind (x11:Z) := 2.
                   
                   Definition P_id_U84 (x11:Z) (x12:Z) (x13:Z) := 2* x13.
                   
                   Definition P_id_s (x11:Z) := 2* x11.
                   
                   Definition P_id_a__U46 (x11:Z) := 1.
                   
                   Definition P_id_U32 (x11:Z) (x12:Z) := 2* x12.
                   
                   Definition P_id_a__U31 (x11:Z) (x12:Z) := 2* x12.
                   
                   Definition P_id_U13 (x11:Z) := 1.
                   
                   Definition P_id_a__U83 (x11:Z) (x12:Z) (x13:Z) := 2* x13.
                   
                   Definition P_id_isNatIList (x11:Z) := 2 + 2* x11.
                   
                   Lemma P_id_U51_monotonic :
                    forall x12 x14 x13 x11, 
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_U51 x12 x14 <= P_id_U51 x11 x13.
                   Proof.
                     intros x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U84_monotonic :
                    forall x16 x12 x14 x13 x11 x15, 
                     (0 <= x16)/\ (x16 <= x15) ->
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_a__U84 x12 x14 x16 <= P_id_a__U84 x11 x13 x15.
                   Proof.
                     intros x16 x15 x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     intros [H_5 H_4].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U32_monotonic :
                    forall x12 x14 x13 x11, 
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_a__U32 x12 x14 <= P_id_a__U32 x11 x13.
                   Proof.
                     intros x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_isNatList_monotonic :
                    forall x12 x11, 
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_isNatList x12 <= P_id_isNatList x11.
                   Proof.
                     intros x12 x11.
                     intros [H_1 H_0].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U13_monotonic :
                    forall x12 x11, 
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_a__U13 x12 <= P_id_a__U13 x11.
                   Proof.
                     intros x12 x11.
                     intros [H_1 H_0].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U85_monotonic :
                    forall x12 x14 x13 x11, 
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_U85 x12 x14 <= P_id_U85 x11 x13.
                   Proof.
                     intros x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__length_monotonic :
                    forall x12 x11, 
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_a__length x12 <= P_id_a__length x11.
                   Proof.
                     intros x12 x11.
                     intros [H_1 H_0].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__isNatIList_monotonic :
                    forall x12 x11, 
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_a__isNatIList x12 <= P_id_a__isNatIList x11.
                   Proof.
                     intros x12 x11.
                     intros [H_1 H_0].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U33_monotonic :
                    forall x12 x11, 
                     (0 <= x12)/\ (x12 <= x11) ->P_id_U33 x12 <= P_id_U33 x11.
                   Proof.
                     intros x12 x11.
                     intros [H_1 H_0].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U11_monotonic :
                    forall x12 x14 x13 x11, 
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_a__U11 x12 x14 <= P_id_a__U11 x11 x13.
                   Proof.
                     intros x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U81_monotonic :
                    forall x16 x12 x14 x13 x11 x15, 
                     (0 <= x16)/\ (x16 <= x15) ->
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_U81 x12 x14 x16 <= P_id_U81 x11 x13 x15.
                   Proof.
                     intros x16 x15 x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     intros [H_5 H_4].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U92_monotonic :
                    forall x16 x12 x14 x13 x11 x15, 
                     (0 <= x16)/\ (x16 <= x15) ->
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_a__U92 x12 x14 x16 <= P_id_a__U92 x11 x13 x15.
                   Proof.
                     intros x16 x15 x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     intros [H_5 H_4].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U43_monotonic :
                    forall x16 x12 x14 x13 x11 x15, 
                     (0 <= x16)/\ (x16 <= x15) ->
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_a__U43 x12 x14 x16 <= P_id_a__U43 x11 x13 x15.
                   Proof.
                     intros x16 x15 x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     intros [H_5 H_4].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U23_monotonic :
                    forall x12 x11, 
                     (0 <= x12)/\ (x12 <= x11) ->P_id_U23 x12 <= P_id_U23 x11.
                   Proof.
                     intros x12 x11.
                     intros [H_1 H_0].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__isNatKind_monotonic :
                    forall x12 x11, 
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_a__isNatKind x12 <= P_id_a__isNatKind x11.
                   Proof.
                     intros x12 x11.
                     intros [H_1 H_0].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U93_monotonic :
                    forall x16 x12 x14 x13 x11 x15, 
                     (0 <= x16)/\ (x16 <= x15) ->
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_U93 x12 x14 x16 <= P_id_U93 x11 x13 x15.
                   Proof.
                     intros x16 x15 x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     intros [H_5 H_4].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U11_monotonic :
                    forall x12 x14 x13 x11, 
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_U11 x12 x14 <= P_id_U11 x11 x13.
                   Proof.
                     intros x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U71_monotonic :
                    forall x12 x11, 
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_a__U71 x12 <= P_id_a__U71 x11.
                   Proof.
                     intros x12 x11.
                     intros [H_1 H_0].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U44_monotonic :
                    forall x16 x12 x14 x13 x11 x15, 
                     (0 <= x16)/\ (x16 <= x15) ->
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_U44 x12 x14 x16 <= P_id_U44 x11 x13 x15.
                   Proof.
                     intros x16 x15 x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     intros [H_5 H_4].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U61_monotonic :
                    forall x12 x11, 
                     (0 <= x12)/\ (x12 <= x11) ->P_id_U61 x12 <= P_id_U61 x11.
                   Proof.
                     intros x12 x11.
                     intros [H_1 H_0].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U86_monotonic :
                    forall x12 x11, 
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_a__U86 x12 <= P_id_a__U86 x11.
                   Proof.
                     intros x12 x11.
                     intros [H_1 H_0].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U41_monotonic :
                    forall x16 x12 x14 x13 x11 x15, 
                     (0 <= x16)/\ (x16 <= x15) ->
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_a__U41 x12 x14 x16 <= P_id_a__U41 x11 x13 x15.
                   Proof.
                     intros x16 x15 x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     intros [H_5 H_4].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U22_monotonic :
                    forall x12 x14 x13 x11, 
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_U22 x12 x14 <= P_id_U22 x11 x13.
                   Proof.
                     intros x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U21_monotonic :
                    forall x12 x14 x13 x11, 
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_a__U21 x12 x14 <= P_id_a__U21 x11 x13.
                   Proof.
                     intros x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U91_monotonic :
                    forall x16 x12 x14 x13 x11 x15, 
                     (0 <= x16)/\ (x16 <= x15) ->
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_U91 x12 x14 x16 <= P_id_U91 x11 x13 x15.
                   Proof.
                     intros x16 x15 x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     intros [H_5 H_4].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_length_monotonic :
                    forall x12 x11, 
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_length x12 <= P_id_length x11.
                   Proof.
                     intros x12 x11.
                     intros [H_1 H_0].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U52_monotonic :
                    forall x12 x11, 
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_a__U52 x12 <= P_id_a__U52 x11.
                   Proof.
                     intros x12 x11.
                     intros [H_1 H_0].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U42_monotonic :
                    forall x16 x12 x14 x13 x11 x15, 
                     (0 <= x16)/\ (x16 <= x15) ->
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_U42 x12 x14 x16 <= P_id_U42 x11 x13 x15.
                   Proof.
                     intros x16 x15 x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     intros [H_5 H_4].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U12_monotonic :
                    forall x12 x14 x13 x11, 
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_a__U12 x12 x14 <= P_id_a__U12 x11 x13.
                   Proof.
                     intros x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U83_monotonic :
                    forall x16 x12 x14 x13 x11 x15, 
                     (0 <= x16)/\ (x16 <= x15) ->
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_U83 x12 x14 x16 <= P_id_U83 x11 x13 x15.
                   Proof.
                     intros x16 x15 x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     intros [H_5 H_4].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U94_monotonic :
                    forall x12 x14 x13 x11, 
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_a__U94 x12 x14 <= P_id_a__U94 x11 x13.
                   Proof.
                     intros x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U45_monotonic :
                    forall x12 x14 x13 x11, 
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_a__U45 x12 x14 <= P_id_a__U45 x11 x13.
                   Proof.
                     intros x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U31_monotonic :
                    forall x12 x14 x13 x11, 
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_U31 x12 x14 <= P_id_U31 x11 x13.
                   Proof.
                     intros x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__isNat_monotonic :
                    forall x12 x11, 
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_a__isNat x12 <= P_id_a__isNat x11.
                   Proof.
                     intros x12 x11.
                     intros [H_1 H_0].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_isNatIListKind_monotonic :
                    forall x12 x11, 
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_isNatIListKind x12 <= P_id_isNatIListKind x11.
                   Proof.
                     intros x12 x11.
                     intros [H_1 H_0].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U82_monotonic :
                    forall x16 x12 x14 x13 x11 x15, 
                     (0 <= x16)/\ (x16 <= x15) ->
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_a__U82 x12 x14 x16 <= P_id_a__U82 x11 x13 x15.
                   Proof.
                     intros x16 x15 x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     intros [H_5 H_4].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U46_monotonic :
                    forall x12 x11, 
                     (0 <= x12)/\ (x12 <= x11) ->P_id_U46 x12 <= P_id_U46 x11.
                   Proof.
                     intros x12 x11.
                     intros [H_1 H_0].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_cons_monotonic :
                    forall x12 x14 x13 x11, 
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_cons x12 x14 <= P_id_cons x11 x13.
                   Proof.
                     intros x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U52_monotonic :
                    forall x12 x11, 
                     (0 <= x12)/\ (x12 <= x11) ->P_id_U52 x12 <= P_id_U52 x11.
                   Proof.
                     intros x12 x11.
                     intros [H_1 H_0].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U85_monotonic :
                    forall x12 x14 x13 x11, 
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_a__U85 x12 x14 <= P_id_a__U85 x11 x13.
                   Proof.
                     intros x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U33_monotonic :
                    forall x12 x11, 
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_a__U33 x12 <= P_id_a__U33 x11.
                   Proof.
                     intros x12 x11.
                     intros [H_1 H_0].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U21_monotonic :
                    forall x12 x14 x13 x11, 
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_U21 x12 x14 <= P_id_U21 x11 x13.
                   Proof.
                     intros x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__isNatList_monotonic :
                    forall x12 x11, 
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_a__isNatList x12 <= P_id_a__isNatList x11.
                   Proof.
                     intros x12 x11.
                     intros [H_1 H_0].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U86_monotonic :
                    forall x12 x11, 
                     (0 <= x12)/\ (x12 <= x11) ->P_id_U86 x12 <= P_id_U86 x11.
                   Proof.
                     intros x12 x11.
                     intros [H_1 H_0].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_mark_monotonic :
                    forall x12 x11, 
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_mark x12 <= P_id_mark x11.
                   Proof.
                     intros x12 x11.
                     intros [H_1 H_0].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U51_monotonic :
                    forall x12 x14 x13 x11, 
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_a__U51 x12 x14 <= P_id_a__U51 x11 x13.
                   Proof.
                     intros x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U41_monotonic :
                    forall x16 x12 x14 x13 x11 x15, 
                     (0 <= x16)/\ (x16 <= x15) ->
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_U41 x12 x14 x16 <= P_id_U41 x11 x13 x15.
                   Proof.
                     intros x16 x15 x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     intros [H_5 H_4].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U82_monotonic :
                    forall x16 x12 x14 x13 x11 x15, 
                     (0 <= x16)/\ (x16 <= x15) ->
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_U82 x12 x14 x16 <= P_id_U82 x11 x13 x15.
                   Proof.
                     intros x16 x15 x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     intros [H_5 H_4].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U93_monotonic :
                    forall x16 x12 x14 x13 x11 x15, 
                     (0 <= x16)/\ (x16 <= x15) ->
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_a__U93 x12 x14 x16 <= P_id_a__U93 x11 x13 x15.
                   Proof.
                     intros x16 x15 x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     intros [H_5 H_4].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U44_monotonic :
                    forall x16 x12 x14 x13 x11 x15, 
                     (0 <= x16)/\ (x16 <= x15) ->
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_a__U44 x12 x14 x16 <= P_id_a__U44 x11 x13 x15.
                   Proof.
                     intros x16 x15 x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     intros [H_5 H_4].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_isNat_monotonic :
                    forall x12 x11, 
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_isNat x12 <= P_id_isNat x11.
                   Proof.
                     intros x12 x11.
                     intros [H_1 H_0].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U23_monotonic :
                    forall x12 x11, 
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_a__U23 x12 <= P_id_a__U23 x11.
                   Proof.
                     intros x12 x11.
                     intros [H_1 H_0].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U94_monotonic :
                    forall x12 x14 x13 x11, 
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_U94 x12 x14 <= P_id_U94 x11 x13.
                   Proof.
                     intros x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U12_monotonic :
                    forall x12 x14 x13 x11, 
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_U12 x12 x14 <= P_id_U12 x11 x13.
                   Proof.
                     intros x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U81_monotonic :
                    forall x16 x12 x14 x13 x11 x15, 
                     (0 <= x16)/\ (x16 <= x15) ->
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_a__U81 x12 x14 x16 <= P_id_a__U81 x11 x13 x15.
                   Proof.
                     intros x16 x15 x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     intros [H_5 H_4].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U45_monotonic :
                    forall x12 x14 x13 x11, 
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_U45 x12 x14 <= P_id_U45 x11 x13.
                   Proof.
                     intros x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U71_monotonic :
                    forall x12 x11, 
                     (0 <= x12)/\ (x12 <= x11) ->P_id_U71 x12 <= P_id_U71 x11.
                   Proof.
                     intros x12 x11.
                     intros [H_1 H_0].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U91_monotonic :
                    forall x16 x12 x14 x13 x11 x15, 
                     (0 <= x16)/\ (x16 <= x15) ->
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_a__U91 x12 x14 x16 <= P_id_a__U91 x11 x13 x15.
                   Proof.
                     intros x16 x15 x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     intros [H_5 H_4].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U42_monotonic :
                    forall x16 x12 x14 x13 x11 x15, 
                     (0 <= x16)/\ (x16 <= x15) ->
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_a__U42 x12 x14 x16 <= P_id_a__U42 x11 x13 x15.
                   Proof.
                     intros x16 x15 x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     intros [H_5 H_4].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_isNatKind_monotonic :
                    forall x12 x11, 
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_isNatKind x12 <= P_id_isNatKind x11.
                   Proof.
                     intros x12 x11.
                     intros [H_1 H_0].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U22_monotonic :
                    forall x12 x14 x13 x11, 
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_a__U22 x12 x14 <= P_id_a__U22 x11 x13.
                   Proof.
                     intros x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U92_monotonic :
                    forall x16 x12 x14 x13 x11 x15, 
                     (0 <= x16)/\ (x16 <= x15) ->
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_U92 x12 x14 x16 <= P_id_U92 x11 x13 x15.
                   Proof.
                     intros x16 x15 x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     intros [H_5 H_4].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U61_monotonic :
                    forall x12 x11, 
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_a__U61 x12 <= P_id_a__U61 x11.
                   Proof.
                     intros x12 x11.
                     intros [H_1 H_0].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U43_monotonic :
                    forall x16 x12 x14 x13 x11 x15, 
                     (0 <= x16)/\ (x16 <= x15) ->
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_U43 x12 x14 x16 <= P_id_U43 x11 x13 x15.
                   Proof.
                     intros x16 x15 x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     intros [H_5 H_4].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__isNatIListKind_monotonic :
                    forall x12 x11, 
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_a__isNatIListKind x12 <= P_id_a__isNatIListKind 
                                                     x11.
                   Proof.
                     intros x12 x11.
                     intros [H_1 H_0].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U84_monotonic :
                    forall x16 x12 x14 x13 x11 x15, 
                     (0 <= x16)/\ (x16 <= x15) ->
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_U84 x12 x14 x16 <= P_id_U84 x11 x13 x15.
                   Proof.
                     intros x16 x15 x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     intros [H_5 H_4].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_s_monotonic :
                    forall x12 x11, 
                     (0 <= x12)/\ (x12 <= x11) ->P_id_s x12 <= P_id_s x11.
                   Proof.
                     intros x12 x11.
                     intros [H_1 H_0].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U46_monotonic :
                    forall x12 x11, 
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_a__U46 x12 <= P_id_a__U46 x11.
                   Proof.
                     intros x12 x11.
                     intros [H_1 H_0].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U32_monotonic :
                    forall x12 x14 x13 x11, 
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_U32 x12 x14 <= P_id_U32 x11 x13.
                   Proof.
                     intros x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U31_monotonic :
                    forall x12 x14 x13 x11, 
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_a__U31 x12 x14 <= P_id_a__U31 x11 x13.
                   Proof.
                     intros x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U13_monotonic :
                    forall x12 x11, 
                     (0 <= x12)/\ (x12 <= x11) ->P_id_U13 x12 <= P_id_U13 x11.
                   Proof.
                     intros x12 x11.
                     intros [H_1 H_0].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U83_monotonic :
                    forall x16 x12 x14 x13 x11 x15, 
                     (0 <= x16)/\ (x16 <= x15) ->
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_a__U83 x12 x14 x16 <= P_id_a__U83 x11 x13 x15.
                   Proof.
                     intros x16 x15 x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     intros [H_5 H_4].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_isNatIList_monotonic :
                    forall x12 x11, 
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_isNatIList x12 <= P_id_isNatIList x11.
                   Proof.
                     intros x12 x11.
                     intros [H_1 H_0].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__zeros_bounded : 0 <= P_id_a__zeros .
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U51_bounded :
                    forall x12 x11, 
                     (0 <= x11) ->(0 <= x12) ->0 <= P_id_U51 x12 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U84_bounded :
                    forall x12 x13 x11, 
                     (0 <= x11) ->
                      (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U84 x13 x12 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U32_bounded :
                    forall x12 x11, 
                     (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U32 x12 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_isNatList_bounded :
                    forall x11, (0 <= x11) ->0 <= P_id_isNatList x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U13_bounded :
                    forall x11, (0 <= x11) ->0 <= P_id_a__U13 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U85_bounded :
                    forall x12 x11, 
                     (0 <= x11) ->(0 <= x12) ->0 <= P_id_U85 x12 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__length_bounded :
                    forall x11, (0 <= x11) ->0 <= P_id_a__length x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__isNatIList_bounded :
                    forall x11, (0 <= x11) ->0 <= P_id_a__isNatIList x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U33_bounded :
                    forall x11, (0 <= x11) ->0 <= P_id_U33 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U11_bounded :
                    forall x12 x11, 
                     (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U11 x12 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U81_bounded :
                    forall x12 x13 x11, 
                     (0 <= x11) ->
                      (0 <= x12) ->(0 <= x13) ->0 <= P_id_U81 x13 x12 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U92_bounded :
                    forall x12 x13 x11, 
                     (0 <= x11) ->
                      (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U92 x13 x12 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U43_bounded :
                    forall x12 x13 x11, 
                     (0 <= x11) ->
                      (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U43 x13 x12 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U23_bounded :
                    forall x11, (0 <= x11) ->0 <= P_id_U23 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__isNatKind_bounded :
                    forall x11, (0 <= x11) ->0 <= P_id_a__isNatKind x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U93_bounded :
                    forall x12 x13 x11, 
                     (0 <= x11) ->
                      (0 <= x12) ->(0 <= x13) ->0 <= P_id_U93 x13 x12 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U11_bounded :
                    forall x12 x11, 
                     (0 <= x11) ->(0 <= x12) ->0 <= P_id_U11 x12 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U71_bounded :
                    forall x11, (0 <= x11) ->0 <= P_id_a__U71 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U44_bounded :
                    forall x12 x13 x11, 
                     (0 <= x11) ->
                      (0 <= x12) ->(0 <= x13) ->0 <= P_id_U44 x13 x12 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_0_bounded : 0 <= P_id_0 .
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U61_bounded :
                    forall x11, (0 <= x11) ->0 <= P_id_U61 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U86_bounded :
                    forall x11, (0 <= x11) ->0 <= P_id_a__U86 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U41_bounded :
                    forall x12 x13 x11, 
                     (0 <= x11) ->
                      (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U41 x13 x12 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U22_bounded :
                    forall x12 x11, 
                     (0 <= x11) ->(0 <= x12) ->0 <= P_id_U22 x12 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U21_bounded :
                    forall x12 x11, 
                     (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U21 x12 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U91_bounded :
                    forall x12 x13 x11, 
                     (0 <= x11) ->
                      (0 <= x12) ->(0 <= x13) ->0 <= P_id_U91 x13 x12 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_length_bounded :
                    forall x11, (0 <= x11) ->0 <= P_id_length x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U52_bounded :
                    forall x11, (0 <= x11) ->0 <= P_id_a__U52 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U42_bounded :
                    forall x12 x13 x11, 
                     (0 <= x11) ->
                      (0 <= x12) ->(0 <= x13) ->0 <= P_id_U42 x13 x12 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U12_bounded :
                    forall x12 x11, 
                     (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U12 x12 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U83_bounded :
                    forall x12 x13 x11, 
                     (0 <= x11) ->
                      (0 <= x12) ->(0 <= x13) ->0 <= P_id_U83 x13 x12 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U94_bounded :
                    forall x12 x11, 
                     (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U94 x12 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U45_bounded :
                    forall x12 x11, 
                     (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U45 x12 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U31_bounded :
                    forall x12 x11, 
                     (0 <= x11) ->(0 <= x12) ->0 <= P_id_U31 x12 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__isNat_bounded :
                    forall x11, (0 <= x11) ->0 <= P_id_a__isNat x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_isNatIListKind_bounded :
                    forall x11, (0 <= x11) ->0 <= P_id_isNatIListKind x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U82_bounded :
                    forall x12 x13 x11, 
                     (0 <= x11) ->
                      (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U82 x13 x12 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U46_bounded :
                    forall x11, (0 <= x11) ->0 <= P_id_U46 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_cons_bounded :
                    forall x12 x11, 
                     (0 <= x11) ->(0 <= x12) ->0 <= P_id_cons x12 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U52_bounded :
                    forall x11, (0 <= x11) ->0 <= P_id_U52 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U85_bounded :
                    forall x12 x11, 
                     (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U85 x12 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U33_bounded :
                    forall x11, (0 <= x11) ->0 <= P_id_a__U33 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U21_bounded :
                    forall x12 x11, 
                     (0 <= x11) ->(0 <= x12) ->0 <= P_id_U21 x12 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__isNatList_bounded :
                    forall x11, (0 <= x11) ->0 <= P_id_a__isNatList x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U86_bounded :
                    forall x11, (0 <= x11) ->0 <= P_id_U86 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_mark_bounded :
                    forall x11, (0 <= x11) ->0 <= P_id_mark x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U51_bounded :
                    forall x12 x11, 
                     (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U51 x12 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U41_bounded :
                    forall x12 x13 x11, 
                     (0 <= x11) ->
                      (0 <= x12) ->(0 <= x13) ->0 <= P_id_U41 x13 x12 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_tt_bounded : 0 <= P_id_tt .
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U82_bounded :
                    forall x12 x13 x11, 
                     (0 <= x11) ->
                      (0 <= x12) ->(0 <= x13) ->0 <= P_id_U82 x13 x12 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U93_bounded :
                    forall x12 x13 x11, 
                     (0 <= x11) ->
                      (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U93 x13 x12 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U44_bounded :
                    forall x12 x13 x11, 
                     (0 <= x11) ->
                      (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U44 x13 x12 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_isNat_bounded :
                    forall x11, (0 <= x11) ->0 <= P_id_isNat x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U23_bounded :
                    forall x11, (0 <= x11) ->0 <= P_id_a__U23 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U94_bounded :
                    forall x12 x11, 
                     (0 <= x11) ->(0 <= x12) ->0 <= P_id_U94 x12 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U12_bounded :
                    forall x12 x11, 
                     (0 <= x11) ->(0 <= x12) ->0 <= P_id_U12 x12 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U81_bounded :
                    forall x12 x13 x11, 
                     (0 <= x11) ->
                      (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U81 x13 x12 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U45_bounded :
                    forall x12 x11, 
                     (0 <= x11) ->(0 <= x12) ->0 <= P_id_U45 x12 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_zeros_bounded : 0 <= P_id_zeros .
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U71_bounded :
                    forall x11, (0 <= x11) ->0 <= P_id_U71 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U91_bounded :
                    forall x12 x13 x11, 
                     (0 <= x11) ->
                      (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U91 x13 x12 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U42_bounded :
                    forall x12 x13 x11, 
                     (0 <= x11) ->
                      (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U42 x13 x12 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_isNatKind_bounded :
                    forall x11, (0 <= x11) ->0 <= P_id_isNatKind x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U22_bounded :
                    forall x12 x11, 
                     (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U22 x12 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U92_bounded :
                    forall x12 x13 x11, 
                     (0 <= x11) ->
                      (0 <= x12) ->(0 <= x13) ->0 <= P_id_U92 x13 x12 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_nil_bounded : 0 <= P_id_nil .
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U61_bounded :
                    forall x11, (0 <= x11) ->0 <= P_id_a__U61 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U43_bounded :
                    forall x12 x13 x11, 
                     (0 <= x11) ->
                      (0 <= x12) ->(0 <= x13) ->0 <= P_id_U43 x13 x12 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__isNatIListKind_bounded :
                    forall x11, (0 <= x11) ->0 <= P_id_a__isNatIListKind x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U84_bounded :
                    forall x12 x13 x11, 
                     (0 <= x11) ->
                      (0 <= x12) ->(0 <= x13) ->0 <= P_id_U84 x13 x12 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_s_bounded :
                    forall x11, (0 <= x11) ->0 <= P_id_s x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U46_bounded :
                    forall x11, (0 <= x11) ->0 <= P_id_a__U46 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U32_bounded :
                    forall x12 x11, 
                     (0 <= x11) ->(0 <= x12) ->0 <= P_id_U32 x12 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U31_bounded :
                    forall x12 x11, 
                     (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U31 x12 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_U13_bounded :
                    forall x11, (0 <= x11) ->0 <= P_id_U13 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_a__U83_bounded :
                    forall x12 x13 x11, 
                     (0 <= x11) ->
                      (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U83 x13 x12 x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_isNatIList_bounded :
                    forall x11, (0 <= x11) ->0 <= P_id_isNatIList x11.
                   Proof.
                     intros .
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Definition measure  := 
                     InterpZ.measure 0 P_id_a__zeros P_id_U51 P_id_a__U84 
                      P_id_a__U32 P_id_isNatList P_id_a__U13 P_id_U85 
                      P_id_a__length P_id_a__isNatIList P_id_U33 P_id_a__U11 
                      P_id_U81 P_id_a__U92 P_id_a__U43 P_id_U23 
                      P_id_a__isNatKind P_id_U93 P_id_U11 P_id_a__U71 
                      P_id_U44 P_id_0 P_id_U61 P_id_a__U86 P_id_a__U41 
                      P_id_U22 P_id_a__U21 P_id_U91 P_id_length P_id_a__U52 
                      P_id_U42 P_id_a__U12 P_id_U83 P_id_a__U94 P_id_a__U45 
                      P_id_U31 P_id_a__isNat P_id_isNatIListKind P_id_a__U82 
                      P_id_U46 P_id_cons P_id_U52 P_id_a__U85 P_id_a__U33 
                      P_id_U21 P_id_a__isNatList P_id_U86 P_id_mark 
                      P_id_a__U51 P_id_U41 P_id_tt P_id_U82 P_id_a__U93 
                      P_id_a__U44 P_id_isNat P_id_a__U23 P_id_U94 P_id_U12 
                      P_id_a__U81 P_id_U45 P_id_zeros P_id_U71 P_id_a__U91 
                      P_id_a__U42 P_id_isNatKind P_id_a__U22 P_id_U92 
                      P_id_nil P_id_a__U61 P_id_U43 P_id_a__isNatIListKind 
                      P_id_U84 P_id_s P_id_a__U46 P_id_U32 P_id_a__U31 
                      P_id_U13 P_id_a__U83 P_id_isNatIList.
                   
                   Lemma measure_equation :
                    forall t, 
                     measure t = match t with
                                   | (algebra.Alg.Term algebra.F.id_a__zeros 
                                      nil) =>
                                    P_id_a__zeros 
                                   | (algebra.Alg.Term algebra.F.id_U51 
                                      (x12::x11::nil)) =>
                                    P_id_U51 (measure x12) (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U84 
                                      (x13::x12::x11::nil)) =>
                                    P_id_a__U84 (measure x13) (measure x12) 
                                     (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U32 
                                      (x12::x11::nil)) =>
                                    P_id_a__U32 (measure x12) (measure x11)
                                   | (algebra.Alg.Term 
                                      algebra.F.id_isNatList (x11::nil)) =>
                                    P_id_isNatList (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U13 
                                      (x11::nil)) =>
                                    P_id_a__U13 (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_U85 
                                      (x12::x11::nil)) =>
                                    P_id_U85 (measure x12) (measure x11)
                                   | (algebra.Alg.Term 
                                      algebra.F.id_a__length (x11::nil)) =>
                                    P_id_a__length (measure x11)
                                   | (algebra.Alg.Term 
                                      algebra.F.id_a__isNatIList (x11::nil)) =>
                                    P_id_a__isNatIList (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_U33 
                                      (x11::nil)) =>
                                    P_id_U33 (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U11 
                                      (x12::x11::nil)) =>
                                    P_id_a__U11 (measure x12) (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_U81 
                                      (x13::x12::x11::nil)) =>
                                    P_id_U81 (measure x13) (measure x12) 
                                     (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U92 
                                      (x13::x12::x11::nil)) =>
                                    P_id_a__U92 (measure x13) (measure x12) 
                                     (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U43 
                                      (x13::x12::x11::nil)) =>
                                    P_id_a__U43 (measure x13) (measure x12) 
                                     (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_U23 
                                      (x11::nil)) =>
                                    P_id_U23 (measure x11)
                                   | (algebra.Alg.Term 
                                      algebra.F.id_a__isNatKind (x11::nil)) =>
                                    P_id_a__isNatKind (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_U93 
                                      (x13::x12::x11::nil)) =>
                                    P_id_U93 (measure x13) (measure x12) 
                                     (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_U11 
                                      (x12::x11::nil)) =>
                                    P_id_U11 (measure x12) (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U71 
                                      (x11::nil)) =>
                                    P_id_a__U71 (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_U44 
                                      (x13::x12::x11::nil)) =>
                                    P_id_U44 (measure x13) (measure x12) 
                                     (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_0 nil) =>
                                    P_id_0 
                                   | (algebra.Alg.Term algebra.F.id_U61 
                                      (x11::nil)) =>
                                    P_id_U61 (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U86 
                                      (x11::nil)) =>
                                    P_id_a__U86 (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U41 
                                      (x13::x12::x11::nil)) =>
                                    P_id_a__U41 (measure x13) (measure x12) 
                                     (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_U22 
                                      (x12::x11::nil)) =>
                                    P_id_U22 (measure x12) (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U21 
                                      (x12::x11::nil)) =>
                                    P_id_a__U21 (measure x12) (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_U91 
                                      (x13::x12::x11::nil)) =>
                                    P_id_U91 (measure x13) (measure x12) 
                                     (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_length 
                                      (x11::nil)) =>
                                    P_id_length (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U52 
                                      (x11::nil)) =>
                                    P_id_a__U52 (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_U42 
                                      (x13::x12::x11::nil)) =>
                                    P_id_U42 (measure x13) (measure x12) 
                                     (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U12 
                                      (x12::x11::nil)) =>
                                    P_id_a__U12 (measure x12) (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_U83 
                                      (x13::x12::x11::nil)) =>
                                    P_id_U83 (measure x13) (measure x12) 
                                     (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U94 
                                      (x12::x11::nil)) =>
                                    P_id_a__U94 (measure x12) (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U45 
                                      (x12::x11::nil)) =>
                                    P_id_a__U45 (measure x12) (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_U31 
                                      (x12::x11::nil)) =>
                                    P_id_U31 (measure x12) (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__isNat 
                                      (x11::nil)) =>
                                    P_id_a__isNat (measure x11)
                                   | (algebra.Alg.Term 
                                      algebra.F.id_isNatIListKind (x11::nil)) =>
                                    P_id_isNatIListKind (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U82 
                                      (x13::x12::x11::nil)) =>
                                    P_id_a__U82 (measure x13) (measure x12) 
                                     (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_U46 
                                      (x11::nil)) =>
                                    P_id_U46 (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_cons 
                                      (x12::x11::nil)) =>
                                    P_id_cons (measure x12) (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_U52 
                                      (x11::nil)) =>
                                    P_id_U52 (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U85 
                                      (x12::x11::nil)) =>
                                    P_id_a__U85 (measure x12) (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U33 
                                      (x11::nil)) =>
                                    P_id_a__U33 (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_U21 
                                      (x12::x11::nil)) =>
                                    P_id_U21 (measure x12) (measure x11)
                                   | (algebra.Alg.Term 
                                      algebra.F.id_a__isNatList (x11::nil)) =>
                                    P_id_a__isNatList (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_U86 
                                      (x11::nil)) =>
                                    P_id_U86 (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_mark 
                                      (x11::nil)) =>
                                    P_id_mark (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U51 
                                      (x12::x11::nil)) =>
                                    P_id_a__U51 (measure x12) (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_U41 
                                      (x13::x12::x11::nil)) =>
                                    P_id_U41 (measure x13) (measure x12) 
                                     (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_tt nil) =>
                                    P_id_tt 
                                   | (algebra.Alg.Term algebra.F.id_U82 
                                      (x13::x12::x11::nil)) =>
                                    P_id_U82 (measure x13) (measure x12) 
                                     (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U93 
                                      (x13::x12::x11::nil)) =>
                                    P_id_a__U93 (measure x13) (measure x12) 
                                     (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U44 
                                      (x13::x12::x11::nil)) =>
                                    P_id_a__U44 (measure x13) (measure x12) 
                                     (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_isNat 
                                      (x11::nil)) =>
                                    P_id_isNat (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U23 
                                      (x11::nil)) =>
                                    P_id_a__U23 (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_U94 
                                      (x12::x11::nil)) =>
                                    P_id_U94 (measure x12) (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_U12 
                                      (x12::x11::nil)) =>
                                    P_id_U12 (measure x12) (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U81 
                                      (x13::x12::x11::nil)) =>
                                    P_id_a__U81 (measure x13) (measure x12) 
                                     (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_U45 
                                      (x12::x11::nil)) =>
                                    P_id_U45 (measure x12) (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_zeros 
                                      nil) =>
                                    P_id_zeros 
                                   | (algebra.Alg.Term algebra.F.id_U71 
                                      (x11::nil)) =>
                                    P_id_U71 (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U91 
                                      (x13::x12::x11::nil)) =>
                                    P_id_a__U91 (measure x13) (measure x12) 
                                     (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U42 
                                      (x13::x12::x11::nil)) =>
                                    P_id_a__U42 (measure x13) (measure x12) 
                                     (measure x11)
                                   | (algebra.Alg.Term 
                                      algebra.F.id_isNatKind (x11::nil)) =>
                                    P_id_isNatKind (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U22 
                                      (x12::x11::nil)) =>
                                    P_id_a__U22 (measure x12) (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_U92 
                                      (x13::x12::x11::nil)) =>
                                    P_id_U92 (measure x13) (measure x12) 
                                     (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_nil nil) =>
                                    P_id_nil 
                                   | (algebra.Alg.Term algebra.F.id_a__U61 
                                      (x11::nil)) =>
                                    P_id_a__U61 (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_U43 
                                      (x13::x12::x11::nil)) =>
                                    P_id_U43 (measure x13) (measure x12) 
                                     (measure x11)
                                   | (algebra.Alg.Term 
                                      algebra.F.id_a__isNatIListKind 
                                      (x11::nil)) =>
                                    P_id_a__isNatIListKind (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_U84 
                                      (x13::x12::x11::nil)) =>
                                    P_id_U84 (measure x13) (measure x12) 
                                     (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_s 
                                      (x11::nil)) =>
                                    P_id_s (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U46 
                                      (x11::nil)) =>
                                    P_id_a__U46 (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_U32 
                                      (x12::x11::nil)) =>
                                    P_id_U32 (measure x12) (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U31 
                                      (x12::x11::nil)) =>
                                    P_id_a__U31 (measure x12) (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_U13 
                                      (x11::nil)) =>
                                    P_id_U13 (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U83 
                                      (x13::x12::x11::nil)) =>
                                    P_id_a__U83 (measure x13) (measure x12) 
                                     (measure x11)
                                   | (algebra.Alg.Term 
                                      algebra.F.id_isNatIList (x11::nil)) =>
                                    P_id_isNatIList (measure x11)
                                   | _ => 0
                                   end.
                   Proof.
                     intros t;case t;intros ;apply refl_equal.
                   Qed.
                   
                   Lemma measure_bounded : forall t, 0 <= measure t.
                   Proof.
                     unfold measure in |-*.
                     
                     apply InterpZ.measure_bounded;
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Ltac generate_pos_hyp  :=
                    match goal with
                      | H:context [measure ?x] |- _ =>
                       let v := fresh "v" in 
                        (let H := fresh "h" in 
                          (set (H:=measure_bounded x) in *;
                            set (v:=measure x) in *;clearbody H;clearbody v))
                      |  |- context [measure ?x] =>
                       let v := fresh "v" in 
                        (let H := fresh "h" in 
                          (set (H:=measure_bounded x) in *;
                            set (v:=measure x) in *;clearbody H;clearbody v))
                      end
                    .
                   
                   Lemma rules_monotonic :
                    forall l r, 
                     (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) ->
                      measure r <= measure l.
                   Proof.
                     intros l r H.
                     fold measure in |-*.
                     
                     inversion H;clear H;subst;inversion H0;clear H0;
                      subst;simpl algebra.EQT.T.apply_subst in |-*;
                      repeat (
                      match goal with
                        |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
                         rewrite (measure_equation (algebra.Alg.Term f t))
                        end
                      );repeat (generate_pos_hyp );
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma measure_star_monotonic :
                    forall l r, 
                     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                                r l) ->measure r <= measure l.
                   Proof.
                     unfold measure in *.
                     apply InterpZ.measure_star_monotonic.
                     intros ;apply P_id_U51_monotonic;assumption.
                     intros ;apply P_id_a__U84_monotonic;assumption.
                     intros ;apply P_id_a__U32_monotonic;assumption.
                     intros ;apply P_id_isNatList_monotonic;assumption.
                     intros ;apply P_id_a__U13_monotonic;assumption.
                     intros ;apply P_id_U85_monotonic;assumption.
                     intros ;apply P_id_a__length_monotonic;assumption.
                     intros ;apply P_id_a__isNatIList_monotonic;assumption.
                     intros ;apply P_id_U33_monotonic;assumption.
                     intros ;apply P_id_a__U11_monotonic;assumption.
                     intros ;apply P_id_U81_monotonic;assumption.
                     intros ;apply P_id_a__U92_monotonic;assumption.
                     intros ;apply P_id_a__U43_monotonic;assumption.
                     intros ;apply P_id_U23_monotonic;assumption.
                     intros ;apply P_id_a__isNatKind_monotonic;assumption.
                     intros ;apply P_id_U93_monotonic;assumption.
                     intros ;apply P_id_U11_monotonic;assumption.
                     intros ;apply P_id_a__U71_monotonic;assumption.
                     intros ;apply P_id_U44_monotonic;assumption.
                     intros ;apply P_id_U61_monotonic;assumption.
                     intros ;apply P_id_a__U86_monotonic;assumption.
                     intros ;apply P_id_a__U41_monotonic;assumption.
                     intros ;apply P_id_U22_monotonic;assumption.
                     intros ;apply P_id_a__U21_monotonic;assumption.
                     intros ;apply P_id_U91_monotonic;assumption.
                     intros ;apply P_id_length_monotonic;assumption.
                     intros ;apply P_id_a__U52_monotonic;assumption.
                     intros ;apply P_id_U42_monotonic;assumption.
                     intros ;apply P_id_a__U12_monotonic;assumption.
                     intros ;apply P_id_U83_monotonic;assumption.
                     intros ;apply P_id_a__U94_monotonic;assumption.
                     intros ;apply P_id_a__U45_monotonic;assumption.
                     intros ;apply P_id_U31_monotonic;assumption.
                     intros ;apply P_id_a__isNat_monotonic;assumption.
                     intros ;apply P_id_isNatIListKind_monotonic;assumption.
                     intros ;apply P_id_a__U82_monotonic;assumption.
                     intros ;apply P_id_U46_monotonic;assumption.
                     intros ;apply P_id_cons_monotonic;assumption.
                     intros ;apply P_id_U52_monotonic;assumption.
                     intros ;apply P_id_a__U85_monotonic;assumption.
                     intros ;apply P_id_a__U33_monotonic;assumption.
                     intros ;apply P_id_U21_monotonic;assumption.
                     intros ;apply P_id_a__isNatList_monotonic;assumption.
                     intros ;apply P_id_U86_monotonic;assumption.
                     intros ;apply P_id_mark_monotonic;assumption.
                     intros ;apply P_id_a__U51_monotonic;assumption.
                     intros ;apply P_id_U41_monotonic;assumption.
                     intros ;apply P_id_U82_monotonic;assumption.
                     intros ;apply P_id_a__U93_monotonic;assumption.
                     intros ;apply P_id_a__U44_monotonic;assumption.
                     intros ;apply P_id_isNat_monotonic;assumption.
                     intros ;apply P_id_a__U23_monotonic;assumption.
                     intros ;apply P_id_U94_monotonic;assumption.
                     intros ;apply P_id_U12_monotonic;assumption.
                     intros ;apply P_id_a__U81_monotonic;assumption.
                     intros ;apply P_id_U45_monotonic;assumption.
                     intros ;apply P_id_U71_monotonic;assumption.
                     intros ;apply P_id_a__U91_monotonic;assumption.
                     intros ;apply P_id_a__U42_monotonic;assumption.
                     intros ;apply P_id_isNatKind_monotonic;assumption.
                     intros ;apply P_id_a__U22_monotonic;assumption.
                     intros ;apply P_id_U92_monotonic;assumption.
                     intros ;apply P_id_a__U61_monotonic;assumption.
                     intros ;apply P_id_U43_monotonic;assumption.
                     
                     intros ;apply P_id_a__isNatIListKind_monotonic;
                      assumption.
                     intros ;apply P_id_U84_monotonic;assumption.
                     intros ;apply P_id_s_monotonic;assumption.
                     intros ;apply P_id_a__U46_monotonic;assumption.
                     intros ;apply P_id_U32_monotonic;assumption.
                     intros ;apply P_id_a__U31_monotonic;assumption.
                     intros ;apply P_id_U13_monotonic;assumption.
                     intros ;apply P_id_a__U83_monotonic;assumption.
                     intros ;apply P_id_isNatIList_monotonic;assumption.
                     intros ;apply P_id_a__zeros_bounded;assumption.
                     intros ;apply P_id_U51_bounded;assumption.
                     intros ;apply P_id_a__U84_bounded;assumption.
                     intros ;apply P_id_a__U32_bounded;assumption.
                     intros ;apply P_id_isNatList_bounded;assumption.
                     intros ;apply P_id_a__U13_bounded;assumption.
                     intros ;apply P_id_U85_bounded;assumption.
                     intros ;apply P_id_a__length_bounded;assumption.
                     intros ;apply P_id_a__isNatIList_bounded;assumption.
                     intros ;apply P_id_U33_bounded;assumption.
                     intros ;apply P_id_a__U11_bounded;assumption.
                     intros ;apply P_id_U81_bounded;assumption.
                     intros ;apply P_id_a__U92_bounded;assumption.
                     intros ;apply P_id_a__U43_bounded;assumption.
                     intros ;apply P_id_U23_bounded;assumption.
                     intros ;apply P_id_a__isNatKind_bounded;assumption.
                     intros ;apply P_id_U93_bounded;assumption.
                     intros ;apply P_id_U11_bounded;assumption.
                     intros ;apply P_id_a__U71_bounded;assumption.
                     intros ;apply P_id_U44_bounded;assumption.
                     intros ;apply P_id_0_bounded;assumption.
                     intros ;apply P_id_U61_bounded;assumption.
                     intros ;apply P_id_a__U86_bounded;assumption.
                     intros ;apply P_id_a__U41_bounded;assumption.
                     intros ;apply P_id_U22_bounded;assumption.
                     intros ;apply P_id_a__U21_bounded;assumption.
                     intros ;apply P_id_U91_bounded;assumption.
                     intros ;apply P_id_length_bounded;assumption.
                     intros ;apply P_id_a__U52_bounded;assumption.
                     intros ;apply P_id_U42_bounded;assumption.
                     intros ;apply P_id_a__U12_bounded;assumption.
                     intros ;apply P_id_U83_bounded;assumption.
                     intros ;apply P_id_a__U94_bounded;assumption.
                     intros ;apply P_id_a__U45_bounded;assumption.
                     intros ;apply P_id_U31_bounded;assumption.
                     intros ;apply P_id_a__isNat_bounded;assumption.
                     intros ;apply P_id_isNatIListKind_bounded;assumption.
                     intros ;apply P_id_a__U82_bounded;assumption.
                     intros ;apply P_id_U46_bounded;assumption.
                     intros ;apply P_id_cons_bounded;assumption.
                     intros ;apply P_id_U52_bounded;assumption.
                     intros ;apply P_id_a__U85_bounded;assumption.
                     intros ;apply P_id_a__U33_bounded;assumption.
                     intros ;apply P_id_U21_bounded;assumption.
                     intros ;apply P_id_a__isNatList_bounded;assumption.
                     intros ;apply P_id_U86_bounded;assumption.
                     intros ;apply P_id_mark_bounded;assumption.
                     intros ;apply P_id_a__U51_bounded;assumption.
                     intros ;apply P_id_U41_bounded;assumption.
                     intros ;apply P_id_tt_bounded;assumption.
                     intros ;apply P_id_U82_bounded;assumption.
                     intros ;apply P_id_a__U93_bounded;assumption.
                     intros ;apply P_id_a__U44_bounded;assumption.
                     intros ;apply P_id_isNat_bounded;assumption.
                     intros ;apply P_id_a__U23_bounded;assumption.
                     intros ;apply P_id_U94_bounded;assumption.
                     intros ;apply P_id_U12_bounded;assumption.
                     intros ;apply P_id_a__U81_bounded;assumption.
                     intros ;apply P_id_U45_bounded;assumption.
                     intros ;apply P_id_zeros_bounded;assumption.
                     intros ;apply P_id_U71_bounded;assumption.
                     intros ;apply P_id_a__U91_bounded;assumption.
                     intros ;apply P_id_a__U42_bounded;assumption.
                     intros ;apply P_id_isNatKind_bounded;assumption.
                     intros ;apply P_id_a__U22_bounded;assumption.
                     intros ;apply P_id_U92_bounded;assumption.
                     intros ;apply P_id_nil_bounded;assumption.
                     intros ;apply P_id_a__U61_bounded;assumption.
                     intros ;apply P_id_U43_bounded;assumption.
                     intros ;apply P_id_a__isNatIListKind_bounded;assumption.
                     intros ;apply P_id_U84_bounded;assumption.
                     intros ;apply P_id_s_bounded;assumption.
                     intros ;apply P_id_a__U46_bounded;assumption.
                     intros ;apply P_id_U32_bounded;assumption.
                     intros ;apply P_id_a__U31_bounded;assumption.
                     intros ;apply P_id_U13_bounded;assumption.
                     intros ;apply P_id_a__U83_bounded;assumption.
                     intros ;apply P_id_isNatIList_bounded;assumption.
                     apply rules_monotonic.
                   Qed.
                   
                   Definition P_id_A__U46 (x11:Z) := 0.
                   
                   Definition P_id_A__ISNATILISTKIND (x11:Z) := 0.
                   
                   Definition P_id_A__U61 (x11:Z) := 0.
                   
                   Definition P_id_A__U85 (x11:Z) (x12:Z) := 0.
                   
                   Definition P_id_A__U31 (x11:Z) (x12:Z) := 0.
                   
                   Definition P_id_A__U81 (x11:Z) (x12:Z) (x13:Z) := 0.
                   
                   Definition P_id_A__U22 (x11:Z) (x12:Z) := 0.
                   
                   Definition P_id_A__U93 (x11:Z) (x12:Z) (x13:Z) := 0.
                   
                   Definition P_id_A__U42 (x11:Z) (x12:Z) (x13:Z) := 0.
                   
                   Definition P_id_A__U51 (x11:Z) (x12:Z) := 0.
                   
                   Definition P_id_A__ISNATLIST (x11:Z) := 0.
                   
                   Definition P_id_A__ZEROS  := 0.
                   
                   Definition P_id_A__U91 (x11:Z) (x12:Z) (x13:Z) := 0.
                   
                   Definition P_id_A__U33 (x11:Z) := 0.
                   
                   Definition P_id_A__U83 (x11:Z) (x12:Z) (x13:Z) := 0.
                   
                   Definition P_id_A__U23 (x11:Z) := 0.
                   
                   Definition P_id_A__U11 (x11:Z) (x12:Z) := 0.
                   
                   Definition P_id_A__LENGTH (x11:Z) := 0.
                   
                   Definition P_id_A__U44 (x11:Z) (x12:Z) (x13:Z) := 0.
                   
                   Definition P_id_A__ISNATILIST (x11:Z) := 0.
                   
                   Definition P_id_A__U13 (x11:Z) := 0.
                   
                   Definition P_id_A__U71 (x11:Z) := 0.
                   
                   Definition P_id_A__U86 (x11:Z) := 0.
                   
                   Definition P_id_A__U32 (x11:Z) (x12:Z) := 0.
                   
                   Definition P_id_A__U82 (x11:Z) (x12:Z) (x13:Z) := 0.
                   
                   Definition P_id_A__ISNATKIND (x11:Z) := 0.
                   
                   Definition P_id_A__U94 (x11:Z) (x12:Z) := 0.
                   
                   Definition P_id_A__U43 (x11:Z) (x12:Z) (x13:Z) := 0.
                   
                   Definition P_id_A__U52 (x11:Z) := 0.
                   
                   Definition P_id_A__U21 (x11:Z) (x12:Z) := 0.
                   
                   Definition P_id_A__U92 (x11:Z) (x12:Z) (x13:Z) := 0.
                   
                   Definition P_id_A__U41 (x11:Z) (x12:Z) (x13:Z) := 0.
                   
                   Definition P_id_A__U84 (x11:Z) (x12:Z) (x13:Z) := 0.
                   
                   Definition P_id_A__ISNAT (x11:Z) := 0.
                   
                   Definition P_id_A__U12 (x11:Z) (x12:Z) := 0.
                   
                   Definition P_id_MARK (x11:Z) := 2* x11.
                   
                   Definition P_id_A__U45 (x11:Z) (x12:Z) := 0.
                   
                   Lemma P_id_A__U46_monotonic :
                    forall x12 x11, 
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_A__U46 x12 <= P_id_A__U46 x11.
                   Proof.
                     intros x12 x11.
                     intros [H_1 H_0].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_A__ISNATILISTKIND_monotonic :
                    forall x12 x11, 
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_A__ISNATILISTKIND x12 <= P_id_A__ISNATILISTKIND 
                                                     x11.
                   Proof.
                     intros x12 x11.
                     intros [H_1 H_0].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_A__U61_monotonic :
                    forall x12 x11, 
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_A__U61 x12 <= P_id_A__U61 x11.
                   Proof.
                     intros x12 x11.
                     intros [H_1 H_0].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_A__U85_monotonic :
                    forall x12 x14 x13 x11, 
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_A__U85 x12 x14 <= P_id_A__U85 x11 x13.
                   Proof.
                     intros x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_A__U31_monotonic :
                    forall x12 x14 x13 x11, 
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_A__U31 x12 x14 <= P_id_A__U31 x11 x13.
                   Proof.
                     intros x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_A__U81_monotonic :
                    forall x16 x12 x14 x13 x11 x15, 
                     (0 <= x16)/\ (x16 <= x15) ->
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_A__U81 x12 x14 x16 <= P_id_A__U81 x11 x13 x15.
                   Proof.
                     intros x16 x15 x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     intros [H_5 H_4].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_A__U22_monotonic :
                    forall x12 x14 x13 x11, 
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_A__U22 x12 x14 <= P_id_A__U22 x11 x13.
                   Proof.
                     intros x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_A__U93_monotonic :
                    forall x16 x12 x14 x13 x11 x15, 
                     (0 <= x16)/\ (x16 <= x15) ->
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_A__U93 x12 x14 x16 <= P_id_A__U93 x11 x13 x15.
                   Proof.
                     intros x16 x15 x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     intros [H_5 H_4].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_A__U42_monotonic :
                    forall x16 x12 x14 x13 x11 x15, 
                     (0 <= x16)/\ (x16 <= x15) ->
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_A__U42 x12 x14 x16 <= P_id_A__U42 x11 x13 x15.
                   Proof.
                     intros x16 x15 x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     intros [H_5 H_4].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_A__U51_monotonic :
                    forall x12 x14 x13 x11, 
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_A__U51 x12 x14 <= P_id_A__U51 x11 x13.
                   Proof.
                     intros x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_A__ISNATLIST_monotonic :
                    forall x12 x11, 
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_A__ISNATLIST x12 <= P_id_A__ISNATLIST x11.
                   Proof.
                     intros x12 x11.
                     intros [H_1 H_0].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_A__U91_monotonic :
                    forall x16 x12 x14 x13 x11 x15, 
                     (0 <= x16)/\ (x16 <= x15) ->
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_A__U91 x12 x14 x16 <= P_id_A__U91 x11 x13 x15.
                   Proof.
                     intros x16 x15 x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     intros [H_5 H_4].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_A__U33_monotonic :
                    forall x12 x11, 
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_A__U33 x12 <= P_id_A__U33 x11.
                   Proof.
                     intros x12 x11.
                     intros [H_1 H_0].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_A__U83_monotonic :
                    forall x16 x12 x14 x13 x11 x15, 
                     (0 <= x16)/\ (x16 <= x15) ->
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_A__U83 x12 x14 x16 <= P_id_A__U83 x11 x13 x15.
                   Proof.
                     intros x16 x15 x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     intros [H_5 H_4].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_A__U23_monotonic :
                    forall x12 x11, 
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_A__U23 x12 <= P_id_A__U23 x11.
                   Proof.
                     intros x12 x11.
                     intros [H_1 H_0].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_A__U11_monotonic :
                    forall x12 x14 x13 x11, 
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_A__U11 x12 x14 <= P_id_A__U11 x11 x13.
                   Proof.
                     intros x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_A__LENGTH_monotonic :
                    forall x12 x11, 
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_A__LENGTH x12 <= P_id_A__LENGTH x11.
                   Proof.
                     intros x12 x11.
                     intros [H_1 H_0].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_A__U44_monotonic :
                    forall x16 x12 x14 x13 x11 x15, 
                     (0 <= x16)/\ (x16 <= x15) ->
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_A__U44 x12 x14 x16 <= P_id_A__U44 x11 x13 x15.
                   Proof.
                     intros x16 x15 x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     intros [H_5 H_4].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_A__ISNATILIST_monotonic :
                    forall x12 x11, 
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_A__ISNATILIST x12 <= P_id_A__ISNATILIST x11.
                   Proof.
                     intros x12 x11.
                     intros [H_1 H_0].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_A__U13_monotonic :
                    forall x12 x11, 
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_A__U13 x12 <= P_id_A__U13 x11.
                   Proof.
                     intros x12 x11.
                     intros [H_1 H_0].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_A__U71_monotonic :
                    forall x12 x11, 
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_A__U71 x12 <= P_id_A__U71 x11.
                   Proof.
                     intros x12 x11.
                     intros [H_1 H_0].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_A__U86_monotonic :
                    forall x12 x11, 
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_A__U86 x12 <= P_id_A__U86 x11.
                   Proof.
                     intros x12 x11.
                     intros [H_1 H_0].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_A__U32_monotonic :
                    forall x12 x14 x13 x11, 
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_A__U32 x12 x14 <= P_id_A__U32 x11 x13.
                   Proof.
                     intros x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_A__U82_monotonic :
                    forall x16 x12 x14 x13 x11 x15, 
                     (0 <= x16)/\ (x16 <= x15) ->
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_A__U82 x12 x14 x16 <= P_id_A__U82 x11 x13 x15.
                   Proof.
                     intros x16 x15 x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     intros [H_5 H_4].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_A__ISNATKIND_monotonic :
                    forall x12 x11, 
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_A__ISNATKIND x12 <= P_id_A__ISNATKIND x11.
                   Proof.
                     intros x12 x11.
                     intros [H_1 H_0].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_A__U94_monotonic :
                    forall x12 x14 x13 x11, 
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_A__U94 x12 x14 <= P_id_A__U94 x11 x13.
                   Proof.
                     intros x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_A__U43_monotonic :
                    forall x16 x12 x14 x13 x11 x15, 
                     (0 <= x16)/\ (x16 <= x15) ->
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_A__U43 x12 x14 x16 <= P_id_A__U43 x11 x13 x15.
                   Proof.
                     intros x16 x15 x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     intros [H_5 H_4].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_A__U52_monotonic :
                    forall x12 x11, 
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_A__U52 x12 <= P_id_A__U52 x11.
                   Proof.
                     intros x12 x11.
                     intros [H_1 H_0].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_A__U21_monotonic :
                    forall x12 x14 x13 x11, 
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_A__U21 x12 x14 <= P_id_A__U21 x11 x13.
                   Proof.
                     intros x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_A__U92_monotonic :
                    forall x16 x12 x14 x13 x11 x15, 
                     (0 <= x16)/\ (x16 <= x15) ->
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_A__U92 x12 x14 x16 <= P_id_A__U92 x11 x13 x15.
                   Proof.
                     intros x16 x15 x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     intros [H_5 H_4].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_A__U41_monotonic :
                    forall x16 x12 x14 x13 x11 x15, 
                     (0 <= x16)/\ (x16 <= x15) ->
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_A__U41 x12 x14 x16 <= P_id_A__U41 x11 x13 x15.
                   Proof.
                     intros x16 x15 x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     intros [H_5 H_4].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_A__U84_monotonic :
                    forall x16 x12 x14 x13 x11 x15, 
                     (0 <= x16)/\ (x16 <= x15) ->
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_A__U84 x12 x14 x16 <= P_id_A__U84 x11 x13 x15.
                   Proof.
                     intros x16 x15 x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     intros [H_5 H_4].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_A__ISNAT_monotonic :
                    forall x12 x11, 
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_A__ISNAT x12 <= P_id_A__ISNAT x11.
                   Proof.
                     intros x12 x11.
                     intros [H_1 H_0].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_A__U12_monotonic :
                    forall x12 x14 x13 x11, 
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_A__U12 x12 x14 <= P_id_A__U12 x11 x13.
                   Proof.
                     intros x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_MARK_monotonic :
                    forall x12 x11, 
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_MARK x12 <= P_id_MARK x11.
                   Proof.
                     intros x12 x11.
                     intros [H_1 H_0].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma P_id_A__U45_monotonic :
                    forall x12 x14 x13 x11, 
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_A__U45 x12 x14 <= P_id_A__U45 x11 x13.
                   Proof.
                     intros x14 x13 x12 x11.
                     intros [H_1 H_0].
                     intros [H_3 H_2].
                     
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Definition marked_measure  := 
                     InterpZ.marked_measure 0 P_id_a__zeros P_id_U51 
                      P_id_a__U84 P_id_a__U32 P_id_isNatList P_id_a__U13 
                      P_id_U85 P_id_a__length P_id_a__isNatIList P_id_U33 
                      P_id_a__U11 P_id_U81 P_id_a__U92 P_id_a__U43 P_id_U23 
                      P_id_a__isNatKind P_id_U93 P_id_U11 P_id_a__U71 
                      P_id_U44 P_id_0 P_id_U61 P_id_a__U86 P_id_a__U41 
                      P_id_U22 P_id_a__U21 P_id_U91 P_id_length P_id_a__U52 
                      P_id_U42 P_id_a__U12 P_id_U83 P_id_a__U94 P_id_a__U45 
                      P_id_U31 P_id_a__isNat P_id_isNatIListKind P_id_a__U82 
                      P_id_U46 P_id_cons P_id_U52 P_id_a__U85 P_id_a__U33 
                      P_id_U21 P_id_a__isNatList P_id_U86 P_id_mark 
                      P_id_a__U51 P_id_U41 P_id_tt P_id_U82 P_id_a__U93 
                      P_id_a__U44 P_id_isNat P_id_a__U23 P_id_U94 P_id_U12 
                      P_id_a__U81 P_id_U45 P_id_zeros P_id_U71 P_id_a__U91 
                      P_id_a__U42 P_id_isNatKind P_id_a__U22 P_id_U92 
                      P_id_nil P_id_a__U61 P_id_U43 P_id_a__isNatIListKind 
                      P_id_U84 P_id_s P_id_a__U46 P_id_U32 P_id_a__U31 
                      P_id_U13 P_id_a__U83 P_id_isNatIList P_id_A__U46 
                      P_id_A__ISNATILISTKIND P_id_A__U61 P_id_A__U85 
                      P_id_A__U31 P_id_A__U81 P_id_A__U22 P_id_A__U93 
                      P_id_A__U42 P_id_A__U51 P_id_A__ISNATLIST 
                      P_id_A__ZEROS P_id_A__U91 P_id_A__U33 P_id_A__U83 
                      P_id_A__U23 P_id_A__U11 P_id_A__LENGTH P_id_A__U44 
                      P_id_A__ISNATILIST P_id_A__U13 P_id_A__U71 P_id_A__U86 
                      P_id_A__U32 P_id_A__U82 P_id_A__ISNATKIND P_id_A__U94 
                      P_id_A__U43 P_id_A__U52 P_id_A__U21 P_id_A__U92 
                      P_id_A__U41 P_id_A__U84 P_id_A__ISNAT P_id_A__U12 
                      P_id_MARK P_id_A__U45.
                   
                   Lemma marked_measure_equation :
                    forall t, 
                     marked_measure t = match t with
                                          | (algebra.Alg.Term 
                                             algebra.F.id_a__U46 (x11::nil)) =>
                                           P_id_A__U46 (measure x11)
                                          | (algebra.Alg.Term 
                                             algebra.F.id_a__isNatIListKind 
                                             (x11::nil)) =>
                                           P_id_A__ISNATILISTKIND (measure 
                                                                    x11)
                                          | (algebra.Alg.Term 
                                             algebra.F.id_a__U61 (x11::nil)) =>
                                           P_id_A__U61 (measure x11)
                                          | (algebra.Alg.Term 
                                             algebra.F.id_a__U85 (x12::
                                             x11::nil)) =>
                                           P_id_A__U85 (measure x12) 
                                            (measure x11)
                                          | (algebra.Alg.Term 
                                             algebra.F.id_a__U31 (x12::
                                             x11::nil)) =>
                                           P_id_A__U31 (measure x12) 
                                            (measure x11)
                                          | (algebra.Alg.Term 
                                             algebra.F.id_a__U81 (x13::x12::
                                             x11::nil)) =>
                                           P_id_A__U81 (measure x13) 
                                            (measure x12) (measure x11)
                                          | (algebra.Alg.Term 
                                             algebra.F.id_a__U22 (x12::
                                             x11::nil)) =>
                                           P_id_A__U22 (measure x12) 
                                            (measure x11)
                                          | (algebra.Alg.Term 
                                             algebra.F.id_a__U93 (x13::x12::
                                             x11::nil)) =>
                                           P_id_A__U93 (measure x13) 
                                            (measure x12) (measure x11)
                                          | (algebra.Alg.Term 
                                             algebra.F.id_a__U42 (x13::x12::
                                             x11::nil)) =>
                                           P_id_A__U42 (measure x13) 
                                            (measure x12) (measure x11)
                                          | (algebra.Alg.Term 
                                             algebra.F.id_a__U51 (x12::
                                             x11::nil)) =>
                                           P_id_A__U51 (measure x12) 
                                            (measure x11)
                                          | (algebra.Alg.Term 
                                             algebra.F.id_a__isNatList 
                                             (x11::nil)) =>
                                           P_id_A__ISNATLIST (measure x11)
                                          | (algebra.Alg.Term 
                                             algebra.F.id_a__zeros nil) =>
                                           P_id_A__ZEROS 
                                          | (algebra.Alg.Term 
                                             algebra.F.id_a__U91 (x13::x12::
                                             x11::nil)) =>
                                           P_id_A__U91 (measure x13) 
                                            (measure x12) (measure x11)
                                          | (algebra.Alg.Term 
                                             algebra.F.id_a__U33 (x11::nil)) =>
                                           P_id_A__U33 (measure x11)
                                          | (algebra.Alg.Term 
                                             algebra.F.id_a__U83 (x13::x12::
                                             x11::nil)) =>
                                           P_id_A__U83 (measure x13) 
                                            (measure x12) (measure x11)
                                          | (algebra.Alg.Term 
                                             algebra.F.id_a__U23 (x11::nil)) =>
                                           P_id_A__U23 (measure x11)
                                          | (algebra.Alg.Term 
                                             algebra.F.id_a__U11 (x12::
                                             x11::nil)) =>
                                           P_id_A__U11 (measure x12) 
                                            (measure x11)
                                          | (algebra.Alg.Term 
                                             algebra.F.id_a__length 
                                             (x11::nil)) =>
                                           P_id_A__LENGTH (measure x11)
                                          | (algebra.Alg.Term 
                                             algebra.F.id_a__U44 (x13::x12::
                                             x11::nil)) =>
                                           P_id_A__U44 (measure x13) 
                                            (measure x12) (measure x11)
                                          | (algebra.Alg.Term 
                                             algebra.F.id_a__isNatIList 
                                             (x11::nil)) =>
                                           P_id_A__ISNATILIST (measure x11)
                                          | (algebra.Alg.Term 
                                             algebra.F.id_a__U13 (x11::nil)) =>
                                           P_id_A__U13 (measure x11)
                                          | (algebra.Alg.Term 
                                             algebra.F.id_a__U71 (x11::nil)) =>
                                           P_id_A__U71 (measure x11)
                                          | (algebra.Alg.Term 
                                             algebra.F.id_a__U86 (x11::nil)) =>
                                           P_id_A__U86 (measure x11)
                                          | (algebra.Alg.Term 
                                             algebra.F.id_a__U32 (x12::
                                             x11::nil)) =>
                                           P_id_A__U32 (measure x12) 
                                            (measure x11)
                                          | (algebra.Alg.Term 
                                             algebra.F.id_a__U82 (x13::x12::
                                             x11::nil)) =>
                                           P_id_A__U82 (measure x13) 
                                            (measure x12) (measure x11)
                                          | (algebra.Alg.Term 
                                             algebra.F.id_a__isNatKind 
                                             (x11::nil)) =>
                                           P_id_A__ISNATKIND (measure x11)
                                          | (algebra.Alg.Term 
                                             algebra.F.id_a__U94 (x12::
                                             x11::nil)) =>
                                           P_id_A__U94 (measure x12) 
                                            (measure x11)
                                          | (algebra.Alg.Term 
                                             algebra.F.id_a__U43 (x13::x12::
                                             x11::nil)) =>
                                           P_id_A__U43 (measure x13) 
                                            (measure x12) (measure x11)
                                          | (algebra.Alg.Term 
                                             algebra.F.id_a__U52 (x11::nil)) =>
                                           P_id_A__U52 (measure x11)
                                          | (algebra.Alg.Term 
                                             algebra.F.id_a__U21 (x12::
                                             x11::nil)) =>
                                           P_id_A__U21 (measure x12) 
                                            (measure x11)
                                          | (algebra.Alg.Term 
                                             algebra.F.id_a__U92 (x13::x12::
                                             x11::nil)) =>
                                           P_id_A__U92 (measure x13) 
                                            (measure x12) (measure x11)
                                          | (algebra.Alg.Term 
                                             algebra.F.id_a__U41 (x13::x12::
                                             x11::nil)) =>
                                           P_id_A__U41 (measure x13) 
                                            (measure x12) (measure x11)
                                          | (algebra.Alg.Term 
                                             algebra.F.id_a__U84 (x13::x12::
                                             x11::nil)) =>
                                           P_id_A__U84 (measure x13) 
                                            (measure x12) (measure x11)
                                          | (algebra.Alg.Term 
                                             algebra.F.id_a__isNat 
                                             (x11::nil)) =>
                                           P_id_A__ISNAT (measure x11)
                                          | (algebra.Alg.Term 
                                             algebra.F.id_a__U12 (x12::
                                             x11::nil)) =>
                                           P_id_A__U12 (measure x12) 
                                            (measure x11)
                                          | (algebra.Alg.Term 
                                             algebra.F.id_mark (x11::nil)) =>
                                           P_id_MARK (measure x11)
                                          | (algebra.Alg.Term 
                                             algebra.F.id_a__U45 (x12::
                                             x11::nil)) =>
                                           P_id_A__U45 (measure x12) 
                                            (measure x11)
                                          | _ => measure t
                                          end.
                   Proof.
                     reflexivity .
                   Qed.
                   
                   Lemma marked_measure_star_monotonic :
                    forall f l1 l2, 
                     (closure.refl_trans_clos (closure.one_step_list 
                                                (algebra.EQT.one_step 
                                                  R_xml_0_deep_rew.R_xml_0_rules)
                                                 ) l1 l2) ->
                      marked_measure (algebra.Alg.Term f l1) <= marked_measure 
                                                                 (algebra.Alg.Term 
                                                                   f 
                                                                   l2).
                   Proof.
                     unfold marked_measure in *.
                     apply InterpZ.marked_measure_star_monotonic.
                     intros ;apply P_id_U51_monotonic;assumption.
                     intros ;apply P_id_a__U84_monotonic;assumption.
                     intros ;apply P_id_a__U32_monotonic;assumption.
                     intros ;apply P_id_isNatList_monotonic;assumption.
                     intros ;apply P_id_a__U13_monotonic;assumption.
                     intros ;apply P_id_U85_monotonic;assumption.
                     intros ;apply P_id_a__length_monotonic;assumption.
                     intros ;apply P_id_a__isNatIList_monotonic;assumption.
                     intros ;apply P_id_U33_monotonic;assumption.
                     intros ;apply P_id_a__U11_monotonic;assumption.
                     intros ;apply P_id_U81_monotonic;assumption.
                     intros ;apply P_id_a__U92_monotonic;assumption.
                     intros ;apply P_id_a__U43_monotonic;assumption.
                     intros ;apply P_id_U23_monotonic;assumption.
                     intros ;apply P_id_a__isNatKind_monotonic;assumption.
                     intros ;apply P_id_U93_monotonic;assumption.
                     intros ;apply P_id_U11_monotonic;assumption.
                     intros ;apply P_id_a__U71_monotonic;assumption.
                     intros ;apply P_id_U44_monotonic;assumption.
                     intros ;apply P_id_U61_monotonic;assumption.
                     intros ;apply P_id_a__U86_monotonic;assumption.
                     intros ;apply P_id_a__U41_monotonic;assumption.
                     intros ;apply P_id_U22_monotonic;assumption.
                     intros ;apply P_id_a__U21_monotonic;assumption.
                     intros ;apply P_id_U91_monotonic;assumption.
                     intros ;apply P_id_length_monotonic;assumption.
                     intros ;apply P_id_a__U52_monotonic;assumption.
                     intros ;apply P_id_U42_monotonic;assumption.
                     intros ;apply P_id_a__U12_monotonic;assumption.
                     intros ;apply P_id_U83_monotonic;assumption.
                     intros ;apply P_id_a__U94_monotonic;assumption.
                     intros ;apply P_id_a__U45_monotonic;assumption.
                     intros ;apply P_id_U31_monotonic;assumption.
                     intros ;apply P_id_a__isNat_monotonic;assumption.
                     intros ;apply P_id_isNatIListKind_monotonic;assumption.
                     intros ;apply P_id_a__U82_monotonic;assumption.
                     intros ;apply P_id_U46_monotonic;assumption.
                     intros ;apply P_id_cons_monotonic;assumption.
                     intros ;apply P_id_U52_monotonic;assumption.
                     intros ;apply P_id_a__U85_monotonic;assumption.
                     intros ;apply P_id_a__U33_monotonic;assumption.
                     intros ;apply P_id_U21_monotonic;assumption.
                     intros ;apply P_id_a__isNatList_monotonic;assumption.
                     intros ;apply P_id_U86_monotonic;assumption.
                     intros ;apply P_id_mark_monotonic;assumption.
                     intros ;apply P_id_a__U51_monotonic;assumption.
                     intros ;apply P_id_U41_monotonic;assumption.
                     intros ;apply P_id_U82_monotonic;assumption.
                     intros ;apply P_id_a__U93_monotonic;assumption.
                     intros ;apply P_id_a__U44_monotonic;assumption.
                     intros ;apply P_id_isNat_monotonic;assumption.
                     intros ;apply P_id_a__U23_monotonic;assumption.
                     intros ;apply P_id_U94_monotonic;assumption.
                     intros ;apply P_id_U12_monotonic;assumption.
                     intros ;apply P_id_a__U81_monotonic;assumption.
                     intros ;apply P_id_U45_monotonic;assumption.
                     intros ;apply P_id_U71_monotonic;assumption.
                     intros ;apply P_id_a__U91_monotonic;assumption.
                     intros ;apply P_id_a__U42_monotonic;assumption.
                     intros ;apply P_id_isNatKind_monotonic;assumption.
                     intros ;apply P_id_a__U22_monotonic;assumption.
                     intros ;apply P_id_U92_monotonic;assumption.
                     intros ;apply P_id_a__U61_monotonic;assumption.
                     intros ;apply P_id_U43_monotonic;assumption.
                     
                     intros ;apply P_id_a__isNatIListKind_monotonic;
                      assumption.
                     intros ;apply P_id_U84_monotonic;assumption.
                     intros ;apply P_id_s_monotonic;assumption.
                     intros ;apply P_id_a__U46_monotonic;assumption.
                     intros ;apply P_id_U32_monotonic;assumption.
                     intros ;apply P_id_a__U31_monotonic;assumption.
                     intros ;apply P_id_U13_monotonic;assumption.
                     intros ;apply P_id_a__U83_monotonic;assumption.
                     intros ;apply P_id_isNatIList_monotonic;assumption.
                     intros ;apply P_id_a__zeros_bounded;assumption.
                     intros ;apply P_id_U51_bounded;assumption.
                     intros ;apply P_id_a__U84_bounded;assumption.
                     intros ;apply P_id_a__U32_bounded;assumption.
                     intros ;apply P_id_isNatList_bounded;assumption.
                     intros ;apply P_id_a__U13_bounded;assumption.
                     intros ;apply P_id_U85_bounded;assumption.
                     intros ;apply P_id_a__length_bounded;assumption.
                     intros ;apply P_id_a__isNatIList_bounded;assumption.
                     intros ;apply P_id_U33_bounded;assumption.
                     intros ;apply P_id_a__U11_bounded;assumption.
                     intros ;apply P_id_U81_bounded;assumption.
                     intros ;apply P_id_a__U92_bounded;assumption.
                     intros ;apply P_id_a__U43_bounded;assumption.
                     intros ;apply P_id_U23_bounded;assumption.
                     intros ;apply P_id_a__isNatKind_bounded;assumption.
                     intros ;apply P_id_U93_bounded;assumption.
                     intros ;apply P_id_U11_bounded;assumption.
                     intros ;apply P_id_a__U71_bounded;assumption.
                     intros ;apply P_id_U44_bounded;assumption.
                     intros ;apply P_id_0_bounded;assumption.
                     intros ;apply P_id_U61_bounded;assumption.
                     intros ;apply P_id_a__U86_bounded;assumption.
                     intros ;apply P_id_a__U41_bounded;assumption.
                     intros ;apply P_id_U22_bounded;assumption.
                     intros ;apply P_id_a__U21_bounded;assumption.
                     intros ;apply P_id_U91_bounded;assumption.
                     intros ;apply P_id_length_bounded;assumption.
                     intros ;apply P_id_a__U52_bounded;assumption.
                     intros ;apply P_id_U42_bounded;assumption.
                     intros ;apply P_id_a__U12_bounded;assumption.
                     intros ;apply P_id_U83_bounded;assumption.
                     intros ;apply P_id_a__U94_bounded;assumption.
                     intros ;apply P_id_a__U45_bounded;assumption.
                     intros ;apply P_id_U31_bounded;assumption.
                     intros ;apply P_id_a__isNat_bounded;assumption.
                     intros ;apply P_id_isNatIListKind_bounded;assumption.
                     intros ;apply P_id_a__U82_bounded;assumption.
                     intros ;apply P_id_U46_bounded;assumption.
                     intros ;apply P_id_cons_bounded;assumption.
                     intros ;apply P_id_U52_bounded;assumption.
                     intros ;apply P_id_a__U85_bounded;assumption.
                     intros ;apply P_id_a__U33_bounded;assumption.
                     intros ;apply P_id_U21_bounded;assumption.
                     intros ;apply P_id_a__isNatList_bounded;assumption.
                     intros ;apply P_id_U86_bounded;assumption.
                     intros ;apply P_id_mark_bounded;assumption.
                     intros ;apply P_id_a__U51_bounded;assumption.
                     intros ;apply P_id_U41_bounded;assumption.
                     intros ;apply P_id_tt_bounded;assumption.
                     intros ;apply P_id_U82_bounded;assumption.
                     intros ;apply P_id_a__U93_bounded;assumption.
                     intros ;apply P_id_a__U44_bounded;assumption.
                     intros ;apply P_id_isNat_bounded;assumption.
                     intros ;apply P_id_a__U23_bounded;assumption.
                     intros ;apply P_id_U94_bounded;assumption.
                     intros ;apply P_id_U12_bounded;assumption.
                     intros ;apply P_id_a__U81_bounded;assumption.
                     intros ;apply P_id_U45_bounded;assumption.
                     intros ;apply P_id_zeros_bounded;assumption.
                     intros ;apply P_id_U71_bounded;assumption.
                     intros ;apply P_id_a__U91_bounded;assumption.
                     intros ;apply P_id_a__U42_bounded;assumption.
                     intros ;apply P_id_isNatKind_bounded;assumption.
                     intros ;apply P_id_a__U22_bounded;assumption.
                     intros ;apply P_id_U92_bounded;assumption.
                     intros ;apply P_id_nil_bounded;assumption.
                     intros ;apply P_id_a__U61_bounded;assumption.
                     intros ;apply P_id_U43_bounded;assumption.
                     intros ;apply P_id_a__isNatIListKind_bounded;assumption.
                     intros ;apply P_id_U84_bounded;assumption.
                     intros ;apply P_id_s_bounded;assumption.
                     intros ;apply P_id_a__U46_bounded;assumption.
                     intros ;apply P_id_U32_bounded;assumption.
                     intros ;apply P_id_a__U31_bounded;assumption.
                     intros ;apply P_id_U13_bounded;assumption.
                     intros ;apply P_id_a__U83_bounded;assumption.
                     intros ;apply P_id_isNatIList_bounded;assumption.
                     apply rules_monotonic.
                     intros ;apply P_id_A__U46_monotonic;assumption.
                     
                     intros ;apply P_id_A__ISNATILISTKIND_monotonic;
                      assumption.
                     intros ;apply P_id_A__U61_monotonic;assumption.
                     intros ;apply P_id_A__U85_monotonic;assumption.
                     intros ;apply P_id_A__U31_monotonic;assumption.
                     intros ;apply P_id_A__U81_monotonic;assumption.
                     intros ;apply P_id_A__U22_monotonic;assumption.
                     intros ;apply P_id_A__U93_monotonic;assumption.
                     intros ;apply P_id_A__U42_monotonic;assumption.
                     intros ;apply P_id_A__U51_monotonic;assumption.
                     intros ;apply P_id_A__ISNATLIST_monotonic;assumption.
                     intros ;apply P_id_A__U91_monotonic;assumption.
                     intros ;apply P_id_A__U33_monotonic;assumption.
                     intros ;apply P_id_A__U83_monotonic;assumption.
                     intros ;apply P_id_A__U23_monotonic;assumption.
                     intros ;apply P_id_A__U11_monotonic;assumption.
                     intros ;apply P_id_A__LENGTH_monotonic;assumption.
                     intros ;apply P_id_A__U44_monotonic;assumption.
                     intros ;apply P_id_A__ISNATILIST_monotonic;assumption.
                     intros ;apply P_id_A__U13_monotonic;assumption.
                     intros ;apply P_id_A__U71_monotonic;assumption.
                     intros ;apply P_id_A__U86_monotonic;assumption.
                     intros ;apply P_id_A__U32_monotonic;assumption.
                     intros ;apply P_id_A__U82_monotonic;assumption.
                     intros ;apply P_id_A__ISNATKIND_monotonic;assumption.
                     intros ;apply P_id_A__U94_monotonic;assumption.
                     intros ;apply P_id_A__U43_monotonic;assumption.
                     intros ;apply P_id_A__U52_monotonic;assumption.
                     intros ;apply P_id_A__U21_monotonic;assumption.
                     intros ;apply P_id_A__U92_monotonic;assumption.
                     intros ;apply P_id_A__U41_monotonic;assumption.
                     intros ;apply P_id_A__U84_monotonic;assumption.
                     intros ;apply P_id_A__ISNAT_monotonic;assumption.
                     intros ;apply P_id_A__U12_monotonic;assumption.
                     intros ;apply P_id_MARK_monotonic;assumption.
                     intros ;apply P_id_A__U45_monotonic;assumption.
                   Qed.
                   
                   Ltac rewrite_and_unfold  :=
                    do 2 (rewrite marked_measure_equation);
                     repeat (
                     match goal with
                       |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
                        rewrite (measure_equation (algebra.Alg.Term f t))
                       | H:context [measure (algebra.Alg.Term ?f ?t)] |- 
                       _ =>
                        rewrite (measure_equation (algebra.Alg.Term f t)) in 
                        H|-
                       end
                     ).
                   
                   Definition lt a b := 
                     (Zwf.Zwf 0) (marked_measure a) (marked_measure b).
                   
                   Definition le a b := marked_measure a <= marked_measure b.
                   
                   Lemma lt_le_compat :
                    forall a b c, (lt a b) ->(le b c) ->lt a c.
                   Proof.
                     unfold lt, le in *.
                     intros a b c.
                     apply (interp.le_lt_compat_right (interp.o_Z 0)).
                   Qed.
                   
                   Lemma wf_lt : well_founded lt.
                   Proof.
                     unfold lt in *.
                     apply Inverse_Image.wf_inverse_image with  (B:=Z).
                     apply Zwf.Zwf_well_founded.
                   Qed.
                   
                   Lemma DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_large_strict_in_lt :
                    Relation_Definitions.inclusion _ 
                     DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_large_strict 
                     lt.
                   Proof.
                     unfold Relation_Definitions.inclusion, lt in *.
                     
                     intros a b H;destruct H;
                      match goal with
                        |  |- (Zwf.Zwf 0) _ 
                               (marked_measure (algebra.Alg.Term ?f ?l)) =>
                         let l'' := algebra.Alg_ext.find_replacement l  in 
                          ((apply (interp.le_lt_compat_right (interp.o_Z 0)) with
                             (marked_measure (algebra.Alg.Term f l''));
                            [idtac|
                            apply marked_measure_star_monotonic;
                             repeat (
                             apply algebra.EQT_ext.one_step_list_refl_trans_clos
                             );(assumption)||(constructor 1)]))
                        end
                      ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Lemma DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_large_large_in_le :
                    Relation_Definitions.inclusion _ 
                     DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_large_large 
                     le.
                   Proof.
                     unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *.
                     
                     intros a b H;destruct H;
                      match goal with
                        |  |- _ <= marked_measure (algebra.Alg.Term ?f ?l) =>
                         let l'' := algebra.Alg_ext.find_replacement l  in 
                          ((apply (interp.le_trans (interp.o_Z 0)) with
                             (marked_measure (algebra.Alg.Term f l''));
                            [idtac|
                            apply marked_measure_star_monotonic;
                             repeat (
                             apply algebra.EQT_ext.one_step_list_refl_trans_clos
                             );(assumption)||(constructor 1)]))
                        end
                      ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                   Qed.
                   
                   Definition wf_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_large_large  := 
                     WF_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_large_large.wf
                     .
                   
                   
                   Lemma wf :
                    well_founded WF_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large.DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_large
                     .
                   Proof.
                     intros x.
                     apply (well_founded_ind wf_lt).
                     clear x.
                     intros x.
                     pattern x.
                     
                     apply (@Acc_ind _ 
                             DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_large_large)
                            .
                     clear x.
                     intros x _ IHx IHx'.
                     constructor.
                     intros y H.
                     
                     destruct H;
                      (apply IHx';
                        apply DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_large_strict_in_lt;
                        econstructor eassumption)||
                      ((apply IHx;[econstructor eassumption|
                        intros y' Hlt;apply IHx';
                         apply lt_le_compat with (1:=Hlt) ;
                         apply DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_large_large_in_le;
                         econstructor eassumption])).
                     
                     apply wf_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_large_large.
                   Qed.
                  End WF_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_large.
                  
                  Open Scope Z_scope.
                  
                  Import ring_extention.
                  
                  Notation Local "a <= b" := (Zle a b).
                  
                  Notation Local "a < b" := (Zlt a b).
                  
                  Definition P_id_a__zeros  := 0.
                  
                  Definition P_id_U51 (x11:Z) (x12:Z) := 2.
                  
                  Definition P_id_a__U84 (x11:Z) (x12:Z) (x13:Z) := 2* x13.
                  
                  Definition P_id_a__U32 (x11:Z) (x12:Z) := 2.
                  
                  Definition P_id_isNatList (x11:Z) := 2* x11.
                  
                  Definition P_id_a__U13 (x11:Z) := 1* x11.
                  
                  Definition P_id_U85 (x11:Z) (x12:Z) := 2* x12.
                  
                  Definition P_id_a__length (x11:Z) := 1 + 2* x11.
                  
                  Definition P_id_a__isNatIList (x11:Z) := 2 + 1* x11.
                  
                  Definition P_id_U33 (x11:Z) := 2.
                  
                  Definition P_id_a__U11 (x11:Z) (x12:Z) := 2* x12.
                  
                  Definition P_id_U81 (x11:Z) (x12:Z) (x13:Z) := 2* x13.
                  
                  Definition P_id_a__U92 (x11:Z) (x12:Z) (x13:Z) := 
                    3 + 2* x12.
                  
                  Definition P_id_a__U43 (x11:Z) (x12:Z) (x13:Z) := 
                    1* x11 + 1* x13.
                  
                  Definition P_id_U23 (x11:Z) := 1* x11.
                  
                  Definition P_id_a__isNatKind (x11:Z) := 2* x11.
                  
                  Definition P_id_U93 (x11:Z) (x12:Z) (x13:Z) := 3 + 2* x12.
                  
                  Definition P_id_U11 (x11:Z) (x12:Z) := 2* x12.
                  
                  Definition P_id_a__U71 (x11:Z) := 1* x11.
                  
                  Definition P_id_U44 (x11:Z) (x12:Z) (x13:Z) := 2 + 1* x13.
                  
                  Definition P_id_0  := 1.
                  
                  Definition P_id_U61 (x11:Z) := 1* x11.
                  
                  Definition P_id_a__U86 (x11:Z) := 1* x11.
                  
                  Definition P_id_a__U41 (x11:Z) (x12:Z) (x13:Z) := 
                    2 + 2* x13.
                  
                  Definition P_id_U22 (x11:Z) (x12:Z) := 1 + 1* x12.
                  
                  Definition P_id_a__U21 (x11:Z) (x12:Z) := 1 + 1* x12.
                  
                  Definition P_id_U91 (x11:Z) (x12:Z) (x13:Z) := 
                    1 + 1* x11 + 2* x12.
                  
                  Definition P_id_length (x11:Z) := 1 + 2* x11.
                  
                  Definition P_id_a__U52 (x11:Z) := 2.
                  
                  Definition P_id_U42 (x11:Z) (x12:Z) (x13:Z) := 2 + 1* x13.
                  
                  Definition P_id_a__U12 (x11:Z) (x12:Z) := 2* x12.
                  
                  Definition P_id_U83 (x11:Z) (x12:Z) (x13:Z) := 2* x13.
                  
                  Definition P_id_a__U94 (x11:Z) (x12:Z) := 3 + 2* x12.
                  
                  Definition P_id_a__U45 (x11:Z) (x12:Z) := 2 + 1* x12.
                  
                  Definition P_id_U31 (x11:Z) (x12:Z) := 1* x11.
                  
                  Definition P_id_a__isNat (x11:Z) := 1 + 1* x11.
                  
                  Definition P_id_isNatIListKind (x11:Z) := 2.
                  
                  Definition P_id_a__U82 (x11:Z) (x12:Z) (x13:Z) := 2* x13.
                  
                  Definition P_id_U46 (x11:Z) := 1* x11.
                  
                  Definition P_id_cons (x11:Z) (x12:Z) := 2* x12.
                  
                  Definition P_id_U52 (x11:Z) := 2.
                  
                  Definition P_id_a__U85 (x11:Z) (x12:Z) := 2* x12.
                  
                  Definition P_id_a__U33 (x11:Z) := 2.
                  
                  Definition P_id_U21 (x11:Z) (x12:Z) := 1 + 1* x12.
                  
                  Definition P_id_a__isNatList (x11:Z) := 2* x11.
                  
                  Definition P_id_U86 (x11:Z) := 1* x11.
                  
                  Definition P_id_mark (x11:Z) := 1* x11.
                  
                  Definition P_id_a__U51 (x11:Z) (x12:Z) := 2.
                  
                  Definition P_id_U41 (x11:Z) (x12:Z) (x13:Z) := 2 + 2* x13.
                  
                  Definition P_id_tt  := 2.
                  
                  Definition P_id_U82 (x11:Z) (x12:Z) (x13:Z) := 2* x13.
                  
                  Definition P_id_a__U93 (x11:Z) (x12:Z) (x13:Z) := 
                    3 + 2* x12.
                  
                  Definition P_id_a__U44 (x11:Z) (x12:Z) (x13:Z) := 
                    2 + 1* x13.
                  
                  Definition P_id_isNat (x11:Z) := 1 + 1* x11.
                  
                  Definition P_id_a__U23 (x11:Z) := 1* x11.
                  
                  Definition P_id_U94 (x11:Z) (x12:Z) := 3 + 2* x12.
                  
                  Definition P_id_U12 (x11:Z) (x12:Z) := 2* x12.
                  
                  Definition P_id_a__U81 (x11:Z) (x12:Z) (x13:Z) := 2* x13.
                  
                  Definition P_id_U45 (x11:Z) (x12:Z) := 2 + 1* x12.
                  
                  Definition P_id_zeros  := 0.
                  
                  Definition P_id_U71 (x11:Z) := 1* x11.
                  
                  Definition P_id_a__U91 (x11:Z) (x12:Z) (x13:Z) := 
                    1 + 1* x11 + 2* x12.
                  
                  Definition P_id_a__U42 (x11:Z) (x12:Z) (x13:Z) := 
                    2 + 1* x13.
                  
                  Definition P_id_isNatKind (x11:Z) := 2* x11.
                  
                  Definition P_id_a__U22 (x11:Z) (x12:Z) := 1 + 1* x12.
                  
                  Definition P_id_U92 (x11:Z) (x12:Z) (x13:Z) := 3 + 2* x12.
                  
                  Definition P_id_nil  := 1.
                  
                  Definition P_id_a__U61 (x11:Z) := 1* x11.
                  
                  Definition P_id_U43 (x11:Z) (x12:Z) (x13:Z) := 
                    1* x11 + 1* x13.
                  
                  Definition P_id_a__isNatIListKind (x11:Z) := 2.
                  
                  Definition P_id_U84 (x11:Z) (x12:Z) (x13:Z) := 2* x13.
                  
                  Definition P_id_s (x11:Z) := 2 + 1* x11.
                  
                  Definition P_id_a__U46 (x11:Z) := 1* x11.
                  
                  Definition P_id_U32 (x11:Z) (x12:Z) := 2.
                  
                  Definition P_id_a__U31 (x11:Z) (x12:Z) := 1* x11.
                  
                  Definition P_id_U13 (x11:Z) := 1* x11.
                  
                  Definition P_id_a__U83 (x11:Z) (x12:Z) (x13:Z) := 2* x13.
                  
                  Definition P_id_isNatIList (x11:Z) := 2 + 1* x11.
                  
                  Lemma P_id_U51_monotonic :
                   forall x12 x14 x13 x11, 
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_U51 x12 x14 <= P_id_U51 x11 x13.
                  Proof.
                    intros x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U84_monotonic :
                   forall x16 x12 x14 x13 x11 x15, 
                    (0 <= x16)/\ (x16 <= x15) ->
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_a__U84 x12 x14 x16 <= P_id_a__U84 x11 x13 x15.
                  Proof.
                    intros x16 x15 x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    intros [H_5 H_4].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U32_monotonic :
                   forall x12 x14 x13 x11, 
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_a__U32 x12 x14 <= P_id_a__U32 x11 x13.
                  Proof.
                    intros x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_isNatList_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_isNatList x12 <= P_id_isNatList x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_0].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U13_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_a__U13 x12 <= P_id_a__U13 x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_0].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U85_monotonic :
                   forall x12 x14 x13 x11, 
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_U85 x12 x14 <= P_id_U85 x11 x13.
                  Proof.
                    intros x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__length_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_a__length x12 <= P_id_a__length x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_0].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__isNatIList_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_a__isNatIList x12 <= P_id_a__isNatIList x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_0].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U33_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->P_id_U33 x12 <= P_id_U33 x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_0].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U11_monotonic :
                   forall x12 x14 x13 x11, 
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_a__U11 x12 x14 <= P_id_a__U11 x11 x13.
                  Proof.
                    intros x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U81_monotonic :
                   forall x16 x12 x14 x13 x11 x15, 
                    (0 <= x16)/\ (x16 <= x15) ->
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_U81 x12 x14 x16 <= P_id_U81 x11 x13 x15.
                  Proof.
                    intros x16 x15 x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    intros [H_5 H_4].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U92_monotonic :
                   forall x16 x12 x14 x13 x11 x15, 
                    (0 <= x16)/\ (x16 <= x15) ->
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_a__U92 x12 x14 x16 <= P_id_a__U92 x11 x13 x15.
                  Proof.
                    intros x16 x15 x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    intros [H_5 H_4].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U43_monotonic :
                   forall x16 x12 x14 x13 x11 x15, 
                    (0 <= x16)/\ (x16 <= x15) ->
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_a__U43 x12 x14 x16 <= P_id_a__U43 x11 x13 x15.
                  Proof.
                    intros x16 x15 x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    intros [H_5 H_4].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U23_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->P_id_U23 x12 <= P_id_U23 x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_0].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__isNatKind_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_a__isNatKind x12 <= P_id_a__isNatKind x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_0].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U93_monotonic :
                   forall x16 x12 x14 x13 x11 x15, 
                    (0 <= x16)/\ (x16 <= x15) ->
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_U93 x12 x14 x16 <= P_id_U93 x11 x13 x15.
                  Proof.
                    intros x16 x15 x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    intros [H_5 H_4].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U11_monotonic :
                   forall x12 x14 x13 x11, 
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_U11 x12 x14 <= P_id_U11 x11 x13.
                  Proof.
                    intros x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U71_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_a__U71 x12 <= P_id_a__U71 x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_0].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U44_monotonic :
                   forall x16 x12 x14 x13 x11 x15, 
                    (0 <= x16)/\ (x16 <= x15) ->
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_U44 x12 x14 x16 <= P_id_U44 x11 x13 x15.
                  Proof.
                    intros x16 x15 x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    intros [H_5 H_4].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U61_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->P_id_U61 x12 <= P_id_U61 x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_0].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U86_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_a__U86 x12 <= P_id_a__U86 x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_0].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U41_monotonic :
                   forall x16 x12 x14 x13 x11 x15, 
                    (0 <= x16)/\ (x16 <= x15) ->
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_a__U41 x12 x14 x16 <= P_id_a__U41 x11 x13 x15.
                  Proof.
                    intros x16 x15 x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    intros [H_5 H_4].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U22_monotonic :
                   forall x12 x14 x13 x11, 
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_U22 x12 x14 <= P_id_U22 x11 x13.
                  Proof.
                    intros x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U21_monotonic :
                   forall x12 x14 x13 x11, 
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_a__U21 x12 x14 <= P_id_a__U21 x11 x13.
                  Proof.
                    intros x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U91_monotonic :
                   forall x16 x12 x14 x13 x11 x15, 
                    (0 <= x16)/\ (x16 <= x15) ->
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_U91 x12 x14 x16 <= P_id_U91 x11 x13 x15.
                  Proof.
                    intros x16 x15 x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    intros [H_5 H_4].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_length_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_length x12 <= P_id_length x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_0].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U52_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_a__U52 x12 <= P_id_a__U52 x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_0].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U42_monotonic :
                   forall x16 x12 x14 x13 x11 x15, 
                    (0 <= x16)/\ (x16 <= x15) ->
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_U42 x12 x14 x16 <= P_id_U42 x11 x13 x15.
                  Proof.
                    intros x16 x15 x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    intros [H_5 H_4].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U12_monotonic :
                   forall x12 x14 x13 x11, 
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_a__U12 x12 x14 <= P_id_a__U12 x11 x13.
                  Proof.
                    intros x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U83_monotonic :
                   forall x16 x12 x14 x13 x11 x15, 
                    (0 <= x16)/\ (x16 <= x15) ->
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_U83 x12 x14 x16 <= P_id_U83 x11 x13 x15.
                  Proof.
                    intros x16 x15 x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    intros [H_5 H_4].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U94_monotonic :
                   forall x12 x14 x13 x11, 
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_a__U94 x12 x14 <= P_id_a__U94 x11 x13.
                  Proof.
                    intros x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U45_monotonic :
                   forall x12 x14 x13 x11, 
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_a__U45 x12 x14 <= P_id_a__U45 x11 x13.
                  Proof.
                    intros x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U31_monotonic :
                   forall x12 x14 x13 x11, 
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_U31 x12 x14 <= P_id_U31 x11 x13.
                  Proof.
                    intros x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__isNat_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_a__isNat x12 <= P_id_a__isNat x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_0].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_isNatIListKind_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_isNatIListKind x12 <= P_id_isNatIListKind x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_0].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U82_monotonic :
                   forall x16 x12 x14 x13 x11 x15, 
                    (0 <= x16)/\ (x16 <= x15) ->
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_a__U82 x12 x14 x16 <= P_id_a__U82 x11 x13 x15.
                  Proof.
                    intros x16 x15 x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    intros [H_5 H_4].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U46_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->P_id_U46 x12 <= P_id_U46 x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_0].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_cons_monotonic :
                   forall x12 x14 x13 x11, 
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_cons x12 x14 <= P_id_cons x11 x13.
                  Proof.
                    intros x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U52_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->P_id_U52 x12 <= P_id_U52 x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_0].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U85_monotonic :
                   forall x12 x14 x13 x11, 
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_a__U85 x12 x14 <= P_id_a__U85 x11 x13.
                  Proof.
                    intros x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U33_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_a__U33 x12 <= P_id_a__U33 x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_0].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U21_monotonic :
                   forall x12 x14 x13 x11, 
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_U21 x12 x14 <= P_id_U21 x11 x13.
                  Proof.
                    intros x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__isNatList_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_a__isNatList x12 <= P_id_a__isNatList x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_0].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U86_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->P_id_U86 x12 <= P_id_U86 x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_0].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_mark_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_mark x12 <= P_id_mark x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_0].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U51_monotonic :
                   forall x12 x14 x13 x11, 
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_a__U51 x12 x14 <= P_id_a__U51 x11 x13.
                  Proof.
                    intros x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U41_monotonic :
                   forall x16 x12 x14 x13 x11 x15, 
                    (0 <= x16)/\ (x16 <= x15) ->
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_U41 x12 x14 x16 <= P_id_U41 x11 x13 x15.
                  Proof.
                    intros x16 x15 x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    intros [H_5 H_4].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U82_monotonic :
                   forall x16 x12 x14 x13 x11 x15, 
                    (0 <= x16)/\ (x16 <= x15) ->
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_U82 x12 x14 x16 <= P_id_U82 x11 x13 x15.
                  Proof.
                    intros x16 x15 x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    intros [H_5 H_4].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U93_monotonic :
                   forall x16 x12 x14 x13 x11 x15, 
                    (0 <= x16)/\ (x16 <= x15) ->
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_a__U93 x12 x14 x16 <= P_id_a__U93 x11 x13 x15.
                  Proof.
                    intros x16 x15 x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    intros [H_5 H_4].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U44_monotonic :
                   forall x16 x12 x14 x13 x11 x15, 
                    (0 <= x16)/\ (x16 <= x15) ->
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_a__U44 x12 x14 x16 <= P_id_a__U44 x11 x13 x15.
                  Proof.
                    intros x16 x15 x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    intros [H_5 H_4].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_isNat_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_isNat x12 <= P_id_isNat x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_0].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U23_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_a__U23 x12 <= P_id_a__U23 x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_0].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U94_monotonic :
                   forall x12 x14 x13 x11, 
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_U94 x12 x14 <= P_id_U94 x11 x13.
                  Proof.
                    intros x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U12_monotonic :
                   forall x12 x14 x13 x11, 
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_U12 x12 x14 <= P_id_U12 x11 x13.
                  Proof.
                    intros x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U81_monotonic :
                   forall x16 x12 x14 x13 x11 x15, 
                    (0 <= x16)/\ (x16 <= x15) ->
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_a__U81 x12 x14 x16 <= P_id_a__U81 x11 x13 x15.
                  Proof.
                    intros x16 x15 x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    intros [H_5 H_4].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U45_monotonic :
                   forall x12 x14 x13 x11, 
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_U45 x12 x14 <= P_id_U45 x11 x13.
                  Proof.
                    intros x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U71_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->P_id_U71 x12 <= P_id_U71 x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_0].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U91_monotonic :
                   forall x16 x12 x14 x13 x11 x15, 
                    (0 <= x16)/\ (x16 <= x15) ->
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_a__U91 x12 x14 x16 <= P_id_a__U91 x11 x13 x15.
                  Proof.
                    intros x16 x15 x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    intros [H_5 H_4].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U42_monotonic :
                   forall x16 x12 x14 x13 x11 x15, 
                    (0 <= x16)/\ (x16 <= x15) ->
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_a__U42 x12 x14 x16 <= P_id_a__U42 x11 x13 x15.
                  Proof.
                    intros x16 x15 x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    intros [H_5 H_4].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_isNatKind_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_isNatKind x12 <= P_id_isNatKind x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_0].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U22_monotonic :
                   forall x12 x14 x13 x11, 
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_a__U22 x12 x14 <= P_id_a__U22 x11 x13.
                  Proof.
                    intros x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U92_monotonic :
                   forall x16 x12 x14 x13 x11 x15, 
                    (0 <= x16)/\ (x16 <= x15) ->
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_U92 x12 x14 x16 <= P_id_U92 x11 x13 x15.
                  Proof.
                    intros x16 x15 x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    intros [H_5 H_4].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U61_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_a__U61 x12 <= P_id_a__U61 x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_0].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U43_monotonic :
                   forall x16 x12 x14 x13 x11 x15, 
                    (0 <= x16)/\ (x16 <= x15) ->
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_U43 x12 x14 x16 <= P_id_U43 x11 x13 x15.
                  Proof.
                    intros x16 x15 x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    intros [H_5 H_4].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__isNatIListKind_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_a__isNatIListKind x12 <= P_id_a__isNatIListKind x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_0].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U84_monotonic :
                   forall x16 x12 x14 x13 x11 x15, 
                    (0 <= x16)/\ (x16 <= x15) ->
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_U84 x12 x14 x16 <= P_id_U84 x11 x13 x15.
                  Proof.
                    intros x16 x15 x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    intros [H_5 H_4].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_s_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->P_id_s x12 <= P_id_s x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_0].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U46_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_a__U46 x12 <= P_id_a__U46 x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_0].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U32_monotonic :
                   forall x12 x14 x13 x11, 
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_U32 x12 x14 <= P_id_U32 x11 x13.
                  Proof.
                    intros x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U31_monotonic :
                   forall x12 x14 x13 x11, 
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_a__U31 x12 x14 <= P_id_a__U31 x11 x13.
                  Proof.
                    intros x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U13_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->P_id_U13 x12 <= P_id_U13 x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_0].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U83_monotonic :
                   forall x16 x12 x14 x13 x11 x15, 
                    (0 <= x16)/\ (x16 <= x15) ->
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_a__U83 x12 x14 x16 <= P_id_a__U83 x11 x13 x15.
                  Proof.
                    intros x16 x15 x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    intros [H_5 H_4].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_isNatIList_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_isNatIList x12 <= P_id_isNatIList x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_0].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__zeros_bounded : 0 <= P_id_a__zeros .
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U51_bounded :
                   forall x12 x11, 
                    (0 <= x11) ->(0 <= x12) ->0 <= P_id_U51 x12 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U84_bounded :
                   forall x12 x13 x11, 
                    (0 <= x11) ->
                     (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U84 x13 x12 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U32_bounded :
                   forall x12 x11, 
                    (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U32 x12 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_isNatList_bounded :
                   forall x11, (0 <= x11) ->0 <= P_id_isNatList x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U13_bounded :
                   forall x11, (0 <= x11) ->0 <= P_id_a__U13 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U85_bounded :
                   forall x12 x11, 
                    (0 <= x11) ->(0 <= x12) ->0 <= P_id_U85 x12 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__length_bounded :
                   forall x11, (0 <= x11) ->0 <= P_id_a__length x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__isNatIList_bounded :
                   forall x11, (0 <= x11) ->0 <= P_id_a__isNatIList x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U33_bounded :
                   forall x11, (0 <= x11) ->0 <= P_id_U33 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U11_bounded :
                   forall x12 x11, 
                    (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U11 x12 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U81_bounded :
                   forall x12 x13 x11, 
                    (0 <= x11) ->
                     (0 <= x12) ->(0 <= x13) ->0 <= P_id_U81 x13 x12 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U92_bounded :
                   forall x12 x13 x11, 
                    (0 <= x11) ->
                     (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U92 x13 x12 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U43_bounded :
                   forall x12 x13 x11, 
                    (0 <= x11) ->
                     (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U43 x13 x12 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U23_bounded :
                   forall x11, (0 <= x11) ->0 <= P_id_U23 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__isNatKind_bounded :
                   forall x11, (0 <= x11) ->0 <= P_id_a__isNatKind x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U93_bounded :
                   forall x12 x13 x11, 
                    (0 <= x11) ->
                     (0 <= x12) ->(0 <= x13) ->0 <= P_id_U93 x13 x12 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U11_bounded :
                   forall x12 x11, 
                    (0 <= x11) ->(0 <= x12) ->0 <= P_id_U11 x12 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U71_bounded :
                   forall x11, (0 <= x11) ->0 <= P_id_a__U71 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U44_bounded :
                   forall x12 x13 x11, 
                    (0 <= x11) ->
                     (0 <= x12) ->(0 <= x13) ->0 <= P_id_U44 x13 x12 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_0_bounded : 0 <= P_id_0 .
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U61_bounded :
                   forall x11, (0 <= x11) ->0 <= P_id_U61 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U86_bounded :
                   forall x11, (0 <= x11) ->0 <= P_id_a__U86 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U41_bounded :
                   forall x12 x13 x11, 
                    (0 <= x11) ->
                     (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U41 x13 x12 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U22_bounded :
                   forall x12 x11, 
                    (0 <= x11) ->(0 <= x12) ->0 <= P_id_U22 x12 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U21_bounded :
                   forall x12 x11, 
                    (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U21 x12 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U91_bounded :
                   forall x12 x13 x11, 
                    (0 <= x11) ->
                     (0 <= x12) ->(0 <= x13) ->0 <= P_id_U91 x13 x12 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_length_bounded :
                   forall x11, (0 <= x11) ->0 <= P_id_length x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U52_bounded :
                   forall x11, (0 <= x11) ->0 <= P_id_a__U52 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U42_bounded :
                   forall x12 x13 x11, 
                    (0 <= x11) ->
                     (0 <= x12) ->(0 <= x13) ->0 <= P_id_U42 x13 x12 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U12_bounded :
                   forall x12 x11, 
                    (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U12 x12 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U83_bounded :
                   forall x12 x13 x11, 
                    (0 <= x11) ->
                     (0 <= x12) ->(0 <= x13) ->0 <= P_id_U83 x13 x12 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U94_bounded :
                   forall x12 x11, 
                    (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U94 x12 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U45_bounded :
                   forall x12 x11, 
                    (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U45 x12 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U31_bounded :
                   forall x12 x11, 
                    (0 <= x11) ->(0 <= x12) ->0 <= P_id_U31 x12 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__isNat_bounded :
                   forall x11, (0 <= x11) ->0 <= P_id_a__isNat x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_isNatIListKind_bounded :
                   forall x11, (0 <= x11) ->0 <= P_id_isNatIListKind x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U82_bounded :
                   forall x12 x13 x11, 
                    (0 <= x11) ->
                     (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U82 x13 x12 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U46_bounded :
                   forall x11, (0 <= x11) ->0 <= P_id_U46 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_cons_bounded :
                   forall x12 x11, 
                    (0 <= x11) ->(0 <= x12) ->0 <= P_id_cons x12 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U52_bounded :
                   forall x11, (0 <= x11) ->0 <= P_id_U52 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U85_bounded :
                   forall x12 x11, 
                    (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U85 x12 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U33_bounded :
                   forall x11, (0 <= x11) ->0 <= P_id_a__U33 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U21_bounded :
                   forall x12 x11, 
                    (0 <= x11) ->(0 <= x12) ->0 <= P_id_U21 x12 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__isNatList_bounded :
                   forall x11, (0 <= x11) ->0 <= P_id_a__isNatList x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U86_bounded :
                   forall x11, (0 <= x11) ->0 <= P_id_U86 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_mark_bounded :
                   forall x11, (0 <= x11) ->0 <= P_id_mark x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U51_bounded :
                   forall x12 x11, 
                    (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U51 x12 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U41_bounded :
                   forall x12 x13 x11, 
                    (0 <= x11) ->
                     (0 <= x12) ->(0 <= x13) ->0 <= P_id_U41 x13 x12 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_tt_bounded : 0 <= P_id_tt .
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U82_bounded :
                   forall x12 x13 x11, 
                    (0 <= x11) ->
                     (0 <= x12) ->(0 <= x13) ->0 <= P_id_U82 x13 x12 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U93_bounded :
                   forall x12 x13 x11, 
                    (0 <= x11) ->
                     (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U93 x13 x12 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U44_bounded :
                   forall x12 x13 x11, 
                    (0 <= x11) ->
                     (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U44 x13 x12 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_isNat_bounded :
                   forall x11, (0 <= x11) ->0 <= P_id_isNat x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U23_bounded :
                   forall x11, (0 <= x11) ->0 <= P_id_a__U23 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U94_bounded :
                   forall x12 x11, 
                    (0 <= x11) ->(0 <= x12) ->0 <= P_id_U94 x12 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U12_bounded :
                   forall x12 x11, 
                    (0 <= x11) ->(0 <= x12) ->0 <= P_id_U12 x12 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U81_bounded :
                   forall x12 x13 x11, 
                    (0 <= x11) ->
                     (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U81 x13 x12 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U45_bounded :
                   forall x12 x11, 
                    (0 <= x11) ->(0 <= x12) ->0 <= P_id_U45 x12 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_zeros_bounded : 0 <= P_id_zeros .
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U71_bounded :
                   forall x11, (0 <= x11) ->0 <= P_id_U71 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U91_bounded :
                   forall x12 x13 x11, 
                    (0 <= x11) ->
                     (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U91 x13 x12 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U42_bounded :
                   forall x12 x13 x11, 
                    (0 <= x11) ->
                     (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U42 x13 x12 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_isNatKind_bounded :
                   forall x11, (0 <= x11) ->0 <= P_id_isNatKind x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U22_bounded :
                   forall x12 x11, 
                    (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U22 x12 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U92_bounded :
                   forall x12 x13 x11, 
                    (0 <= x11) ->
                     (0 <= x12) ->(0 <= x13) ->0 <= P_id_U92 x13 x12 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_nil_bounded : 0 <= P_id_nil .
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U61_bounded :
                   forall x11, (0 <= x11) ->0 <= P_id_a__U61 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U43_bounded :
                   forall x12 x13 x11, 
                    (0 <= x11) ->
                     (0 <= x12) ->(0 <= x13) ->0 <= P_id_U43 x13 x12 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__isNatIListKind_bounded :
                   forall x11, (0 <= x11) ->0 <= P_id_a__isNatIListKind x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U84_bounded :
                   forall x12 x13 x11, 
                    (0 <= x11) ->
                     (0 <= x12) ->(0 <= x13) ->0 <= P_id_U84 x13 x12 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_s_bounded :
                   forall x11, (0 <= x11) ->0 <= P_id_s x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U46_bounded :
                   forall x11, (0 <= x11) ->0 <= P_id_a__U46 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U32_bounded :
                   forall x12 x11, 
                    (0 <= x11) ->(0 <= x12) ->0 <= P_id_U32 x12 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U31_bounded :
                   forall x12 x11, 
                    (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U31 x12 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U13_bounded :
                   forall x11, (0 <= x11) ->0 <= P_id_U13 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_a__U83_bounded :
                   forall x12 x13 x11, 
                    (0 <= x11) ->
                     (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U83 x13 x12 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_isNatIList_bounded :
                   forall x11, (0 <= x11) ->0 <= P_id_isNatIList x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Definition measure  := 
                    InterpZ.measure 0 P_id_a__zeros P_id_U51 P_id_a__U84 
                     P_id_a__U32 P_id_isNatList P_id_a__U13 P_id_U85 
                     P_id_a__length P_id_a__isNatIList P_id_U33 P_id_a__U11 
                     P_id_U81 P_id_a__U92 P_id_a__U43 P_id_U23 
                     P_id_a__isNatKind P_id_U93 P_id_U11 P_id_a__U71 
                     P_id_U44 P_id_0 P_id_U61 P_id_a__U86 P_id_a__U41 
                     P_id_U22 P_id_a__U21 P_id_U91 P_id_length P_id_a__U52 
                     P_id_U42 P_id_a__U12 P_id_U83 P_id_a__U94 P_id_a__U45 
                     P_id_U31 P_id_a__isNat P_id_isNatIListKind P_id_a__U82 
                     P_id_U46 P_id_cons P_id_U52 P_id_a__U85 P_id_a__U33 
                     P_id_U21 P_id_a__isNatList P_id_U86 P_id_mark 
                     P_id_a__U51 P_id_U41 P_id_tt P_id_U82 P_id_a__U93 
                     P_id_a__U44 P_id_isNat P_id_a__U23 P_id_U94 P_id_U12 
                     P_id_a__U81 P_id_U45 P_id_zeros P_id_U71 P_id_a__U91 
                     P_id_a__U42 P_id_isNatKind P_id_a__U22 P_id_U92 
                     P_id_nil P_id_a__U61 P_id_U43 P_id_a__isNatIListKind 
                     P_id_U84 P_id_s P_id_a__U46 P_id_U32 P_id_a__U31 
                     P_id_U13 P_id_a__U83 P_id_isNatIList.
                  
                  Lemma measure_equation :
                   forall t, 
                    measure t = match t with
                                  | (algebra.Alg.Term algebra.F.id_a__zeros 
                                     nil) =>
                                   P_id_a__zeros 
                                  | (algebra.Alg.Term algebra.F.id_U51 (x12::
                                     x11::nil)) =>
                                   P_id_U51 (measure x12) (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U84 
                                     (x13::x12::x11::nil)) =>
                                   P_id_a__U84 (measure x13) (measure x12) 
                                    (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U32 
                                     (x12::x11::nil)) =>
                                   P_id_a__U32 (measure x12) (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_isNatList 
                                     (x11::nil)) =>
                                   P_id_isNatList (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U13 
                                     (x11::nil)) =>
                                   P_id_a__U13 (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_U85 (x12::
                                     x11::nil)) =>
                                   P_id_U85 (measure x12) (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__length 
                                     (x11::nil)) =>
                                   P_id_a__length (measure x11)
                                  | (algebra.Alg.Term 
                                     algebra.F.id_a__isNatIList (x11::nil)) =>
                                   P_id_a__isNatIList (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_U33 
                                     (x11::nil)) =>
                                   P_id_U33 (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U11 
                                     (x12::x11::nil)) =>
                                   P_id_a__U11 (measure x12) (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_U81 (x13::
                                     x12::x11::nil)) =>
                                   P_id_U81 (measure x13) (measure x12) 
                                    (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U92 
                                     (x13::x12::x11::nil)) =>
                                   P_id_a__U92 (measure x13) (measure x12) 
                                    (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U43 
                                     (x13::x12::x11::nil)) =>
                                   P_id_a__U43 (measure x13) (measure x12) 
                                    (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_U23 
                                     (x11::nil)) =>
                                   P_id_U23 (measure x11)
                                  | (algebra.Alg.Term 
                                     algebra.F.id_a__isNatKind (x11::nil)) =>
                                   P_id_a__isNatKind (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_U93 (x13::
                                     x12::x11::nil)) =>
                                   P_id_U93 (measure x13) (measure x12) 
                                    (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_U11 (x12::
                                     x11::nil)) =>
                                   P_id_U11 (measure x12) (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U71 
                                     (x11::nil)) =>
                                   P_id_a__U71 (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_U44 (x13::
                                     x12::x11::nil)) =>
                                   P_id_U44 (measure x13) (measure x12) 
                                    (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_0 nil) =>
                                   P_id_0 
                                  | (algebra.Alg.Term algebra.F.id_U61 
                                     (x11::nil)) =>
                                   P_id_U61 (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U86 
                                     (x11::nil)) =>
                                   P_id_a__U86 (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U41 
                                     (x13::x12::x11::nil)) =>
                                   P_id_a__U41 (measure x13) (measure x12) 
                                    (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_U22 (x12::
                                     x11::nil)) =>
                                   P_id_U22 (measure x12) (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U21 
                                     (x12::x11::nil)) =>
                                   P_id_a__U21 (measure x12) (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_U91 (x13::
                                     x12::x11::nil)) =>
                                   P_id_U91 (measure x13) (measure x12) 
                                    (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_length 
                                     (x11::nil)) =>
                                   P_id_length (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U52 
                                     (x11::nil)) =>
                                   P_id_a__U52 (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_U42 (x13::
                                     x12::x11::nil)) =>
                                   P_id_U42 (measure x13) (measure x12) 
                                    (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U12 
                                     (x12::x11::nil)) =>
                                   P_id_a__U12 (measure x12) (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_U83 (x13::
                                     x12::x11::nil)) =>
                                   P_id_U83 (measure x13) (measure x12) 
                                    (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U94 
                                     (x12::x11::nil)) =>
                                   P_id_a__U94 (measure x12) (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U45 
                                     (x12::x11::nil)) =>
                                   P_id_a__U45 (measure x12) (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_U31 (x12::
                                     x11::nil)) =>
                                   P_id_U31 (measure x12) (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__isNat 
                                     (x11::nil)) =>
                                   P_id_a__isNat (measure x11)
                                  | (algebra.Alg.Term 
                                     algebra.F.id_isNatIListKind (x11::nil)) =>
                                   P_id_isNatIListKind (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U82 
                                     (x13::x12::x11::nil)) =>
                                   P_id_a__U82 (measure x13) (measure x12) 
                                    (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_U46 
                                     (x11::nil)) =>
                                   P_id_U46 (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_cons 
                                     (x12::x11::nil)) =>
                                   P_id_cons (measure x12) (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_U52 
                                     (x11::nil)) =>
                                   P_id_U52 (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U85 
                                     (x12::x11::nil)) =>
                                   P_id_a__U85 (measure x12) (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U33 
                                     (x11::nil)) =>
                                   P_id_a__U33 (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_U21 (x12::
                                     x11::nil)) =>
                                   P_id_U21 (measure x12) (measure x11)
                                  | (algebra.Alg.Term 
                                     algebra.F.id_a__isNatList (x11::nil)) =>
                                   P_id_a__isNatList (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_U86 
                                     (x11::nil)) =>
                                   P_id_U86 (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_mark 
                                     (x11::nil)) =>
                                   P_id_mark (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U51 
                                     (x12::x11::nil)) =>
                                   P_id_a__U51 (measure x12) (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_U41 (x13::
                                     x12::x11::nil)) =>
                                   P_id_U41 (measure x13) (measure x12) 
                                    (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_tt nil) =>
                                   P_id_tt 
                                  | (algebra.Alg.Term algebra.F.id_U82 (x13::
                                     x12::x11::nil)) =>
                                   P_id_U82 (measure x13) (measure x12) 
                                    (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U93 
                                     (x13::x12::x11::nil)) =>
                                   P_id_a__U93 (measure x13) (measure x12) 
                                    (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U44 
                                     (x13::x12::x11::nil)) =>
                                   P_id_a__U44 (measure x13) (measure x12) 
                                    (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_isNat 
                                     (x11::nil)) =>
                                   P_id_isNat (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U23 
                                     (x11::nil)) =>
                                   P_id_a__U23 (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_U94 (x12::
                                     x11::nil)) =>
                                   P_id_U94 (measure x12) (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_U12 (x12::
                                     x11::nil)) =>
                                   P_id_U12 (measure x12) (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U81 
                                     (x13::x12::x11::nil)) =>
                                   P_id_a__U81 (measure x13) (measure x12) 
                                    (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_U45 (x12::
                                     x11::nil)) =>
                                   P_id_U45 (measure x12) (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_zeros nil) =>
                                   P_id_zeros 
                                  | (algebra.Alg.Term algebra.F.id_U71 
                                     (x11::nil)) =>
                                   P_id_U71 (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U91 
                                     (x13::x12::x11::nil)) =>
                                   P_id_a__U91 (measure x13) (measure x12) 
                                    (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U42 
                                     (x13::x12::x11::nil)) =>
                                   P_id_a__U42 (measure x13) (measure x12) 
                                    (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_isNatKind 
                                     (x11::nil)) =>
                                   P_id_isNatKind (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U22 
                                     (x12::x11::nil)) =>
                                   P_id_a__U22 (measure x12) (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_U92 (x13::
                                     x12::x11::nil)) =>
                                   P_id_U92 (measure x13) (measure x12) 
                                    (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_nil nil) =>
                                   P_id_nil 
                                  | (algebra.Alg.Term algebra.F.id_a__U61 
                                     (x11::nil)) =>
                                   P_id_a__U61 (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_U43 (x13::
                                     x12::x11::nil)) =>
                                   P_id_U43 (measure x13) (measure x12) 
                                    (measure x11)
                                  | (algebra.Alg.Term 
                                     algebra.F.id_a__isNatIListKind 
                                     (x11::nil)) =>
                                   P_id_a__isNatIListKind (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_U84 (x13::
                                     x12::x11::nil)) =>
                                   P_id_U84 (measure x13) (measure x12) 
                                    (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_s 
                                     (x11::nil)) =>
                                   P_id_s (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U46 
                                     (x11::nil)) =>
                                   P_id_a__U46 (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_U32 (x12::
                                     x11::nil)) =>
                                   P_id_U32 (measure x12) (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U31 
                                     (x12::x11::nil)) =>
                                   P_id_a__U31 (measure x12) (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_U13 
                                     (x11::nil)) =>
                                   P_id_U13 (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U83 
                                     (x13::x12::x11::nil)) =>
                                   P_id_a__U83 (measure x13) (measure x12) 
                                    (measure x11)
                                  | (algebra.Alg.Term 
                                     algebra.F.id_isNatIList (x11::nil)) =>
                                   P_id_isNatIList (measure x11)
                                  | _ => 0
                                  end.
                  Proof.
                    intros t;case t;intros ;apply refl_equal.
                  Qed.
                  
                  Lemma measure_bounded : forall t, 0 <= measure t.
                  Proof.
                    unfold measure in |-*.
                    
                    apply InterpZ.measure_bounded;
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Ltac generate_pos_hyp  :=
                   match goal with
                     | H:context [measure ?x] |- _ =>
                      let v := fresh "v" in 
                       (let H := fresh "h" in 
                         (set (H:=measure_bounded x) in *;
                           set (v:=measure x) in *;clearbody H;clearbody v))
                     |  |- context [measure ?x] =>
                      let v := fresh "v" in 
                       (let H := fresh "h" in 
                         (set (H:=measure_bounded x) in *;
                           set (v:=measure x) in *;clearbody H;clearbody v))
                     end
                   .
                  
                  Lemma rules_monotonic :
                   forall l r, 
                    (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) ->
                     measure r <= measure l.
                  Proof.
                    intros l r H.
                    fold measure in |-*.
                    
                    inversion H;clear H;subst;inversion H0;clear H0;subst;
                     simpl algebra.EQT.T.apply_subst in |-*;
                     repeat (
                     match goal with
                       |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
                        rewrite (measure_equation (algebra.Alg.Term f t))
                       end
                     );repeat (generate_pos_hyp );
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma measure_star_monotonic :
                   forall l r, 
                    (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                               r l) ->measure r <= measure l.
                  Proof.
                    unfold measure in *.
                    apply InterpZ.measure_star_monotonic.
                    intros ;apply P_id_U51_monotonic;assumption.
                    intros ;apply P_id_a__U84_monotonic;assumption.
                    intros ;apply P_id_a__U32_monotonic;assumption.
                    intros ;apply P_id_isNatList_monotonic;assumption.
                    intros ;apply P_id_a__U13_monotonic;assumption.
                    intros ;apply P_id_U85_monotonic;assumption.
                    intros ;apply P_id_a__length_monotonic;assumption.
                    intros ;apply P_id_a__isNatIList_monotonic;assumption.
                    intros ;apply P_id_U33_monotonic;assumption.
                    intros ;apply P_id_a__U11_monotonic;assumption.
                    intros ;apply P_id_U81_monotonic;assumption.
                    intros ;apply P_id_a__U92_monotonic;assumption.
                    intros ;apply P_id_a__U43_monotonic;assumption.
                    intros ;apply P_id_U23_monotonic;assumption.
                    intros ;apply P_id_a__isNatKind_monotonic;assumption.
                    intros ;apply P_id_U93_monotonic;assumption.
                    intros ;apply P_id_U11_monotonic;assumption.
                    intros ;apply P_id_a__U71_monotonic;assumption.
                    intros ;apply P_id_U44_monotonic;assumption.
                    intros ;apply P_id_U61_monotonic;assumption.
                    intros ;apply P_id_a__U86_monotonic;assumption.
                    intros ;apply P_id_a__U41_monotonic;assumption.
                    intros ;apply P_id_U22_monotonic;assumption.
                    intros ;apply P_id_a__U21_monotonic;assumption.
                    intros ;apply P_id_U91_monotonic;assumption.
                    intros ;apply P_id_length_monotonic;assumption.
                    intros ;apply P_id_a__U52_monotonic;assumption.
                    intros ;apply P_id_U42_monotonic;assumption.
                    intros ;apply P_id_a__U12_monotonic;assumption.
                    intros ;apply P_id_U83_monotonic;assumption.
                    intros ;apply P_id_a__U94_monotonic;assumption.
                    intros ;apply P_id_a__U45_monotonic;assumption.
                    intros ;apply P_id_U31_monotonic;assumption.
                    intros ;apply P_id_a__isNat_monotonic;assumption.
                    intros ;apply P_id_isNatIListKind_monotonic;assumption.
                    intros ;apply P_id_a__U82_monotonic;assumption.
                    intros ;apply P_id_U46_monotonic;assumption.
                    intros ;apply P_id_cons_monotonic;assumption.
                    intros ;apply P_id_U52_monotonic;assumption.
                    intros ;apply P_id_a__U85_monotonic;assumption.
                    intros ;apply P_id_a__U33_monotonic;assumption.
                    intros ;apply P_id_U21_monotonic;assumption.
                    intros ;apply P_id_a__isNatList_monotonic;assumption.
                    intros ;apply P_id_U86_monotonic;assumption.
                    intros ;apply P_id_mark_monotonic;assumption.
                    intros ;apply P_id_a__U51_monotonic;assumption.
                    intros ;apply P_id_U41_monotonic;assumption.
                    intros ;apply P_id_U82_monotonic;assumption.
                    intros ;apply P_id_a__U93_monotonic;assumption.
                    intros ;apply P_id_a__U44_monotonic;assumption.
                    intros ;apply P_id_isNat_monotonic;assumption.
                    intros ;apply P_id_a__U23_monotonic;assumption.
                    intros ;apply P_id_U94_monotonic;assumption.
                    intros ;apply P_id_U12_monotonic;assumption.
                    intros ;apply P_id_a__U81_monotonic;assumption.
                    intros ;apply P_id_U45_monotonic;assumption.
                    intros ;apply P_id_U71_monotonic;assumption.
                    intros ;apply P_id_a__U91_monotonic;assumption.
                    intros ;apply P_id_a__U42_monotonic;assumption.
                    intros ;apply P_id_isNatKind_monotonic;assumption.
                    intros ;apply P_id_a__U22_monotonic;assumption.
                    intros ;apply P_id_U92_monotonic;assumption.
                    intros ;apply P_id_a__U61_monotonic;assumption.
                    intros ;apply P_id_U43_monotonic;assumption.
                    
                    intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
                    intros ;apply P_id_U84_monotonic;assumption.
                    intros ;apply P_id_s_monotonic;assumption.
                    intros ;apply P_id_a__U46_monotonic;assumption.
                    intros ;apply P_id_U32_monotonic;assumption.
                    intros ;apply P_id_a__U31_monotonic;assumption.
                    intros ;apply P_id_U13_monotonic;assumption.
                    intros ;apply P_id_a__U83_monotonic;assumption.
                    intros ;apply P_id_isNatIList_monotonic;assumption.
                    intros ;apply P_id_a__zeros_bounded;assumption.
                    intros ;apply P_id_U51_bounded;assumption.
                    intros ;apply P_id_a__U84_bounded;assumption.
                    intros ;apply P_id_a__U32_bounded;assumption.
                    intros ;apply P_id_isNatList_bounded;assumption.
                    intros ;apply P_id_a__U13_bounded;assumption.
                    intros ;apply P_id_U85_bounded;assumption.
                    intros ;apply P_id_a__length_bounded;assumption.
                    intros ;apply P_id_a__isNatIList_bounded;assumption.
                    intros ;apply P_id_U33_bounded;assumption.
                    intros ;apply P_id_a__U11_bounded;assumption.
                    intros ;apply P_id_U81_bounded;assumption.
                    intros ;apply P_id_a__U92_bounded;assumption.
                    intros ;apply P_id_a__U43_bounded;assumption.
                    intros ;apply P_id_U23_bounded;assumption.
                    intros ;apply P_id_a__isNatKind_bounded;assumption.
                    intros ;apply P_id_U93_bounded;assumption.
                    intros ;apply P_id_U11_bounded;assumption.
                    intros ;apply P_id_a__U71_bounded;assumption.
                    intros ;apply P_id_U44_bounded;assumption.
                    intros ;apply P_id_0_bounded;assumption.
                    intros ;apply P_id_U61_bounded;assumption.
                    intros ;apply P_id_a__U86_bounded;assumption.
                    intros ;apply P_id_a__U41_bounded;assumption.
                    intros ;apply P_id_U22_bounded;assumption.
                    intros ;apply P_id_a__U21_bounded;assumption.
                    intros ;apply P_id_U91_bounded;assumption.
                    intros ;apply P_id_length_bounded;assumption.
                    intros ;apply P_id_a__U52_bounded;assumption.
                    intros ;apply P_id_U42_bounded;assumption.
                    intros ;apply P_id_a__U12_bounded;assumption.
                    intros ;apply P_id_U83_bounded;assumption.
                    intros ;apply P_id_a__U94_bounded;assumption.
                    intros ;apply P_id_a__U45_bounded;assumption.
                    intros ;apply P_id_U31_bounded;assumption.
                    intros ;apply P_id_a__isNat_bounded;assumption.
                    intros ;apply P_id_isNatIListKind_bounded;assumption.
                    intros ;apply P_id_a__U82_bounded;assumption.
                    intros ;apply P_id_U46_bounded;assumption.
                    intros ;apply P_id_cons_bounded;assumption.
                    intros ;apply P_id_U52_bounded;assumption.
                    intros ;apply P_id_a__U85_bounded;assumption.
                    intros ;apply P_id_a__U33_bounded;assumption.
                    intros ;apply P_id_U21_bounded;assumption.
                    intros ;apply P_id_a__isNatList_bounded;assumption.
                    intros ;apply P_id_U86_bounded;assumption.
                    intros ;apply P_id_mark_bounded;assumption.
                    intros ;apply P_id_a__U51_bounded;assumption.
                    intros ;apply P_id_U41_bounded;assumption.
                    intros ;apply P_id_tt_bounded;assumption.
                    intros ;apply P_id_U82_bounded;assumption.
                    intros ;apply P_id_a__U93_bounded;assumption.
                    intros ;apply P_id_a__U44_bounded;assumption.
                    intros ;apply P_id_isNat_bounded;assumption.
                    intros ;apply P_id_a__U23_bounded;assumption.
                    intros ;apply P_id_U94_bounded;assumption.
                    intros ;apply P_id_U12_bounded;assumption.
                    intros ;apply P_id_a__U81_bounded;assumption.
                    intros ;apply P_id_U45_bounded;assumption.
                    intros ;apply P_id_zeros_bounded;assumption.
                    intros ;apply P_id_U71_bounded;assumption.
                    intros ;apply P_id_a__U91_bounded;assumption.
                    intros ;apply P_id_a__U42_bounded;assumption.
                    intros ;apply P_id_isNatKind_bounded;assumption.
                    intros ;apply P_id_a__U22_bounded;assumption.
                    intros ;apply P_id_U92_bounded;assumption.
                    intros ;apply P_id_nil_bounded;assumption.
                    intros ;apply P_id_a__U61_bounded;assumption.
                    intros ;apply P_id_U43_bounded;assumption.
                    intros ;apply P_id_a__isNatIListKind_bounded;assumption.
                    intros ;apply P_id_U84_bounded;assumption.
                    intros ;apply P_id_s_bounded;assumption.
                    intros ;apply P_id_a__U46_bounded;assumption.
                    intros ;apply P_id_U32_bounded;assumption.
                    intros ;apply P_id_a__U31_bounded;assumption.
                    intros ;apply P_id_U13_bounded;assumption.
                    intros ;apply P_id_a__U83_bounded;assumption.
                    intros ;apply P_id_isNatIList_bounded;assumption.
                    apply rules_monotonic.
                  Qed.
                  
                  Definition P_id_A__U46 (x11:Z) := 0.
                  
                  Definition P_id_A__ISNATILISTKIND (x11:Z) := 0.
                  
                  Definition P_id_A__U61 (x11:Z) := 0.
                  
                  Definition P_id_A__U85 (x11:Z) (x12:Z) := 0.
                  
                  Definition P_id_A__U31 (x11:Z) (x12:Z) := 0.
                  
                  Definition P_id_A__U81 (x11:Z) (x12:Z) (x13:Z) := 0.
                  
                  Definition P_id_A__U22 (x11:Z) (x12:Z) := 0.
                  
                  Definition P_id_A__U93 (x11:Z) (x12:Z) (x13:Z) := 0.
                  
                  Definition P_id_A__U42 (x11:Z) (x12:Z) (x13:Z) := 0.
                  
                  Definition P_id_A__U51 (x11:Z) (x12:Z) := 0.
                  
                  Definition P_id_A__ISNATLIST (x11:Z) := 0.
                  
                  Definition P_id_A__ZEROS  := 0.
                  
                  Definition P_id_A__U91 (x11:Z) (x12:Z) (x13:Z) := 0.
                  
                  Definition P_id_A__U33 (x11:Z) := 0.
                  
                  Definition P_id_A__U83 (x11:Z) (x12:Z) (x13:Z) := 0.
                  
                  Definition P_id_A__U23 (x11:Z) := 0.
                  
                  Definition P_id_A__U11 (x11:Z) (x12:Z) := 0.
                  
                  Definition P_id_A__LENGTH (x11:Z) := 0.
                  
                  Definition P_id_A__U44 (x11:Z) (x12:Z) (x13:Z) := 0.
                  
                  Definition P_id_A__ISNATILIST (x11:Z) := 0.
                  
                  Definition P_id_A__U13 (x11:Z) := 0.
                  
                  Definition P_id_A__U71 (x11:Z) := 0.
                  
                  Definition P_id_A__U86 (x11:Z) := 0.
                  
                  Definition P_id_A__U32 (x11:Z) (x12:Z) := 0.
                  
                  Definition P_id_A__U82 (x11:Z) (x12:Z) (x13:Z) := 0.
                  
                  Definition P_id_A__ISNATKIND (x11:Z) := 0.
                  
                  Definition P_id_A__U94 (x11:Z) (x12:Z) := 0.
                  
                  Definition P_id_A__U43 (x11:Z) (x12:Z) (x13:Z) := 0.
                  
                  Definition P_id_A__U52 (x11:Z) := 0.
                  
                  Definition P_id_A__U21 (x11:Z) (x12:Z) := 0.
                  
                  Definition P_id_A__U92 (x11:Z) (x12:Z) (x13:Z) := 0.
                  
                  Definition P_id_A__U41 (x11:Z) (x12:Z) (x13:Z) := 0.
                  
                  Definition P_id_A__U84 (x11:Z) (x12:Z) (x13:Z) := 0.
                  
                  Definition P_id_A__ISNAT (x11:Z) := 0.
                  
                  Definition P_id_A__U12 (x11:Z) (x12:Z) := 0.
                  
                  Definition P_id_MARK (x11:Z) := 3* x11.
                  
                  Definition P_id_A__U45 (x11:Z) (x12:Z) := 0.
                  
                  Lemma P_id_A__U46_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_A__U46 x12 <= P_id_A__U46 x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_0].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_A__ISNATILISTKIND_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_A__ISNATILISTKIND x12 <= P_id_A__ISNATILISTKIND x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_0].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_A__U61_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_A__U61 x12 <= P_id_A__U61 x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_0].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_A__U85_monotonic :
                   forall x12 x14 x13 x11, 
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_A__U85 x12 x14 <= P_id_A__U85 x11 x13.
                  Proof.
                    intros x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_A__U31_monotonic :
                   forall x12 x14 x13 x11, 
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_A__U31 x12 x14 <= P_id_A__U31 x11 x13.
                  Proof.
                    intros x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_A__U81_monotonic :
                   forall x16 x12 x14 x13 x11 x15, 
                    (0 <= x16)/\ (x16 <= x15) ->
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_A__U81 x12 x14 x16 <= P_id_A__U81 x11 x13 x15.
                  Proof.
                    intros x16 x15 x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    intros [H_5 H_4].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_A__U22_monotonic :
                   forall x12 x14 x13 x11, 
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_A__U22 x12 x14 <= P_id_A__U22 x11 x13.
                  Proof.
                    intros x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_A__U93_monotonic :
                   forall x16 x12 x14 x13 x11 x15, 
                    (0 <= x16)/\ (x16 <= x15) ->
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_A__U93 x12 x14 x16 <= P_id_A__U93 x11 x13 x15.
                  Proof.
                    intros x16 x15 x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    intros [H_5 H_4].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_A__U42_monotonic :
                   forall x16 x12 x14 x13 x11 x15, 
                    (0 <= x16)/\ (x16 <= x15) ->
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_A__U42 x12 x14 x16 <= P_id_A__U42 x11 x13 x15.
                  Proof.
                    intros x16 x15 x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    intros [H_5 H_4].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_A__U51_monotonic :
                   forall x12 x14 x13 x11, 
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_A__U51 x12 x14 <= P_id_A__U51 x11 x13.
                  Proof.
                    intros x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_A__ISNATLIST_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_A__ISNATLIST x12 <= P_id_A__ISNATLIST x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_0].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_A__U91_monotonic :
                   forall x16 x12 x14 x13 x11 x15, 
                    (0 <= x16)/\ (x16 <= x15) ->
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_A__U91 x12 x14 x16 <= P_id_A__U91 x11 x13 x15.
                  Proof.
                    intros x16 x15 x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    intros [H_5 H_4].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_A__U33_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_A__U33 x12 <= P_id_A__U33 x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_0].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_A__U83_monotonic :
                   forall x16 x12 x14 x13 x11 x15, 
                    (0 <= x16)/\ (x16 <= x15) ->
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_A__U83 x12 x14 x16 <= P_id_A__U83 x11 x13 x15.
                  Proof.
                    intros x16 x15 x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    intros [H_5 H_4].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_A__U23_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_A__U23 x12 <= P_id_A__U23 x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_0].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_A__U11_monotonic :
                   forall x12 x14 x13 x11, 
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_A__U11 x12 x14 <= P_id_A__U11 x11 x13.
                  Proof.
                    intros x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_A__LENGTH_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_A__LENGTH x12 <= P_id_A__LENGTH x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_0].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_A__U44_monotonic :
                   forall x16 x12 x14 x13 x11 x15, 
                    (0 <= x16)/\ (x16 <= x15) ->
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_A__U44 x12 x14 x16 <= P_id_A__U44 x11 x13 x15.
                  Proof.
                    intros x16 x15 x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    intros [H_5 H_4].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_A__ISNATILIST_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_A__ISNATILIST x12 <= P_id_A__ISNATILIST x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_0].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_A__U13_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_A__U13 x12 <= P_id_A__U13 x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_0].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_A__U71_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_A__U71 x12 <= P_id_A__U71 x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_0].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_A__U86_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_A__U86 x12 <= P_id_A__U86 x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_0].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_A__U32_monotonic :
                   forall x12 x14 x13 x11, 
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_A__U32 x12 x14 <= P_id_A__U32 x11 x13.
                  Proof.
                    intros x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_A__U82_monotonic :
                   forall x16 x12 x14 x13 x11 x15, 
                    (0 <= x16)/\ (x16 <= x15) ->
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_A__U82 x12 x14 x16 <= P_id_A__U82 x11 x13 x15.
                  Proof.
                    intros x16 x15 x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    intros [H_5 H_4].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_A__ISNATKIND_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_A__ISNATKIND x12 <= P_id_A__ISNATKIND x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_0].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_A__U94_monotonic :
                   forall x12 x14 x13 x11, 
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_A__U94 x12 x14 <= P_id_A__U94 x11 x13.
                  Proof.
                    intros x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_A__U43_monotonic :
                   forall x16 x12 x14 x13 x11 x15, 
                    (0 <= x16)/\ (x16 <= x15) ->
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_A__U43 x12 x14 x16 <= P_id_A__U43 x11 x13 x15.
                  Proof.
                    intros x16 x15 x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    intros [H_5 H_4].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_A__U52_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_A__U52 x12 <= P_id_A__U52 x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_0].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_A__U21_monotonic :
                   forall x12 x14 x13 x11, 
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_A__U21 x12 x14 <= P_id_A__U21 x11 x13.
                  Proof.
                    intros x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_A__U92_monotonic :
                   forall x16 x12 x14 x13 x11 x15, 
                    (0 <= x16)/\ (x16 <= x15) ->
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_A__U92 x12 x14 x16 <= P_id_A__U92 x11 x13 x15.
                  Proof.
                    intros x16 x15 x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    intros [H_5 H_4].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_A__U41_monotonic :
                   forall x16 x12 x14 x13 x11 x15, 
                    (0 <= x16)/\ (x16 <= x15) ->
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_A__U41 x12 x14 x16 <= P_id_A__U41 x11 x13 x15.
                  Proof.
                    intros x16 x15 x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    intros [H_5 H_4].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_A__U84_monotonic :
                   forall x16 x12 x14 x13 x11 x15, 
                    (0 <= x16)/\ (x16 <= x15) ->
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_A__U84 x12 x14 x16 <= P_id_A__U84 x11 x13 x15.
                  Proof.
                    intros x16 x15 x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    intros [H_5 H_4].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_A__ISNAT_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_A__ISNAT x12 <= P_id_A__ISNAT x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_0].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_A__U12_monotonic :
                   forall x12 x14 x13 x11, 
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_A__U12 x12 x14 <= P_id_A__U12 x11 x13.
                  Proof.
                    intros x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_MARK_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_MARK x12 <= P_id_MARK x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_0].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_A__U45_monotonic :
                   forall x12 x14 x13 x11, 
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_A__U45 x12 x14 <= P_id_A__U45 x11 x13.
                  Proof.
                    intros x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Definition marked_measure  := 
                    InterpZ.marked_measure 0 P_id_a__zeros P_id_U51 
                     P_id_a__U84 P_id_a__U32 P_id_isNatList P_id_a__U13 
                     P_id_U85 P_id_a__length P_id_a__isNatIList P_id_U33 
                     P_id_a__U11 P_id_U81 P_id_a__U92 P_id_a__U43 P_id_U23 
                     P_id_a__isNatKind P_id_U93 P_id_U11 P_id_a__U71 
                     P_id_U44 P_id_0 P_id_U61 P_id_a__U86 P_id_a__U41 
                     P_id_U22 P_id_a__U21 P_id_U91 P_id_length P_id_a__U52 
                     P_id_U42 P_id_a__U12 P_id_U83 P_id_a__U94 P_id_a__U45 
                     P_id_U31 P_id_a__isNat P_id_isNatIListKind P_id_a__U82 
                     P_id_U46 P_id_cons P_id_U52 P_id_a__U85 P_id_a__U33 
                     P_id_U21 P_id_a__isNatList P_id_U86 P_id_mark 
                     P_id_a__U51 P_id_U41 P_id_tt P_id_U82 P_id_a__U93 
                     P_id_a__U44 P_id_isNat P_id_a__U23 P_id_U94 P_id_U12 
                     P_id_a__U81 P_id_U45 P_id_zeros P_id_U71 P_id_a__U91 
                     P_id_a__U42 P_id_isNatKind P_id_a__U22 P_id_U92 
                     P_id_nil P_id_a__U61 P_id_U43 P_id_a__isNatIListKind 
                     P_id_U84 P_id_s P_id_a__U46 P_id_U32 P_id_a__U31 
                     P_id_U13 P_id_a__U83 P_id_isNatIList P_id_A__U46 
                     P_id_A__ISNATILISTKIND P_id_A__U61 P_id_A__U85 
                     P_id_A__U31 P_id_A__U81 P_id_A__U22 P_id_A__U93 
                     P_id_A__U42 P_id_A__U51 P_id_A__ISNATLIST P_id_A__ZEROS 
                     P_id_A__U91 P_id_A__U33 P_id_A__U83 P_id_A__U23 
                     P_id_A__U11 P_id_A__LENGTH P_id_A__U44 
                     P_id_A__ISNATILIST P_id_A__U13 P_id_A__U71 P_id_A__U86 
                     P_id_A__U32 P_id_A__U82 P_id_A__ISNATKIND P_id_A__U94 
                     P_id_A__U43 P_id_A__U52 P_id_A__U21 P_id_A__U92 
                     P_id_A__U41 P_id_A__U84 P_id_A__ISNAT P_id_A__U12 
                     P_id_MARK P_id_A__U45.
                  
                  Lemma marked_measure_equation :
                   forall t, 
                    marked_measure t = match t with
                                         | (algebra.Alg.Term 
                                            algebra.F.id_a__U46 (x11::nil)) =>
                                          P_id_A__U46 (measure x11)
                                         | (algebra.Alg.Term 
                                            algebra.F.id_a__isNatIListKind 
                                            (x11::nil)) =>
                                          P_id_A__ISNATILISTKIND (measure x11)
                                         | (algebra.Alg.Term 
                                            algebra.F.id_a__U61 (x11::nil)) =>
                                          P_id_A__U61 (measure x11)
                                         | (algebra.Alg.Term 
                                            algebra.F.id_a__U85 (x12::
                                            x11::nil)) =>
                                          P_id_A__U85 (measure x12) 
                                           (measure x11)
                                         | (algebra.Alg.Term 
                                            algebra.F.id_a__U31 (x12::
                                            x11::nil)) =>
                                          P_id_A__U31 (measure x12) 
                                           (measure x11)
                                         | (algebra.Alg.Term 
                                            algebra.F.id_a__U81 (x13::x12::
                                            x11::nil)) =>
                                          P_id_A__U81 (measure x13) 
                                           (measure x12) (measure x11)
                                         | (algebra.Alg.Term 
                                            algebra.F.id_a__U22 (x12::
                                            x11::nil)) =>
                                          P_id_A__U22 (measure x12) 
                                           (measure x11)
                                         | (algebra.Alg.Term 
                                            algebra.F.id_a__U93 (x13::x12::
                                            x11::nil)) =>
                                          P_id_A__U93 (measure x13) 
                                           (measure x12) (measure x11)
                                         | (algebra.Alg.Term 
                                            algebra.F.id_a__U42 (x13::x12::
                                            x11::nil)) =>
                                          P_id_A__U42 (measure x13) 
                                           (measure x12) (measure x11)
                                         | (algebra.Alg.Term 
                                            algebra.F.id_a__U51 (x12::
                                            x11::nil)) =>
                                          P_id_A__U51 (measure x12) 
                                           (measure x11)
                                         | (algebra.Alg.Term 
                                            algebra.F.id_a__isNatList 
                                            (x11::nil)) =>
                                          P_id_A__ISNATLIST (measure x11)
                                         | (algebra.Alg.Term 
                                            algebra.F.id_a__zeros nil) =>
                                          P_id_A__ZEROS 
                                         | (algebra.Alg.Term 
                                            algebra.F.id_a__U91 (x13::x12::
                                            x11::nil)) =>
                                          P_id_A__U91 (measure x13) 
                                           (measure x12) (measure x11)
                                         | (algebra.Alg.Term 
                                            algebra.F.id_a__U33 (x11::nil)) =>
                                          P_id_A__U33 (measure x11)
                                         | (algebra.Alg.Term 
                                            algebra.F.id_a__U83 (x13::x12::
                                            x11::nil)) =>
                                          P_id_A__U83 (measure x13) 
                                           (measure x12) (measure x11)
                                         | (algebra.Alg.Term 
                                            algebra.F.id_a__U23 (x11::nil)) =>
                                          P_id_A__U23 (measure x11)
                                         | (algebra.Alg.Term 
                                            algebra.F.id_a__U11 (x12::
                                            x11::nil)) =>
                                          P_id_A__U11 (measure x12) 
                                           (measure x11)
                                         | (algebra.Alg.Term 
                                            algebra.F.id_a__length 
                                            (x11::nil)) =>
                                          P_id_A__LENGTH (measure x11)
                                         | (algebra.Alg.Term 
                                            algebra.F.id_a__U44 (x13::x12::
                                            x11::nil)) =>
                                          P_id_A__U44 (measure x13) 
                                           (measure x12) (measure x11)
                                         | (algebra.Alg.Term 
                                            algebra.F.id_a__isNatIList 
                                            (x11::nil)) =>
                                          P_id_A__ISNATILIST (measure x11)
                                         | (algebra.Alg.Term 
                                            algebra.F.id_a__U13 (x11::nil)) =>
                                          P_id_A__U13 (measure x11)
                                         | (algebra.Alg.Term 
                                            algebra.F.id_a__U71 (x11::nil)) =>
                                          P_id_A__U71 (measure x11)
                                         | (algebra.Alg.Term 
                                            algebra.F.id_a__U86 (x11::nil)) =>
                                          P_id_A__U86 (measure x11)
                                         | (algebra.Alg.Term 
                                            algebra.F.id_a__U32 (x12::
                                            x11::nil)) =>
                                          P_id_A__U32 (measure x12) 
                                           (measure x11)
                                         | (algebra.Alg.Term 
                                            algebra.F.id_a__U82 (x13::x12::
                                            x11::nil)) =>
                                          P_id_A__U82 (measure x13) 
                                           (measure x12) (measure x11)
                                         | (algebra.Alg.Term 
                                            algebra.F.id_a__isNatKind 
                                            (x11::nil)) =>
                                          P_id_A__ISNATKIND (measure x11)
                                         | (algebra.Alg.Term 
                                            algebra.F.id_a__U94 (x12::
                                            x11::nil)) =>
                                          P_id_A__U94 (measure x12) 
                                           (measure x11)
                                         | (algebra.Alg.Term 
                                            algebra.F.id_a__U43 (x13::x12::
                                            x11::nil)) =>
                                          P_id_A__U43 (measure x13) 
                                           (measure x12) (measure x11)
                                         | (algebra.Alg.Term 
                                            algebra.F.id_a__U52 (x11::nil)) =>
                                          P_id_A__U52 (measure x11)
                                         | (algebra.Alg.Term 
                                            algebra.F.id_a__U21 (x12::
                                            x11::nil)) =>
                                          P_id_A__U21 (measure x12) 
                                           (measure x11)
                                         | (algebra.Alg.Term 
                                            algebra.F.id_a__U92 (x13::x12::
                                            x11::nil)) =>
                                          P_id_A__U92 (measure x13) 
                                           (measure x12) (measure x11)
                                         | (algebra.Alg.Term 
                                            algebra.F.id_a__U41 (x13::x12::
                                            x11::nil)) =>
                                          P_id_A__U41 (measure x13) 
                                           (measure x12) (measure x11)
                                         | (algebra.Alg.Term 
                                            algebra.F.id_a__U84 (x13::x12::
                                            x11::nil)) =>
                                          P_id_A__U84 (measure x13) 
                                           (measure x12) (measure x11)
                                         | (algebra.Alg.Term 
                                            algebra.F.id_a__isNat (x11::nil)) =>
                                          P_id_A__ISNAT (measure x11)
                                         | (algebra.Alg.Term 
                                            algebra.F.id_a__U12 (x12::
                                            x11::nil)) =>
                                          P_id_A__U12 (measure x12) 
                                           (measure x11)
                                         | (algebra.Alg.Term 
                                            algebra.F.id_mark (x11::nil)) =>
                                          P_id_MARK (measure x11)
                                         | (algebra.Alg.Term 
                                            algebra.F.id_a__U45 (x12::
                                            x11::nil)) =>
                                          P_id_A__U45 (measure x12) 
                                           (measure x11)
                                         | _ => measure t
                                         end.
                  Proof.
                    reflexivity .
                  Qed.
                  
                  Lemma marked_measure_star_monotonic :
                   forall f l1 l2, 
                    (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step 
                                                                    R_xml_0_deep_rew.R_xml_0_rules)
                                                                    ) 
                      l1 l2) ->
                     marked_measure (algebra.Alg.Term f l1) <= marked_measure 
                                                                (algebra.Alg.Term 
                                                                  f l2).
                  Proof.
                    unfold marked_measure in *.
                    apply InterpZ.marked_measure_star_monotonic.
                    intros ;apply P_id_U51_monotonic;assumption.
                    intros ;apply P_id_a__U84_monotonic;assumption.
                    intros ;apply P_id_a__U32_monotonic;assumption.
                    intros ;apply P_id_isNatList_monotonic;assumption.
                    intros ;apply P_id_a__U13_monotonic;assumption.
                    intros ;apply P_id_U85_monotonic;assumption.
                    intros ;apply P_id_a__length_monotonic;assumption.
                    intros ;apply P_id_a__isNatIList_monotonic;assumption.
                    intros ;apply P_id_U33_monotonic;assumption.
                    intros ;apply P_id_a__U11_monotonic;assumption.
                    intros ;apply P_id_U81_monotonic;assumption.
                    intros ;apply P_id_a__U92_monotonic;assumption.
                    intros ;apply P_id_a__U43_monotonic;assumption.
                    intros ;apply P_id_U23_monotonic;assumption.
                    intros ;apply P_id_a__isNatKind_monotonic;assumption.
                    intros ;apply P_id_U93_monotonic;assumption.
                    intros ;apply P_id_U11_monotonic;assumption.
                    intros ;apply P_id_a__U71_monotonic;assumption.
                    intros ;apply P_id_U44_monotonic;assumption.
                    intros ;apply P_id_U61_monotonic;assumption.
                    intros ;apply P_id_a__U86_monotonic;assumption.
                    intros ;apply P_id_a__U41_monotonic;assumption.
                    intros ;apply P_id_U22_monotonic;assumption.
                    intros ;apply P_id_a__U21_monotonic;assumption.
                    intros ;apply P_id_U91_monotonic;assumption.
                    intros ;apply P_id_length_monotonic;assumption.
                    intros ;apply P_id_a__U52_monotonic;assumption.
                    intros ;apply P_id_U42_monotonic;assumption.
                    intros ;apply P_id_a__U12_monotonic;assumption.
                    intros ;apply P_id_U83_monotonic;assumption.
                    intros ;apply P_id_a__U94_monotonic;assumption.
                    intros ;apply P_id_a__U45_monotonic;assumption.
                    intros ;apply P_id_U31_monotonic;assumption.
                    intros ;apply P_id_a__isNat_monotonic;assumption.
                    intros ;apply P_id_isNatIListKind_monotonic;assumption.
                    intros ;apply P_id_a__U82_monotonic;assumption.
                    intros ;apply P_id_U46_monotonic;assumption.
                    intros ;apply P_id_cons_monotonic;assumption.
                    intros ;apply P_id_U52_monotonic;assumption.
                    intros ;apply P_id_a__U85_monotonic;assumption.
                    intros ;apply P_id_a__U33_monotonic;assumption.
                    intros ;apply P_id_U21_monotonic;assumption.
                    intros ;apply P_id_a__isNatList_monotonic;assumption.
                    intros ;apply P_id_U86_monotonic;assumption.
                    intros ;apply P_id_mark_monotonic;assumption.
                    intros ;apply P_id_a__U51_monotonic;assumption.
                    intros ;apply P_id_U41_monotonic;assumption.
                    intros ;apply P_id_U82_monotonic;assumption.
                    intros ;apply P_id_a__U93_monotonic;assumption.
                    intros ;apply P_id_a__U44_monotonic;assumption.
                    intros ;apply P_id_isNat_monotonic;assumption.
                    intros ;apply P_id_a__U23_monotonic;assumption.
                    intros ;apply P_id_U94_monotonic;assumption.
                    intros ;apply P_id_U12_monotonic;assumption.
                    intros ;apply P_id_a__U81_monotonic;assumption.
                    intros ;apply P_id_U45_monotonic;assumption.
                    intros ;apply P_id_U71_monotonic;assumption.
                    intros ;apply P_id_a__U91_monotonic;assumption.
                    intros ;apply P_id_a__U42_monotonic;assumption.
                    intros ;apply P_id_isNatKind_monotonic;assumption.
                    intros ;apply P_id_a__U22_monotonic;assumption.
                    intros ;apply P_id_U92_monotonic;assumption.
                    intros ;apply P_id_a__U61_monotonic;assumption.
                    intros ;apply P_id_U43_monotonic;assumption.
                    
                    intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
                    intros ;apply P_id_U84_monotonic;assumption.
                    intros ;apply P_id_s_monotonic;assumption.
                    intros ;apply P_id_a__U46_monotonic;assumption.
                    intros ;apply P_id_U32_monotonic;assumption.
                    intros ;apply P_id_a__U31_monotonic;assumption.
                    intros ;apply P_id_U13_monotonic;assumption.
                    intros ;apply P_id_a__U83_monotonic;assumption.
                    intros ;apply P_id_isNatIList_monotonic;assumption.
                    intros ;apply P_id_a__zeros_bounded;assumption.
                    intros ;apply P_id_U51_bounded;assumption.
                    intros ;apply P_id_a__U84_bounded;assumption.
                    intros ;apply P_id_a__U32_bounded;assumption.
                    intros ;apply P_id_isNatList_bounded;assumption.
                    intros ;apply P_id_a__U13_bounded;assumption.
                    intros ;apply P_id_U85_bounded;assumption.
                    intros ;apply P_id_a__length_bounded;assumption.
                    intros ;apply P_id_a__isNatIList_bounded;assumption.
                    intros ;apply P_id_U33_bounded;assumption.
                    intros ;apply P_id_a__U11_bounded;assumption.
                    intros ;apply P_id_U81_bounded;assumption.
                    intros ;apply P_id_a__U92_bounded;assumption.
                    intros ;apply P_id_a__U43_bounded;assumption.
                    intros ;apply P_id_U23_bounded;assumption.
                    intros ;apply P_id_a__isNatKind_bounded;assumption.
                    intros ;apply P_id_U93_bounded;assumption.
                    intros ;apply P_id_U11_bounded;assumption.
                    intros ;apply P_id_a__U71_bounded;assumption.
                    intros ;apply P_id_U44_bounded;assumption.
                    intros ;apply P_id_0_bounded;assumption.
                    intros ;apply P_id_U61_bounded;assumption.
                    intros ;apply P_id_a__U86_bounded;assumption.
                    intros ;apply P_id_a__U41_bounded;assumption.
                    intros ;apply P_id_U22_bounded;assumption.
                    intros ;apply P_id_a__U21_bounded;assumption.
                    intros ;apply P_id_U91_bounded;assumption.
                    intros ;apply P_id_length_bounded;assumption.
                    intros ;apply P_id_a__U52_bounded;assumption.
                    intros ;apply P_id_U42_bounded;assumption.
                    intros ;apply P_id_a__U12_bounded;assumption.
                    intros ;apply P_id_U83_bounded;assumption.
                    intros ;apply P_id_a__U94_bounded;assumption.
                    intros ;apply P_id_a__U45_bounded;assumption.
                    intros ;apply P_id_U31_bounded;assumption.
                    intros ;apply P_id_a__isNat_bounded;assumption.
                    intros ;apply P_id_isNatIListKind_bounded;assumption.
                    intros ;apply P_id_a__U82_bounded;assumption.
                    intros ;apply P_id_U46_bounded;assumption.
                    intros ;apply P_id_cons_bounded;assumption.
                    intros ;apply P_id_U52_bounded;assumption.
                    intros ;apply P_id_a__U85_bounded;assumption.
                    intros ;apply P_id_a__U33_bounded;assumption.
                    intros ;apply P_id_U21_bounded;assumption.
                    intros ;apply P_id_a__isNatList_bounded;assumption.
                    intros ;apply P_id_U86_bounded;assumption.
                    intros ;apply P_id_mark_bounded;assumption.
                    intros ;apply P_id_a__U51_bounded;assumption.
                    intros ;apply P_id_U41_bounded;assumption.
                    intros ;apply P_id_tt_bounded;assumption.
                    intros ;apply P_id_U82_bounded;assumption.
                    intros ;apply P_id_a__U93_bounded;assumption.
                    intros ;apply P_id_a__U44_bounded;assumption.
                    intros ;apply P_id_isNat_bounded;assumption.
                    intros ;apply P_id_a__U23_bounded;assumption.
                    intros ;apply P_id_U94_bounded;assumption.
                    intros ;apply P_id_U12_bounded;assumption.
                    intros ;apply P_id_a__U81_bounded;assumption.
                    intros ;apply P_id_U45_bounded;assumption.
                    intros ;apply P_id_zeros_bounded;assumption.
                    intros ;apply P_id_U71_bounded;assumption.
                    intros ;apply P_id_a__U91_bounded;assumption.
                    intros ;apply P_id_a__U42_bounded;assumption.
                    intros ;apply P_id_isNatKind_bounded;assumption.
                    intros ;apply P_id_a__U22_bounded;assumption.
                    intros ;apply P_id_U92_bounded;assumption.
                    intros ;apply P_id_nil_bounded;assumption.
                    intros ;apply P_id_a__U61_bounded;assumption.
                    intros ;apply P_id_U43_bounded;assumption.
                    intros ;apply P_id_a__isNatIListKind_bounded;assumption.
                    intros ;apply P_id_U84_bounded;assumption.
                    intros ;apply P_id_s_bounded;assumption.
                    intros ;apply P_id_a__U46_bounded;assumption.
                    intros ;apply P_id_U32_bounded;assumption.
                    intros ;apply P_id_a__U31_bounded;assumption.
                    intros ;apply P_id_U13_bounded;assumption.
                    intros ;apply P_id_a__U83_bounded;assumption.
                    intros ;apply P_id_isNatIList_bounded;assumption.
                    apply rules_monotonic.
                    intros ;apply P_id_A__U46_monotonic;assumption.
                    
                    intros ;apply P_id_A__ISNATILISTKIND_monotonic;assumption.
                    intros ;apply P_id_A__U61_monotonic;assumption.
                    intros ;apply P_id_A__U85_monotonic;assumption.
                    intros ;apply P_id_A__U31_monotonic;assumption.
                    intros ;apply P_id_A__U81_monotonic;assumption.
                    intros ;apply P_id_A__U22_monotonic;assumption.
                    intros ;apply P_id_A__U93_monotonic;assumption.
                    intros ;apply P_id_A__U42_monotonic;assumption.
                    intros ;apply P_id_A__U51_monotonic;assumption.
                    intros ;apply P_id_A__ISNATLIST_monotonic;assumption.
                    intros ;apply P_id_A__U91_monotonic;assumption.
                    intros ;apply P_id_A__U33_monotonic;assumption.
                    intros ;apply P_id_A__U83_monotonic;assumption.
                    intros ;apply P_id_A__U23_monotonic;assumption.
                    intros ;apply P_id_A__U11_monotonic;assumption.
                    intros ;apply P_id_A__LENGTH_monotonic;assumption.
                    intros ;apply P_id_A__U44_monotonic;assumption.
                    intros ;apply P_id_A__ISNATILIST_monotonic;assumption.
                    intros ;apply P_id_A__U13_monotonic;assumption.
                    intros ;apply P_id_A__U71_monotonic;assumption.
                    intros ;apply P_id_A__U86_monotonic;assumption.
                    intros ;apply P_id_A__U32_monotonic;assumption.
                    intros ;apply P_id_A__U82_monotonic;assumption.
                    intros ;apply P_id_A__ISNATKIND_monotonic;assumption.
                    intros ;apply P_id_A__U94_monotonic;assumption.
                    intros ;apply P_id_A__U43_monotonic;assumption.
                    intros ;apply P_id_A__U52_monotonic;assumption.
                    intros ;apply P_id_A__U21_monotonic;assumption.
                    intros ;apply P_id_A__U92_monotonic;assumption.
                    intros ;apply P_id_A__U41_monotonic;assumption.
                    intros ;apply P_id_A__U84_monotonic;assumption.
                    intros ;apply P_id_A__ISNAT_monotonic;assumption.
                    intros ;apply P_id_A__U12_monotonic;assumption.
                    intros ;apply P_id_MARK_monotonic;assumption.
                    intros ;apply P_id_A__U45_monotonic;assumption.
                  Qed.
                  
                  Ltac rewrite_and_unfold  :=
                   do 2 (rewrite marked_measure_equation);
                    repeat (
                    match goal with
                      |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
                       rewrite (measure_equation (algebra.Alg.Term f t))
                      | H:context [measure (algebra.Alg.Term ?f ?t)] |- 
                      _ =>
                       rewrite (measure_equation (algebra.Alg.Term f t)) in 
                       H|-
                      end
                    ).
                  
                  Definition lt a b := 
                    (Zwf.Zwf 0) (marked_measure a) (marked_measure b).
                  
                  Definition le a b := marked_measure a <= marked_measure b.
                  
                  Lemma lt_le_compat :
                   forall a b c, (lt a b) ->(le b c) ->lt a c.
                  Proof.
                    unfold lt, le in *.
                    intros a b c.
                    apply (interp.le_lt_compat_right (interp.o_Z 0)).
                  Qed.
                  
                  Lemma wf_lt : well_founded lt.
                  Proof.
                    unfold lt in *.
                    apply Inverse_Image.wf_inverse_image with  (B:=Z).
                    apply Zwf.Zwf_well_founded.
                  Qed.
                  
                  Lemma DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_strict_in_lt :
                   Relation_Definitions.inclusion _ 
                    DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_strict 
                    lt.
                  Proof.
                    unfold Relation_Definitions.inclusion, lt in *.
                    
                    intros a b H;destruct H;
                     match goal with
                       |  |- (Zwf.Zwf 0) _ 
                              (marked_measure (algebra.Alg.Term ?f ?l)) =>
                        let l'' := algebra.Alg_ext.find_replacement l  in 
                         ((apply (interp.le_lt_compat_right (interp.o_Z 0)) with
                            (marked_measure (algebra.Alg.Term f l''));[
                           idtac|
                           apply marked_measure_star_monotonic;
                            repeat (
                            apply algebra.EQT_ext.one_step_list_refl_trans_clos
                            );(assumption)||(constructor 1)]))
                       end
                     ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_large_in_le :
                   Relation_Definitions.inclusion _ 
                    DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_large 
                    le.
                  Proof.
                    unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *.
                    
                    intros a b H;destruct H;
                     match goal with
                       |  |- _ <= marked_measure (algebra.Alg.Term ?f ?l) =>
                        let l'' := algebra.Alg_ext.find_replacement l  in 
                         ((apply (interp.le_trans (interp.o_Z 0)) with
                            (marked_measure (algebra.Alg.Term f l''));[
                           idtac|
                           apply marked_measure_star_monotonic;
                            repeat (
                            apply algebra.EQT_ext.one_step_list_refl_trans_clos
                            );(assumption)||(constructor 1)]))
                       end
                     ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
                     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                      intuition;
                      (auto with zarith)||
                      (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Definition wf_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_large  := 
                    WF_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_large.wf
                    .
                  
                  
                  Lemma wf :
                   well_founded WF_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large.DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large
                    .
                  Proof.
                    intros x.
                    apply (well_founded_ind wf_lt).
                    clear x.
                    intros x.
                    pattern x.
                    
                    apply (@Acc_ind _ 
                            DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_large)
                           .
                    clear x.
                    intros x _ IHx IHx'.
                    constructor.
                    intros y H.
                    
                    destruct H;
                     (apply IHx';
                       apply DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_strict_in_lt;
                       econstructor eassumption)||
                     ((apply IHx;[econstructor eassumption|
                       intros y' Hlt;apply IHx';
                        apply lt_le_compat with (1:=Hlt) ;
                        apply DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_large_in_le;
                        econstructor eassumption])).
                    
                    apply wf_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_large.
                  Qed.
                 End WF_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large.
                 
                 Open Scope Z_scope.
                 
                 Import ring_extention.
                 
                 Notation Local "a <= b" := (Zle a b).
                 
                 Notation Local "a < b" := (Zlt a b).
                 
                 Definition P_id_a__zeros  := 2.
                 
                 Definition P_id_U51 (x11:Z) (x12:Z) := 1 + 2* x11 + 2* x12.
                 
                 Definition P_id_a__U84 (x11:Z) (x12:Z) (x13:Z) := 0.
                 
                 Definition P_id_a__U32 (x11:Z) (x12:Z) := 2.
                 
                 Definition P_id_isNatList (x11:Z) := 2* x11.
                 
                 Definition P_id_a__U13 (x11:Z) := 0.
                 
                 Definition P_id_U85 (x11:Z) (x12:Z) := 0.
                 
                 Definition P_id_a__length (x11:Z) := 0.
                 
                 Definition P_id_a__isNatIList (x11:Z) := 2 + 2* x11.
                 
                 Definition P_id_U33 (x11:Z) := 0.
                 
                 Definition P_id_a__U11 (x11:Z) (x12:Z) := 0.
                 
                 Definition P_id_U81 (x11:Z) (x12:Z) (x13:Z) := 2 + 2* x13.
                 
                 Definition P_id_a__U92 (x11:Z) (x12:Z) (x13:Z) := 0.
                 
                 Definition P_id_a__U43 (x11:Z) (x12:Z) (x13:Z) := 0.
                 
                 Definition P_id_U23 (x11:Z) := 2* x11.
                 
                 Definition P_id_a__isNatKind (x11:Z) := 0.
                 
                 Definition P_id_U93 (x11:Z) (x12:Z) (x13:Z) := 0.
                 
                 Definition P_id_U11 (x11:Z) (x12:Z) := 0.
                 
                 Definition P_id_a__U71 (x11:Z) := 2* x11.
                 
                 Definition P_id_U44 (x11:Z) (x12:Z) (x13:Z) := 0.
                 
                 Definition P_id_0  := 0.
                 
                 Definition P_id_U61 (x11:Z) := 0.
                 
                 Definition P_id_a__U86 (x11:Z) := 0.
                 
                 Definition P_id_a__U41 (x11:Z) (x12:Z) (x13:Z) := 2* x13.
                 
                 Definition P_id_U22 (x11:Z) (x12:Z) := 0.
                 
                 Definition P_id_a__U21 (x11:Z) (x12:Z) := 0.
                 
                 Definition P_id_U91 (x11:Z) (x12:Z) (x13:Z) := 0.
                 
                 Definition P_id_length (x11:Z) := 0.
                 
                 Definition P_id_a__U52 (x11:Z) := 1 + 1* x11.
                 
                 Definition P_id_U42 (x11:Z) (x12:Z) (x13:Z) := 0.
                 
                 Definition P_id_a__U12 (x11:Z) (x12:Z) := 0.
                 
                 Definition P_id_U83 (x11:Z) (x12:Z) (x13:Z) := 0.
                 
                 Definition P_id_a__U94 (x11:Z) (x12:Z) := 0.
                 
                 Definition P_id_a__U45 (x11:Z) (x12:Z) := 0.
                 
                 Definition P_id_U31 (x11:Z) (x12:Z) := 1.
                 
                 Definition P_id_a__isNat (x11:Z) := 0.
                 
                 Definition P_id_isNatIListKind (x11:Z) := 2* x11.
                 
                 Definition P_id_a__U82 (x11:Z) (x12:Z) (x13:Z) := 2 + 2* x13
                   .
                 
                 Definition P_id_U46 (x11:Z) := 0.
                 
                 Definition P_id_cons (x11:Z) (x12:Z) := 1 + 1* x12.
                 
                 Definition P_id_U52 (x11:Z) := 1 + 1* x11.
                 
                 Definition P_id_a__U85 (x11:Z) (x12:Z) := 0.
                 
                 Definition P_id_a__U33 (x11:Z) := 0.
                 
                 Definition P_id_U21 (x11:Z) (x12:Z) := 0.
                 
                 Definition P_id_a__isNatList (x11:Z) := 3* x11.
                 
                 Definition P_id_U86 (x11:Z) := 0.
                 
                 Definition P_id_mark (x11:Z) := 2* x11.
                 
                 Definition P_id_a__U51 (x11:Z) (x12:Z) := 
                   1 + 2* x11 + 2* x12.
                 
                 Definition P_id_U41 (x11:Z) (x12:Z) (x13:Z) := 1* x13.
                 
                 Definition P_id_tt  := 0.
                 
                 Definition P_id_U82 (x11:Z) (x12:Z) (x13:Z) := 1 + 1* x13.
                 
                 Definition P_id_a__U93 (x11:Z) (x12:Z) (x13:Z) := 0.
                 
                 Definition P_id_a__U44 (x11:Z) (x12:Z) (x13:Z) := 0.
                 
                 Definition P_id_isNat (x11:Z) := 0.
                 
                 Definition P_id_a__U23 (x11:Z) := 2* x11.
                 
                 Definition P_id_U94 (x11:Z) (x12:Z) := 0.
                 
                 Definition P_id_U12 (x11:Z) (x12:Z) := 0.
                 
                 Definition P_id_a__U81 (x11:Z) (x12:Z) (x13:Z) := 2 + 3* x13
                   .
                 
                 Definition P_id_U45 (x11:Z) (x12:Z) := 0.
                 
                 Definition P_id_zeros  := 1.
                 
                 Definition P_id_U71 (x11:Z) := 2* x11.
                 
                 Definition P_id_a__U91 (x11:Z) (x12:Z) (x13:Z) := 0.
                 
                 Definition P_id_a__U42 (x11:Z) (x12:Z) (x13:Z) := 0.
                 
                 Definition P_id_isNatKind (x11:Z) := 0.
                 
                 Definition P_id_a__U22 (x11:Z) (x12:Z) := 0.
                 
                 Definition P_id_U92 (x11:Z) (x12:Z) (x13:Z) := 0.
                 
                 Definition P_id_nil  := 0.
                 
                 Definition P_id_a__U61 (x11:Z) := 0.
                 
                 Definition P_id_U43 (x11:Z) (x12:Z) (x13:Z) := 0.
                 
                 Definition P_id_a__isNatIListKind (x11:Z) := 2* x11.
                 
                 Definition P_id_U84 (x11:Z) (x12:Z) (x13:Z) := 0.
                 
                 Definition P_id_s (x11:Z) := 2* x11.
                 
                 Definition P_id_a__U46 (x11:Z) := 0.
                 
                 Definition P_id_U32 (x11:Z) (x12:Z) := 1.
                 
                 Definition P_id_a__U31 (x11:Z) (x12:Z) := 2.
                 
                 Definition P_id_U13 (x11:Z) := 0.
                 
                 Definition P_id_a__U83 (x11:Z) (x12:Z) (x13:Z) := 0.
                 
                 Definition P_id_isNatIList (x11:Z) := 1 + 1* x11.
                 
                 Lemma P_id_U51_monotonic :
                  forall x12 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_U51 x12 x14 <= P_id_U51 x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U84_monotonic :
                  forall x16 x12 x14 x13 x11 x15, 
                   (0 <= x16)/\ (x16 <= x15) ->
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_a__U84 x12 x14 x16 <= P_id_a__U84 x11 x13 x15.
                 Proof.
                   intros x16 x15 x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   intros [H_5 H_4].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U32_monotonic :
                  forall x12 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_a__U32 x12 x14 <= P_id_a__U32 x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_isNatList_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_isNatList x12 <= P_id_isNatList x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U13_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_a__U13 x12 <= P_id_a__U13 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U85_monotonic :
                  forall x12 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_U85 x12 x14 <= P_id_U85 x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__length_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_a__length x12 <= P_id_a__length x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__isNatIList_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_a__isNatIList x12 <= P_id_a__isNatIList x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U33_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->P_id_U33 x12 <= P_id_U33 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U11_monotonic :
                  forall x12 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_a__U11 x12 x14 <= P_id_a__U11 x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U81_monotonic :
                  forall x16 x12 x14 x13 x11 x15, 
                   (0 <= x16)/\ (x16 <= x15) ->
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_U81 x12 x14 x16 <= P_id_U81 x11 x13 x15.
                 Proof.
                   intros x16 x15 x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   intros [H_5 H_4].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U92_monotonic :
                  forall x16 x12 x14 x13 x11 x15, 
                   (0 <= x16)/\ (x16 <= x15) ->
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_a__U92 x12 x14 x16 <= P_id_a__U92 x11 x13 x15.
                 Proof.
                   intros x16 x15 x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   intros [H_5 H_4].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U43_monotonic :
                  forall x16 x12 x14 x13 x11 x15, 
                   (0 <= x16)/\ (x16 <= x15) ->
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_a__U43 x12 x14 x16 <= P_id_a__U43 x11 x13 x15.
                 Proof.
                   intros x16 x15 x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   intros [H_5 H_4].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U23_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->P_id_U23 x12 <= P_id_U23 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__isNatKind_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_a__isNatKind x12 <= P_id_a__isNatKind x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U93_monotonic :
                  forall x16 x12 x14 x13 x11 x15, 
                   (0 <= x16)/\ (x16 <= x15) ->
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_U93 x12 x14 x16 <= P_id_U93 x11 x13 x15.
                 Proof.
                   intros x16 x15 x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   intros [H_5 H_4].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U11_monotonic :
                  forall x12 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_U11 x12 x14 <= P_id_U11 x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U71_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_a__U71 x12 <= P_id_a__U71 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U44_monotonic :
                  forall x16 x12 x14 x13 x11 x15, 
                   (0 <= x16)/\ (x16 <= x15) ->
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_U44 x12 x14 x16 <= P_id_U44 x11 x13 x15.
                 Proof.
                   intros x16 x15 x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   intros [H_5 H_4].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U61_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->P_id_U61 x12 <= P_id_U61 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U86_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_a__U86 x12 <= P_id_a__U86 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U41_monotonic :
                  forall x16 x12 x14 x13 x11 x15, 
                   (0 <= x16)/\ (x16 <= x15) ->
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_a__U41 x12 x14 x16 <= P_id_a__U41 x11 x13 x15.
                 Proof.
                   intros x16 x15 x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   intros [H_5 H_4].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U22_monotonic :
                  forall x12 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_U22 x12 x14 <= P_id_U22 x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U21_monotonic :
                  forall x12 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_a__U21 x12 x14 <= P_id_a__U21 x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U91_monotonic :
                  forall x16 x12 x14 x13 x11 x15, 
                   (0 <= x16)/\ (x16 <= x15) ->
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_U91 x12 x14 x16 <= P_id_U91 x11 x13 x15.
                 Proof.
                   intros x16 x15 x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   intros [H_5 H_4].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_length_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_length x12 <= P_id_length x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U52_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_a__U52 x12 <= P_id_a__U52 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U42_monotonic :
                  forall x16 x12 x14 x13 x11 x15, 
                   (0 <= x16)/\ (x16 <= x15) ->
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_U42 x12 x14 x16 <= P_id_U42 x11 x13 x15.
                 Proof.
                   intros x16 x15 x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   intros [H_5 H_4].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U12_monotonic :
                  forall x12 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_a__U12 x12 x14 <= P_id_a__U12 x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U83_monotonic :
                  forall x16 x12 x14 x13 x11 x15, 
                   (0 <= x16)/\ (x16 <= x15) ->
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_U83 x12 x14 x16 <= P_id_U83 x11 x13 x15.
                 Proof.
                   intros x16 x15 x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   intros [H_5 H_4].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U94_monotonic :
                  forall x12 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_a__U94 x12 x14 <= P_id_a__U94 x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U45_monotonic :
                  forall x12 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_a__U45 x12 x14 <= P_id_a__U45 x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U31_monotonic :
                  forall x12 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_U31 x12 x14 <= P_id_U31 x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__isNat_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_a__isNat x12 <= P_id_a__isNat x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_isNatIListKind_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_isNatIListKind x12 <= P_id_isNatIListKind x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U82_monotonic :
                  forall x16 x12 x14 x13 x11 x15, 
                   (0 <= x16)/\ (x16 <= x15) ->
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_a__U82 x12 x14 x16 <= P_id_a__U82 x11 x13 x15.
                 Proof.
                   intros x16 x15 x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   intros [H_5 H_4].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U46_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->P_id_U46 x12 <= P_id_U46 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_cons_monotonic :
                  forall x12 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_cons x12 x14 <= P_id_cons x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U52_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->P_id_U52 x12 <= P_id_U52 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U85_monotonic :
                  forall x12 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_a__U85 x12 x14 <= P_id_a__U85 x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U33_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_a__U33 x12 <= P_id_a__U33 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U21_monotonic :
                  forall x12 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_U21 x12 x14 <= P_id_U21 x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__isNatList_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_a__isNatList x12 <= P_id_a__isNatList x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U86_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->P_id_U86 x12 <= P_id_U86 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_mark_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->P_id_mark x12 <= P_id_mark x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U51_monotonic :
                  forall x12 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_a__U51 x12 x14 <= P_id_a__U51 x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U41_monotonic :
                  forall x16 x12 x14 x13 x11 x15, 
                   (0 <= x16)/\ (x16 <= x15) ->
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_U41 x12 x14 x16 <= P_id_U41 x11 x13 x15.
                 Proof.
                   intros x16 x15 x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   intros [H_5 H_4].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U82_monotonic :
                  forall x16 x12 x14 x13 x11 x15, 
                   (0 <= x16)/\ (x16 <= x15) ->
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_U82 x12 x14 x16 <= P_id_U82 x11 x13 x15.
                 Proof.
                   intros x16 x15 x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   intros [H_5 H_4].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U93_monotonic :
                  forall x16 x12 x14 x13 x11 x15, 
                   (0 <= x16)/\ (x16 <= x15) ->
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_a__U93 x12 x14 x16 <= P_id_a__U93 x11 x13 x15.
                 Proof.
                   intros x16 x15 x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   intros [H_5 H_4].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U44_monotonic :
                  forall x16 x12 x14 x13 x11 x15, 
                   (0 <= x16)/\ (x16 <= x15) ->
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_a__U44 x12 x14 x16 <= P_id_a__U44 x11 x13 x15.
                 Proof.
                   intros x16 x15 x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   intros [H_5 H_4].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_isNat_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_isNat x12 <= P_id_isNat x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U23_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_a__U23 x12 <= P_id_a__U23 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U94_monotonic :
                  forall x12 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_U94 x12 x14 <= P_id_U94 x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U12_monotonic :
                  forall x12 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_U12 x12 x14 <= P_id_U12 x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U81_monotonic :
                  forall x16 x12 x14 x13 x11 x15, 
                   (0 <= x16)/\ (x16 <= x15) ->
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_a__U81 x12 x14 x16 <= P_id_a__U81 x11 x13 x15.
                 Proof.
                   intros x16 x15 x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   intros [H_5 H_4].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U45_monotonic :
                  forall x12 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_U45 x12 x14 <= P_id_U45 x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U71_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->P_id_U71 x12 <= P_id_U71 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U91_monotonic :
                  forall x16 x12 x14 x13 x11 x15, 
                   (0 <= x16)/\ (x16 <= x15) ->
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_a__U91 x12 x14 x16 <= P_id_a__U91 x11 x13 x15.
                 Proof.
                   intros x16 x15 x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   intros [H_5 H_4].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U42_monotonic :
                  forall x16 x12 x14 x13 x11 x15, 
                   (0 <= x16)/\ (x16 <= x15) ->
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_a__U42 x12 x14 x16 <= P_id_a__U42 x11 x13 x15.
                 Proof.
                   intros x16 x15 x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   intros [H_5 H_4].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_isNatKind_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_isNatKind x12 <= P_id_isNatKind x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U22_monotonic :
                  forall x12 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_a__U22 x12 x14 <= P_id_a__U22 x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U92_monotonic :
                  forall x16 x12 x14 x13 x11 x15, 
                   (0 <= x16)/\ (x16 <= x15) ->
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_U92 x12 x14 x16 <= P_id_U92 x11 x13 x15.
                 Proof.
                   intros x16 x15 x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   intros [H_5 H_4].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U61_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_a__U61 x12 <= P_id_a__U61 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U43_monotonic :
                  forall x16 x12 x14 x13 x11 x15, 
                   (0 <= x16)/\ (x16 <= x15) ->
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_U43 x12 x14 x16 <= P_id_U43 x11 x13 x15.
                 Proof.
                   intros x16 x15 x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   intros [H_5 H_4].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__isNatIListKind_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_a__isNatIListKind x12 <= P_id_a__isNatIListKind x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U84_monotonic :
                  forall x16 x12 x14 x13 x11 x15, 
                   (0 <= x16)/\ (x16 <= x15) ->
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_U84 x12 x14 x16 <= P_id_U84 x11 x13 x15.
                 Proof.
                   intros x16 x15 x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   intros [H_5 H_4].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_s_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->P_id_s x12 <= P_id_s x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U46_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_a__U46 x12 <= P_id_a__U46 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U32_monotonic :
                  forall x12 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_U32 x12 x14 <= P_id_U32 x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U31_monotonic :
                  forall x12 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_a__U31 x12 x14 <= P_id_a__U31 x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U13_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->P_id_U13 x12 <= P_id_U13 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U83_monotonic :
                  forall x16 x12 x14 x13 x11 x15, 
                   (0 <= x16)/\ (x16 <= x15) ->
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_a__U83 x12 x14 x16 <= P_id_a__U83 x11 x13 x15.
                 Proof.
                   intros x16 x15 x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   intros [H_5 H_4].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_isNatIList_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_isNatIList x12 <= P_id_isNatIList x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__zeros_bounded : 0 <= P_id_a__zeros .
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U51_bounded :
                  forall x12 x11, 
                   (0 <= x11) ->(0 <= x12) ->0 <= P_id_U51 x12 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U84_bounded :
                  forall x12 x13 x11, 
                   (0 <= x11) ->
                    (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U84 x13 x12 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U32_bounded :
                  forall x12 x11, 
                   (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U32 x12 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_isNatList_bounded :
                  forall x11, (0 <= x11) ->0 <= P_id_isNatList x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U13_bounded :
                  forall x11, (0 <= x11) ->0 <= P_id_a__U13 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U85_bounded :
                  forall x12 x11, 
                   (0 <= x11) ->(0 <= x12) ->0 <= P_id_U85 x12 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__length_bounded :
                  forall x11, (0 <= x11) ->0 <= P_id_a__length x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__isNatIList_bounded :
                  forall x11, (0 <= x11) ->0 <= P_id_a__isNatIList x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U33_bounded :
                  forall x11, (0 <= x11) ->0 <= P_id_U33 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U11_bounded :
                  forall x12 x11, 
                   (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U11 x12 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U81_bounded :
                  forall x12 x13 x11, 
                   (0 <= x11) ->
                    (0 <= x12) ->(0 <= x13) ->0 <= P_id_U81 x13 x12 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U92_bounded :
                  forall x12 x13 x11, 
                   (0 <= x11) ->
                    (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U92 x13 x12 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U43_bounded :
                  forall x12 x13 x11, 
                   (0 <= x11) ->
                    (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U43 x13 x12 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U23_bounded :
                  forall x11, (0 <= x11) ->0 <= P_id_U23 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__isNatKind_bounded :
                  forall x11, (0 <= x11) ->0 <= P_id_a__isNatKind x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U93_bounded :
                  forall x12 x13 x11, 
                   (0 <= x11) ->
                    (0 <= x12) ->(0 <= x13) ->0 <= P_id_U93 x13 x12 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U11_bounded :
                  forall x12 x11, 
                   (0 <= x11) ->(0 <= x12) ->0 <= P_id_U11 x12 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U71_bounded :
                  forall x11, (0 <= x11) ->0 <= P_id_a__U71 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U44_bounded :
                  forall x12 x13 x11, 
                   (0 <= x11) ->
                    (0 <= x12) ->(0 <= x13) ->0 <= P_id_U44 x13 x12 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_0_bounded : 0 <= P_id_0 .
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U61_bounded :
                  forall x11, (0 <= x11) ->0 <= P_id_U61 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U86_bounded :
                  forall x11, (0 <= x11) ->0 <= P_id_a__U86 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U41_bounded :
                  forall x12 x13 x11, 
                   (0 <= x11) ->
                    (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U41 x13 x12 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U22_bounded :
                  forall x12 x11, 
                   (0 <= x11) ->(0 <= x12) ->0 <= P_id_U22 x12 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U21_bounded :
                  forall x12 x11, 
                   (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U21 x12 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U91_bounded :
                  forall x12 x13 x11, 
                   (0 <= x11) ->
                    (0 <= x12) ->(0 <= x13) ->0 <= P_id_U91 x13 x12 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_length_bounded :
                  forall x11, (0 <= x11) ->0 <= P_id_length x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U52_bounded :
                  forall x11, (0 <= x11) ->0 <= P_id_a__U52 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U42_bounded :
                  forall x12 x13 x11, 
                   (0 <= x11) ->
                    (0 <= x12) ->(0 <= x13) ->0 <= P_id_U42 x13 x12 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U12_bounded :
                  forall x12 x11, 
                   (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U12 x12 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U83_bounded :
                  forall x12 x13 x11, 
                   (0 <= x11) ->
                    (0 <= x12) ->(0 <= x13) ->0 <= P_id_U83 x13 x12 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U94_bounded :
                  forall x12 x11, 
                   (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U94 x12 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U45_bounded :
                  forall x12 x11, 
                   (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U45 x12 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U31_bounded :
                  forall x12 x11, 
                   (0 <= x11) ->(0 <= x12) ->0 <= P_id_U31 x12 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__isNat_bounded :
                  forall x11, (0 <= x11) ->0 <= P_id_a__isNat x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_isNatIListKind_bounded :
                  forall x11, (0 <= x11) ->0 <= P_id_isNatIListKind x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U82_bounded :
                  forall x12 x13 x11, 
                   (0 <= x11) ->
                    (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U82 x13 x12 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U46_bounded :
                  forall x11, (0 <= x11) ->0 <= P_id_U46 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_cons_bounded :
                  forall x12 x11, 
                   (0 <= x11) ->(0 <= x12) ->0 <= P_id_cons x12 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U52_bounded :
                  forall x11, (0 <= x11) ->0 <= P_id_U52 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U85_bounded :
                  forall x12 x11, 
                   (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U85 x12 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U33_bounded :
                  forall x11, (0 <= x11) ->0 <= P_id_a__U33 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U21_bounded :
                  forall x12 x11, 
                   (0 <= x11) ->(0 <= x12) ->0 <= P_id_U21 x12 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__isNatList_bounded :
                  forall x11, (0 <= x11) ->0 <= P_id_a__isNatList x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U86_bounded :
                  forall x11, (0 <= x11) ->0 <= P_id_U86 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_mark_bounded :
                  forall x11, (0 <= x11) ->0 <= P_id_mark x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U51_bounded :
                  forall x12 x11, 
                   (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U51 x12 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U41_bounded :
                  forall x12 x13 x11, 
                   (0 <= x11) ->
                    (0 <= x12) ->(0 <= x13) ->0 <= P_id_U41 x13 x12 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_tt_bounded : 0 <= P_id_tt .
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U82_bounded :
                  forall x12 x13 x11, 
                   (0 <= x11) ->
                    (0 <= x12) ->(0 <= x13) ->0 <= P_id_U82 x13 x12 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U93_bounded :
                  forall x12 x13 x11, 
                   (0 <= x11) ->
                    (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U93 x13 x12 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U44_bounded :
                  forall x12 x13 x11, 
                   (0 <= x11) ->
                    (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U44 x13 x12 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_isNat_bounded :
                  forall x11, (0 <= x11) ->0 <= P_id_isNat x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U23_bounded :
                  forall x11, (0 <= x11) ->0 <= P_id_a__U23 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U94_bounded :
                  forall x12 x11, 
                   (0 <= x11) ->(0 <= x12) ->0 <= P_id_U94 x12 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U12_bounded :
                  forall x12 x11, 
                   (0 <= x11) ->(0 <= x12) ->0 <= P_id_U12 x12 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U81_bounded :
                  forall x12 x13 x11, 
                   (0 <= x11) ->
                    (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U81 x13 x12 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U45_bounded :
                  forall x12 x11, 
                   (0 <= x11) ->(0 <= x12) ->0 <= P_id_U45 x12 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_zeros_bounded : 0 <= P_id_zeros .
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U71_bounded :
                  forall x11, (0 <= x11) ->0 <= P_id_U71 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U91_bounded :
                  forall x12 x13 x11, 
                   (0 <= x11) ->
                    (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U91 x13 x12 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U42_bounded :
                  forall x12 x13 x11, 
                   (0 <= x11) ->
                    (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U42 x13 x12 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_isNatKind_bounded :
                  forall x11, (0 <= x11) ->0 <= P_id_isNatKind x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U22_bounded :
                  forall x12 x11, 
                   (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U22 x12 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U92_bounded :
                  forall x12 x13 x11, 
                   (0 <= x11) ->
                    (0 <= x12) ->(0 <= x13) ->0 <= P_id_U92 x13 x12 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_nil_bounded : 0 <= P_id_nil .
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U61_bounded :
                  forall x11, (0 <= x11) ->0 <= P_id_a__U61 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U43_bounded :
                  forall x12 x13 x11, 
                   (0 <= x11) ->
                    (0 <= x12) ->(0 <= x13) ->0 <= P_id_U43 x13 x12 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__isNatIListKind_bounded :
                  forall x11, (0 <= x11) ->0 <= P_id_a__isNatIListKind x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U84_bounded :
                  forall x12 x13 x11, 
                   (0 <= x11) ->
                    (0 <= x12) ->(0 <= x13) ->0 <= P_id_U84 x13 x12 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_s_bounded :
                  forall x11, (0 <= x11) ->0 <= P_id_s x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U46_bounded :
                  forall x11, (0 <= x11) ->0 <= P_id_a__U46 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U32_bounded :
                  forall x12 x11, 
                   (0 <= x11) ->(0 <= x12) ->0 <= P_id_U32 x12 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U31_bounded :
                  forall x12 x11, 
                   (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U31 x12 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U13_bounded :
                  forall x11, (0 <= x11) ->0 <= P_id_U13 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_a__U83_bounded :
                  forall x12 x13 x11, 
                   (0 <= x11) ->
                    (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U83 x13 x12 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_isNatIList_bounded :
                  forall x11, (0 <= x11) ->0 <= P_id_isNatIList x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Definition measure  := 
                   InterpZ.measure 0 P_id_a__zeros P_id_U51 P_id_a__U84 
                    P_id_a__U32 P_id_isNatList P_id_a__U13 P_id_U85 
                    P_id_a__length P_id_a__isNatIList P_id_U33 P_id_a__U11 
                    P_id_U81 P_id_a__U92 P_id_a__U43 P_id_U23 
                    P_id_a__isNatKind P_id_U93 P_id_U11 P_id_a__U71 P_id_U44 
                    P_id_0 P_id_U61 P_id_a__U86 P_id_a__U41 P_id_U22 
                    P_id_a__U21 P_id_U91 P_id_length P_id_a__U52 P_id_U42 
                    P_id_a__U12 P_id_U83 P_id_a__U94 P_id_a__U45 P_id_U31 
                    P_id_a__isNat P_id_isNatIListKind P_id_a__U82 P_id_U46 
                    P_id_cons P_id_U52 P_id_a__U85 P_id_a__U33 P_id_U21 
                    P_id_a__isNatList P_id_U86 P_id_mark P_id_a__U51 
                    P_id_U41 P_id_tt P_id_U82 P_id_a__U93 P_id_a__U44 
                    P_id_isNat P_id_a__U23 P_id_U94 P_id_U12 P_id_a__U81 
                    P_id_U45 P_id_zeros P_id_U71 P_id_a__U91 P_id_a__U42 
                    P_id_isNatKind P_id_a__U22 P_id_U92 P_id_nil P_id_a__U61 
                    P_id_U43 P_id_a__isNatIListKind P_id_U84 P_id_s 
                    P_id_a__U46 P_id_U32 P_id_a__U31 P_id_U13 P_id_a__U83 
                    P_id_isNatIList.
                 
                 Lemma measure_equation :
                  forall t, 
                   measure t = match t with
                                 | (algebra.Alg.Term algebra.F.id_a__zeros 
                                    nil) =>
                                  P_id_a__zeros 
                                 | (algebra.Alg.Term algebra.F.id_U51 (x12::
                                    x11::nil)) =>
                                  P_id_U51 (measure x12) (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U84 
                                    (x13::x12::x11::nil)) =>
                                  P_id_a__U84 (measure x13) (measure x12) 
                                   (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U32 
                                    (x12::x11::nil)) =>
                                  P_id_a__U32 (measure x12) (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_isNatList 
                                    (x11::nil)) =>
                                  P_id_isNatList (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U13 
                                    (x11::nil)) =>
                                  P_id_a__U13 (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_U85 (x12::
                                    x11::nil)) =>
                                  P_id_U85 (measure x12) (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__length 
                                    (x11::nil)) =>
                                  P_id_a__length (measure x11)
                                 | (algebra.Alg.Term 
                                    algebra.F.id_a__isNatIList (x11::nil)) =>
                                  P_id_a__isNatIList (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_U33 
                                    (x11::nil)) =>
                                  P_id_U33 (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U11 
                                    (x12::x11::nil)) =>
                                  P_id_a__U11 (measure x12) (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_U81 (x13::
                                    x12::x11::nil)) =>
                                  P_id_U81 (measure x13) (measure x12) 
                                   (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U92 
                                    (x13::x12::x11::nil)) =>
                                  P_id_a__U92 (measure x13) (measure x12) 
                                   (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U43 
                                    (x13::x12::x11::nil)) =>
                                  P_id_a__U43 (measure x13) (measure x12) 
                                   (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_U23 
                                    (x11::nil)) =>
                                  P_id_U23 (measure x11)
                                 | (algebra.Alg.Term 
                                    algebra.F.id_a__isNatKind (x11::nil)) =>
                                  P_id_a__isNatKind (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_U93 (x13::
                                    x12::x11::nil)) =>
                                  P_id_U93 (measure x13) (measure x12) 
                                   (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_U11 (x12::
                                    x11::nil)) =>
                                  P_id_U11 (measure x12) (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U71 
                                    (x11::nil)) =>
                                  P_id_a__U71 (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_U44 (x13::
                                    x12::x11::nil)) =>
                                  P_id_U44 (measure x13) (measure x12) 
                                   (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_0 nil) =>
                                  P_id_0 
                                 | (algebra.Alg.Term algebra.F.id_U61 
                                    (x11::nil)) =>
                                  P_id_U61 (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U86 
                                    (x11::nil)) =>
                                  P_id_a__U86 (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U41 
                                    (x13::x12::x11::nil)) =>
                                  P_id_a__U41 (measure x13) (measure x12) 
                                   (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_U22 (x12::
                                    x11::nil)) =>
                                  P_id_U22 (measure x12) (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U21 
                                    (x12::x11::nil)) =>
                                  P_id_a__U21 (measure x12) (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_U91 (x13::
                                    x12::x11::nil)) =>
                                  P_id_U91 (measure x13) (measure x12) 
                                   (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_length 
                                    (x11::nil)) =>
                                  P_id_length (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U52 
                                    (x11::nil)) =>
                                  P_id_a__U52 (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_U42 (x13::
                                    x12::x11::nil)) =>
                                  P_id_U42 (measure x13) (measure x12) 
                                   (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U12 
                                    (x12::x11::nil)) =>
                                  P_id_a__U12 (measure x12) (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_U83 (x13::
                                    x12::x11::nil)) =>
                                  P_id_U83 (measure x13) (measure x12) 
                                   (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U94 
                                    (x12::x11::nil)) =>
                                  P_id_a__U94 (measure x12) (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U45 
                                    (x12::x11::nil)) =>
                                  P_id_a__U45 (measure x12) (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_U31 (x12::
                                    x11::nil)) =>
                                  P_id_U31 (measure x12) (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__isNat 
                                    (x11::nil)) =>
                                  P_id_a__isNat (measure x11)
                                 | (algebra.Alg.Term 
                                    algebra.F.id_isNatIListKind (x11::nil)) =>
                                  P_id_isNatIListKind (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U82 
                                    (x13::x12::x11::nil)) =>
                                  P_id_a__U82 (measure x13) (measure x12) 
                                   (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_U46 
                                    (x11::nil)) =>
                                  P_id_U46 (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_cons (x12::
                                    x11::nil)) =>
                                  P_id_cons (measure x12) (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_U52 
                                    (x11::nil)) =>
                                  P_id_U52 (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U85 
                                    (x12::x11::nil)) =>
                                  P_id_a__U85 (measure x12) (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U33 
                                    (x11::nil)) =>
                                  P_id_a__U33 (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_U21 (x12::
                                    x11::nil)) =>
                                  P_id_U21 (measure x12) (measure x11)
                                 | (algebra.Alg.Term 
                                    algebra.F.id_a__isNatList (x11::nil)) =>
                                  P_id_a__isNatList (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_U86 
                                    (x11::nil)) =>
                                  P_id_U86 (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_mark 
                                    (x11::nil)) =>
                                  P_id_mark (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U51 
                                    (x12::x11::nil)) =>
                                  P_id_a__U51 (measure x12) (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_U41 (x13::
                                    x12::x11::nil)) =>
                                  P_id_U41 (measure x13) (measure x12) 
                                   (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_tt nil) =>
                                  P_id_tt 
                                 | (algebra.Alg.Term algebra.F.id_U82 (x13::
                                    x12::x11::nil)) =>
                                  P_id_U82 (measure x13) (measure x12) 
                                   (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U93 
                                    (x13::x12::x11::nil)) =>
                                  P_id_a__U93 (measure x13) (measure x12) 
                                   (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U44 
                                    (x13::x12::x11::nil)) =>
                                  P_id_a__U44 (measure x13) (measure x12) 
                                   (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_isNat 
                                    (x11::nil)) =>
                                  P_id_isNat (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U23 
                                    (x11::nil)) =>
                                  P_id_a__U23 (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_U94 (x12::
                                    x11::nil)) =>
                                  P_id_U94 (measure x12) (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_U12 (x12::
                                    x11::nil)) =>
                                  P_id_U12 (measure x12) (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U81 
                                    (x13::x12::x11::nil)) =>
                                  P_id_a__U81 (measure x13) (measure x12) 
                                   (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_U45 (x12::
                                    x11::nil)) =>
                                  P_id_U45 (measure x12) (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_zeros nil) =>
                                  P_id_zeros 
                                 | (algebra.Alg.Term algebra.F.id_U71 
                                    (x11::nil)) =>
                                  P_id_U71 (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U91 
                                    (x13::x12::x11::nil)) =>
                                  P_id_a__U91 (measure x13) (measure x12) 
                                   (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U42 
                                    (x13::x12::x11::nil)) =>
                                  P_id_a__U42 (measure x13) (measure x12) 
                                   (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_isNatKind 
                                    (x11::nil)) =>
                                  P_id_isNatKind (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U22 
                                    (x12::x11::nil)) =>
                                  P_id_a__U22 (measure x12) (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_U92 (x13::
                                    x12::x11::nil)) =>
                                  P_id_U92 (measure x13) (measure x12) 
                                   (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_nil nil) =>
                                  P_id_nil 
                                 | (algebra.Alg.Term algebra.F.id_a__U61 
                                    (x11::nil)) =>
                                  P_id_a__U61 (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_U43 (x13::
                                    x12::x11::nil)) =>
                                  P_id_U43 (measure x13) (measure x12) 
                                   (measure x11)
                                 | (algebra.Alg.Term 
                                    algebra.F.id_a__isNatIListKind 
                                    (x11::nil)) =>
                                  P_id_a__isNatIListKind (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_U84 (x13::
                                    x12::x11::nil)) =>
                                  P_id_U84 (measure x13) (measure x12) 
                                   (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_s 
                                    (x11::nil)) =>
                                  P_id_s (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U46 
                                    (x11::nil)) =>
                                  P_id_a__U46 (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_U32 (x12::
                                    x11::nil)) =>
                                  P_id_U32 (measure x12) (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U31 
                                    (x12::x11::nil)) =>
                                  P_id_a__U31 (measure x12) (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_U13 
                                    (x11::nil)) =>
                                  P_id_U13 (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U83 
                                    (x13::x12::x11::nil)) =>
                                  P_id_a__U83 (measure x13) (measure x12) 
                                   (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_isNatIList 
                                    (x11::nil)) =>
                                  P_id_isNatIList (measure x11)
                                 | _ => 0
                                 end.
                 Proof.
                   intros t;case t;intros ;apply refl_equal.
                 Qed.
                 
                 Lemma measure_bounded : forall t, 0 <= measure t.
                 Proof.
                   unfold measure in |-*.
                   
                   apply InterpZ.measure_bounded;
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Ltac generate_pos_hyp  :=
                  match goal with
                    | H:context [measure ?x] |- _ =>
                     let v := fresh "v" in 
                      (let H := fresh "h" in 
                        (set (H:=measure_bounded x) in *;
                          set (v:=measure x) in *;clearbody H;clearbody v))
                    |  |- context [measure ?x] =>
                     let v := fresh "v" in 
                      (let H := fresh "h" in 
                        (set (H:=measure_bounded x) in *;
                          set (v:=measure x) in *;clearbody H;clearbody v))
                    end
                  .
                 
                 Lemma rules_monotonic :
                  forall l r, 
                   (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) ->
                    measure r <= measure l.
                 Proof.
                   intros l r H.
                   fold measure in |-*.
                   
                   inversion H;clear H;subst;inversion H0;clear H0;subst;
                    simpl algebra.EQT.T.apply_subst in |-*;
                    repeat (
                    match goal with
                      |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
                       rewrite (measure_equation (algebra.Alg.Term f t))
                      end
                    );repeat (generate_pos_hyp );
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma measure_star_monotonic :
                  forall l r, 
                   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                              r l) ->measure r <= measure l.
                 Proof.
                   unfold measure in *.
                   apply InterpZ.measure_star_monotonic.
                   intros ;apply P_id_U51_monotonic;assumption.
                   intros ;apply P_id_a__U84_monotonic;assumption.
                   intros ;apply P_id_a__U32_monotonic;assumption.
                   intros ;apply P_id_isNatList_monotonic;assumption.
                   intros ;apply P_id_a__U13_monotonic;assumption.
                   intros ;apply P_id_U85_monotonic;assumption.
                   intros ;apply P_id_a__length_monotonic;assumption.
                   intros ;apply P_id_a__isNatIList_monotonic;assumption.
                   intros ;apply P_id_U33_monotonic;assumption.
                   intros ;apply P_id_a__U11_monotonic;assumption.
                   intros ;apply P_id_U81_monotonic;assumption.
                   intros ;apply P_id_a__U92_monotonic;assumption.
                   intros ;apply P_id_a__U43_monotonic;assumption.
                   intros ;apply P_id_U23_monotonic;assumption.
                   intros ;apply P_id_a__isNatKind_monotonic;assumption.
                   intros ;apply P_id_U93_monotonic;assumption.
                   intros ;apply P_id_U11_monotonic;assumption.
                   intros ;apply P_id_a__U71_monotonic;assumption.
                   intros ;apply P_id_U44_monotonic;assumption.
                   intros ;apply P_id_U61_monotonic;assumption.
                   intros ;apply P_id_a__U86_monotonic;assumption.
                   intros ;apply P_id_a__U41_monotonic;assumption.
                   intros ;apply P_id_U22_monotonic;assumption.
                   intros ;apply P_id_a__U21_monotonic;assumption.
                   intros ;apply P_id_U91_monotonic;assumption.
                   intros ;apply P_id_length_monotonic;assumption.
                   intros ;apply P_id_a__U52_monotonic;assumption.
                   intros ;apply P_id_U42_monotonic;assumption.
                   intros ;apply P_id_a__U12_monotonic;assumption.
                   intros ;apply P_id_U83_monotonic;assumption.
                   intros ;apply P_id_a__U94_monotonic;assumption.
                   intros ;apply P_id_a__U45_monotonic;assumption.
                   intros ;apply P_id_U31_monotonic;assumption.
                   intros ;apply P_id_a__isNat_monotonic;assumption.
                   intros ;apply P_id_isNatIListKind_monotonic;assumption.
                   intros ;apply P_id_a__U82_monotonic;assumption.
                   intros ;apply P_id_U46_monotonic;assumption.
                   intros ;apply P_id_cons_monotonic;assumption.
                   intros ;apply P_id_U52_monotonic;assumption.
                   intros ;apply P_id_a__U85_monotonic;assumption.
                   intros ;apply P_id_a__U33_monotonic;assumption.
                   intros ;apply P_id_U21_monotonic;assumption.
                   intros ;apply P_id_a__isNatList_monotonic;assumption.
                   intros ;apply P_id_U86_monotonic;assumption.
                   intros ;apply P_id_mark_monotonic;assumption.
                   intros ;apply P_id_a__U51_monotonic;assumption.
                   intros ;apply P_id_U41_monotonic;assumption.
                   intros ;apply P_id_U82_monotonic;assumption.
                   intros ;apply P_id_a__U93_monotonic;assumption.
                   intros ;apply P_id_a__U44_monotonic;assumption.
                   intros ;apply P_id_isNat_monotonic;assumption.
                   intros ;apply P_id_a__U23_monotonic;assumption.
                   intros ;apply P_id_U94_monotonic;assumption.
                   intros ;apply P_id_U12_monotonic;assumption.
                   intros ;apply P_id_a__U81_monotonic;assumption.
                   intros ;apply P_id_U45_monotonic;assumption.
                   intros ;apply P_id_U71_monotonic;assumption.
                   intros ;apply P_id_a__U91_monotonic;assumption.
                   intros ;apply P_id_a__U42_monotonic;assumption.
                   intros ;apply P_id_isNatKind_monotonic;assumption.
                   intros ;apply P_id_a__U22_monotonic;assumption.
                   intros ;apply P_id_U92_monotonic;assumption.
                   intros ;apply P_id_a__U61_monotonic;assumption.
                   intros ;apply P_id_U43_monotonic;assumption.
                   intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
                   intros ;apply P_id_U84_monotonic;assumption.
                   intros ;apply P_id_s_monotonic;assumption.
                   intros ;apply P_id_a__U46_monotonic;assumption.
                   intros ;apply P_id_U32_monotonic;assumption.
                   intros ;apply P_id_a__U31_monotonic;assumption.
                   intros ;apply P_id_U13_monotonic;assumption.
                   intros ;apply P_id_a__U83_monotonic;assumption.
                   intros ;apply P_id_isNatIList_monotonic;assumption.
                   intros ;apply P_id_a__zeros_bounded;assumption.
                   intros ;apply P_id_U51_bounded;assumption.
                   intros ;apply P_id_a__U84_bounded;assumption.
                   intros ;apply P_id_a__U32_bounded;assumption.
                   intros ;apply P_id_isNatList_bounded;assumption.
                   intros ;apply P_id_a__U13_bounded;assumption.
                   intros ;apply P_id_U85_bounded;assumption.
                   intros ;apply P_id_a__length_bounded;assumption.
                   intros ;apply P_id_a__isNatIList_bounded;assumption.
                   intros ;apply P_id_U33_bounded;assumption.
                   intros ;apply P_id_a__U11_bounded;assumption.
                   intros ;apply P_id_U81_bounded;assumption.
                   intros ;apply P_id_a__U92_bounded;assumption.
                   intros ;apply P_id_a__U43_bounded;assumption.
                   intros ;apply P_id_U23_bounded;assumption.
                   intros ;apply P_id_a__isNatKind_bounded;assumption.
                   intros ;apply P_id_U93_bounded;assumption.
                   intros ;apply P_id_U11_bounded;assumption.
                   intros ;apply P_id_a__U71_bounded;assumption.
                   intros ;apply P_id_U44_bounded;assumption.
                   intros ;apply P_id_0_bounded;assumption.
                   intros ;apply P_id_U61_bounded;assumption.
                   intros ;apply P_id_a__U86_bounded;assumption.
                   intros ;apply P_id_a__U41_bounded;assumption.
                   intros ;apply P_id_U22_bounded;assumption.
                   intros ;apply P_id_a__U21_bounded;assumption.
                   intros ;apply P_id_U91_bounded;assumption.
                   intros ;apply P_id_length_bounded;assumption.
                   intros ;apply P_id_a__U52_bounded;assumption.
                   intros ;apply P_id_U42_bounded;assumption.
                   intros ;apply P_id_a__U12_bounded;assumption.
                   intros ;apply P_id_U83_bounded;assumption.
                   intros ;apply P_id_a__U94_bounded;assumption.
                   intros ;apply P_id_a__U45_bounded;assumption.
                   intros ;apply P_id_U31_bounded;assumption.
                   intros ;apply P_id_a__isNat_bounded;assumption.
                   intros ;apply P_id_isNatIListKind_bounded;assumption.
                   intros ;apply P_id_a__U82_bounded;assumption.
                   intros ;apply P_id_U46_bounded;assumption.
                   intros ;apply P_id_cons_bounded;assumption.
                   intros ;apply P_id_U52_bounded;assumption.
                   intros ;apply P_id_a__U85_bounded;assumption.
                   intros ;apply P_id_a__U33_bounded;assumption.
                   intros ;apply P_id_U21_bounded;assumption.
                   intros ;apply P_id_a__isNatList_bounded;assumption.
                   intros ;apply P_id_U86_bounded;assumption.
                   intros ;apply P_id_mark_bounded;assumption.
                   intros ;apply P_id_a__U51_bounded;assumption.
                   intros ;apply P_id_U41_bounded;assumption.
                   intros ;apply P_id_tt_bounded;assumption.
                   intros ;apply P_id_U82_bounded;assumption.
                   intros ;apply P_id_a__U93_bounded;assumption.
                   intros ;apply P_id_a__U44_bounded;assumption.
                   intros ;apply P_id_isNat_bounded;assumption.
                   intros ;apply P_id_a__U23_bounded;assumption.
                   intros ;apply P_id_U94_bounded;assumption.
                   intros ;apply P_id_U12_bounded;assumption.
                   intros ;apply P_id_a__U81_bounded;assumption.
                   intros ;apply P_id_U45_bounded;assumption.
                   intros ;apply P_id_zeros_bounded;assumption.
                   intros ;apply P_id_U71_bounded;assumption.
                   intros ;apply P_id_a__U91_bounded;assumption.
                   intros ;apply P_id_a__U42_bounded;assumption.
                   intros ;apply P_id_isNatKind_bounded;assumption.
                   intros ;apply P_id_a__U22_bounded;assumption.
                   intros ;apply P_id_U92_bounded;assumption.
                   intros ;apply P_id_nil_bounded;assumption.
                   intros ;apply P_id_a__U61_bounded;assumption.
                   intros ;apply P_id_U43_bounded;assumption.
                   intros ;apply P_id_a__isNatIListKind_bounded;assumption.
                   intros ;apply P_id_U84_bounded;assumption.
                   intros ;apply P_id_s_bounded;assumption.
                   intros ;apply P_id_a__U46_bounded;assumption.
                   intros ;apply P_id_U32_bounded;assumption.
                   intros ;apply P_id_a__U31_bounded;assumption.
                   intros ;apply P_id_U13_bounded;assumption.
                   intros ;apply P_id_a__U83_bounded;assumption.
                   intros ;apply P_id_isNatIList_bounded;assumption.
                   apply rules_monotonic.
                 Qed.
                 
                 Definition P_id_A__U46 (x11:Z) := 0.
                 
                 Definition P_id_A__ISNATILISTKIND (x11:Z) := 0.
                 
                 Definition P_id_A__U61 (x11:Z) := 0.
                 
                 Definition P_id_A__U85 (x11:Z) (x12:Z) := 0.
                 
                 Definition P_id_A__U31 (x11:Z) (x12:Z) := 0.
                 
                 Definition P_id_A__U81 (x11:Z) (x12:Z) (x13:Z) := 0.
                 
                 Definition P_id_A__U22 (x11:Z) (x12:Z) := 0.
                 
                 Definition P_id_A__U93 (x11:Z) (x12:Z) (x13:Z) := 0.
                 
                 Definition P_id_A__U42 (x11:Z) (x12:Z) (x13:Z) := 0.
                 
                 Definition P_id_A__U51 (x11:Z) (x12:Z) := 0.
                 
                 Definition P_id_A__ISNATLIST (x11:Z) := 0.
                 
                 Definition P_id_A__ZEROS  := 0.
                 
                 Definition P_id_A__U91 (x11:Z) (x12:Z) (x13:Z) := 0.
                 
                 Definition P_id_A__U33 (x11:Z) := 0.
                 
                 Definition P_id_A__U83 (x11:Z) (x12:Z) (x13:Z) := 0.
                 
                 Definition P_id_A__U23 (x11:Z) := 0.
                 
                 Definition P_id_A__U11 (x11:Z) (x12:Z) := 0.
                 
                 Definition P_id_A__LENGTH (x11:Z) := 0.
                 
                 Definition P_id_A__U44 (x11:Z) (x12:Z) (x13:Z) := 0.
                 
                 Definition P_id_A__ISNATILIST (x11:Z) := 0.
                 
                 Definition P_id_A__U13 (x11:Z) := 0.
                 
                 Definition P_id_A__U71 (x11:Z) := 0.
                 
                 Definition P_id_A__U86 (x11:Z) := 0.
                 
                 Definition P_id_A__U32 (x11:Z) (x12:Z) := 0.
                 
                 Definition P_id_A__U82 (x11:Z) (x12:Z) (x13:Z) := 0.
                 
                 Definition P_id_A__ISNATKIND (x11:Z) := 0.
                 
                 Definition P_id_A__U94 (x11:Z) (x12:Z) := 0.
                 
                 Definition P_id_A__U43 (x11:Z) (x12:Z) (x13:Z) := 0.
                 
                 Definition P_id_A__U52 (x11:Z) := 0.
                 
                 Definition P_id_A__U21 (x11:Z) (x12:Z) := 0.
                 
                 Definition P_id_A__U92 (x11:Z) (x12:Z) (x13:Z) := 0.
                 
                 Definition P_id_A__U41 (x11:Z) (x12:Z) (x13:Z) := 0.
                 
                 Definition P_id_A__U84 (x11:Z) (x12:Z) (x13:Z) := 0.
                 
                 Definition P_id_A__ISNAT (x11:Z) := 0.
                 
                 Definition P_id_A__U12 (x11:Z) (x12:Z) := 0.
                 
                 Definition P_id_MARK (x11:Z) := 1* x11.
                 
                 Definition P_id_A__U45 (x11:Z) (x12:Z) := 0.
                 
                 Lemma P_id_A__U46_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_A__U46 x12 <= P_id_A__U46 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_A__ISNATILISTKIND_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_A__ISNATILISTKIND x12 <= P_id_A__ISNATILISTKIND x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_A__U61_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_A__U61 x12 <= P_id_A__U61 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_A__U85_monotonic :
                  forall x12 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_A__U85 x12 x14 <= P_id_A__U85 x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_A__U31_monotonic :
                  forall x12 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_A__U31 x12 x14 <= P_id_A__U31 x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_A__U81_monotonic :
                  forall x16 x12 x14 x13 x11 x15, 
                   (0 <= x16)/\ (x16 <= x15) ->
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_A__U81 x12 x14 x16 <= P_id_A__U81 x11 x13 x15.
                 Proof.
                   intros x16 x15 x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   intros [H_5 H_4].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_A__U22_monotonic :
                  forall x12 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_A__U22 x12 x14 <= P_id_A__U22 x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_A__U93_monotonic :
                  forall x16 x12 x14 x13 x11 x15, 
                   (0 <= x16)/\ (x16 <= x15) ->
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_A__U93 x12 x14 x16 <= P_id_A__U93 x11 x13 x15.
                 Proof.
                   intros x16 x15 x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   intros [H_5 H_4].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_A__U42_monotonic :
                  forall x16 x12 x14 x13 x11 x15, 
                   (0 <= x16)/\ (x16 <= x15) ->
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_A__U42 x12 x14 x16 <= P_id_A__U42 x11 x13 x15.
                 Proof.
                   intros x16 x15 x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   intros [H_5 H_4].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_A__U51_monotonic :
                  forall x12 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_A__U51 x12 x14 <= P_id_A__U51 x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_A__ISNATLIST_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_A__ISNATLIST x12 <= P_id_A__ISNATLIST x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_A__U91_monotonic :
                  forall x16 x12 x14 x13 x11 x15, 
                   (0 <= x16)/\ (x16 <= x15) ->
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_A__U91 x12 x14 x16 <= P_id_A__U91 x11 x13 x15.
                 Proof.
                   intros x16 x15 x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   intros [H_5 H_4].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_A__U33_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_A__U33 x12 <= P_id_A__U33 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_A__U83_monotonic :
                  forall x16 x12 x14 x13 x11 x15, 
                   (0 <= x16)/\ (x16 <= x15) ->
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_A__U83 x12 x14 x16 <= P_id_A__U83 x11 x13 x15.
                 Proof.
                   intros x16 x15 x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   intros [H_5 H_4].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_A__U23_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_A__U23 x12 <= P_id_A__U23 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_A__U11_monotonic :
                  forall x12 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_A__U11 x12 x14 <= P_id_A__U11 x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_A__LENGTH_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_A__LENGTH x12 <= P_id_A__LENGTH x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_A__U44_monotonic :
                  forall x16 x12 x14 x13 x11 x15, 
                   (0 <= x16)/\ (x16 <= x15) ->
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_A__U44 x12 x14 x16 <= P_id_A__U44 x11 x13 x15.
                 Proof.
                   intros x16 x15 x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   intros [H_5 H_4].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_A__ISNATILIST_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_A__ISNATILIST x12 <= P_id_A__ISNATILIST x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_A__U13_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_A__U13 x12 <= P_id_A__U13 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_A__U71_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_A__U71 x12 <= P_id_A__U71 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_A__U86_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_A__U86 x12 <= P_id_A__U86 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_A__U32_monotonic :
                  forall x12 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_A__U32 x12 x14 <= P_id_A__U32 x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_A__U82_monotonic :
                  forall x16 x12 x14 x13 x11 x15, 
                   (0 <= x16)/\ (x16 <= x15) ->
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_A__U82 x12 x14 x16 <= P_id_A__U82 x11 x13 x15.
                 Proof.
                   intros x16 x15 x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   intros [H_5 H_4].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_A__ISNATKIND_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_A__ISNATKIND x12 <= P_id_A__ISNATKIND x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_A__U94_monotonic :
                  forall x12 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_A__U94 x12 x14 <= P_id_A__U94 x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_A__U43_monotonic :
                  forall x16 x12 x14 x13 x11 x15, 
                   (0 <= x16)/\ (x16 <= x15) ->
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_A__U43 x12 x14 x16 <= P_id_A__U43 x11 x13 x15.
                 Proof.
                   intros x16 x15 x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   intros [H_5 H_4].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_A__U52_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_A__U52 x12 <= P_id_A__U52 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_A__U21_monotonic :
                  forall x12 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_A__U21 x12 x14 <= P_id_A__U21 x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_A__U92_monotonic :
                  forall x16 x12 x14 x13 x11 x15, 
                   (0 <= x16)/\ (x16 <= x15) ->
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_A__U92 x12 x14 x16 <= P_id_A__U92 x11 x13 x15.
                 Proof.
                   intros x16 x15 x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   intros [H_5 H_4].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_A__U41_monotonic :
                  forall x16 x12 x14 x13 x11 x15, 
                   (0 <= x16)/\ (x16 <= x15) ->
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_A__U41 x12 x14 x16 <= P_id_A__U41 x11 x13 x15.
                 Proof.
                   intros x16 x15 x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   intros [H_5 H_4].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_A__U84_monotonic :
                  forall x16 x12 x14 x13 x11 x15, 
                   (0 <= x16)/\ (x16 <= x15) ->
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_A__U84 x12 x14 x16 <= P_id_A__U84 x11 x13 x15.
                 Proof.
                   intros x16 x15 x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   intros [H_5 H_4].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_A__ISNAT_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_A__ISNAT x12 <= P_id_A__ISNAT x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_A__U12_monotonic :
                  forall x12 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_A__U12 x12 x14 <= P_id_A__U12 x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_MARK_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->P_id_MARK x12 <= P_id_MARK x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_A__U45_monotonic :
                  forall x12 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_A__U45 x12 x14 <= P_id_A__U45 x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Definition marked_measure  := 
                   InterpZ.marked_measure 0 P_id_a__zeros P_id_U51 
                    P_id_a__U84 P_id_a__U32 P_id_isNatList P_id_a__U13 
                    P_id_U85 P_id_a__length P_id_a__isNatIList P_id_U33 
                    P_id_a__U11 P_id_U81 P_id_a__U92 P_id_a__U43 P_id_U23 
                    P_id_a__isNatKind P_id_U93 P_id_U11 P_id_a__U71 P_id_U44 
                    P_id_0 P_id_U61 P_id_a__U86 P_id_a__U41 P_id_U22 
                    P_id_a__U21 P_id_U91 P_id_length P_id_a__U52 P_id_U42 
                    P_id_a__U12 P_id_U83 P_id_a__U94 P_id_a__U45 P_id_U31 
                    P_id_a__isNat P_id_isNatIListKind P_id_a__U82 P_id_U46 
                    P_id_cons P_id_U52 P_id_a__U85 P_id_a__U33 P_id_U21 
                    P_id_a__isNatList P_id_U86 P_id_mark P_id_a__U51 
                    P_id_U41 P_id_tt P_id_U82 P_id_a__U93 P_id_a__U44 
                    P_id_isNat P_id_a__U23 P_id_U94 P_id_U12 P_id_a__U81 
                    P_id_U45 P_id_zeros P_id_U71 P_id_a__U91 P_id_a__U42 
                    P_id_isNatKind P_id_a__U22 P_id_U92 P_id_nil P_id_a__U61 
                    P_id_U43 P_id_a__isNatIListKind P_id_U84 P_id_s 
                    P_id_a__U46 P_id_U32 P_id_a__U31 P_id_U13 P_id_a__U83 
                    P_id_isNatIList P_id_A__U46 P_id_A__ISNATILISTKIND 
                    P_id_A__U61 P_id_A__U85 P_id_A__U31 P_id_A__U81 
                    P_id_A__U22 P_id_A__U93 P_id_A__U42 P_id_A__U51 
                    P_id_A__ISNATLIST P_id_A__ZEROS P_id_A__U91 P_id_A__U33 
                    P_id_A__U83 P_id_A__U23 P_id_A__U11 P_id_A__LENGTH 
                    P_id_A__U44 P_id_A__ISNATILIST P_id_A__U13 P_id_A__U71 
                    P_id_A__U86 P_id_A__U32 P_id_A__U82 P_id_A__ISNATKIND 
                    P_id_A__U94 P_id_A__U43 P_id_A__U52 P_id_A__U21 
                    P_id_A__U92 P_id_A__U41 P_id_A__U84 P_id_A__ISNAT 
                    P_id_A__U12 P_id_MARK P_id_A__U45.
                 
                 Lemma marked_measure_equation :
                  forall t, 
                   marked_measure t = match t with
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__U46 (x11::nil)) =>
                                         P_id_A__U46 (measure x11)
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__isNatIListKind 
                                           (x11::nil)) =>
                                         P_id_A__ISNATILISTKIND (measure x11)
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__U61 (x11::nil)) =>
                                         P_id_A__U61 (measure x11)
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__U85 (x12::
                                           x11::nil)) =>
                                         P_id_A__U85 (measure x12) 
                                          (measure x11)
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__U31 (x12::
                                           x11::nil)) =>
                                         P_id_A__U31 (measure x12) 
                                          (measure x11)
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__U81 (x13::x12::
                                           x11::nil)) =>
                                         P_id_A__U81 (measure x13) 
                                          (measure x12) (measure x11)
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__U22 (x12::
                                           x11::nil)) =>
                                         P_id_A__U22 (measure x12) 
                                          (measure x11)
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__U93 (x13::x12::
                                           x11::nil)) =>
                                         P_id_A__U93 (measure x13) 
                                          (measure x12) (measure x11)
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__U42 (x13::x12::
                                           x11::nil)) =>
                                         P_id_A__U42 (measure x13) 
                                          (measure x12) (measure x11)
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__U51 (x12::
                                           x11::nil)) =>
                                         P_id_A__U51 (measure x12) 
                                          (measure x11)
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__isNatList 
                                           (x11::nil)) =>
                                         P_id_A__ISNATLIST (measure x11)
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__zeros nil) =>
                                         P_id_A__ZEROS 
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__U91 (x13::x12::
                                           x11::nil)) =>
                                         P_id_A__U91 (measure x13) 
                                          (measure x12) (measure x11)
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__U33 (x11::nil)) =>
                                         P_id_A__U33 (measure x11)
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__U83 (x13::x12::
                                           x11::nil)) =>
                                         P_id_A__U83 (measure x13) 
                                          (measure x12) (measure x11)
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__U23 (x11::nil)) =>
                                         P_id_A__U23 (measure x11)
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__U11 (x12::
                                           x11::nil)) =>
                                         P_id_A__U11 (measure x12) 
                                          (measure x11)
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__length (x11::nil)) =>
                                         P_id_A__LENGTH (measure x11)
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__U44 (x13::x12::
                                           x11::nil)) =>
                                         P_id_A__U44 (measure x13) 
                                          (measure x12) (measure x11)
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__isNatIList 
                                           (x11::nil)) =>
                                         P_id_A__ISNATILIST (measure x11)
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__U13 (x11::nil)) =>
                                         P_id_A__U13 (measure x11)
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__U71 (x11::nil)) =>
                                         P_id_A__U71 (measure x11)
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__U86 (x11::nil)) =>
                                         P_id_A__U86 (measure x11)
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__U32 (x12::
                                           x11::nil)) =>
                                         P_id_A__U32 (measure x12) 
                                          (measure x11)
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__U82 (x13::x12::
                                           x11::nil)) =>
                                         P_id_A__U82 (measure x13) 
                                          (measure x12) (measure x11)
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__isNatKind 
                                           (x11::nil)) =>
                                         P_id_A__ISNATKIND (measure x11)
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__U94 (x12::
                                           x11::nil)) =>
                                         P_id_A__U94 (measure x12) 
                                          (measure x11)
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__U43 (x13::x12::
                                           x11::nil)) =>
                                         P_id_A__U43 (measure x13) 
                                          (measure x12) (measure x11)
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__U52 (x11::nil)) =>
                                         P_id_A__U52 (measure x11)
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__U21 (x12::
                                           x11::nil)) =>
                                         P_id_A__U21 (measure x12) 
                                          (measure x11)
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__U92 (x13::x12::
                                           x11::nil)) =>
                                         P_id_A__U92 (measure x13) 
                                          (measure x12) (measure x11)
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__U41 (x13::x12::
                                           x11::nil)) =>
                                         P_id_A__U41 (measure x13) 
                                          (measure x12) (measure x11)
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__U84 (x13::x12::
                                           x11::nil)) =>
                                         P_id_A__U84 (measure x13) 
                                          (measure x12) (measure x11)
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__isNat (x11::nil)) =>
                                         P_id_A__ISNAT (measure x11)
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__U12 (x12::
                                           x11::nil)) =>
                                         P_id_A__U12 (measure x12) 
                                          (measure x11)
                                        | (algebra.Alg.Term 
                                           algebra.F.id_mark (x11::nil)) =>
                                         P_id_MARK (measure x11)
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__U45 (x12::
                                           x11::nil)) =>
                                         P_id_A__U45 (measure x12) 
                                          (measure x11)
                                        | _ => measure t
                                        end.
                 Proof.
                   reflexivity .
                 Qed.
                 
                 Lemma marked_measure_star_monotonic :
                  forall f l1 l2, 
                   (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step 
                                                                    R_xml_0_deep_rew.R_xml_0_rules)
                                                                    ) 
                     l1 l2) ->
                    marked_measure (algebra.Alg.Term f l1) <= marked_measure 
                                                               (algebra.Alg.Term 
                                                                 f l2).
                 Proof.
                   unfold marked_measure in *.
                   apply InterpZ.marked_measure_star_monotonic.
                   intros ;apply P_id_U51_monotonic;assumption.
                   intros ;apply P_id_a__U84_monotonic;assumption.
                   intros ;apply P_id_a__U32_monotonic;assumption.
                   intros ;apply P_id_isNatList_monotonic;assumption.
                   intros ;apply P_id_a__U13_monotonic;assumption.
                   intros ;apply P_id_U85_monotonic;assumption.
                   intros ;apply P_id_a__length_monotonic;assumption.
                   intros ;apply P_id_a__isNatIList_monotonic;assumption.
                   intros ;apply P_id_U33_monotonic;assumption.
                   intros ;apply P_id_a__U11_monotonic;assumption.
                   intros ;apply P_id_U81_monotonic;assumption.
                   intros ;apply P_id_a__U92_monotonic;assumption.
                   intros ;apply P_id_a__U43_monotonic;assumption.
                   intros ;apply P_id_U23_monotonic;assumption.
                   intros ;apply P_id_a__isNatKind_monotonic;assumption.
                   intros ;apply P_id_U93_monotonic;assumption.
                   intros ;apply P_id_U11_monotonic;assumption.
                   intros ;apply P_id_a__U71_monotonic;assumption.
                   intros ;apply P_id_U44_monotonic;assumption.
                   intros ;apply P_id_U61_monotonic;assumption.
                   intros ;apply P_id_a__U86_monotonic;assumption.
                   intros ;apply P_id_a__U41_monotonic;assumption.
                   intros ;apply P_id_U22_monotonic;assumption.
                   intros ;apply P_id_a__U21_monotonic;assumption.
                   intros ;apply P_id_U91_monotonic;assumption.
                   intros ;apply P_id_length_monotonic;assumption.
                   intros ;apply P_id_a__U52_monotonic;assumption.
                   intros ;apply P_id_U42_monotonic;assumption.
                   intros ;apply P_id_a__U12_monotonic;assumption.
                   intros ;apply P_id_U83_monotonic;assumption.
                   intros ;apply P_id_a__U94_monotonic;assumption.
                   intros ;apply P_id_a__U45_monotonic;assumption.
                   intros ;apply P_id_U31_monotonic;assumption.
                   intros ;apply P_id_a__isNat_monotonic;assumption.
                   intros ;apply P_id_isNatIListKind_monotonic;assumption.
                   intros ;apply P_id_a__U82_monotonic;assumption.
                   intros ;apply P_id_U46_monotonic;assumption.
                   intros ;apply P_id_cons_monotonic;assumption.
                   intros ;apply P_id_U52_monotonic;assumption.
                   intros ;apply P_id_a__U85_monotonic;assumption.
                   intros ;apply P_id_a__U33_monotonic;assumption.
                   intros ;apply P_id_U21_monotonic;assumption.
                   intros ;apply P_id_a__isNatList_monotonic;assumption.
                   intros ;apply P_id_U86_monotonic;assumption.
                   intros ;apply P_id_mark_monotonic;assumption.
                   intros ;apply P_id_a__U51_monotonic;assumption.
                   intros ;apply P_id_U41_monotonic;assumption.
                   intros ;apply P_id_U82_monotonic;assumption.
                   intros ;apply P_id_a__U93_monotonic;assumption.
                   intros ;apply P_id_a__U44_monotonic;assumption.
                   intros ;apply P_id_isNat_monotonic;assumption.
                   intros ;apply P_id_a__U23_monotonic;assumption.
                   intros ;apply P_id_U94_monotonic;assumption.
                   intros ;apply P_id_U12_monotonic;assumption.
                   intros ;apply P_id_a__U81_monotonic;assumption.
                   intros ;apply P_id_U45_monotonic;assumption.
                   intros ;apply P_id_U71_monotonic;assumption.
                   intros ;apply P_id_a__U91_monotonic;assumption.
                   intros ;apply P_id_a__U42_monotonic;assumption.
                   intros ;apply P_id_isNatKind_monotonic;assumption.
                   intros ;apply P_id_a__U22_monotonic;assumption.
                   intros ;apply P_id_U92_monotonic;assumption.
                   intros ;apply P_id_a__U61_monotonic;assumption.
                   intros ;apply P_id_U43_monotonic;assumption.
                   intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
                   intros ;apply P_id_U84_monotonic;assumption.
                   intros ;apply P_id_s_monotonic;assumption.
                   intros ;apply P_id_a__U46_monotonic;assumption.
                   intros ;apply P_id_U32_monotonic;assumption.
                   intros ;apply P_id_a__U31_monotonic;assumption.
                   intros ;apply P_id_U13_monotonic;assumption.
                   intros ;apply P_id_a__U83_monotonic;assumption.
                   intros ;apply P_id_isNatIList_monotonic;assumption.
                   intros ;apply P_id_a__zeros_bounded;assumption.
                   intros ;apply P_id_U51_bounded;assumption.
                   intros ;apply P_id_a__U84_bounded;assumption.
                   intros ;apply P_id_a__U32_bounded;assumption.
                   intros ;apply P_id_isNatList_bounded;assumption.
                   intros ;apply P_id_a__U13_bounded;assumption.
                   intros ;apply P_id_U85_bounded;assumption.
                   intros ;apply P_id_a__length_bounded;assumption.
                   intros ;apply P_id_a__isNatIList_bounded;assumption.
                   intros ;apply P_id_U33_bounded;assumption.
                   intros ;apply P_id_a__U11_bounded;assumption.
                   intros ;apply P_id_U81_bounded;assumption.
                   intros ;apply P_id_a__U92_bounded;assumption.
                   intros ;apply P_id_a__U43_bounded;assumption.
                   intros ;apply P_id_U23_bounded;assumption.
                   intros ;apply P_id_a__isNatKind_bounded;assumption.
                   intros ;apply P_id_U93_bounded;assumption.
                   intros ;apply P_id_U11_bounded;assumption.
                   intros ;apply P_id_a__U71_bounded;assumption.
                   intros ;apply P_id_U44_bounded;assumption.
                   intros ;apply P_id_0_bounded;assumption.
                   intros ;apply P_id_U61_bounded;assumption.
                   intros ;apply P_id_a__U86_bounded;assumption.
                   intros ;apply P_id_a__U41_bounded;assumption.
                   intros ;apply P_id_U22_bounded;assumption.
                   intros ;apply P_id_a__U21_bounded;assumption.
                   intros ;apply P_id_U91_bounded;assumption.
                   intros ;apply P_id_length_bounded;assumption.
                   intros ;apply P_id_a__U52_bounded;assumption.
                   intros ;apply P_id_U42_bounded;assumption.
                   intros ;apply P_id_a__U12_bounded;assumption.
                   intros ;apply P_id_U83_bounded;assumption.
                   intros ;apply P_id_a__U94_bounded;assumption.
                   intros ;apply P_id_a__U45_bounded;assumption.
                   intros ;apply P_id_U31_bounded;assumption.
                   intros ;apply P_id_a__isNat_bounded;assumption.
                   intros ;apply P_id_isNatIListKind_bounded;assumption.
                   intros ;apply P_id_a__U82_bounded;assumption.
                   intros ;apply P_id_U46_bounded;assumption.
                   intros ;apply P_id_cons_bounded;assumption.
                   intros ;apply P_id_U52_bounded;assumption.
                   intros ;apply P_id_a__U85_bounded;assumption.
                   intros ;apply P_id_a__U33_bounded;assumption.
                   intros ;apply P_id_U21_bounded;assumption.
                   intros ;apply P_id_a__isNatList_bounded;assumption.
                   intros ;apply P_id_U86_bounded;assumption.
                   intros ;apply P_id_mark_bounded;assumption.
                   intros ;apply P_id_a__U51_bounded;assumption.
                   intros ;apply P_id_U41_bounded;assumption.
                   intros ;apply P_id_tt_bounded;assumption.
                   intros ;apply P_id_U82_bounded;assumption.
                   intros ;apply P_id_a__U93_bounded;assumption.
                   intros ;apply P_id_a__U44_bounded;assumption.
                   intros ;apply P_id_isNat_bounded;assumption.
                   intros ;apply P_id_a__U23_bounded;assumption.
                   intros ;apply P_id_U94_bounded;assumption.
                   intros ;apply P_id_U12_bounded;assumption.
                   intros ;apply P_id_a__U81_bounded;assumption.
                   intros ;apply P_id_U45_bounded;assumption.
                   intros ;apply P_id_zeros_bounded;assumption.
                   intros ;apply P_id_U71_bounded;assumption.
                   intros ;apply P_id_a__U91_bounded;assumption.
                   intros ;apply P_id_a__U42_bounded;assumption.
                   intros ;apply P_id_isNatKind_bounded;assumption.
                   intros ;apply P_id_a__U22_bounded;assumption.
                   intros ;apply P_id_U92_bounded;assumption.
                   intros ;apply P_id_nil_bounded;assumption.
                   intros ;apply P_id_a__U61_bounded;assumption.
                   intros ;apply P_id_U43_bounded;assumption.
                   intros ;apply P_id_a__isNatIListKind_bounded;assumption.
                   intros ;apply P_id_U84_bounded;assumption.
                   intros ;apply P_id_s_bounded;assumption.
                   intros ;apply P_id_a__U46_bounded;assumption.
                   intros ;apply P_id_U32_bounded;assumption.
                   intros ;apply P_id_a__U31_bounded;assumption.
                   intros ;apply P_id_U13_bounded;assumption.
                   intros ;apply P_id_a__U83_bounded;assumption.
                   intros ;apply P_id_isNatIList_bounded;assumption.
                   apply rules_monotonic.
                   intros ;apply P_id_A__U46_monotonic;assumption.
                   intros ;apply P_id_A__ISNATILISTKIND_monotonic;assumption.
                   intros ;apply P_id_A__U61_monotonic;assumption.
                   intros ;apply P_id_A__U85_monotonic;assumption.
                   intros ;apply P_id_A__U31_monotonic;assumption.
                   intros ;apply P_id_A__U81_monotonic;assumption.
                   intros ;apply P_id_A__U22_monotonic;assumption.
                   intros ;apply P_id_A__U93_monotonic;assumption.
                   intros ;apply P_id_A__U42_monotonic;assumption.
                   intros ;apply P_id_A__U51_monotonic;assumption.
                   intros ;apply P_id_A__ISNATLIST_monotonic;assumption.
                   intros ;apply P_id_A__U91_monotonic;assumption.
                   intros ;apply P_id_A__U33_monotonic;assumption.
                   intros ;apply P_id_A__U83_monotonic;assumption.
                   intros ;apply P_id_A__U23_monotonic;assumption.
                   intros ;apply P_id_A__U11_monotonic;assumption.
                   intros ;apply P_id_A__LENGTH_monotonic;assumption.
                   intros ;apply P_id_A__U44_monotonic;assumption.
                   intros ;apply P_id_A__ISNATILIST_monotonic;assumption.
                   intros ;apply P_id_A__U13_monotonic;assumption.
                   intros ;apply P_id_A__U71_monotonic;assumption.
                   intros ;apply P_id_A__U86_monotonic;assumption.
                   intros ;apply P_id_A__U32_monotonic;assumption.
                   intros ;apply P_id_A__U82_monotonic;assumption.
                   intros ;apply P_id_A__ISNATKIND_monotonic;assumption.
                   intros ;apply P_id_A__U94_monotonic;assumption.
                   intros ;apply P_id_A__U43_monotonic;assumption.
                   intros ;apply P_id_A__U52_monotonic;assumption.
                   intros ;apply P_id_A__U21_monotonic;assumption.
                   intros ;apply P_id_A__U92_monotonic;assumption.
                   intros ;apply P_id_A__U41_monotonic;assumption.
                   intros ;apply P_id_A__U84_monotonic;assumption.
                   intros ;apply P_id_A__ISNAT_monotonic;assumption.
                   intros ;apply P_id_A__U12_monotonic;assumption.
                   intros ;apply P_id_MARK_monotonic;assumption.
                   intros ;apply P_id_A__U45_monotonic;assumption.
                 Qed.
                 
                 Ltac rewrite_and_unfold  :=
                  do 2 (rewrite marked_measure_equation);
                   repeat (
                   match goal with
                     |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
                      rewrite (measure_equation (algebra.Alg.Term f t))
                     | H:context [measure (algebra.Alg.Term ?f ?t)] |- 
                     _ =>
                      rewrite (measure_equation (algebra.Alg.Term f t)) in 
                      H|-
                     end
                   ).
                 
                 Definition lt a b := 
                   (Zwf.Zwf 0) (marked_measure a) (marked_measure b).
                 
                 Definition le a b := marked_measure a <= marked_measure b.
                 
                 Lemma lt_le_compat :
                  forall a b c, (lt a b) ->(le b c) ->lt a c.
                 Proof.
                   unfold lt, le in *.
                   intros a b c.
                   apply (interp.le_lt_compat_right (interp.o_Z 0)).
                 Qed.
                 
                 Lemma wf_lt : well_founded lt.
                 Proof.
                   unfold lt in *.
                   apply Inverse_Image.wf_inverse_image with  (B:=Z).
                   apply Zwf.Zwf_well_founded.
                 Qed.
                 
                 Lemma DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_strict_in_lt :
                  Relation_Definitions.inclusion _ 
                   DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_strict 
                   lt.
                 Proof.
                   unfold Relation_Definitions.inclusion, lt in *.
                   
                   intros a b H;destruct H;
                    match goal with
                      |  |- (Zwf.Zwf 0) _ 
                             (marked_measure (algebra.Alg.Term ?f ?l)) =>
                       let l'' := algebra.Alg_ext.find_replacement l  in 
                        ((apply (interp.le_lt_compat_right (interp.o_Z 0)) with
                           (marked_measure (algebra.Alg.Term f l''));[
                          idtac|
                          apply marked_measure_star_monotonic;
                           repeat (
                           apply algebra.EQT_ext.one_step_list_refl_trans_clos
                           );(assumption)||(constructor 1)]))
                      end
                    ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_in_le :
                  Relation_Definitions.inclusion _ 
                   DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large 
                   le.
                 Proof.
                   unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *.
                   
                   intros a b H;destruct H;
                    match goal with
                      |  |- _ <= marked_measure (algebra.Alg.Term ?f ?l) =>
                       let l'' := algebra.Alg_ext.find_replacement l  in 
                        ((apply (interp.le_trans (interp.o_Z 0)) with
                           (marked_measure (algebra.Alg.Term f l''));[
                          idtac|
                          apply marked_measure_star_monotonic;
                           repeat (
                           apply algebra.EQT_ext.one_step_list_refl_trans_clos
                           );(assumption)||(constructor 1)]))
                      end
                    ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Definition wf_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large  := 
                   WF_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large.wf
                   .
                 
                 
                 Lemma wf :
                  well_founded WF_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large.DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large
                   .
                 Proof.
                   intros x.
                   apply (well_founded_ind wf_lt).
                   clear x.
                   intros x.
                   pattern x.
                   
                   apply (@Acc_ind _ 
                           DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large)
                          .
                   clear x.
                   intros x _ IHx IHx'.
                   constructor.
                   intros y H.
                   
                   destruct H;
                    (apply IHx';
                      apply DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_strict_in_lt;
                      econstructor eassumption)||
                    ((apply IHx;[econstructor eassumption|
                      intros y' Hlt;apply IHx';
                       apply lt_le_compat with (1:=Hlt) ;
                       apply DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_in_le;
                       econstructor eassumption])).
                   
                   apply wf_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large.
                 Qed.
                End WF_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large.
                
                Open Scope Z_scope.
                
                Import ring_extention.
                
                Notation Local "a <= b" := (Zle a b).
                
                Notation Local "a < b" := (Zlt a b).
                
                Definition P_id_a__zeros  := 0.
                
                Definition P_id_U51 (x11:Z) (x12:Z) := 1* x11 + 1* x12.
                
                Definition P_id_a__U84 (x11:Z) (x12:Z) (x13:Z) := 0.
                
                Definition P_id_a__U32 (x11:Z) (x12:Z) := 1* x12.
                
                Definition P_id_isNatList (x11:Z) := 1.
                
                Definition P_id_a__U13 (x11:Z) := 0.
                
                Definition P_id_U85 (x11:Z) (x12:Z) := 0.
                
                Definition P_id_a__length (x11:Z) := 1 + 1* x11.
                
                Definition P_id_a__isNatIList (x11:Z) := 2 + 2* x11.
                
                Definition P_id_U33 (x11:Z) := 0.
                
                Definition P_id_a__U11 (x11:Z) (x12:Z) := 0.
                
                Definition P_id_U81 (x11:Z) (x12:Z) (x13:Z) := 1.
                
                Definition P_id_a__U92 (x11:Z) (x12:Z) (x13:Z) := 
                  1 + 2* x12 + 1* x13.
                
                Definition P_id_a__U43 (x11:Z) (x12:Z) (x13:Z) := 0.
                
                Definition P_id_U23 (x11:Z) := 2* x11.
                
                Definition P_id_a__isNatKind (x11:Z) := 2* x11.
                
                Definition P_id_U93 (x11:Z) (x12:Z) (x13:Z) := 1 + 1* x12.
                
                Definition P_id_U11 (x11:Z) (x12:Z) := 0.
                
                Definition P_id_a__U71 (x11:Z) := 1* x11.
                
                Definition P_id_U44 (x11:Z) (x12:Z) (x13:Z) := 0.
                
                Definition P_id_0  := 0.
                
                Definition P_id_U61 (x11:Z) := 1 + 1* x11.
                
                Definition P_id_a__U86 (x11:Z) := 0.
                
                Definition P_id_a__U41 (x11:Z) (x12:Z) (x13:Z) := 1 + 2* x12.
                
                Definition P_id_U22 (x11:Z) (x12:Z) := 0.
                
                Definition P_id_a__U21 (x11:Z) (x12:Z) := 0.
                
                Definition P_id_U91 (x11:Z) (x12:Z) (x13:Z) := 
                  1 + 2* x12 + 1* x13.
                
                Definition P_id_length (x11:Z) := 1 + 1* x11.
                
                Definition P_id_a__U52 (x11:Z) := 1* x11.
                
                Definition P_id_U42 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
                
                Definition P_id_a__U12 (x11:Z) (x12:Z) := 0.
                
                Definition P_id_U83 (x11:Z) (x12:Z) (x13:Z) := 1.
                
                Definition P_id_a__U94 (x11:Z) (x12:Z) := 1 + 2* x12.
                
                Definition P_id_a__U45 (x11:Z) (x12:Z) := 0.
                
                Definition P_id_U31 (x11:Z) (x12:Z) := 1 + 1* x12.
                
                Definition P_id_a__isNat (x11:Z) := 0.
                
                Definition P_id_isNatIListKind (x11:Z) := 1* x11.
                
                Definition P_id_a__U82 (x11:Z) (x12:Z) (x13:Z) := 1.
                
                Definition P_id_U46 (x11:Z) := 0.
                
                Definition P_id_cons (x11:Z) (x12:Z) := 2* x11 + 3* x12.
                
                Definition P_id_U52 (x11:Z) := 1* x11.
                
                Definition P_id_a__U85 (x11:Z) (x12:Z) := 0.
                
                Definition P_id_a__U33 (x11:Z) := 0.
                
                Definition P_id_U21 (x11:Z) (x12:Z) := 0.
                
                Definition P_id_a__isNatList (x11:Z) := 2.
                
                Definition P_id_U86 (x11:Z) := 0.
                
                Definition P_id_mark (x11:Z) := 2* x11.
                
                Definition P_id_a__U51 (x11:Z) (x12:Z) := 1* x11 + 1* x12.
                
                Definition P_id_U41 (x11:Z) (x12:Z) (x13:Z) := 1 + 2* x12.
                
                Definition P_id_tt  := 0.
                
                Definition P_id_U82 (x11:Z) (x12:Z) (x13:Z) := 1.
                
                Definition P_id_a__U93 (x11:Z) (x12:Z) (x13:Z) := 1 + 2* x12.
                
                Definition P_id_a__U44 (x11:Z) (x12:Z) (x13:Z) := 0.
                
                Definition P_id_isNat (x11:Z) := 0.
                
                Definition P_id_a__U23 (x11:Z) := 2* x11.
                
                Definition P_id_U94 (x11:Z) (x12:Z) := 1 + 2* x12.
                
                Definition P_id_U12 (x11:Z) (x12:Z) := 0.
                
                Definition P_id_a__U81 (x11:Z) (x12:Z) (x13:Z) := 2.
                
                Definition P_id_U45 (x11:Z) (x12:Z) := 0.
                
                Definition P_id_zeros  := 0.
                
                Definition P_id_U71 (x11:Z) := 1* x11.
                
                Definition P_id_a__U91 (x11:Z) (x12:Z) (x13:Z) := 
                  1 + 2* x12 + 1* x13.
                
                Definition P_id_a__U42 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
                
                Definition P_id_isNatKind (x11:Z) := 2* x11.
                
                Definition P_id_a__U22 (x11:Z) (x12:Z) := 0.
                
                Definition P_id_U92 (x11:Z) (x12:Z) (x13:Z) := 
                  1 + 2* x12 + 1* x13.
                
                Definition P_id_nil  := 0.
                
                Definition P_id_a__U61 (x11:Z) := 2 + 1* x11.
                
                Definition P_id_U43 (x11:Z) (x12:Z) (x13:Z) := 0.
                
                Definition P_id_a__isNatIListKind (x11:Z) := 1* x11.
                
                Definition P_id_U84 (x11:Z) (x12:Z) (x13:Z) := 0.
                
                Definition P_id_s (x11:Z) := 1* x11.
                
                Definition P_id_a__U46 (x11:Z) := 0.
                
                Definition P_id_U32 (x11:Z) (x12:Z) := 1* x12.
                
                Definition P_id_a__U31 (x11:Z) (x12:Z) := 2 + 2* x12.
                
                Definition P_id_U13 (x11:Z) := 0.
                
                Definition P_id_a__U83 (x11:Z) (x12:Z) (x13:Z) := 1.
                
                Definition P_id_isNatIList (x11:Z) := 1 + 2* x11.
                
                Lemma P_id_U51_monotonic :
                 forall x12 x14 x13 x11, 
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_U51 x12 x14 <= P_id_U51 x11 x13.
                Proof.
                  intros x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U84_monotonic :
                 forall x16 x12 x14 x13 x11 x15, 
                  (0 <= x16)/\ (x16 <= x15) ->
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_a__U84 x12 x14 x16 <= P_id_a__U84 x11 x13 x15.
                Proof.
                  intros x16 x15 x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  intros [H_5 H_4].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U32_monotonic :
                 forall x12 x14 x13 x11, 
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_a__U32 x12 x14 <= P_id_a__U32 x11 x13.
                Proof.
                  intros x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_isNatList_monotonic :
                 forall x12 x11, 
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_isNatList x12 <= P_id_isNatList x11.
                Proof.
                  intros x12 x11.
                  intros [H_1 H_0].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U13_monotonic :
                 forall x12 x11, 
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_a__U13 x12 <= P_id_a__U13 x11.
                Proof.
                  intros x12 x11.
                  intros [H_1 H_0].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U85_monotonic :
                 forall x12 x14 x13 x11, 
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_U85 x12 x14 <= P_id_U85 x11 x13.
                Proof.
                  intros x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__length_monotonic :
                 forall x12 x11, 
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_a__length x12 <= P_id_a__length x11.
                Proof.
                  intros x12 x11.
                  intros [H_1 H_0].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__isNatIList_monotonic :
                 forall x12 x11, 
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_a__isNatIList x12 <= P_id_a__isNatIList x11.
                Proof.
                  intros x12 x11.
                  intros [H_1 H_0].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U33_monotonic :
                 forall x12 x11, 
                  (0 <= x12)/\ (x12 <= x11) ->P_id_U33 x12 <= P_id_U33 x11.
                Proof.
                  intros x12 x11.
                  intros [H_1 H_0].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U11_monotonic :
                 forall x12 x14 x13 x11, 
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_a__U11 x12 x14 <= P_id_a__U11 x11 x13.
                Proof.
                  intros x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U81_monotonic :
                 forall x16 x12 x14 x13 x11 x15, 
                  (0 <= x16)/\ (x16 <= x15) ->
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_U81 x12 x14 x16 <= P_id_U81 x11 x13 x15.
                Proof.
                  intros x16 x15 x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  intros [H_5 H_4].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U92_monotonic :
                 forall x16 x12 x14 x13 x11 x15, 
                  (0 <= x16)/\ (x16 <= x15) ->
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_a__U92 x12 x14 x16 <= P_id_a__U92 x11 x13 x15.
                Proof.
                  intros x16 x15 x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  intros [H_5 H_4].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U43_monotonic :
                 forall x16 x12 x14 x13 x11 x15, 
                  (0 <= x16)/\ (x16 <= x15) ->
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_a__U43 x12 x14 x16 <= P_id_a__U43 x11 x13 x15.
                Proof.
                  intros x16 x15 x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  intros [H_5 H_4].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U23_monotonic :
                 forall x12 x11, 
                  (0 <= x12)/\ (x12 <= x11) ->P_id_U23 x12 <= P_id_U23 x11.
                Proof.
                  intros x12 x11.
                  intros [H_1 H_0].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__isNatKind_monotonic :
                 forall x12 x11, 
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_a__isNatKind x12 <= P_id_a__isNatKind x11.
                Proof.
                  intros x12 x11.
                  intros [H_1 H_0].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U93_monotonic :
                 forall x16 x12 x14 x13 x11 x15, 
                  (0 <= x16)/\ (x16 <= x15) ->
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_U93 x12 x14 x16 <= P_id_U93 x11 x13 x15.
                Proof.
                  intros x16 x15 x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  intros [H_5 H_4].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U11_monotonic :
                 forall x12 x14 x13 x11, 
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_U11 x12 x14 <= P_id_U11 x11 x13.
                Proof.
                  intros x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U71_monotonic :
                 forall x12 x11, 
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_a__U71 x12 <= P_id_a__U71 x11.
                Proof.
                  intros x12 x11.
                  intros [H_1 H_0].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U44_monotonic :
                 forall x16 x12 x14 x13 x11 x15, 
                  (0 <= x16)/\ (x16 <= x15) ->
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_U44 x12 x14 x16 <= P_id_U44 x11 x13 x15.
                Proof.
                  intros x16 x15 x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  intros [H_5 H_4].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U61_monotonic :
                 forall x12 x11, 
                  (0 <= x12)/\ (x12 <= x11) ->P_id_U61 x12 <= P_id_U61 x11.
                Proof.
                  intros x12 x11.
                  intros [H_1 H_0].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U86_monotonic :
                 forall x12 x11, 
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_a__U86 x12 <= P_id_a__U86 x11.
                Proof.
                  intros x12 x11.
                  intros [H_1 H_0].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U41_monotonic :
                 forall x16 x12 x14 x13 x11 x15, 
                  (0 <= x16)/\ (x16 <= x15) ->
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_a__U41 x12 x14 x16 <= P_id_a__U41 x11 x13 x15.
                Proof.
                  intros x16 x15 x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  intros [H_5 H_4].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U22_monotonic :
                 forall x12 x14 x13 x11, 
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_U22 x12 x14 <= P_id_U22 x11 x13.
                Proof.
                  intros x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U21_monotonic :
                 forall x12 x14 x13 x11, 
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_a__U21 x12 x14 <= P_id_a__U21 x11 x13.
                Proof.
                  intros x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U91_monotonic :
                 forall x16 x12 x14 x13 x11 x15, 
                  (0 <= x16)/\ (x16 <= x15) ->
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_U91 x12 x14 x16 <= P_id_U91 x11 x13 x15.
                Proof.
                  intros x16 x15 x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  intros [H_5 H_4].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_length_monotonic :
                 forall x12 x11, 
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_length x12 <= P_id_length x11.
                Proof.
                  intros x12 x11.
                  intros [H_1 H_0].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U52_monotonic :
                 forall x12 x11, 
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_a__U52 x12 <= P_id_a__U52 x11.
                Proof.
                  intros x12 x11.
                  intros [H_1 H_0].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U42_monotonic :
                 forall x16 x12 x14 x13 x11 x15, 
                  (0 <= x16)/\ (x16 <= x15) ->
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_U42 x12 x14 x16 <= P_id_U42 x11 x13 x15.
                Proof.
                  intros x16 x15 x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  intros [H_5 H_4].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U12_monotonic :
                 forall x12 x14 x13 x11, 
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_a__U12 x12 x14 <= P_id_a__U12 x11 x13.
                Proof.
                  intros x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U83_monotonic :
                 forall x16 x12 x14 x13 x11 x15, 
                  (0 <= x16)/\ (x16 <= x15) ->
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_U83 x12 x14 x16 <= P_id_U83 x11 x13 x15.
                Proof.
                  intros x16 x15 x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  intros [H_5 H_4].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U94_monotonic :
                 forall x12 x14 x13 x11, 
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_a__U94 x12 x14 <= P_id_a__U94 x11 x13.
                Proof.
                  intros x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U45_monotonic :
                 forall x12 x14 x13 x11, 
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_a__U45 x12 x14 <= P_id_a__U45 x11 x13.
                Proof.
                  intros x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U31_monotonic :
                 forall x12 x14 x13 x11, 
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_U31 x12 x14 <= P_id_U31 x11 x13.
                Proof.
                  intros x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__isNat_monotonic :
                 forall x12 x11, 
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_a__isNat x12 <= P_id_a__isNat x11.
                Proof.
                  intros x12 x11.
                  intros [H_1 H_0].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_isNatIListKind_monotonic :
                 forall x12 x11, 
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_isNatIListKind x12 <= P_id_isNatIListKind x11.
                Proof.
                  intros x12 x11.
                  intros [H_1 H_0].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U82_monotonic :
                 forall x16 x12 x14 x13 x11 x15, 
                  (0 <= x16)/\ (x16 <= x15) ->
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_a__U82 x12 x14 x16 <= P_id_a__U82 x11 x13 x15.
                Proof.
                  intros x16 x15 x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  intros [H_5 H_4].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U46_monotonic :
                 forall x12 x11, 
                  (0 <= x12)/\ (x12 <= x11) ->P_id_U46 x12 <= P_id_U46 x11.
                Proof.
                  intros x12 x11.
                  intros [H_1 H_0].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_cons_monotonic :
                 forall x12 x14 x13 x11, 
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_cons x12 x14 <= P_id_cons x11 x13.
                Proof.
                  intros x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U52_monotonic :
                 forall x12 x11, 
                  (0 <= x12)/\ (x12 <= x11) ->P_id_U52 x12 <= P_id_U52 x11.
                Proof.
                  intros x12 x11.
                  intros [H_1 H_0].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U85_monotonic :
                 forall x12 x14 x13 x11, 
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_a__U85 x12 x14 <= P_id_a__U85 x11 x13.
                Proof.
                  intros x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U33_monotonic :
                 forall x12 x11, 
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_a__U33 x12 <= P_id_a__U33 x11.
                Proof.
                  intros x12 x11.
                  intros [H_1 H_0].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U21_monotonic :
                 forall x12 x14 x13 x11, 
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_U21 x12 x14 <= P_id_U21 x11 x13.
                Proof.
                  intros x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__isNatList_monotonic :
                 forall x12 x11, 
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_a__isNatList x12 <= P_id_a__isNatList x11.
                Proof.
                  intros x12 x11.
                  intros [H_1 H_0].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U86_monotonic :
                 forall x12 x11, 
                  (0 <= x12)/\ (x12 <= x11) ->P_id_U86 x12 <= P_id_U86 x11.
                Proof.
                  intros x12 x11.
                  intros [H_1 H_0].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_mark_monotonic :
                 forall x12 x11, 
                  (0 <= x12)/\ (x12 <= x11) ->P_id_mark x12 <= P_id_mark x11.
                Proof.
                  intros x12 x11.
                  intros [H_1 H_0].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U51_monotonic :
                 forall x12 x14 x13 x11, 
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_a__U51 x12 x14 <= P_id_a__U51 x11 x13.
                Proof.
                  intros x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U41_monotonic :
                 forall x16 x12 x14 x13 x11 x15, 
                  (0 <= x16)/\ (x16 <= x15) ->
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_U41 x12 x14 x16 <= P_id_U41 x11 x13 x15.
                Proof.
                  intros x16 x15 x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  intros [H_5 H_4].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U82_monotonic :
                 forall x16 x12 x14 x13 x11 x15, 
                  (0 <= x16)/\ (x16 <= x15) ->
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_U82 x12 x14 x16 <= P_id_U82 x11 x13 x15.
                Proof.
                  intros x16 x15 x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  intros [H_5 H_4].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U93_monotonic :
                 forall x16 x12 x14 x13 x11 x15, 
                  (0 <= x16)/\ (x16 <= x15) ->
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_a__U93 x12 x14 x16 <= P_id_a__U93 x11 x13 x15.
                Proof.
                  intros x16 x15 x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  intros [H_5 H_4].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U44_monotonic :
                 forall x16 x12 x14 x13 x11 x15, 
                  (0 <= x16)/\ (x16 <= x15) ->
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_a__U44 x12 x14 x16 <= P_id_a__U44 x11 x13 x15.
                Proof.
                  intros x16 x15 x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  intros [H_5 H_4].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_isNat_monotonic :
                 forall x12 x11, 
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_isNat x12 <= P_id_isNat x11.
                Proof.
                  intros x12 x11.
                  intros [H_1 H_0].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U23_monotonic :
                 forall x12 x11, 
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_a__U23 x12 <= P_id_a__U23 x11.
                Proof.
                  intros x12 x11.
                  intros [H_1 H_0].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U94_monotonic :
                 forall x12 x14 x13 x11, 
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_U94 x12 x14 <= P_id_U94 x11 x13.
                Proof.
                  intros x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U12_monotonic :
                 forall x12 x14 x13 x11, 
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_U12 x12 x14 <= P_id_U12 x11 x13.
                Proof.
                  intros x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U81_monotonic :
                 forall x16 x12 x14 x13 x11 x15, 
                  (0 <= x16)/\ (x16 <= x15) ->
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_a__U81 x12 x14 x16 <= P_id_a__U81 x11 x13 x15.
                Proof.
                  intros x16 x15 x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  intros [H_5 H_4].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U45_monotonic :
                 forall x12 x14 x13 x11, 
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_U45 x12 x14 <= P_id_U45 x11 x13.
                Proof.
                  intros x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U71_monotonic :
                 forall x12 x11, 
                  (0 <= x12)/\ (x12 <= x11) ->P_id_U71 x12 <= P_id_U71 x11.
                Proof.
                  intros x12 x11.
                  intros [H_1 H_0].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U91_monotonic :
                 forall x16 x12 x14 x13 x11 x15, 
                  (0 <= x16)/\ (x16 <= x15) ->
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_a__U91 x12 x14 x16 <= P_id_a__U91 x11 x13 x15.
                Proof.
                  intros x16 x15 x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  intros [H_5 H_4].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U42_monotonic :
                 forall x16 x12 x14 x13 x11 x15, 
                  (0 <= x16)/\ (x16 <= x15) ->
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_a__U42 x12 x14 x16 <= P_id_a__U42 x11 x13 x15.
                Proof.
                  intros x16 x15 x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  intros [H_5 H_4].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_isNatKind_monotonic :
                 forall x12 x11, 
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_isNatKind x12 <= P_id_isNatKind x11.
                Proof.
                  intros x12 x11.
                  intros [H_1 H_0].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U22_monotonic :
                 forall x12 x14 x13 x11, 
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_a__U22 x12 x14 <= P_id_a__U22 x11 x13.
                Proof.
                  intros x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U92_monotonic :
                 forall x16 x12 x14 x13 x11 x15, 
                  (0 <= x16)/\ (x16 <= x15) ->
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_U92 x12 x14 x16 <= P_id_U92 x11 x13 x15.
                Proof.
                  intros x16 x15 x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  intros [H_5 H_4].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U61_monotonic :
                 forall x12 x11, 
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_a__U61 x12 <= P_id_a__U61 x11.
                Proof.
                  intros x12 x11.
                  intros [H_1 H_0].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U43_monotonic :
                 forall x16 x12 x14 x13 x11 x15, 
                  (0 <= x16)/\ (x16 <= x15) ->
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_U43 x12 x14 x16 <= P_id_U43 x11 x13 x15.
                Proof.
                  intros x16 x15 x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  intros [H_5 H_4].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__isNatIListKind_monotonic :
                 forall x12 x11, 
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_a__isNatIListKind x12 <= P_id_a__isNatIListKind x11.
                Proof.
                  intros x12 x11.
                  intros [H_1 H_0].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U84_monotonic :
                 forall x16 x12 x14 x13 x11 x15, 
                  (0 <= x16)/\ (x16 <= x15) ->
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_U84 x12 x14 x16 <= P_id_U84 x11 x13 x15.
                Proof.
                  intros x16 x15 x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  intros [H_5 H_4].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_s_monotonic :
                 forall x12 x11, 
                  (0 <= x12)/\ (x12 <= x11) ->P_id_s x12 <= P_id_s x11.
                Proof.
                  intros x12 x11.
                  intros [H_1 H_0].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U46_monotonic :
                 forall x12 x11, 
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_a__U46 x12 <= P_id_a__U46 x11.
                Proof.
                  intros x12 x11.
                  intros [H_1 H_0].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U32_monotonic :
                 forall x12 x14 x13 x11, 
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_U32 x12 x14 <= P_id_U32 x11 x13.
                Proof.
                  intros x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U31_monotonic :
                 forall x12 x14 x13 x11, 
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_a__U31 x12 x14 <= P_id_a__U31 x11 x13.
                Proof.
                  intros x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U13_monotonic :
                 forall x12 x11, 
                  (0 <= x12)/\ (x12 <= x11) ->P_id_U13 x12 <= P_id_U13 x11.
                Proof.
                  intros x12 x11.
                  intros [H_1 H_0].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U83_monotonic :
                 forall x16 x12 x14 x13 x11 x15, 
                  (0 <= x16)/\ (x16 <= x15) ->
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_a__U83 x12 x14 x16 <= P_id_a__U83 x11 x13 x15.
                Proof.
                  intros x16 x15 x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  intros [H_5 H_4].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_isNatIList_monotonic :
                 forall x12 x11, 
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_isNatIList x12 <= P_id_isNatIList x11.
                Proof.
                  intros x12 x11.
                  intros [H_1 H_0].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__zeros_bounded : 0 <= P_id_a__zeros .
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U51_bounded :
                 forall x12 x11, 
                  (0 <= x11) ->(0 <= x12) ->0 <= P_id_U51 x12 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U84_bounded :
                 forall x12 x13 x11, 
                  (0 <= x11) ->
                   (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U84 x13 x12 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U32_bounded :
                 forall x12 x11, 
                  (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U32 x12 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_isNatList_bounded :
                 forall x11, (0 <= x11) ->0 <= P_id_isNatList x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U13_bounded :
                 forall x11, (0 <= x11) ->0 <= P_id_a__U13 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U85_bounded :
                 forall x12 x11, 
                  (0 <= x11) ->(0 <= x12) ->0 <= P_id_U85 x12 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__length_bounded :
                 forall x11, (0 <= x11) ->0 <= P_id_a__length x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__isNatIList_bounded :
                 forall x11, (0 <= x11) ->0 <= P_id_a__isNatIList x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U33_bounded :
                 forall x11, (0 <= x11) ->0 <= P_id_U33 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U11_bounded :
                 forall x12 x11, 
                  (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U11 x12 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U81_bounded :
                 forall x12 x13 x11, 
                  (0 <= x11) ->
                   (0 <= x12) ->(0 <= x13) ->0 <= P_id_U81 x13 x12 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U92_bounded :
                 forall x12 x13 x11, 
                  (0 <= x11) ->
                   (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U92 x13 x12 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U43_bounded :
                 forall x12 x13 x11, 
                  (0 <= x11) ->
                   (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U43 x13 x12 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U23_bounded :
                 forall x11, (0 <= x11) ->0 <= P_id_U23 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__isNatKind_bounded :
                 forall x11, (0 <= x11) ->0 <= P_id_a__isNatKind x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U93_bounded :
                 forall x12 x13 x11, 
                  (0 <= x11) ->
                   (0 <= x12) ->(0 <= x13) ->0 <= P_id_U93 x13 x12 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U11_bounded :
                 forall x12 x11, 
                  (0 <= x11) ->(0 <= x12) ->0 <= P_id_U11 x12 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U71_bounded :
                 forall x11, (0 <= x11) ->0 <= P_id_a__U71 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U44_bounded :
                 forall x12 x13 x11, 
                  (0 <= x11) ->
                   (0 <= x12) ->(0 <= x13) ->0 <= P_id_U44 x13 x12 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_0_bounded : 0 <= P_id_0 .
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U61_bounded :
                 forall x11, (0 <= x11) ->0 <= P_id_U61 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U86_bounded :
                 forall x11, (0 <= x11) ->0 <= P_id_a__U86 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U41_bounded :
                 forall x12 x13 x11, 
                  (0 <= x11) ->
                   (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U41 x13 x12 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U22_bounded :
                 forall x12 x11, 
                  (0 <= x11) ->(0 <= x12) ->0 <= P_id_U22 x12 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U21_bounded :
                 forall x12 x11, 
                  (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U21 x12 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U91_bounded :
                 forall x12 x13 x11, 
                  (0 <= x11) ->
                   (0 <= x12) ->(0 <= x13) ->0 <= P_id_U91 x13 x12 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_length_bounded :
                 forall x11, (0 <= x11) ->0 <= P_id_length x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U52_bounded :
                 forall x11, (0 <= x11) ->0 <= P_id_a__U52 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U42_bounded :
                 forall x12 x13 x11, 
                  (0 <= x11) ->
                   (0 <= x12) ->(0 <= x13) ->0 <= P_id_U42 x13 x12 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U12_bounded :
                 forall x12 x11, 
                  (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U12 x12 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U83_bounded :
                 forall x12 x13 x11, 
                  (0 <= x11) ->
                   (0 <= x12) ->(0 <= x13) ->0 <= P_id_U83 x13 x12 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U94_bounded :
                 forall x12 x11, 
                  (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U94 x12 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U45_bounded :
                 forall x12 x11, 
                  (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U45 x12 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U31_bounded :
                 forall x12 x11, 
                  (0 <= x11) ->(0 <= x12) ->0 <= P_id_U31 x12 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__isNat_bounded :
                 forall x11, (0 <= x11) ->0 <= P_id_a__isNat x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_isNatIListKind_bounded :
                 forall x11, (0 <= x11) ->0 <= P_id_isNatIListKind x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U82_bounded :
                 forall x12 x13 x11, 
                  (0 <= x11) ->
                   (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U82 x13 x12 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U46_bounded :
                 forall x11, (0 <= x11) ->0 <= P_id_U46 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_cons_bounded :
                 forall x12 x11, 
                  (0 <= x11) ->(0 <= x12) ->0 <= P_id_cons x12 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U52_bounded :
                 forall x11, (0 <= x11) ->0 <= P_id_U52 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U85_bounded :
                 forall x12 x11, 
                  (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U85 x12 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U33_bounded :
                 forall x11, (0 <= x11) ->0 <= P_id_a__U33 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U21_bounded :
                 forall x12 x11, 
                  (0 <= x11) ->(0 <= x12) ->0 <= P_id_U21 x12 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__isNatList_bounded :
                 forall x11, (0 <= x11) ->0 <= P_id_a__isNatList x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U86_bounded :
                 forall x11, (0 <= x11) ->0 <= P_id_U86 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_mark_bounded :
                 forall x11, (0 <= x11) ->0 <= P_id_mark x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U51_bounded :
                 forall x12 x11, 
                  (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U51 x12 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U41_bounded :
                 forall x12 x13 x11, 
                  (0 <= x11) ->
                   (0 <= x12) ->(0 <= x13) ->0 <= P_id_U41 x13 x12 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_tt_bounded : 0 <= P_id_tt .
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U82_bounded :
                 forall x12 x13 x11, 
                  (0 <= x11) ->
                   (0 <= x12) ->(0 <= x13) ->0 <= P_id_U82 x13 x12 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U93_bounded :
                 forall x12 x13 x11, 
                  (0 <= x11) ->
                   (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U93 x13 x12 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U44_bounded :
                 forall x12 x13 x11, 
                  (0 <= x11) ->
                   (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U44 x13 x12 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_isNat_bounded :
                 forall x11, (0 <= x11) ->0 <= P_id_isNat x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U23_bounded :
                 forall x11, (0 <= x11) ->0 <= P_id_a__U23 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U94_bounded :
                 forall x12 x11, 
                  (0 <= x11) ->(0 <= x12) ->0 <= P_id_U94 x12 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U12_bounded :
                 forall x12 x11, 
                  (0 <= x11) ->(0 <= x12) ->0 <= P_id_U12 x12 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U81_bounded :
                 forall x12 x13 x11, 
                  (0 <= x11) ->
                   (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U81 x13 x12 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U45_bounded :
                 forall x12 x11, 
                  (0 <= x11) ->(0 <= x12) ->0 <= P_id_U45 x12 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_zeros_bounded : 0 <= P_id_zeros .
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U71_bounded :
                 forall x11, (0 <= x11) ->0 <= P_id_U71 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U91_bounded :
                 forall x12 x13 x11, 
                  (0 <= x11) ->
                   (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U91 x13 x12 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U42_bounded :
                 forall x12 x13 x11, 
                  (0 <= x11) ->
                   (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U42 x13 x12 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_isNatKind_bounded :
                 forall x11, (0 <= x11) ->0 <= P_id_isNatKind x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U22_bounded :
                 forall x12 x11, 
                  (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U22 x12 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U92_bounded :
                 forall x12 x13 x11, 
                  (0 <= x11) ->
                   (0 <= x12) ->(0 <= x13) ->0 <= P_id_U92 x13 x12 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_nil_bounded : 0 <= P_id_nil .
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U61_bounded :
                 forall x11, (0 <= x11) ->0 <= P_id_a__U61 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U43_bounded :
                 forall x12 x13 x11, 
                  (0 <= x11) ->
                   (0 <= x12) ->(0 <= x13) ->0 <= P_id_U43 x13 x12 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__isNatIListKind_bounded :
                 forall x11, (0 <= x11) ->0 <= P_id_a__isNatIListKind x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U84_bounded :
                 forall x12 x13 x11, 
                  (0 <= x11) ->
                   (0 <= x12) ->(0 <= x13) ->0 <= P_id_U84 x13 x12 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_s_bounded :
                 forall x11, (0 <= x11) ->0 <= P_id_s x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U46_bounded :
                 forall x11, (0 <= x11) ->0 <= P_id_a__U46 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U32_bounded :
                 forall x12 x11, 
                  (0 <= x11) ->(0 <= x12) ->0 <= P_id_U32 x12 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U31_bounded :
                 forall x12 x11, 
                  (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U31 x12 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_U13_bounded :
                 forall x11, (0 <= x11) ->0 <= P_id_U13 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_a__U83_bounded :
                 forall x12 x13 x11, 
                  (0 <= x11) ->
                   (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U83 x13 x12 x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_isNatIList_bounded :
                 forall x11, (0 <= x11) ->0 <= P_id_isNatIList x11.
                Proof.
                  intros .
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Definition measure  := 
                  InterpZ.measure 0 P_id_a__zeros P_id_U51 P_id_a__U84 
                   P_id_a__U32 P_id_isNatList P_id_a__U13 P_id_U85 
                   P_id_a__length P_id_a__isNatIList P_id_U33 P_id_a__U11 
                   P_id_U81 P_id_a__U92 P_id_a__U43 P_id_U23 
                   P_id_a__isNatKind P_id_U93 P_id_U11 P_id_a__U71 P_id_U44 
                   P_id_0 P_id_U61 P_id_a__U86 P_id_a__U41 P_id_U22 
                   P_id_a__U21 P_id_U91 P_id_length P_id_a__U52 P_id_U42 
                   P_id_a__U12 P_id_U83 P_id_a__U94 P_id_a__U45 P_id_U31 
                   P_id_a__isNat P_id_isNatIListKind P_id_a__U82 P_id_U46 
                   P_id_cons P_id_U52 P_id_a__U85 P_id_a__U33 P_id_U21 
                   P_id_a__isNatList P_id_U86 P_id_mark P_id_a__U51 P_id_U41 
                   P_id_tt P_id_U82 P_id_a__U93 P_id_a__U44 P_id_isNat 
                   P_id_a__U23 P_id_U94 P_id_U12 P_id_a__U81 P_id_U45 
                   P_id_zeros P_id_U71 P_id_a__U91 P_id_a__U42 
                   P_id_isNatKind P_id_a__U22 P_id_U92 P_id_nil P_id_a__U61 
                   P_id_U43 P_id_a__isNatIListKind P_id_U84 P_id_s 
                   P_id_a__U46 P_id_U32 P_id_a__U31 P_id_U13 P_id_a__U83 
                   P_id_isNatIList.
                
                Lemma measure_equation :
                 forall t, 
                  measure t = match t with
                                | (algebra.Alg.Term algebra.F.id_a__zeros 
                                   nil) =>
                                 P_id_a__zeros 
                                | (algebra.Alg.Term algebra.F.id_U51 (x12::
                                   x11::nil)) =>
                                 P_id_U51 (measure x12) (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U84 
                                   (x13::x12::x11::nil)) =>
                                 P_id_a__U84 (measure x13) (measure x12) 
                                  (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U32 
                                   (x12::x11::nil)) =>
                                 P_id_a__U32 (measure x12) (measure x11)
                                | (algebra.Alg.Term algebra.F.id_isNatList 
                                   (x11::nil)) =>
                                 P_id_isNatList (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U13 
                                   (x11::nil)) =>
                                 P_id_a__U13 (measure x11)
                                | (algebra.Alg.Term algebra.F.id_U85 (x12::
                                   x11::nil)) =>
                                 P_id_U85 (measure x12) (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__length 
                                   (x11::nil)) =>
                                 P_id_a__length (measure x11)
                                | (algebra.Alg.Term 
                                   algebra.F.id_a__isNatIList (x11::nil)) =>
                                 P_id_a__isNatIList (measure x11)
                                | (algebra.Alg.Term algebra.F.id_U33 
                                   (x11::nil)) =>
                                 P_id_U33 (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U11 
                                   (x12::x11::nil)) =>
                                 P_id_a__U11 (measure x12) (measure x11)
                                | (algebra.Alg.Term algebra.F.id_U81 (x13::
                                   x12::x11::nil)) =>
                                 P_id_U81 (measure x13) (measure x12) 
                                  (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U92 
                                   (x13::x12::x11::nil)) =>
                                 P_id_a__U92 (measure x13) (measure x12) 
                                  (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U43 
                                   (x13::x12::x11::nil)) =>
                                 P_id_a__U43 (measure x13) (measure x12) 
                                  (measure x11)
                                | (algebra.Alg.Term algebra.F.id_U23 
                                   (x11::nil)) =>
                                 P_id_U23 (measure x11)
                                | (algebra.Alg.Term 
                                   algebra.F.id_a__isNatKind (x11::nil)) =>
                                 P_id_a__isNatKind (measure x11)
                                | (algebra.Alg.Term algebra.F.id_U93 (x13::
                                   x12::x11::nil)) =>
                                 P_id_U93 (measure x13) (measure x12) 
                                  (measure x11)
                                | (algebra.Alg.Term algebra.F.id_U11 (x12::
                                   x11::nil)) =>
                                 P_id_U11 (measure x12) (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U71 
                                   (x11::nil)) =>
                                 P_id_a__U71 (measure x11)
                                | (algebra.Alg.Term algebra.F.id_U44 (x13::
                                   x12::x11::nil)) =>
                                 P_id_U44 (measure x13) (measure x12) 
                                  (measure x11)
                                | (algebra.Alg.Term algebra.F.id_0 nil) =>
                                 P_id_0 
                                | (algebra.Alg.Term algebra.F.id_U61 
                                   (x11::nil)) =>
                                 P_id_U61 (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U86 
                                   (x11::nil)) =>
                                 P_id_a__U86 (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U41 
                                   (x13::x12::x11::nil)) =>
                                 P_id_a__U41 (measure x13) (measure x12) 
                                  (measure x11)
                                | (algebra.Alg.Term algebra.F.id_U22 (x12::
                                   x11::nil)) =>
                                 P_id_U22 (measure x12) (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U21 
                                   (x12::x11::nil)) =>
                                 P_id_a__U21 (measure x12) (measure x11)
                                | (algebra.Alg.Term algebra.F.id_U91 (x13::
                                   x12::x11::nil)) =>
                                 P_id_U91 (measure x13) (measure x12) 
                                  (measure x11)
                                | (algebra.Alg.Term algebra.F.id_length 
                                   (x11::nil)) =>
                                 P_id_length (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U52 
                                   (x11::nil)) =>
                                 P_id_a__U52 (measure x11)
                                | (algebra.Alg.Term algebra.F.id_U42 (x13::
                                   x12::x11::nil)) =>
                                 P_id_U42 (measure x13) (measure x12) 
                                  (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U12 
                                   (x12::x11::nil)) =>
                                 P_id_a__U12 (measure x12) (measure x11)
                                | (algebra.Alg.Term algebra.F.id_U83 (x13::
                                   x12::x11::nil)) =>
                                 P_id_U83 (measure x13) (measure x12) 
                                  (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U94 
                                   (x12::x11::nil)) =>
                                 P_id_a__U94 (measure x12) (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U45 
                                   (x12::x11::nil)) =>
                                 P_id_a__U45 (measure x12) (measure x11)
                                | (algebra.Alg.Term algebra.F.id_U31 (x12::
                                   x11::nil)) =>
                                 P_id_U31 (measure x12) (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__isNat 
                                   (x11::nil)) =>
                                 P_id_a__isNat (measure x11)
                                | (algebra.Alg.Term 
                                   algebra.F.id_isNatIListKind (x11::nil)) =>
                                 P_id_isNatIListKind (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U82 
                                   (x13::x12::x11::nil)) =>
                                 P_id_a__U82 (measure x13) (measure x12) 
                                  (measure x11)
                                | (algebra.Alg.Term algebra.F.id_U46 
                                   (x11::nil)) =>
                                 P_id_U46 (measure x11)
                                | (algebra.Alg.Term algebra.F.id_cons (x12::
                                   x11::nil)) =>
                                 P_id_cons (measure x12) (measure x11)
                                | (algebra.Alg.Term algebra.F.id_U52 
                                   (x11::nil)) =>
                                 P_id_U52 (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U85 
                                   (x12::x11::nil)) =>
                                 P_id_a__U85 (measure x12) (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U33 
                                   (x11::nil)) =>
                                 P_id_a__U33 (measure x11)
                                | (algebra.Alg.Term algebra.F.id_U21 (x12::
                                   x11::nil)) =>
                                 P_id_U21 (measure x12) (measure x11)
                                | (algebra.Alg.Term 
                                   algebra.F.id_a__isNatList (x11::nil)) =>
                                 P_id_a__isNatList (measure x11)
                                | (algebra.Alg.Term algebra.F.id_U86 
                                   (x11::nil)) =>
                                 P_id_U86 (measure x11)
                                | (algebra.Alg.Term algebra.F.id_mark 
                                   (x11::nil)) =>
                                 P_id_mark (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U51 
                                   (x12::x11::nil)) =>
                                 P_id_a__U51 (measure x12) (measure x11)
                                | (algebra.Alg.Term algebra.F.id_U41 (x13::
                                   x12::x11::nil)) =>
                                 P_id_U41 (measure x13) (measure x12) 
                                  (measure x11)
                                | (algebra.Alg.Term algebra.F.id_tt nil) =>
                                 P_id_tt 
                                | (algebra.Alg.Term algebra.F.id_U82 (x13::
                                   x12::x11::nil)) =>
                                 P_id_U82 (measure x13) (measure x12) 
                                  (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U93 
                                   (x13::x12::x11::nil)) =>
                                 P_id_a__U93 (measure x13) (measure x12) 
                                  (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U44 
                                   (x13::x12::x11::nil)) =>
                                 P_id_a__U44 (measure x13) (measure x12) 
                                  (measure x11)
                                | (algebra.Alg.Term algebra.F.id_isNat 
                                   (x11::nil)) =>
                                 P_id_isNat (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U23 
                                   (x11::nil)) =>
                                 P_id_a__U23 (measure x11)
                                | (algebra.Alg.Term algebra.F.id_U94 (x12::
                                   x11::nil)) =>
                                 P_id_U94 (measure x12) (measure x11)
                                | (algebra.Alg.Term algebra.F.id_U12 (x12::
                                   x11::nil)) =>
                                 P_id_U12 (measure x12) (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U81 
                                   (x13::x12::x11::nil)) =>
                                 P_id_a__U81 (measure x13) (measure x12) 
                                  (measure x11)
                                | (algebra.Alg.Term algebra.F.id_U45 (x12::
                                   x11::nil)) =>
                                 P_id_U45 (measure x12) (measure x11)
                                | (algebra.Alg.Term algebra.F.id_zeros nil) =>
                                 P_id_zeros 
                                | (algebra.Alg.Term algebra.F.id_U71 
                                   (x11::nil)) =>
                                 P_id_U71 (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U91 
                                   (x13::x12::x11::nil)) =>
                                 P_id_a__U91 (measure x13) (measure x12) 
                                  (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U42 
                                   (x13::x12::x11::nil)) =>
                                 P_id_a__U42 (measure x13) (measure x12) 
                                  (measure x11)
                                | (algebra.Alg.Term algebra.F.id_isNatKind 
                                   (x11::nil)) =>
                                 P_id_isNatKind (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U22 
                                   (x12::x11::nil)) =>
                                 P_id_a__U22 (measure x12) (measure x11)
                                | (algebra.Alg.Term algebra.F.id_U92 (x13::
                                   x12::x11::nil)) =>
                                 P_id_U92 (measure x13) (measure x12) 
                                  (measure x11)
                                | (algebra.Alg.Term algebra.F.id_nil nil) =>
                                 P_id_nil 
                                | (algebra.Alg.Term algebra.F.id_a__U61 
                                   (x11::nil)) =>
                                 P_id_a__U61 (measure x11)
                                | (algebra.Alg.Term algebra.F.id_U43 (x13::
                                   x12::x11::nil)) =>
                                 P_id_U43 (measure x13) (measure x12) 
                                  (measure x11)
                                | (algebra.Alg.Term 
                                   algebra.F.id_a__isNatIListKind (x11::nil)) =>
                                 P_id_a__isNatIListKind (measure x11)
                                | (algebra.Alg.Term algebra.F.id_U84 (x13::
                                   x12::x11::nil)) =>
                                 P_id_U84 (measure x13) (measure x12) 
                                  (measure x11)
                                | (algebra.Alg.Term algebra.F.id_s 
                                   (x11::nil)) =>
                                 P_id_s (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U46 
                                   (x11::nil)) =>
                                 P_id_a__U46 (measure x11)
                                | (algebra.Alg.Term algebra.F.id_U32 (x12::
                                   x11::nil)) =>
                                 P_id_U32 (measure x12) (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U31 
                                   (x12::x11::nil)) =>
                                 P_id_a__U31 (measure x12) (measure x11)
                                | (algebra.Alg.Term algebra.F.id_U13 
                                   (x11::nil)) =>
                                 P_id_U13 (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U83 
                                   (x13::x12::x11::nil)) =>
                                 P_id_a__U83 (measure x13) (measure x12) 
                                  (measure x11)
                                | (algebra.Alg.Term algebra.F.id_isNatIList 
                                   (x11::nil)) =>
                                 P_id_isNatIList (measure x11)
                                | _ => 0
                                end.
                Proof.
                  intros t;case t;intros ;apply refl_equal.
                Qed.
                
                Lemma measure_bounded : forall t, 0 <= measure t.
                Proof.
                  unfold measure in |-*.
                  
                  apply InterpZ.measure_bounded;
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                Qed.
                
                Ltac generate_pos_hyp  :=
                 match goal with
                   | H:context [measure ?x] |- _ =>
                    let v := fresh "v" in 
                     (let H := fresh "h" in 
                       (set (H:=measure_bounded x) in *;
                         set (v:=measure x) in *;clearbody H;clearbody v))
                   |  |- context [measure ?x] =>
                    let v := fresh "v" in 
                     (let H := fresh "h" in 
                       (set (H:=measure_bounded x) in *;
                         set (v:=measure x) in *;clearbody H;clearbody v))
                   end
                 .
                
                Lemma rules_monotonic :
                 forall l r, 
                  (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) ->
                   measure r <= measure l.
                Proof.
                  intros l r H.
                  fold measure in |-*.
                  
                  inversion H;clear H;subst;inversion H0;clear H0;subst;
                   simpl algebra.EQT.T.apply_subst in |-*;
                   repeat (
                   match goal with
                     |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
                      rewrite (measure_equation (algebra.Alg.Term f t))
                     end
                   );repeat (generate_pos_hyp );
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma measure_star_monotonic :
                 forall l r, 
                  (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                             r l) ->measure r <= measure l.
                Proof.
                  unfold measure in *.
                  apply InterpZ.measure_star_monotonic.
                  intros ;apply P_id_U51_monotonic;assumption.
                  intros ;apply P_id_a__U84_monotonic;assumption.
                  intros ;apply P_id_a__U32_monotonic;assumption.
                  intros ;apply P_id_isNatList_monotonic;assumption.
                  intros ;apply P_id_a__U13_monotonic;assumption.
                  intros ;apply P_id_U85_monotonic;assumption.
                  intros ;apply P_id_a__length_monotonic;assumption.
                  intros ;apply P_id_a__isNatIList_monotonic;assumption.
                  intros ;apply P_id_U33_monotonic;assumption.
                  intros ;apply P_id_a__U11_monotonic;assumption.
                  intros ;apply P_id_U81_monotonic;assumption.
                  intros ;apply P_id_a__U92_monotonic;assumption.
                  intros ;apply P_id_a__U43_monotonic;assumption.
                  intros ;apply P_id_U23_monotonic;assumption.
                  intros ;apply P_id_a__isNatKind_monotonic;assumption.
                  intros ;apply P_id_U93_monotonic;assumption.
                  intros ;apply P_id_U11_monotonic;assumption.
                  intros ;apply P_id_a__U71_monotonic;assumption.
                  intros ;apply P_id_U44_monotonic;assumption.
                  intros ;apply P_id_U61_monotonic;assumption.
                  intros ;apply P_id_a__U86_monotonic;assumption.
                  intros ;apply P_id_a__U41_monotonic;assumption.
                  intros ;apply P_id_U22_monotonic;assumption.
                  intros ;apply P_id_a__U21_monotonic;assumption.
                  intros ;apply P_id_U91_monotonic;assumption.
                  intros ;apply P_id_length_monotonic;assumption.
                  intros ;apply P_id_a__U52_monotonic;assumption.
                  intros ;apply P_id_U42_monotonic;assumption.
                  intros ;apply P_id_a__U12_monotonic;assumption.
                  intros ;apply P_id_U83_monotonic;assumption.
                  intros ;apply P_id_a__U94_monotonic;assumption.
                  intros ;apply P_id_a__U45_monotonic;assumption.
                  intros ;apply P_id_U31_monotonic;assumption.
                  intros ;apply P_id_a__isNat_monotonic;assumption.
                  intros ;apply P_id_isNatIListKind_monotonic;assumption.
                  intros ;apply P_id_a__U82_monotonic;assumption.
                  intros ;apply P_id_U46_monotonic;assumption.
                  intros ;apply P_id_cons_monotonic;assumption.
                  intros ;apply P_id_U52_monotonic;assumption.
                  intros ;apply P_id_a__U85_monotonic;assumption.
                  intros ;apply P_id_a__U33_monotonic;assumption.
                  intros ;apply P_id_U21_monotonic;assumption.
                  intros ;apply P_id_a__isNatList_monotonic;assumption.
                  intros ;apply P_id_U86_monotonic;assumption.
                  intros ;apply P_id_mark_monotonic;assumption.
                  intros ;apply P_id_a__U51_monotonic;assumption.
                  intros ;apply P_id_U41_monotonic;assumption.
                  intros ;apply P_id_U82_monotonic;assumption.
                  intros ;apply P_id_a__U93_monotonic;assumption.
                  intros ;apply P_id_a__U44_monotonic;assumption.
                  intros ;apply P_id_isNat_monotonic;assumption.
                  intros ;apply P_id_a__U23_monotonic;assumption.
                  intros ;apply P_id_U94_monotonic;assumption.
                  intros ;apply P_id_U12_monotonic;assumption.
                  intros ;apply P_id_a__U81_monotonic;assumption.
                  intros ;apply P_id_U45_monotonic;assumption.
                  intros ;apply P_id_U71_monotonic;assumption.
                  intros ;apply P_id_a__U91_monotonic;assumption.
                  intros ;apply P_id_a__U42_monotonic;assumption.
                  intros ;apply P_id_isNatKind_monotonic;assumption.
                  intros ;apply P_id_a__U22_monotonic;assumption.
                  intros ;apply P_id_U92_monotonic;assumption.
                  intros ;apply P_id_a__U61_monotonic;assumption.
                  intros ;apply P_id_U43_monotonic;assumption.
                  intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
                  intros ;apply P_id_U84_monotonic;assumption.
                  intros ;apply P_id_s_monotonic;assumption.
                  intros ;apply P_id_a__U46_monotonic;assumption.
                  intros ;apply P_id_U32_monotonic;assumption.
                  intros ;apply P_id_a__U31_monotonic;assumption.
                  intros ;apply P_id_U13_monotonic;assumption.
                  intros ;apply P_id_a__U83_monotonic;assumption.
                  intros ;apply P_id_isNatIList_monotonic;assumption.
                  intros ;apply P_id_a__zeros_bounded;assumption.
                  intros ;apply P_id_U51_bounded;assumption.
                  intros ;apply P_id_a__U84_bounded;assumption.
                  intros ;apply P_id_a__U32_bounded;assumption.
                  intros ;apply P_id_isNatList_bounded;assumption.
                  intros ;apply P_id_a__U13_bounded;assumption.
                  intros ;apply P_id_U85_bounded;assumption.
                  intros ;apply P_id_a__length_bounded;assumption.
                  intros ;apply P_id_a__isNatIList_bounded;assumption.
                  intros ;apply P_id_U33_bounded;assumption.
                  intros ;apply P_id_a__U11_bounded;assumption.
                  intros ;apply P_id_U81_bounded;assumption.
                  intros ;apply P_id_a__U92_bounded;assumption.
                  intros ;apply P_id_a__U43_bounded;assumption.
                  intros ;apply P_id_U23_bounded;assumption.
                  intros ;apply P_id_a__isNatKind_bounded;assumption.
                  intros ;apply P_id_U93_bounded;assumption.
                  intros ;apply P_id_U11_bounded;assumption.
                  intros ;apply P_id_a__U71_bounded;assumption.
                  intros ;apply P_id_U44_bounded;assumption.
                  intros ;apply P_id_0_bounded;assumption.
                  intros ;apply P_id_U61_bounded;assumption.
                  intros ;apply P_id_a__U86_bounded;assumption.
                  intros ;apply P_id_a__U41_bounded;assumption.
                  intros ;apply P_id_U22_bounded;assumption.
                  intros ;apply P_id_a__U21_bounded;assumption.
                  intros ;apply P_id_U91_bounded;assumption.
                  intros ;apply P_id_length_bounded;assumption.
                  intros ;apply P_id_a__U52_bounded;assumption.
                  intros ;apply P_id_U42_bounded;assumption.
                  intros ;apply P_id_a__U12_bounded;assumption.
                  intros ;apply P_id_U83_bounded;assumption.
                  intros ;apply P_id_a__U94_bounded;assumption.
                  intros ;apply P_id_a__U45_bounded;assumption.
                  intros ;apply P_id_U31_bounded;assumption.
                  intros ;apply P_id_a__isNat_bounded;assumption.
                  intros ;apply P_id_isNatIListKind_bounded;assumption.
                  intros ;apply P_id_a__U82_bounded;assumption.
                  intros ;apply P_id_U46_bounded;assumption.
                  intros ;apply P_id_cons_bounded;assumption.
                  intros ;apply P_id_U52_bounded;assumption.
                  intros ;apply P_id_a__U85_bounded;assumption.
                  intros ;apply P_id_a__U33_bounded;assumption.
                  intros ;apply P_id_U21_bounded;assumption.
                  intros ;apply P_id_a__isNatList_bounded;assumption.
                  intros ;apply P_id_U86_bounded;assumption.
                  intros ;apply P_id_mark_bounded;assumption.
                  intros ;apply P_id_a__U51_bounded;assumption.
                  intros ;apply P_id_U41_bounded;assumption.
                  intros ;apply P_id_tt_bounded;assumption.
                  intros ;apply P_id_U82_bounded;assumption.
                  intros ;apply P_id_a__U93_bounded;assumption.
                  intros ;apply P_id_a__U44_bounded;assumption.
                  intros ;apply P_id_isNat_bounded;assumption.
                  intros ;apply P_id_a__U23_bounded;assumption.
                  intros ;apply P_id_U94_bounded;assumption.
                  intros ;apply P_id_U12_bounded;assumption.
                  intros ;apply P_id_a__U81_bounded;assumption.
                  intros ;apply P_id_U45_bounded;assumption.
                  intros ;apply P_id_zeros_bounded;assumption.
                  intros ;apply P_id_U71_bounded;assumption.
                  intros ;apply P_id_a__U91_bounded;assumption.
                  intros ;apply P_id_a__U42_bounded;assumption.
                  intros ;apply P_id_isNatKind_bounded;assumption.
                  intros ;apply P_id_a__U22_bounded;assumption.
                  intros ;apply P_id_U92_bounded;assumption.
                  intros ;apply P_id_nil_bounded;assumption.
                  intros ;apply P_id_a__U61_bounded;assumption.
                  intros ;apply P_id_U43_bounded;assumption.
                  intros ;apply P_id_a__isNatIListKind_bounded;assumption.
                  intros ;apply P_id_U84_bounded;assumption.
                  intros ;apply P_id_s_bounded;assumption.
                  intros ;apply P_id_a__U46_bounded;assumption.
                  intros ;apply P_id_U32_bounded;assumption.
                  intros ;apply P_id_a__U31_bounded;assumption.
                  intros ;apply P_id_U13_bounded;assumption.
                  intros ;apply P_id_a__U83_bounded;assumption.
                  intros ;apply P_id_isNatIList_bounded;assumption.
                  apply rules_monotonic.
                Qed.
                
                Definition P_id_A__U46 (x11:Z) := 0.
                
                Definition P_id_A__ISNATILISTKIND (x11:Z) := 0.
                
                Definition P_id_A__U61 (x11:Z) := 0.
                
                Definition P_id_A__U85 (x11:Z) (x12:Z) := 0.
                
                Definition P_id_A__U31 (x11:Z) (x12:Z) := 0.
                
                Definition P_id_A__U81 (x11:Z) (x12:Z) (x13:Z) := 0.
                
                Definition P_id_A__U22 (x11:Z) (x12:Z) := 0.
                
                Definition P_id_A__U93 (x11:Z) (x12:Z) (x13:Z) := 0.
                
                Definition P_id_A__U42 (x11:Z) (x12:Z) (x13:Z) := 0.
                
                Definition P_id_A__U51 (x11:Z) (x12:Z) := 0.
                
                Definition P_id_A__ISNATLIST (x11:Z) := 0.
                
                Definition P_id_A__ZEROS  := 0.
                
                Definition P_id_A__U91 (x11:Z) (x12:Z) (x13:Z) := 0.
                
                Definition P_id_A__U33 (x11:Z) := 0.
                
                Definition P_id_A__U83 (x11:Z) (x12:Z) (x13:Z) := 0.
                
                Definition P_id_A__U23 (x11:Z) := 0.
                
                Definition P_id_A__U11 (x11:Z) (x12:Z) := 0.
                
                Definition P_id_A__LENGTH (x11:Z) := 0.
                
                Definition P_id_A__U44 (x11:Z) (x12:Z) (x13:Z) := 0.
                
                Definition P_id_A__ISNATILIST (x11:Z) := 0.
                
                Definition P_id_A__U13 (x11:Z) := 0.
                
                Definition P_id_A__U71 (x11:Z) := 0.
                
                Definition P_id_A__U86 (x11:Z) := 0.
                
                Definition P_id_A__U32 (x11:Z) (x12:Z) := 0.
                
                Definition P_id_A__U82 (x11:Z) (x12:Z) (x13:Z) := 0.
                
                Definition P_id_A__ISNATKIND (x11:Z) := 0.
                
                Definition P_id_A__U94 (x11:Z) (x12:Z) := 0.
                
                Definition P_id_A__U43 (x11:Z) (x12:Z) (x13:Z) := 0.
                
                Definition P_id_A__U52 (x11:Z) := 0.
                
                Definition P_id_A__U21 (x11:Z) (x12:Z) := 0.
                
                Definition P_id_A__U92 (x11:Z) (x12:Z) (x13:Z) := 0.
                
                Definition P_id_A__U41 (x11:Z) (x12:Z) (x13:Z) := 0.
                
                Definition P_id_A__U84 (x11:Z) (x12:Z) (x13:Z) := 0.
                
                Definition P_id_A__ISNAT (x11:Z) := 0.
                
                Definition P_id_A__U12 (x11:Z) (x12:Z) := 0.
                
                Definition P_id_MARK (x11:Z) := 1* x11.
                
                Definition P_id_A__U45 (x11:Z) (x12:Z) := 0.
                
                Lemma P_id_A__U46_monotonic :
                 forall x12 x11, 
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_A__U46 x12 <= P_id_A__U46 x11.
                Proof.
                  intros x12 x11.
                  intros [H_1 H_0].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_A__ISNATILISTKIND_monotonic :
                 forall x12 x11, 
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_A__ISNATILISTKIND x12 <= P_id_A__ISNATILISTKIND x11.
                Proof.
                  intros x12 x11.
                  intros [H_1 H_0].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_A__U61_monotonic :
                 forall x12 x11, 
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_A__U61 x12 <= P_id_A__U61 x11.
                Proof.
                  intros x12 x11.
                  intros [H_1 H_0].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_A__U85_monotonic :
                 forall x12 x14 x13 x11, 
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_A__U85 x12 x14 <= P_id_A__U85 x11 x13.
                Proof.
                  intros x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_A__U31_monotonic :
                 forall x12 x14 x13 x11, 
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_A__U31 x12 x14 <= P_id_A__U31 x11 x13.
                Proof.
                  intros x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_A__U81_monotonic :
                 forall x16 x12 x14 x13 x11 x15, 
                  (0 <= x16)/\ (x16 <= x15) ->
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_A__U81 x12 x14 x16 <= P_id_A__U81 x11 x13 x15.
                Proof.
                  intros x16 x15 x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  intros [H_5 H_4].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_A__U22_monotonic :
                 forall x12 x14 x13 x11, 
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_A__U22 x12 x14 <= P_id_A__U22 x11 x13.
                Proof.
                  intros x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_A__U93_monotonic :
                 forall x16 x12 x14 x13 x11 x15, 
                  (0 <= x16)/\ (x16 <= x15) ->
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_A__U93 x12 x14 x16 <= P_id_A__U93 x11 x13 x15.
                Proof.
                  intros x16 x15 x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  intros [H_5 H_4].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_A__U42_monotonic :
                 forall x16 x12 x14 x13 x11 x15, 
                  (0 <= x16)/\ (x16 <= x15) ->
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_A__U42 x12 x14 x16 <= P_id_A__U42 x11 x13 x15.
                Proof.
                  intros x16 x15 x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  intros [H_5 H_4].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_A__U51_monotonic :
                 forall x12 x14 x13 x11, 
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_A__U51 x12 x14 <= P_id_A__U51 x11 x13.
                Proof.
                  intros x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_A__ISNATLIST_monotonic :
                 forall x12 x11, 
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_A__ISNATLIST x12 <= P_id_A__ISNATLIST x11.
                Proof.
                  intros x12 x11.
                  intros [H_1 H_0].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_A__U91_monotonic :
                 forall x16 x12 x14 x13 x11 x15, 
                  (0 <= x16)/\ (x16 <= x15) ->
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_A__U91 x12 x14 x16 <= P_id_A__U91 x11 x13 x15.
                Proof.
                  intros x16 x15 x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  intros [H_5 H_4].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_A__U33_monotonic :
                 forall x12 x11, 
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_A__U33 x12 <= P_id_A__U33 x11.
                Proof.
                  intros x12 x11.
                  intros [H_1 H_0].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_A__U83_monotonic :
                 forall x16 x12 x14 x13 x11 x15, 
                  (0 <= x16)/\ (x16 <= x15) ->
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_A__U83 x12 x14 x16 <= P_id_A__U83 x11 x13 x15.
                Proof.
                  intros x16 x15 x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  intros [H_5 H_4].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_A__U23_monotonic :
                 forall x12 x11, 
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_A__U23 x12 <= P_id_A__U23 x11.
                Proof.
                  intros x12 x11.
                  intros [H_1 H_0].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_A__U11_monotonic :
                 forall x12 x14 x13 x11, 
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_A__U11 x12 x14 <= P_id_A__U11 x11 x13.
                Proof.
                  intros x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_A__LENGTH_monotonic :
                 forall x12 x11, 
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_A__LENGTH x12 <= P_id_A__LENGTH x11.
                Proof.
                  intros x12 x11.
                  intros [H_1 H_0].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_A__U44_monotonic :
                 forall x16 x12 x14 x13 x11 x15, 
                  (0 <= x16)/\ (x16 <= x15) ->
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_A__U44 x12 x14 x16 <= P_id_A__U44 x11 x13 x15.
                Proof.
                  intros x16 x15 x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  intros [H_5 H_4].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_A__ISNATILIST_monotonic :
                 forall x12 x11, 
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_A__ISNATILIST x12 <= P_id_A__ISNATILIST x11.
                Proof.
                  intros x12 x11.
                  intros [H_1 H_0].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_A__U13_monotonic :
                 forall x12 x11, 
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_A__U13 x12 <= P_id_A__U13 x11.
                Proof.
                  intros x12 x11.
                  intros [H_1 H_0].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_A__U71_monotonic :
                 forall x12 x11, 
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_A__U71 x12 <= P_id_A__U71 x11.
                Proof.
                  intros x12 x11.
                  intros [H_1 H_0].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_A__U86_monotonic :
                 forall x12 x11, 
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_A__U86 x12 <= P_id_A__U86 x11.
                Proof.
                  intros x12 x11.
                  intros [H_1 H_0].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_A__U32_monotonic :
                 forall x12 x14 x13 x11, 
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_A__U32 x12 x14 <= P_id_A__U32 x11 x13.
                Proof.
                  intros x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_A__U82_monotonic :
                 forall x16 x12 x14 x13 x11 x15, 
                  (0 <= x16)/\ (x16 <= x15) ->
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_A__U82 x12 x14 x16 <= P_id_A__U82 x11 x13 x15.
                Proof.
                  intros x16 x15 x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  intros [H_5 H_4].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_A__ISNATKIND_monotonic :
                 forall x12 x11, 
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_A__ISNATKIND x12 <= P_id_A__ISNATKIND x11.
                Proof.
                  intros x12 x11.
                  intros [H_1 H_0].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_A__U94_monotonic :
                 forall x12 x14 x13 x11, 
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_A__U94 x12 x14 <= P_id_A__U94 x11 x13.
                Proof.
                  intros x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_A__U43_monotonic :
                 forall x16 x12 x14 x13 x11 x15, 
                  (0 <= x16)/\ (x16 <= x15) ->
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_A__U43 x12 x14 x16 <= P_id_A__U43 x11 x13 x15.
                Proof.
                  intros x16 x15 x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  intros [H_5 H_4].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_A__U52_monotonic :
                 forall x12 x11, 
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_A__U52 x12 <= P_id_A__U52 x11.
                Proof.
                  intros x12 x11.
                  intros [H_1 H_0].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_A__U21_monotonic :
                 forall x12 x14 x13 x11, 
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_A__U21 x12 x14 <= P_id_A__U21 x11 x13.
                Proof.
                  intros x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_A__U92_monotonic :
                 forall x16 x12 x14 x13 x11 x15, 
                  (0 <= x16)/\ (x16 <= x15) ->
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_A__U92 x12 x14 x16 <= P_id_A__U92 x11 x13 x15.
                Proof.
                  intros x16 x15 x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  intros [H_5 H_4].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_A__U41_monotonic :
                 forall x16 x12 x14 x13 x11 x15, 
                  (0 <= x16)/\ (x16 <= x15) ->
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_A__U41 x12 x14 x16 <= P_id_A__U41 x11 x13 x15.
                Proof.
                  intros x16 x15 x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  intros [H_5 H_4].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_A__U84_monotonic :
                 forall x16 x12 x14 x13 x11 x15, 
                  (0 <= x16)/\ (x16 <= x15) ->
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_A__U84 x12 x14 x16 <= P_id_A__U84 x11 x13 x15.
                Proof.
                  intros x16 x15 x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  intros [H_5 H_4].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_A__ISNAT_monotonic :
                 forall x12 x11, 
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_A__ISNAT x12 <= P_id_A__ISNAT x11.
                Proof.
                  intros x12 x11.
                  intros [H_1 H_0].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_A__U12_monotonic :
                 forall x12 x14 x13 x11, 
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_A__U12 x12 x14 <= P_id_A__U12 x11 x13.
                Proof.
                  intros x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_MARK_monotonic :
                 forall x12 x11, 
                  (0 <= x12)/\ (x12 <= x11) ->P_id_MARK x12 <= P_id_MARK x11.
                Proof.
                  intros x12 x11.
                  intros [H_1 H_0].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma P_id_A__U45_monotonic :
                 forall x12 x14 x13 x11, 
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_A__U45 x12 x14 <= P_id_A__U45 x11 x13.
                Proof.
                  intros x14 x13 x12 x11.
                  intros [H_1 H_0].
                  intros [H_3 H_2].
                  
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
                Qed.
                
                Definition marked_measure  := 
                  InterpZ.marked_measure 0 P_id_a__zeros P_id_U51 
                   P_id_a__U84 P_id_a__U32 P_id_isNatList P_id_a__U13 
                   P_id_U85 P_id_a__length P_id_a__isNatIList P_id_U33 
                   P_id_a__U11 P_id_U81 P_id_a__U92 P_id_a__U43 P_id_U23 
                   P_id_a__isNatKind P_id_U93 P_id_U11 P_id_a__U71 P_id_U44 
                   P_id_0 P_id_U61 P_id_a__U86 P_id_a__U41 P_id_U22 
                   P_id_a__U21 P_id_U91 P_id_length P_id_a__U52 P_id_U42 
                   P_id_a__U12 P_id_U83 P_id_a__U94 P_id_a__U45 P_id_U31 
                   P_id_a__isNat P_id_isNatIListKind P_id_a__U82 P_id_U46 
                   P_id_cons P_id_U52 P_id_a__U85 P_id_a__U33 P_id_U21 
                   P_id_a__isNatList P_id_U86 P_id_mark P_id_a__U51 P_id_U41 
                   P_id_tt P_id_U82 P_id_a__U93 P_id_a__U44 P_id_isNat 
                   P_id_a__U23 P_id_U94 P_id_U12 P_id_a__U81 P_id_U45 
                   P_id_zeros P_id_U71 P_id_a__U91 P_id_a__U42 
                   P_id_isNatKind P_id_a__U22 P_id_U92 P_id_nil P_id_a__U61 
                   P_id_U43 P_id_a__isNatIListKind P_id_U84 P_id_s 
                   P_id_a__U46 P_id_U32 P_id_a__U31 P_id_U13 P_id_a__U83 
                   P_id_isNatIList P_id_A__U46 P_id_A__ISNATILISTKIND 
                   P_id_A__U61 P_id_A__U85 P_id_A__U31 P_id_A__U81 
                   P_id_A__U22 P_id_A__U93 P_id_A__U42 P_id_A__U51 
                   P_id_A__ISNATLIST P_id_A__ZEROS P_id_A__U91 P_id_A__U33 
                   P_id_A__U83 P_id_A__U23 P_id_A__U11 P_id_A__LENGTH 
                   P_id_A__U44 P_id_A__ISNATILIST P_id_A__U13 P_id_A__U71 
                   P_id_A__U86 P_id_A__U32 P_id_A__U82 P_id_A__ISNATKIND 
                   P_id_A__U94 P_id_A__U43 P_id_A__U52 P_id_A__U21 
                   P_id_A__U92 P_id_A__U41 P_id_A__U84 P_id_A__ISNAT 
                   P_id_A__U12 P_id_MARK P_id_A__U45.
                
                Lemma marked_measure_equation :
                 forall t, 
                  marked_measure t = match t with
                                       | (algebra.Alg.Term 
                                          algebra.F.id_a__U46 (x11::nil)) =>
                                        P_id_A__U46 (measure x11)
                                       | (algebra.Alg.Term 
                                          algebra.F.id_a__isNatIListKind 
                                          (x11::nil)) =>
                                        P_id_A__ISNATILISTKIND (measure x11)
                                       | (algebra.Alg.Term 
                                          algebra.F.id_a__U61 (x11::nil)) =>
                                        P_id_A__U61 (measure x11)
                                       | (algebra.Alg.Term 
                                          algebra.F.id_a__U85 (x12::
                                          x11::nil)) =>
                                        P_id_A__U85 (measure x12) 
                                         (measure x11)
                                       | (algebra.Alg.Term 
                                          algebra.F.id_a__U31 (x12::
                                          x11::nil)) =>
                                        P_id_A__U31 (measure x12) 
                                         (measure x11)
                                       | (algebra.Alg.Term 
                                          algebra.F.id_a__U81 (x13::x12::
                                          x11::nil)) =>
                                        P_id_A__U81 (measure x13) 
                                         (measure x12) (measure x11)
                                       | (algebra.Alg.Term 
                                          algebra.F.id_a__U22 (x12::
                                          x11::nil)) =>
                                        P_id_A__U22 (measure x12) 
                                         (measure x11)
                                       | (algebra.Alg.Term 
                                          algebra.F.id_a__U93 (x13::x12::
                                          x11::nil)) =>
                                        P_id_A__U93 (measure x13) 
                                         (measure x12) (measure x11)
                                       | (algebra.Alg.Term 
                                          algebra.F.id_a__U42 (x13::x12::
                                          x11::nil)) =>
                                        P_id_A__U42 (measure x13) 
                                         (measure x12) (measure x11)
                                       | (algebra.Alg.Term 
                                          algebra.F.id_a__U51 (x12::
                                          x11::nil)) =>
                                        P_id_A__U51 (measure x12) 
                                         (measure x11)
                                       | (algebra.Alg.Term 
                                          algebra.F.id_a__isNatList 
                                          (x11::nil)) =>
                                        P_id_A__ISNATLIST (measure x11)
                                       | (algebra.Alg.Term 
                                          algebra.F.id_a__zeros nil) =>
                                        P_id_A__ZEROS 
                                       | (algebra.Alg.Term 
                                          algebra.F.id_a__U91 (x13::x12::
                                          x11::nil)) =>
                                        P_id_A__U91 (measure x13) 
                                         (measure x12) (measure x11)
                                       | (algebra.Alg.Term 
                                          algebra.F.id_a__U33 (x11::nil)) =>
                                        P_id_A__U33 (measure x11)
                                       | (algebra.Alg.Term 
                                          algebra.F.id_a__U83 (x13::x12::
                                          x11::nil)) =>
                                        P_id_A__U83 (measure x13) 
                                         (measure x12) (measure x11)
                                       | (algebra.Alg.Term 
                                          algebra.F.id_a__U23 (x11::nil)) =>
                                        P_id_A__U23 (measure x11)
                                       | (algebra.Alg.Term 
                                          algebra.F.id_a__U11 (x12::
                                          x11::nil)) =>
                                        P_id_A__U11 (measure x12) 
                                         (measure x11)
                                       | (algebra.Alg.Term 
                                          algebra.F.id_a__length (x11::nil)) =>
                                        P_id_A__LENGTH (measure x11)
                                       | (algebra.Alg.Term 
                                          algebra.F.id_a__U44 (x13::x12::
                                          x11::nil)) =>
                                        P_id_A__U44 (measure x13) 
                                         (measure x12) (measure x11)
                                       | (algebra.Alg.Term 
                                          algebra.F.id_a__isNatIList 
                                          (x11::nil)) =>
                                        P_id_A__ISNATILIST (measure x11)
                                       | (algebra.Alg.Term 
                                          algebra.F.id_a__U13 (x11::nil)) =>
                                        P_id_A__U13 (measure x11)
                                       | (algebra.Alg.Term 
                                          algebra.F.id_a__U71 (x11::nil)) =>
                                        P_id_A__U71 (measure x11)
                                       | (algebra.Alg.Term 
                                          algebra.F.id_a__U86 (x11::nil)) =>
                                        P_id_A__U86 (measure x11)
                                       | (algebra.Alg.Term 
                                          algebra.F.id_a__U32 (x12::
                                          x11::nil)) =>
                                        P_id_A__U32 (measure x12) 
                                         (measure x11)
                                       | (algebra.Alg.Term 
                                          algebra.F.id_a__U82 (x13::x12::
                                          x11::nil)) =>
                                        P_id_A__U82 (measure x13) 
                                         (measure x12) (measure x11)
                                       | (algebra.Alg.Term 
                                          algebra.F.id_a__isNatKind 
                                          (x11::nil)) =>
                                        P_id_A__ISNATKIND (measure x11)
                                       | (algebra.Alg.Term 
                                          algebra.F.id_a__U94 (x12::
                                          x11::nil)) =>
                                        P_id_A__U94 (measure x12) 
                                         (measure x11)
                                       | (algebra.Alg.Term 
                                          algebra.F.id_a__U43 (x13::x12::
                                          x11::nil)) =>
                                        P_id_A__U43 (measure x13) 
                                         (measure x12) (measure x11)
                                       | (algebra.Alg.Term 
                                          algebra.F.id_a__U52 (x11::nil)) =>
                                        P_id_A__U52 (measure x11)
                                       | (algebra.Alg.Term 
                                          algebra.F.id_a__U21 (x12::
                                          x11::nil)) =>
                                        P_id_A__U21 (measure x12) 
                                         (measure x11)
                                       | (algebra.Alg.Term 
                                          algebra.F.id_a__U92 (x13::x12::
                                          x11::nil)) =>
                                        P_id_A__U92 (measure x13) 
                                         (measure x12) (measure x11)
                                       | (algebra.Alg.Term 
                                          algebra.F.id_a__U41 (x13::x12::
                                          x11::nil)) =>
                                        P_id_A__U41 (measure x13) 
                                         (measure x12) (measure x11)
                                       | (algebra.Alg.Term 
                                          algebra.F.id_a__U84 (x13::x12::
                                          x11::nil)) =>
                                        P_id_A__U84 (measure x13) 
                                         (measure x12) (measure x11)
                                       | (algebra.Alg.Term 
                                          algebra.F.id_a__isNat (x11::nil)) =>
                                        P_id_A__ISNAT (measure x11)
                                       | (algebra.Alg.Term 
                                          algebra.F.id_a__U12 (x12::
                                          x11::nil)) =>
                                        P_id_A__U12 (measure x12) 
                                         (measure x11)
                                       | (algebra.Alg.Term algebra.F.id_mark 
                                          (x11::nil)) =>
                                        P_id_MARK (measure x11)
                                       | (algebra.Alg.Term 
                                          algebra.F.id_a__U45 (x12::
                                          x11::nil)) =>
                                        P_id_A__U45 (measure x12) 
                                         (measure x11)
                                       | _ => measure t
                                       end.
                Proof.
                  reflexivity .
                Qed.
                
                Lemma marked_measure_star_monotonic :
                 forall f l1 l2, 
                  (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step 
                                                                    R_xml_0_deep_rew.R_xml_0_rules)
                                                                   ) 
                    l1 l2) ->
                   marked_measure (algebra.Alg.Term f l1) <= marked_measure 
                                                              (algebra.Alg.Term 
                                                                f l2).
                Proof.
                  unfold marked_measure in *.
                  apply InterpZ.marked_measure_star_monotonic.
                  intros ;apply P_id_U51_monotonic;assumption.
                  intros ;apply P_id_a__U84_monotonic;assumption.
                  intros ;apply P_id_a__U32_monotonic;assumption.
                  intros ;apply P_id_isNatList_monotonic;assumption.
                  intros ;apply P_id_a__U13_monotonic;assumption.
                  intros ;apply P_id_U85_monotonic;assumption.
                  intros ;apply P_id_a__length_monotonic;assumption.
                  intros ;apply P_id_a__isNatIList_monotonic;assumption.
                  intros ;apply P_id_U33_monotonic;assumption.
                  intros ;apply P_id_a__U11_monotonic;assumption.
                  intros ;apply P_id_U81_monotonic;assumption.
                  intros ;apply P_id_a__U92_monotonic;assumption.
                  intros ;apply P_id_a__U43_monotonic;assumption.
                  intros ;apply P_id_U23_monotonic;assumption.
                  intros ;apply P_id_a__isNatKind_monotonic;assumption.
                  intros ;apply P_id_U93_monotonic;assumption.
                  intros ;apply P_id_U11_monotonic;assumption.
                  intros ;apply P_id_a__U71_monotonic;assumption.
                  intros ;apply P_id_U44_monotonic;assumption.
                  intros ;apply P_id_U61_monotonic;assumption.
                  intros ;apply P_id_a__U86_monotonic;assumption.
                  intros ;apply P_id_a__U41_monotonic;assumption.
                  intros ;apply P_id_U22_monotonic;assumption.
                  intros ;apply P_id_a__U21_monotonic;assumption.
                  intros ;apply P_id_U91_monotonic;assumption.
                  intros ;apply P_id_length_monotonic;assumption.
                  intros ;apply P_id_a__U52_monotonic;assumption.
                  intros ;apply P_id_U42_monotonic;assumption.
                  intros ;apply P_id_a__U12_monotonic;assumption.
                  intros ;apply P_id_U83_monotonic;assumption.
                  intros ;apply P_id_a__U94_monotonic;assumption.
                  intros ;apply P_id_a__U45_monotonic;assumption.
                  intros ;apply P_id_U31_monotonic;assumption.
                  intros ;apply P_id_a__isNat_monotonic;assumption.
                  intros ;apply P_id_isNatIListKind_monotonic;assumption.
                  intros ;apply P_id_a__U82_monotonic;assumption.
                  intros ;apply P_id_U46_monotonic;assumption.
                  intros ;apply P_id_cons_monotonic;assumption.
                  intros ;apply P_id_U52_monotonic;assumption.
                  intros ;apply P_id_a__U85_monotonic;assumption.
                  intros ;apply P_id_a__U33_monotonic;assumption.
                  intros ;apply P_id_U21_monotonic;assumption.
                  intros ;apply P_id_a__isNatList_monotonic;assumption.
                  intros ;apply P_id_U86_monotonic;assumption.
                  intros ;apply P_id_mark_monotonic;assumption.
                  intros ;apply P_id_a__U51_monotonic;assumption.
                  intros ;apply P_id_U41_monotonic;assumption.
                  intros ;apply P_id_U82_monotonic;assumption.
                  intros ;apply P_id_a__U93_monotonic;assumption.
                  intros ;apply P_id_a__U44_monotonic;assumption.
                  intros ;apply P_id_isNat_monotonic;assumption.
                  intros ;apply P_id_a__U23_monotonic;assumption.
                  intros ;apply P_id_U94_monotonic;assumption.
                  intros ;apply P_id_U12_monotonic;assumption.
                  intros ;apply P_id_a__U81_monotonic;assumption.
                  intros ;apply P_id_U45_monotonic;assumption.
                  intros ;apply P_id_U71_monotonic;assumption.
                  intros ;apply P_id_a__U91_monotonic;assumption.
                  intros ;apply P_id_a__U42_monotonic;assumption.
                  intros ;apply P_id_isNatKind_monotonic;assumption.
                  intros ;apply P_id_a__U22_monotonic;assumption.
                  intros ;apply P_id_U92_monotonic;assumption.
                  intros ;apply P_id_a__U61_monotonic;assumption.
                  intros ;apply P_id_U43_monotonic;assumption.
                  intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
                  intros ;apply P_id_U84_monotonic;assumption.
                  intros ;apply P_id_s_monotonic;assumption.
                  intros ;apply P_id_a__U46_monotonic;assumption.
                  intros ;apply P_id_U32_monotonic;assumption.
                  intros ;apply P_id_a__U31_monotonic;assumption.
                  intros ;apply P_id_U13_monotonic;assumption.
                  intros ;apply P_id_a__U83_monotonic;assumption.
                  intros ;apply P_id_isNatIList_monotonic;assumption.
                  intros ;apply P_id_a__zeros_bounded;assumption.
                  intros ;apply P_id_U51_bounded;assumption.
                  intros ;apply P_id_a__U84_bounded;assumption.
                  intros ;apply P_id_a__U32_bounded;assumption.
                  intros ;apply P_id_isNatList_bounded;assumption.
                  intros ;apply P_id_a__U13_bounded;assumption.
                  intros ;apply P_id_U85_bounded;assumption.
                  intros ;apply P_id_a__length_bounded;assumption.
                  intros ;apply P_id_a__isNatIList_bounded;assumption.
                  intros ;apply P_id_U33_bounded;assumption.
                  intros ;apply P_id_a__U11_bounded;assumption.
                  intros ;apply P_id_U81_bounded;assumption.
                  intros ;apply P_id_a__U92_bounded;assumption.
                  intros ;apply P_id_a__U43_bounded;assumption.
                  intros ;apply P_id_U23_bounded;assumption.
                  intros ;apply P_id_a__isNatKind_bounded;assumption.
                  intros ;apply P_id_U93_bounded;assumption.
                  intros ;apply P_id_U11_bounded;assumption.
                  intros ;apply P_id_a__U71_bounded;assumption.
                  intros ;apply P_id_U44_bounded;assumption.
                  intros ;apply P_id_0_bounded;assumption.
                  intros ;apply P_id_U61_bounded;assumption.
                  intros ;apply P_id_a__U86_bounded;assumption.
                  intros ;apply P_id_a__U41_bounded;assumption.
                  intros ;apply P_id_U22_bounded;assumption.
                  intros ;apply P_id_a__U21_bounded;assumption.
                  intros ;apply P_id_U91_bounded;assumption.
                  intros ;apply P_id_length_bounded;assumption.
                  intros ;apply P_id_a__U52_bounded;assumption.
                  intros ;apply P_id_U42_bounded;assumption.
                  intros ;apply P_id_a__U12_bounded;assumption.
                  intros ;apply P_id_U83_bounded;assumption.
                  intros ;apply P_id_a__U94_bounded;assumption.
                  intros ;apply P_id_a__U45_bounded;assumption.
                  intros ;apply P_id_U31_bounded;assumption.
                  intros ;apply P_id_a__isNat_bounded;assumption.
                  intros ;apply P_id_isNatIListKind_bounded;assumption.
                  intros ;apply P_id_a__U82_bounded;assumption.
                  intros ;apply P_id_U46_bounded;assumption.
                  intros ;apply P_id_cons_bounded;assumption.
                  intros ;apply P_id_U52_bounded;assumption.
                  intros ;apply P_id_a__U85_bounded;assumption.
                  intros ;apply P_id_a__U33_bounded;assumption.
                  intros ;apply P_id_U21_bounded;assumption.
                  intros ;apply P_id_a__isNatList_bounded;assumption.
                  intros ;apply P_id_U86_bounded;assumption.
                  intros ;apply P_id_mark_bounded;assumption.
                  intros ;apply P_id_a__U51_bounded;assumption.
                  intros ;apply P_id_U41_bounded;assumption.
                  intros ;apply P_id_tt_bounded;assumption.
                  intros ;apply P_id_U82_bounded;assumption.
                  intros ;apply P_id_a__U93_bounded;assumption.
                  intros ;apply P_id_a__U44_bounded;assumption.
                  intros ;apply P_id_isNat_bounded;assumption.
                  intros ;apply P_id_a__U23_bounded;assumption.
                  intros ;apply P_id_U94_bounded;assumption.
                  intros ;apply P_id_U12_bounded;assumption.
                  intros ;apply P_id_a__U81_bounded;assumption.
                  intros ;apply P_id_U45_bounded;assumption.
                  intros ;apply P_id_zeros_bounded;assumption.
                  intros ;apply P_id_U71_bounded;assumption.
                  intros ;apply P_id_a__U91_bounded;assumption.
                  intros ;apply P_id_a__U42_bounded;assumption.
                  intros ;apply P_id_isNatKind_bounded;assumption.
                  intros ;apply P_id_a__U22_bounded;assumption.
                  intros ;apply P_id_U92_bounded;assumption.
                  intros ;apply P_id_nil_bounded;assumption.
                  intros ;apply P_id_a__U61_bounded;assumption.
                  intros ;apply P_id_U43_bounded;assumption.
                  intros ;apply P_id_a__isNatIListKind_bounded;assumption.
                  intros ;apply P_id_U84_bounded;assumption.
                  intros ;apply P_id_s_bounded;assumption.
                  intros ;apply P_id_a__U46_bounded;assumption.
                  intros ;apply P_id_U32_bounded;assumption.
                  intros ;apply P_id_a__U31_bounded;assumption.
                  intros ;apply P_id_U13_bounded;assumption.
                  intros ;apply P_id_a__U83_bounded;assumption.
                  intros ;apply P_id_isNatIList_bounded;assumption.
                  apply rules_monotonic.
                  intros ;apply P_id_A__U46_monotonic;assumption.
                  intros ;apply P_id_A__ISNATILISTKIND_monotonic;assumption.
                  intros ;apply P_id_A__U61_monotonic;assumption.
                  intros ;apply P_id_A__U85_monotonic;assumption.
                  intros ;apply P_id_A__U31_monotonic;assumption.
                  intros ;apply P_id_A__U81_monotonic;assumption.
                  intros ;apply P_id_A__U22_monotonic;assumption.
                  intros ;apply P_id_A__U93_monotonic;assumption.
                  intros ;apply P_id_A__U42_monotonic;assumption.
                  intros ;apply P_id_A__U51_monotonic;assumption.
                  intros ;apply P_id_A__ISNATLIST_monotonic;assumption.
                  intros ;apply P_id_A__U91_monotonic;assumption.
                  intros ;apply P_id_A__U33_monotonic;assumption.
                  intros ;apply P_id_A__U83_monotonic;assumption.
                  intros ;apply P_id_A__U23_monotonic;assumption.
                  intros ;apply P_id_A__U11_monotonic;assumption.
                  intros ;apply P_id_A__LENGTH_monotonic;assumption.
                  intros ;apply P_id_A__U44_monotonic;assumption.
                  intros ;apply P_id_A__ISNATILIST_monotonic;assumption.
                  intros ;apply P_id_A__U13_monotonic;assumption.
                  intros ;apply P_id_A__U71_monotonic;assumption.
                  intros ;apply P_id_A__U86_monotonic;assumption.
                  intros ;apply P_id_A__U32_monotonic;assumption.
                  intros ;apply P_id_A__U82_monotonic;assumption.
                  intros ;apply P_id_A__ISNATKIND_monotonic;assumption.
                  intros ;apply P_id_A__U94_monotonic;assumption.
                  intros ;apply P_id_A__U43_monotonic;assumption.
                  intros ;apply P_id_A__U52_monotonic;assumption.
                  intros ;apply P_id_A__U21_monotonic;assumption.
                  intros ;apply P_id_A__U92_monotonic;assumption.
                  intros ;apply P_id_A__U41_monotonic;assumption.
                  intros ;apply P_id_A__U84_monotonic;assumption.
                  intros ;apply P_id_A__ISNAT_monotonic;assumption.
                  intros ;apply P_id_A__U12_monotonic;assumption.
                  intros ;apply P_id_MARK_monotonic;assumption.
                  intros ;apply P_id_A__U45_monotonic;assumption.
                Qed.
                
                Ltac rewrite_and_unfold  :=
                 do 2 (rewrite marked_measure_equation);
                  repeat (
                  match goal with
                    |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
                     rewrite (measure_equation (algebra.Alg.Term f t))
                    | H:context [measure (algebra.Alg.Term ?f ?t)] |- 
                    _ =>
                     rewrite (measure_equation (algebra.Alg.Term f t)) in H|-
                    end
                  ).
                
                Definition lt a b := 
                  (Zwf.Zwf 0) (marked_measure a) (marked_measure b).
                
                Definition le a b := marked_measure a <= marked_measure b.
                
                Lemma lt_le_compat :
                 forall a b c, (lt a b) ->(le b c) ->lt a c.
                Proof.
                  unfold lt, le in *.
                  intros a b c.
                  apply (interp.le_lt_compat_right (interp.o_Z 0)).
                Qed.
                
                Lemma wf_lt : well_founded lt.
                Proof.
                  unfold lt in *.
                  apply Inverse_Image.wf_inverse_image with  (B:=Z).
                  apply Zwf.Zwf_well_founded.
                Qed.
                
                Lemma DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_strict_in_lt :
                 Relation_Definitions.inclusion _ 
                  DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_strict 
                  lt.
                Proof.
                  unfold Relation_Definitions.inclusion, lt in *.
                  
                  intros a b H;destruct H;
                   match goal with
                     |  |- (Zwf.Zwf 0) _ 
                            (marked_measure (algebra.Alg.Term ?f ?l)) =>
                      let l'' := algebra.Alg_ext.find_replacement l  in 
                       ((apply (interp.le_lt_compat_right (interp.o_Z 0)) with
                          (marked_measure (algebra.Alg.Term f l''));[
                         idtac|
                         apply marked_measure_star_monotonic;
                          repeat (
                          apply algebra.EQT_ext.one_step_list_refl_trans_clos
                          );(assumption)||(constructor 1)]))
                     end
                   ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                Qed.
                
                Lemma DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_in_le :
                 Relation_Definitions.inclusion _ 
                  DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large 
                  le.
                Proof.
                  unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *.
                  
                  intros a b H;destruct H;
                   match goal with
                     |  |- _ <= marked_measure (algebra.Alg.Term ?f ?l) =>
                      let l'' := algebra.Alg_ext.find_replacement l  in 
                       ((apply (interp.le_trans (interp.o_Z 0)) with
                          (marked_measure (algebra.Alg.Term f l''));[
                         idtac|
                         apply marked_measure_star_monotonic;
                          repeat (
                          apply algebra.EQT_ext.one_step_list_refl_trans_clos
                          );(assumption)||(constructor 1)]))
                     end
                   ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                Qed.
                
                Definition wf_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large  := 
                  WF_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large.wf
                  .
                
                
                Lemma wf :
                 well_founded WF_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large.DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large
                  .
                Proof.
                  intros x.
                  apply (well_founded_ind wf_lt).
                  clear x.
                  intros x.
                  pattern x.
                  
                  apply (@Acc_ind _ 
                          DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large)
                         .
                  clear x.
                  intros x _ IHx IHx'.
                  constructor.
                  intros y H.
                  
                  destruct H;
                   (apply IHx';
                     apply DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_strict_in_lt;
                     econstructor eassumption)||
                   ((apply IHx;[econstructor eassumption|
                     intros y' Hlt;apply IHx';
                      apply lt_le_compat with (1:=Hlt) ;
                      apply DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_in_le;
                      econstructor eassumption])).
                  
                  apply wf_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large.
                Qed.
               End WF_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large.
               
               Open Scope Z_scope.
               
               Import ring_extention.
               
               Notation Local "a <= b" := (Zle a b).
               
               Notation Local "a < b" := (Zlt a b).
               
               Definition P_id_a__zeros  := 2.
               
               Definition P_id_U51 (x11:Z) (x12:Z) := 2* x11.
               
               Definition P_id_a__U84 (x11:Z) (x12:Z) (x13:Z) := 1 + 1* x13.
               
               Definition P_id_a__U32 (x11:Z) (x12:Z) := 0.
               
               Definition P_id_isNatList (x11:Z) := 1* x11.
               
               Definition P_id_a__U13 (x11:Z) := 0.
               
               Definition P_id_U85 (x11:Z) (x12:Z) := 1 + 1* x12.
               
               Definition P_id_a__length (x11:Z) := 0.
               
               Definition P_id_a__isNatIList (x11:Z) := 0.
               
               Definition P_id_U33 (x11:Z) := 0.
               
               Definition P_id_a__U11 (x11:Z) (x12:Z) := 0.
               
               Definition P_id_U81 (x11:Z) (x12:Z) (x13:Z) := 1 + 1* x13.
               
               Definition P_id_a__U92 (x11:Z) (x12:Z) (x13:Z) := 0.
               
               Definition P_id_a__U43 (x11:Z) (x12:Z) (x13:Z) := 0.
               
               Definition P_id_U23 (x11:Z) := 2* x11.
               
               Definition P_id_a__isNatKind (x11:Z) := 0.
               
               Definition P_id_U93 (x11:Z) (x12:Z) (x13:Z) := 0.
               
               Definition P_id_U11 (x11:Z) (x12:Z) := 0.
               
               Definition P_id_a__U71 (x11:Z) := 1* x11.
               
               Definition P_id_U44 (x11:Z) (x12:Z) (x13:Z) := 0.
               
               Definition P_id_0  := 0.
               
               Definition P_id_U61 (x11:Z) := 2* x11.
               
               Definition P_id_a__U86 (x11:Z) := 1 + 1* x11.
               
               Definition P_id_a__U41 (x11:Z) (x12:Z) (x13:Z) := 0.
               
               Definition P_id_U22 (x11:Z) (x12:Z) := 0.
               
               Definition P_id_a__U21 (x11:Z) (x12:Z) := 0.
               
               Definition P_id_U91 (x11:Z) (x12:Z) (x13:Z) := 0.
               
               Definition P_id_length (x11:Z) := 0.
               
               Definition P_id_a__U52 (x11:Z) := 1* x11.
               
               Definition P_id_U42 (x11:Z) (x12:Z) (x13:Z) := 0.
               
               Definition P_id_a__U12 (x11:Z) (x12:Z) := 0.
               
               Definition P_id_U83 (x11:Z) (x12:Z) (x13:Z) := 1 + 1* x13.
               
               Definition P_id_a__U94 (x11:Z) (x12:Z) := 0.
               
               Definition P_id_a__U45 (x11:Z) (x12:Z) := 0.
               
               Definition P_id_U31 (x11:Z) (x12:Z) := 0.
               
               Definition P_id_a__isNat (x11:Z) := 0.
               
               Definition P_id_isNatIListKind (x11:Z) := 0.
               
               Definition P_id_a__U82 (x11:Z) (x12:Z) (x13:Z) := 1 + 1* x13.
               
               Definition P_id_U46 (x11:Z) := 0.
               
               Definition P_id_cons (x11:Z) (x12:Z) := 1 + 1* x12.
               
               Definition P_id_U52 (x11:Z) := 1* x11.
               
               Definition P_id_a__U85 (x11:Z) (x12:Z) := 1 + 1* x12.
               
               Definition P_id_a__U33 (x11:Z) := 0.
               
               Definition P_id_U21 (x11:Z) (x12:Z) := 0.
               
               Definition P_id_a__isNatList (x11:Z) := 1* x11.
               
               Definition P_id_U86 (x11:Z) := 1 + 1* x11.
               
               Definition P_id_mark (x11:Z) := 2* x11.
               
               Definition P_id_a__U51 (x11:Z) (x12:Z) := 2* x11.
               
               Definition P_id_U41 (x11:Z) (x12:Z) (x13:Z) := 0.
               
               Definition P_id_tt  := 0.
               
               Definition P_id_U82 (x11:Z) (x12:Z) (x13:Z) := 1 + 1* x13.
               
               Definition P_id_a__U93 (x11:Z) (x12:Z) (x13:Z) := 0.
               
               Definition P_id_a__U44 (x11:Z) (x12:Z) (x13:Z) := 0.
               
               Definition P_id_isNat (x11:Z) := 0.
               
               Definition P_id_a__U23 (x11:Z) := 2* x11.
               
               Definition P_id_U94 (x11:Z) (x12:Z) := 0.
               
               Definition P_id_U12 (x11:Z) (x12:Z) := 0.
               
               Definition P_id_a__U81 (x11:Z) (x12:Z) (x13:Z) := 1 + 1* x13.
               
               Definition P_id_U45 (x11:Z) (x12:Z) := 0.
               
               Definition P_id_zeros  := 1.
               
               Definition P_id_U71 (x11:Z) := 1* x11.
               
               Definition P_id_a__U91 (x11:Z) (x12:Z) (x13:Z) := 0.
               
               Definition P_id_a__U42 (x11:Z) (x12:Z) (x13:Z) := 0.
               
               Definition P_id_isNatKind (x11:Z) := 0.
               
               Definition P_id_a__U22 (x11:Z) (x12:Z) := 0.
               
               Definition P_id_U92 (x11:Z) (x12:Z) (x13:Z) := 0.
               
               Definition P_id_nil  := 0.
               
               Definition P_id_a__U61 (x11:Z) := 2* x11.
               
               Definition P_id_U43 (x11:Z) (x12:Z) (x13:Z) := 0.
               
               Definition P_id_a__isNatIListKind (x11:Z) := 0.
               
               Definition P_id_U84 (x11:Z) (x12:Z) (x13:Z) := 1 + 1* x13.
               
               Definition P_id_s (x11:Z) := 2* x11.
               
               Definition P_id_a__U46 (x11:Z) := 0.
               
               Definition P_id_U32 (x11:Z) (x12:Z) := 0.
               
               Definition P_id_a__U31 (x11:Z) (x12:Z) := 0.
               
               Definition P_id_U13 (x11:Z) := 0.
               
               Definition P_id_a__U83 (x11:Z) (x12:Z) (x13:Z) := 1 + 1* x13.
               
               Definition P_id_isNatIList (x11:Z) := 0.
               
               Lemma P_id_U51_monotonic :
                forall x12 x14 x13 x11, 
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_U51 x12 x14 <= P_id_U51 x11 x13.
               Proof.
                 intros x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U84_monotonic :
                forall x16 x12 x14 x13 x11 x15, 
                 (0 <= x16)/\ (x16 <= x15) ->
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_a__U84 x12 x14 x16 <= P_id_a__U84 x11 x13 x15.
               Proof.
                 intros x16 x15 x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 intros [H_5 H_4].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U32_monotonic :
                forall x12 x14 x13 x11, 
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_a__U32 x12 x14 <= P_id_a__U32 x11 x13.
               Proof.
                 intros x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_isNatList_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_isNatList x12 <= P_id_isNatList x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_0].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U13_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_a__U13 x12 <= P_id_a__U13 x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_0].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U85_monotonic :
                forall x12 x14 x13 x11, 
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_U85 x12 x14 <= P_id_U85 x11 x13.
               Proof.
                 intros x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__length_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_a__length x12 <= P_id_a__length x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_0].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__isNatIList_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_a__isNatIList x12 <= P_id_a__isNatIList x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_0].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U33_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->P_id_U33 x12 <= P_id_U33 x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_0].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U11_monotonic :
                forall x12 x14 x13 x11, 
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_a__U11 x12 x14 <= P_id_a__U11 x11 x13.
               Proof.
                 intros x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U81_monotonic :
                forall x16 x12 x14 x13 x11 x15, 
                 (0 <= x16)/\ (x16 <= x15) ->
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_U81 x12 x14 x16 <= P_id_U81 x11 x13 x15.
               Proof.
                 intros x16 x15 x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 intros [H_5 H_4].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U92_monotonic :
                forall x16 x12 x14 x13 x11 x15, 
                 (0 <= x16)/\ (x16 <= x15) ->
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_a__U92 x12 x14 x16 <= P_id_a__U92 x11 x13 x15.
               Proof.
                 intros x16 x15 x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 intros [H_5 H_4].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U43_monotonic :
                forall x16 x12 x14 x13 x11 x15, 
                 (0 <= x16)/\ (x16 <= x15) ->
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_a__U43 x12 x14 x16 <= P_id_a__U43 x11 x13 x15.
               Proof.
                 intros x16 x15 x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 intros [H_5 H_4].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U23_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->P_id_U23 x12 <= P_id_U23 x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_0].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__isNatKind_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_a__isNatKind x12 <= P_id_a__isNatKind x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_0].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U93_monotonic :
                forall x16 x12 x14 x13 x11 x15, 
                 (0 <= x16)/\ (x16 <= x15) ->
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_U93 x12 x14 x16 <= P_id_U93 x11 x13 x15.
               Proof.
                 intros x16 x15 x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 intros [H_5 H_4].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U11_monotonic :
                forall x12 x14 x13 x11, 
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_U11 x12 x14 <= P_id_U11 x11 x13.
               Proof.
                 intros x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U71_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_a__U71 x12 <= P_id_a__U71 x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_0].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U44_monotonic :
                forall x16 x12 x14 x13 x11 x15, 
                 (0 <= x16)/\ (x16 <= x15) ->
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_U44 x12 x14 x16 <= P_id_U44 x11 x13 x15.
               Proof.
                 intros x16 x15 x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 intros [H_5 H_4].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U61_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->P_id_U61 x12 <= P_id_U61 x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_0].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U86_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_a__U86 x12 <= P_id_a__U86 x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_0].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U41_monotonic :
                forall x16 x12 x14 x13 x11 x15, 
                 (0 <= x16)/\ (x16 <= x15) ->
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_a__U41 x12 x14 x16 <= P_id_a__U41 x11 x13 x15.
               Proof.
                 intros x16 x15 x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 intros [H_5 H_4].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U22_monotonic :
                forall x12 x14 x13 x11, 
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_U22 x12 x14 <= P_id_U22 x11 x13.
               Proof.
                 intros x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U21_monotonic :
                forall x12 x14 x13 x11, 
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_a__U21 x12 x14 <= P_id_a__U21 x11 x13.
               Proof.
                 intros x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U91_monotonic :
                forall x16 x12 x14 x13 x11 x15, 
                 (0 <= x16)/\ (x16 <= x15) ->
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_U91 x12 x14 x16 <= P_id_U91 x11 x13 x15.
               Proof.
                 intros x16 x15 x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 intros [H_5 H_4].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_length_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_length x12 <= P_id_length x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_0].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U52_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_a__U52 x12 <= P_id_a__U52 x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_0].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U42_monotonic :
                forall x16 x12 x14 x13 x11 x15, 
                 (0 <= x16)/\ (x16 <= x15) ->
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_U42 x12 x14 x16 <= P_id_U42 x11 x13 x15.
               Proof.
                 intros x16 x15 x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 intros [H_5 H_4].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U12_monotonic :
                forall x12 x14 x13 x11, 
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_a__U12 x12 x14 <= P_id_a__U12 x11 x13.
               Proof.
                 intros x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U83_monotonic :
                forall x16 x12 x14 x13 x11 x15, 
                 (0 <= x16)/\ (x16 <= x15) ->
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_U83 x12 x14 x16 <= P_id_U83 x11 x13 x15.
               Proof.
                 intros x16 x15 x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 intros [H_5 H_4].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U94_monotonic :
                forall x12 x14 x13 x11, 
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_a__U94 x12 x14 <= P_id_a__U94 x11 x13.
               Proof.
                 intros x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U45_monotonic :
                forall x12 x14 x13 x11, 
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_a__U45 x12 x14 <= P_id_a__U45 x11 x13.
               Proof.
                 intros x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U31_monotonic :
                forall x12 x14 x13 x11, 
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_U31 x12 x14 <= P_id_U31 x11 x13.
               Proof.
                 intros x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__isNat_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_a__isNat x12 <= P_id_a__isNat x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_0].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_isNatIListKind_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_isNatIListKind x12 <= P_id_isNatIListKind x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_0].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U82_monotonic :
                forall x16 x12 x14 x13 x11 x15, 
                 (0 <= x16)/\ (x16 <= x15) ->
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_a__U82 x12 x14 x16 <= P_id_a__U82 x11 x13 x15.
               Proof.
                 intros x16 x15 x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 intros [H_5 H_4].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U46_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->P_id_U46 x12 <= P_id_U46 x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_0].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_cons_monotonic :
                forall x12 x14 x13 x11, 
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_cons x12 x14 <= P_id_cons x11 x13.
               Proof.
                 intros x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U52_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->P_id_U52 x12 <= P_id_U52 x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_0].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U85_monotonic :
                forall x12 x14 x13 x11, 
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_a__U85 x12 x14 <= P_id_a__U85 x11 x13.
               Proof.
                 intros x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U33_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_a__U33 x12 <= P_id_a__U33 x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_0].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U21_monotonic :
                forall x12 x14 x13 x11, 
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_U21 x12 x14 <= P_id_U21 x11 x13.
               Proof.
                 intros x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__isNatList_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_a__isNatList x12 <= P_id_a__isNatList x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_0].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U86_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->P_id_U86 x12 <= P_id_U86 x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_0].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_mark_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->P_id_mark x12 <= P_id_mark x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_0].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U51_monotonic :
                forall x12 x14 x13 x11, 
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_a__U51 x12 x14 <= P_id_a__U51 x11 x13.
               Proof.
                 intros x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U41_monotonic :
                forall x16 x12 x14 x13 x11 x15, 
                 (0 <= x16)/\ (x16 <= x15) ->
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_U41 x12 x14 x16 <= P_id_U41 x11 x13 x15.
               Proof.
                 intros x16 x15 x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 intros [H_5 H_4].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U82_monotonic :
                forall x16 x12 x14 x13 x11 x15, 
                 (0 <= x16)/\ (x16 <= x15) ->
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_U82 x12 x14 x16 <= P_id_U82 x11 x13 x15.
               Proof.
                 intros x16 x15 x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 intros [H_5 H_4].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U93_monotonic :
                forall x16 x12 x14 x13 x11 x15, 
                 (0 <= x16)/\ (x16 <= x15) ->
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_a__U93 x12 x14 x16 <= P_id_a__U93 x11 x13 x15.
               Proof.
                 intros x16 x15 x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 intros [H_5 H_4].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U44_monotonic :
                forall x16 x12 x14 x13 x11 x15, 
                 (0 <= x16)/\ (x16 <= x15) ->
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_a__U44 x12 x14 x16 <= P_id_a__U44 x11 x13 x15.
               Proof.
                 intros x16 x15 x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 intros [H_5 H_4].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_isNat_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->P_id_isNat x12 <= P_id_isNat x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_0].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U23_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_a__U23 x12 <= P_id_a__U23 x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_0].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U94_monotonic :
                forall x12 x14 x13 x11, 
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_U94 x12 x14 <= P_id_U94 x11 x13.
               Proof.
                 intros x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U12_monotonic :
                forall x12 x14 x13 x11, 
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_U12 x12 x14 <= P_id_U12 x11 x13.
               Proof.
                 intros x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U81_monotonic :
                forall x16 x12 x14 x13 x11 x15, 
                 (0 <= x16)/\ (x16 <= x15) ->
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_a__U81 x12 x14 x16 <= P_id_a__U81 x11 x13 x15.
               Proof.
                 intros x16 x15 x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 intros [H_5 H_4].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U45_monotonic :
                forall x12 x14 x13 x11, 
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_U45 x12 x14 <= P_id_U45 x11 x13.
               Proof.
                 intros x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U71_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->P_id_U71 x12 <= P_id_U71 x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_0].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U91_monotonic :
                forall x16 x12 x14 x13 x11 x15, 
                 (0 <= x16)/\ (x16 <= x15) ->
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_a__U91 x12 x14 x16 <= P_id_a__U91 x11 x13 x15.
               Proof.
                 intros x16 x15 x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 intros [H_5 H_4].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U42_monotonic :
                forall x16 x12 x14 x13 x11 x15, 
                 (0 <= x16)/\ (x16 <= x15) ->
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_a__U42 x12 x14 x16 <= P_id_a__U42 x11 x13 x15.
               Proof.
                 intros x16 x15 x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 intros [H_5 H_4].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_isNatKind_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_isNatKind x12 <= P_id_isNatKind x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_0].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U22_monotonic :
                forall x12 x14 x13 x11, 
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_a__U22 x12 x14 <= P_id_a__U22 x11 x13.
               Proof.
                 intros x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U92_monotonic :
                forall x16 x12 x14 x13 x11 x15, 
                 (0 <= x16)/\ (x16 <= x15) ->
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_U92 x12 x14 x16 <= P_id_U92 x11 x13 x15.
               Proof.
                 intros x16 x15 x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 intros [H_5 H_4].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U61_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_a__U61 x12 <= P_id_a__U61 x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_0].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U43_monotonic :
                forall x16 x12 x14 x13 x11 x15, 
                 (0 <= x16)/\ (x16 <= x15) ->
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_U43 x12 x14 x16 <= P_id_U43 x11 x13 x15.
               Proof.
                 intros x16 x15 x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 intros [H_5 H_4].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__isNatIListKind_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_a__isNatIListKind x12 <= P_id_a__isNatIListKind x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_0].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U84_monotonic :
                forall x16 x12 x14 x13 x11 x15, 
                 (0 <= x16)/\ (x16 <= x15) ->
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_U84 x12 x14 x16 <= P_id_U84 x11 x13 x15.
               Proof.
                 intros x16 x15 x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 intros [H_5 H_4].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_s_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->P_id_s x12 <= P_id_s x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_0].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U46_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_a__U46 x12 <= P_id_a__U46 x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_0].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U32_monotonic :
                forall x12 x14 x13 x11, 
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_U32 x12 x14 <= P_id_U32 x11 x13.
               Proof.
                 intros x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U31_monotonic :
                forall x12 x14 x13 x11, 
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_a__U31 x12 x14 <= P_id_a__U31 x11 x13.
               Proof.
                 intros x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U13_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->P_id_U13 x12 <= P_id_U13 x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_0].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U83_monotonic :
                forall x16 x12 x14 x13 x11 x15, 
                 (0 <= x16)/\ (x16 <= x15) ->
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_a__U83 x12 x14 x16 <= P_id_a__U83 x11 x13 x15.
               Proof.
                 intros x16 x15 x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 intros [H_5 H_4].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_isNatIList_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_isNatIList x12 <= P_id_isNatIList x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_0].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__zeros_bounded : 0 <= P_id_a__zeros .
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U51_bounded :
                forall x12 x11, 
                 (0 <= x11) ->(0 <= x12) ->0 <= P_id_U51 x12 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U84_bounded :
                forall x12 x13 x11, 
                 (0 <= x11) ->
                  (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U84 x13 x12 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U32_bounded :
                forall x12 x11, 
                 (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U32 x12 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_isNatList_bounded :
                forall x11, (0 <= x11) ->0 <= P_id_isNatList x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U13_bounded :
                forall x11, (0 <= x11) ->0 <= P_id_a__U13 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U85_bounded :
                forall x12 x11, 
                 (0 <= x11) ->(0 <= x12) ->0 <= P_id_U85 x12 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__length_bounded :
                forall x11, (0 <= x11) ->0 <= P_id_a__length x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__isNatIList_bounded :
                forall x11, (0 <= x11) ->0 <= P_id_a__isNatIList x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U33_bounded :
                forall x11, (0 <= x11) ->0 <= P_id_U33 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U11_bounded :
                forall x12 x11, 
                 (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U11 x12 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U81_bounded :
                forall x12 x13 x11, 
                 (0 <= x11) ->
                  (0 <= x12) ->(0 <= x13) ->0 <= P_id_U81 x13 x12 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U92_bounded :
                forall x12 x13 x11, 
                 (0 <= x11) ->
                  (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U92 x13 x12 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U43_bounded :
                forall x12 x13 x11, 
                 (0 <= x11) ->
                  (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U43 x13 x12 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U23_bounded :
                forall x11, (0 <= x11) ->0 <= P_id_U23 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__isNatKind_bounded :
                forall x11, (0 <= x11) ->0 <= P_id_a__isNatKind x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U93_bounded :
                forall x12 x13 x11, 
                 (0 <= x11) ->
                  (0 <= x12) ->(0 <= x13) ->0 <= P_id_U93 x13 x12 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U11_bounded :
                forall x12 x11, 
                 (0 <= x11) ->(0 <= x12) ->0 <= P_id_U11 x12 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U71_bounded :
                forall x11, (0 <= x11) ->0 <= P_id_a__U71 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U44_bounded :
                forall x12 x13 x11, 
                 (0 <= x11) ->
                  (0 <= x12) ->(0 <= x13) ->0 <= P_id_U44 x13 x12 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_0_bounded : 0 <= P_id_0 .
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U61_bounded :
                forall x11, (0 <= x11) ->0 <= P_id_U61 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U86_bounded :
                forall x11, (0 <= x11) ->0 <= P_id_a__U86 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U41_bounded :
                forall x12 x13 x11, 
                 (0 <= x11) ->
                  (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U41 x13 x12 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U22_bounded :
                forall x12 x11, 
                 (0 <= x11) ->(0 <= x12) ->0 <= P_id_U22 x12 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U21_bounded :
                forall x12 x11, 
                 (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U21 x12 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U91_bounded :
                forall x12 x13 x11, 
                 (0 <= x11) ->
                  (0 <= x12) ->(0 <= x13) ->0 <= P_id_U91 x13 x12 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_length_bounded :
                forall x11, (0 <= x11) ->0 <= P_id_length x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U52_bounded :
                forall x11, (0 <= x11) ->0 <= P_id_a__U52 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U42_bounded :
                forall x12 x13 x11, 
                 (0 <= x11) ->
                  (0 <= x12) ->(0 <= x13) ->0 <= P_id_U42 x13 x12 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U12_bounded :
                forall x12 x11, 
                 (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U12 x12 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U83_bounded :
                forall x12 x13 x11, 
                 (0 <= x11) ->
                  (0 <= x12) ->(0 <= x13) ->0 <= P_id_U83 x13 x12 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U94_bounded :
                forall x12 x11, 
                 (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U94 x12 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U45_bounded :
                forall x12 x11, 
                 (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U45 x12 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U31_bounded :
                forall x12 x11, 
                 (0 <= x11) ->(0 <= x12) ->0 <= P_id_U31 x12 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__isNat_bounded :
                forall x11, (0 <= x11) ->0 <= P_id_a__isNat x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_isNatIListKind_bounded :
                forall x11, (0 <= x11) ->0 <= P_id_isNatIListKind x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U82_bounded :
                forall x12 x13 x11, 
                 (0 <= x11) ->
                  (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U82 x13 x12 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U46_bounded :
                forall x11, (0 <= x11) ->0 <= P_id_U46 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_cons_bounded :
                forall x12 x11, 
                 (0 <= x11) ->(0 <= x12) ->0 <= P_id_cons x12 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U52_bounded :
                forall x11, (0 <= x11) ->0 <= P_id_U52 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U85_bounded :
                forall x12 x11, 
                 (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U85 x12 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U33_bounded :
                forall x11, (0 <= x11) ->0 <= P_id_a__U33 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U21_bounded :
                forall x12 x11, 
                 (0 <= x11) ->(0 <= x12) ->0 <= P_id_U21 x12 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__isNatList_bounded :
                forall x11, (0 <= x11) ->0 <= P_id_a__isNatList x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U86_bounded :
                forall x11, (0 <= x11) ->0 <= P_id_U86 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_mark_bounded :
                forall x11, (0 <= x11) ->0 <= P_id_mark x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U51_bounded :
                forall x12 x11, 
                 (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U51 x12 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U41_bounded :
                forall x12 x13 x11, 
                 (0 <= x11) ->
                  (0 <= x12) ->(0 <= x13) ->0 <= P_id_U41 x13 x12 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_tt_bounded : 0 <= P_id_tt .
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U82_bounded :
                forall x12 x13 x11, 
                 (0 <= x11) ->
                  (0 <= x12) ->(0 <= x13) ->0 <= P_id_U82 x13 x12 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U93_bounded :
                forall x12 x13 x11, 
                 (0 <= x11) ->
                  (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U93 x13 x12 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U44_bounded :
                forall x12 x13 x11, 
                 (0 <= x11) ->
                  (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U44 x13 x12 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_isNat_bounded :
                forall x11, (0 <= x11) ->0 <= P_id_isNat x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U23_bounded :
                forall x11, (0 <= x11) ->0 <= P_id_a__U23 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U94_bounded :
                forall x12 x11, 
                 (0 <= x11) ->(0 <= x12) ->0 <= P_id_U94 x12 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U12_bounded :
                forall x12 x11, 
                 (0 <= x11) ->(0 <= x12) ->0 <= P_id_U12 x12 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U81_bounded :
                forall x12 x13 x11, 
                 (0 <= x11) ->
                  (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U81 x13 x12 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U45_bounded :
                forall x12 x11, 
                 (0 <= x11) ->(0 <= x12) ->0 <= P_id_U45 x12 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_zeros_bounded : 0 <= P_id_zeros .
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U71_bounded :
                forall x11, (0 <= x11) ->0 <= P_id_U71 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U91_bounded :
                forall x12 x13 x11, 
                 (0 <= x11) ->
                  (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U91 x13 x12 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U42_bounded :
                forall x12 x13 x11, 
                 (0 <= x11) ->
                  (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U42 x13 x12 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_isNatKind_bounded :
                forall x11, (0 <= x11) ->0 <= P_id_isNatKind x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U22_bounded :
                forall x12 x11, 
                 (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U22 x12 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U92_bounded :
                forall x12 x13 x11, 
                 (0 <= x11) ->
                  (0 <= x12) ->(0 <= x13) ->0 <= P_id_U92 x13 x12 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_nil_bounded : 0 <= P_id_nil .
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U61_bounded :
                forall x11, (0 <= x11) ->0 <= P_id_a__U61 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U43_bounded :
                forall x12 x13 x11, 
                 (0 <= x11) ->
                  (0 <= x12) ->(0 <= x13) ->0 <= P_id_U43 x13 x12 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__isNatIListKind_bounded :
                forall x11, (0 <= x11) ->0 <= P_id_a__isNatIListKind x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U84_bounded :
                forall x12 x13 x11, 
                 (0 <= x11) ->
                  (0 <= x12) ->(0 <= x13) ->0 <= P_id_U84 x13 x12 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_s_bounded :
                forall x11, (0 <= x11) ->0 <= P_id_s x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U46_bounded :
                forall x11, (0 <= x11) ->0 <= P_id_a__U46 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U32_bounded :
                forall x12 x11, 
                 (0 <= x11) ->(0 <= x12) ->0 <= P_id_U32 x12 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U31_bounded :
                forall x12 x11, 
                 (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U31 x12 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U13_bounded :
                forall x11, (0 <= x11) ->0 <= P_id_U13 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_a__U83_bounded :
                forall x12 x13 x11, 
                 (0 <= x11) ->
                  (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U83 x13 x12 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_isNatIList_bounded :
                forall x11, (0 <= x11) ->0 <= P_id_isNatIList x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Definition measure  := 
                 InterpZ.measure 0 P_id_a__zeros P_id_U51 P_id_a__U84 
                  P_id_a__U32 P_id_isNatList P_id_a__U13 P_id_U85 
                  P_id_a__length P_id_a__isNatIList P_id_U33 P_id_a__U11 
                  P_id_U81 P_id_a__U92 P_id_a__U43 P_id_U23 
                  P_id_a__isNatKind P_id_U93 P_id_U11 P_id_a__U71 P_id_U44 
                  P_id_0 P_id_U61 P_id_a__U86 P_id_a__U41 P_id_U22 
                  P_id_a__U21 P_id_U91 P_id_length P_id_a__U52 P_id_U42 
                  P_id_a__U12 P_id_U83 P_id_a__U94 P_id_a__U45 P_id_U31 
                  P_id_a__isNat P_id_isNatIListKind P_id_a__U82 P_id_U46 
                  P_id_cons P_id_U52 P_id_a__U85 P_id_a__U33 P_id_U21 
                  P_id_a__isNatList P_id_U86 P_id_mark P_id_a__U51 P_id_U41 
                  P_id_tt P_id_U82 P_id_a__U93 P_id_a__U44 P_id_isNat 
                  P_id_a__U23 P_id_U94 P_id_U12 P_id_a__U81 P_id_U45 
                  P_id_zeros P_id_U71 P_id_a__U91 P_id_a__U42 P_id_isNatKind 
                  P_id_a__U22 P_id_U92 P_id_nil P_id_a__U61 P_id_U43 
                  P_id_a__isNatIListKind P_id_U84 P_id_s P_id_a__U46 
                  P_id_U32 P_id_a__U31 P_id_U13 P_id_a__U83 P_id_isNatIList.
               
               Lemma measure_equation :
                forall t, 
                 measure t = match t with
                               | (algebra.Alg.Term algebra.F.id_a__zeros nil) =>
                                P_id_a__zeros 
                               | (algebra.Alg.Term algebra.F.id_U51 (x12::
                                  x11::nil)) =>
                                P_id_U51 (measure x12) (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U84 (x13::
                                  x12::x11::nil)) =>
                                P_id_a__U84 (measure x13) (measure x12) 
                                 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U32 (x12::
                                  x11::nil)) =>
                                P_id_a__U32 (measure x12) (measure x11)
                               | (algebra.Alg.Term algebra.F.id_isNatList 
                                  (x11::nil)) =>
                                P_id_isNatList (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U13 
                                  (x11::nil)) =>
                                P_id_a__U13 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_U85 (x12::
                                  x11::nil)) =>
                                P_id_U85 (measure x12) (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__length 
                                  (x11::nil)) =>
                                P_id_a__length (measure x11)
                               | (algebra.Alg.Term 
                                  algebra.F.id_a__isNatIList (x11::nil)) =>
                                P_id_a__isNatIList (measure x11)
                               | (algebra.Alg.Term algebra.F.id_U33 
                                  (x11::nil)) =>
                                P_id_U33 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U11 (x12::
                                  x11::nil)) =>
                                P_id_a__U11 (measure x12) (measure x11)
                               | (algebra.Alg.Term algebra.F.id_U81 (x13::
                                  x12::x11::nil)) =>
                                P_id_U81 (measure x13) (measure x12) 
                                 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U92 (x13::
                                  x12::x11::nil)) =>
                                P_id_a__U92 (measure x13) (measure x12) 
                                 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U43 (x13::
                                  x12::x11::nil)) =>
                                P_id_a__U43 (measure x13) (measure x12) 
                                 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_U23 
                                  (x11::nil)) =>
                                P_id_U23 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__isNatKind 
                                  (x11::nil)) =>
                                P_id_a__isNatKind (measure x11)
                               | (algebra.Alg.Term algebra.F.id_U93 (x13::
                                  x12::x11::nil)) =>
                                P_id_U93 (measure x13) (measure x12) 
                                 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_U11 (x12::
                                  x11::nil)) =>
                                P_id_U11 (measure x12) (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U71 
                                  (x11::nil)) =>
                                P_id_a__U71 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_U44 (x13::
                                  x12::x11::nil)) =>
                                P_id_U44 (measure x13) (measure x12) 
                                 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_0 nil) =>
                                P_id_0 
                               | (algebra.Alg.Term algebra.F.id_U61 
                                  (x11::nil)) =>
                                P_id_U61 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U86 
                                  (x11::nil)) =>
                                P_id_a__U86 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U41 (x13::
                                  x12::x11::nil)) =>
                                P_id_a__U41 (measure x13) (measure x12) 
                                 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_U22 (x12::
                                  x11::nil)) =>
                                P_id_U22 (measure x12) (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U21 (x12::
                                  x11::nil)) =>
                                P_id_a__U21 (measure x12) (measure x11)
                               | (algebra.Alg.Term algebra.F.id_U91 (x13::
                                  x12::x11::nil)) =>
                                P_id_U91 (measure x13) (measure x12) 
                                 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_length 
                                  (x11::nil)) =>
                                P_id_length (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U52 
                                  (x11::nil)) =>
                                P_id_a__U52 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_U42 (x13::
                                  x12::x11::nil)) =>
                                P_id_U42 (measure x13) (measure x12) 
                                 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U12 (x12::
                                  x11::nil)) =>
                                P_id_a__U12 (measure x12) (measure x11)
                               | (algebra.Alg.Term algebra.F.id_U83 (x13::
                                  x12::x11::nil)) =>
                                P_id_U83 (measure x13) (measure x12) 
                                 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U94 (x12::
                                  x11::nil)) =>
                                P_id_a__U94 (measure x12) (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U45 (x12::
                                  x11::nil)) =>
                                P_id_a__U45 (measure x12) (measure x11)
                               | (algebra.Alg.Term algebra.F.id_U31 (x12::
                                  x11::nil)) =>
                                P_id_U31 (measure x12) (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__isNat 
                                  (x11::nil)) =>
                                P_id_a__isNat (measure x11)
                               | (algebra.Alg.Term 
                                  algebra.F.id_isNatIListKind (x11::nil)) =>
                                P_id_isNatIListKind (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U82 (x13::
                                  x12::x11::nil)) =>
                                P_id_a__U82 (measure x13) (measure x12) 
                                 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_U46 
                                  (x11::nil)) =>
                                P_id_U46 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_cons (x12::
                                  x11::nil)) =>
                                P_id_cons (measure x12) (measure x11)
                               | (algebra.Alg.Term algebra.F.id_U52 
                                  (x11::nil)) =>
                                P_id_U52 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U85 (x12::
                                  x11::nil)) =>
                                P_id_a__U85 (measure x12) (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U33 
                                  (x11::nil)) =>
                                P_id_a__U33 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_U21 (x12::
                                  x11::nil)) =>
                                P_id_U21 (measure x12) (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__isNatList 
                                  (x11::nil)) =>
                                P_id_a__isNatList (measure x11)
                               | (algebra.Alg.Term algebra.F.id_U86 
                                  (x11::nil)) =>
                                P_id_U86 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_mark 
                                  (x11::nil)) =>
                                P_id_mark (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U51 (x12::
                                  x11::nil)) =>
                                P_id_a__U51 (measure x12) (measure x11)
                               | (algebra.Alg.Term algebra.F.id_U41 (x13::
                                  x12::x11::nil)) =>
                                P_id_U41 (measure x13) (measure x12) 
                                 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_tt nil) =>
                                P_id_tt 
                               | (algebra.Alg.Term algebra.F.id_U82 (x13::
                                  x12::x11::nil)) =>
                                P_id_U82 (measure x13) (measure x12) 
                                 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U93 (x13::
                                  x12::x11::nil)) =>
                                P_id_a__U93 (measure x13) (measure x12) 
                                 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U44 (x13::
                                  x12::x11::nil)) =>
                                P_id_a__U44 (measure x13) (measure x12) 
                                 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_isNat 
                                  (x11::nil)) =>
                                P_id_isNat (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U23 
                                  (x11::nil)) =>
                                P_id_a__U23 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_U94 (x12::
                                  x11::nil)) =>
                                P_id_U94 (measure x12) (measure x11)
                               | (algebra.Alg.Term algebra.F.id_U12 (x12::
                                  x11::nil)) =>
                                P_id_U12 (measure x12) (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U81 (x13::
                                  x12::x11::nil)) =>
                                P_id_a__U81 (measure x13) (measure x12) 
                                 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_U45 (x12::
                                  x11::nil)) =>
                                P_id_U45 (measure x12) (measure x11)
                               | (algebra.Alg.Term algebra.F.id_zeros nil) =>
                                P_id_zeros 
                               | (algebra.Alg.Term algebra.F.id_U71 
                                  (x11::nil)) =>
                                P_id_U71 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U91 (x13::
                                  x12::x11::nil)) =>
                                P_id_a__U91 (measure x13) (measure x12) 
                                 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U42 (x13::
                                  x12::x11::nil)) =>
                                P_id_a__U42 (measure x13) (measure x12) 
                                 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_isNatKind 
                                  (x11::nil)) =>
                                P_id_isNatKind (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U22 (x12::
                                  x11::nil)) =>
                                P_id_a__U22 (measure x12) (measure x11)
                               | (algebra.Alg.Term algebra.F.id_U92 (x13::
                                  x12::x11::nil)) =>
                                P_id_U92 (measure x13) (measure x12) 
                                 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_nil nil) =>
                                P_id_nil 
                               | (algebra.Alg.Term algebra.F.id_a__U61 
                                  (x11::nil)) =>
                                P_id_a__U61 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_U43 (x13::
                                  x12::x11::nil)) =>
                                P_id_U43 (measure x13) (measure x12) 
                                 (measure x11)
                               | (algebra.Alg.Term 
                                  algebra.F.id_a__isNatIListKind (x11::nil)) =>
                                P_id_a__isNatIListKind (measure x11)
                               | (algebra.Alg.Term algebra.F.id_U84 (x13::
                                  x12::x11::nil)) =>
                                P_id_U84 (measure x13) (measure x12) 
                                 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_s (x11::nil)) =>
                                P_id_s (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U46 
                                  (x11::nil)) =>
                                P_id_a__U46 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_U32 (x12::
                                  x11::nil)) =>
                                P_id_U32 (measure x12) (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U31 (x12::
                                  x11::nil)) =>
                                P_id_a__U31 (measure x12) (measure x11)
                               | (algebra.Alg.Term algebra.F.id_U13 
                                  (x11::nil)) =>
                                P_id_U13 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U83 (x13::
                                  x12::x11::nil)) =>
                                P_id_a__U83 (measure x13) (measure x12) 
                                 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_isNatIList 
                                  (x11::nil)) =>
                                P_id_isNatIList (measure x11)
                               | _ => 0
                               end.
               Proof.
                 intros t;case t;intros ;apply refl_equal.
               Qed.
               
               Lemma measure_bounded : forall t, 0 <= measure t.
               Proof.
                 unfold measure in |-*.
                 
                 apply InterpZ.measure_bounded;
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Ltac generate_pos_hyp  :=
                match goal with
                  | H:context [measure ?x] |- _ =>
                   let v := fresh "v" in 
                    (let H := fresh "h" in 
                      (set (H:=measure_bounded x) in *;
                        set (v:=measure x) in *;clearbody H;clearbody v))
                  |  |- context [measure ?x] =>
                   let v := fresh "v" in 
                    (let H := fresh "h" in 
                      (set (H:=measure_bounded x) in *;
                        set (v:=measure x) in *;clearbody H;clearbody v))
                  end
                .
               
               Lemma rules_monotonic :
                forall l r, 
                 (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) ->
                  measure r <= measure l.
               Proof.
                 intros l r H.
                 fold measure in |-*.
                 
                 inversion H;clear H;subst;inversion H0;clear H0;subst;
                  simpl algebra.EQT.T.apply_subst in |-*;
                  repeat (
                  match goal with
                    |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
                     rewrite (measure_equation (algebra.Alg.Term f t))
                    end
                  );repeat (generate_pos_hyp );
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma measure_star_monotonic :
                forall l r, 
                 (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                            r l) ->measure r <= measure l.
               Proof.
                 unfold measure in *.
                 apply InterpZ.measure_star_monotonic.
                 intros ;apply P_id_U51_monotonic;assumption.
                 intros ;apply P_id_a__U84_monotonic;assumption.
                 intros ;apply P_id_a__U32_monotonic;assumption.
                 intros ;apply P_id_isNatList_monotonic;assumption.
                 intros ;apply P_id_a__U13_monotonic;assumption.
                 intros ;apply P_id_U85_monotonic;assumption.
                 intros ;apply P_id_a__length_monotonic;assumption.
                 intros ;apply P_id_a__isNatIList_monotonic;assumption.
                 intros ;apply P_id_U33_monotonic;assumption.
                 intros ;apply P_id_a__U11_monotonic;assumption.
                 intros ;apply P_id_U81_monotonic;assumption.
                 intros ;apply P_id_a__U92_monotonic;assumption.
                 intros ;apply P_id_a__U43_monotonic;assumption.
                 intros ;apply P_id_U23_monotonic;assumption.
                 intros ;apply P_id_a__isNatKind_monotonic;assumption.
                 intros ;apply P_id_U93_monotonic;assumption.
                 intros ;apply P_id_U11_monotonic;assumption.
                 intros ;apply P_id_a__U71_monotonic;assumption.
                 intros ;apply P_id_U44_monotonic;assumption.
                 intros ;apply P_id_U61_monotonic;assumption.
                 intros ;apply P_id_a__U86_monotonic;assumption.
                 intros ;apply P_id_a__U41_monotonic;assumption.
                 intros ;apply P_id_U22_monotonic;assumption.
                 intros ;apply P_id_a__U21_monotonic;assumption.
                 intros ;apply P_id_U91_monotonic;assumption.
                 intros ;apply P_id_length_monotonic;assumption.
                 intros ;apply P_id_a__U52_monotonic;assumption.
                 intros ;apply P_id_U42_monotonic;assumption.
                 intros ;apply P_id_a__U12_monotonic;assumption.
                 intros ;apply P_id_U83_monotonic;assumption.
                 intros ;apply P_id_a__U94_monotonic;assumption.
                 intros ;apply P_id_a__U45_monotonic;assumption.
                 intros ;apply P_id_U31_monotonic;assumption.
                 intros ;apply P_id_a__isNat_monotonic;assumption.
                 intros ;apply P_id_isNatIListKind_monotonic;assumption.
                 intros ;apply P_id_a__U82_monotonic;assumption.
                 intros ;apply P_id_U46_monotonic;assumption.
                 intros ;apply P_id_cons_monotonic;assumption.
                 intros ;apply P_id_U52_monotonic;assumption.
                 intros ;apply P_id_a__U85_monotonic;assumption.
                 intros ;apply P_id_a__U33_monotonic;assumption.
                 intros ;apply P_id_U21_monotonic;assumption.
                 intros ;apply P_id_a__isNatList_monotonic;assumption.
                 intros ;apply P_id_U86_monotonic;assumption.
                 intros ;apply P_id_mark_monotonic;assumption.
                 intros ;apply P_id_a__U51_monotonic;assumption.
                 intros ;apply P_id_U41_monotonic;assumption.
                 intros ;apply P_id_U82_monotonic;assumption.
                 intros ;apply P_id_a__U93_monotonic;assumption.
                 intros ;apply P_id_a__U44_monotonic;assumption.
                 intros ;apply P_id_isNat_monotonic;assumption.
                 intros ;apply P_id_a__U23_monotonic;assumption.
                 intros ;apply P_id_U94_monotonic;assumption.
                 intros ;apply P_id_U12_monotonic;assumption.
                 intros ;apply P_id_a__U81_monotonic;assumption.
                 intros ;apply P_id_U45_monotonic;assumption.
                 intros ;apply P_id_U71_monotonic;assumption.
                 intros ;apply P_id_a__U91_monotonic;assumption.
                 intros ;apply P_id_a__U42_monotonic;assumption.
                 intros ;apply P_id_isNatKind_monotonic;assumption.
                 intros ;apply P_id_a__U22_monotonic;assumption.
                 intros ;apply P_id_U92_monotonic;assumption.
                 intros ;apply P_id_a__U61_monotonic;assumption.
                 intros ;apply P_id_U43_monotonic;assumption.
                 intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
                 intros ;apply P_id_U84_monotonic;assumption.
                 intros ;apply P_id_s_monotonic;assumption.
                 intros ;apply P_id_a__U46_monotonic;assumption.
                 intros ;apply P_id_U32_monotonic;assumption.
                 intros ;apply P_id_a__U31_monotonic;assumption.
                 intros ;apply P_id_U13_monotonic;assumption.
                 intros ;apply P_id_a__U83_monotonic;assumption.
                 intros ;apply P_id_isNatIList_monotonic;assumption.
                 intros ;apply P_id_a__zeros_bounded;assumption.
                 intros ;apply P_id_U51_bounded;assumption.
                 intros ;apply P_id_a__U84_bounded;assumption.
                 intros ;apply P_id_a__U32_bounded;assumption.
                 intros ;apply P_id_isNatList_bounded;assumption.
                 intros ;apply P_id_a__U13_bounded;assumption.
                 intros ;apply P_id_U85_bounded;assumption.
                 intros ;apply P_id_a__length_bounded;assumption.
                 intros ;apply P_id_a__isNatIList_bounded;assumption.
                 intros ;apply P_id_U33_bounded;assumption.
                 intros ;apply P_id_a__U11_bounded;assumption.
                 intros ;apply P_id_U81_bounded;assumption.
                 intros ;apply P_id_a__U92_bounded;assumption.
                 intros ;apply P_id_a__U43_bounded;assumption.
                 intros ;apply P_id_U23_bounded;assumption.
                 intros ;apply P_id_a__isNatKind_bounded;assumption.
                 intros ;apply P_id_U93_bounded;assumption.
                 intros ;apply P_id_U11_bounded;assumption.
                 intros ;apply P_id_a__U71_bounded;assumption.
                 intros ;apply P_id_U44_bounded;assumption.
                 intros ;apply P_id_0_bounded;assumption.
                 intros ;apply P_id_U61_bounded;assumption.
                 intros ;apply P_id_a__U86_bounded;assumption.
                 intros ;apply P_id_a__U41_bounded;assumption.
                 intros ;apply P_id_U22_bounded;assumption.
                 intros ;apply P_id_a__U21_bounded;assumption.
                 intros ;apply P_id_U91_bounded;assumption.
                 intros ;apply P_id_length_bounded;assumption.
                 intros ;apply P_id_a__U52_bounded;assumption.
                 intros ;apply P_id_U42_bounded;assumption.
                 intros ;apply P_id_a__U12_bounded;assumption.
                 intros ;apply P_id_U83_bounded;assumption.
                 intros ;apply P_id_a__U94_bounded;assumption.
                 intros ;apply P_id_a__U45_bounded;assumption.
                 intros ;apply P_id_U31_bounded;assumption.
                 intros ;apply P_id_a__isNat_bounded;assumption.
                 intros ;apply P_id_isNatIListKind_bounded;assumption.
                 intros ;apply P_id_a__U82_bounded;assumption.
                 intros ;apply P_id_U46_bounded;assumption.
                 intros ;apply P_id_cons_bounded;assumption.
                 intros ;apply P_id_U52_bounded;assumption.
                 intros ;apply P_id_a__U85_bounded;assumption.
                 intros ;apply P_id_a__U33_bounded;assumption.
                 intros ;apply P_id_U21_bounded;assumption.
                 intros ;apply P_id_a__isNatList_bounded;assumption.
                 intros ;apply P_id_U86_bounded;assumption.
                 intros ;apply P_id_mark_bounded;assumption.
                 intros ;apply P_id_a__U51_bounded;assumption.
                 intros ;apply P_id_U41_bounded;assumption.
                 intros ;apply P_id_tt_bounded;assumption.
                 intros ;apply P_id_U82_bounded;assumption.
                 intros ;apply P_id_a__U93_bounded;assumption.
                 intros ;apply P_id_a__U44_bounded;assumption.
                 intros ;apply P_id_isNat_bounded;assumption.
                 intros ;apply P_id_a__U23_bounded;assumption.
                 intros ;apply P_id_U94_bounded;assumption.
                 intros ;apply P_id_U12_bounded;assumption.
                 intros ;apply P_id_a__U81_bounded;assumption.
                 intros ;apply P_id_U45_bounded;assumption.
                 intros ;apply P_id_zeros_bounded;assumption.
                 intros ;apply P_id_U71_bounded;assumption.
                 intros ;apply P_id_a__U91_bounded;assumption.
                 intros ;apply P_id_a__U42_bounded;assumption.
                 intros ;apply P_id_isNatKind_bounded;assumption.
                 intros ;apply P_id_a__U22_bounded;assumption.
                 intros ;apply P_id_U92_bounded;assumption.
                 intros ;apply P_id_nil_bounded;assumption.
                 intros ;apply P_id_a__U61_bounded;assumption.
                 intros ;apply P_id_U43_bounded;assumption.
                 intros ;apply P_id_a__isNatIListKind_bounded;assumption.
                 intros ;apply P_id_U84_bounded;assumption.
                 intros ;apply P_id_s_bounded;assumption.
                 intros ;apply P_id_a__U46_bounded;assumption.
                 intros ;apply P_id_U32_bounded;assumption.
                 intros ;apply P_id_a__U31_bounded;assumption.
                 intros ;apply P_id_U13_bounded;assumption.
                 intros ;apply P_id_a__U83_bounded;assumption.
                 intros ;apply P_id_isNatIList_bounded;assumption.
                 apply rules_monotonic.
               Qed.
               
               Definition P_id_A__U46 (x11:Z) := 0.
               
               Definition P_id_A__ISNATILISTKIND (x11:Z) := 0.
               
               Definition P_id_A__U61 (x11:Z) := 0.
               
               Definition P_id_A__U85 (x11:Z) (x12:Z) := 0.
               
               Definition P_id_A__U31 (x11:Z) (x12:Z) := 0.
               
               Definition P_id_A__U81 (x11:Z) (x12:Z) (x13:Z) := 0.
               
               Definition P_id_A__U22 (x11:Z) (x12:Z) := 0.
               
               Definition P_id_A__U93 (x11:Z) (x12:Z) (x13:Z) := 0.
               
               Definition P_id_A__U42 (x11:Z) (x12:Z) (x13:Z) := 0.
               
               Definition P_id_A__U51 (x11:Z) (x12:Z) := 0.
               
               Definition P_id_A__ISNATLIST (x11:Z) := 0.
               
               Definition P_id_A__ZEROS  := 0.
               
               Definition P_id_A__U91 (x11:Z) (x12:Z) (x13:Z) := 0.
               
               Definition P_id_A__U33 (x11:Z) := 0.
               
               Definition P_id_A__U83 (x11:Z) (x12:Z) (x13:Z) := 0.
               
               Definition P_id_A__U23 (x11:Z) := 0.
               
               Definition P_id_A__U11 (x11:Z) (x12:Z) := 0.
               
               Definition P_id_A__LENGTH (x11:Z) := 0.
               
               Definition P_id_A__U44 (x11:Z) (x12:Z) (x13:Z) := 0.
               
               Definition P_id_A__ISNATILIST (x11:Z) := 0.
               
               Definition P_id_A__U13 (x11:Z) := 0.
               
               Definition P_id_A__U71 (x11:Z) := 0.
               
               Definition P_id_A__U86 (x11:Z) := 0.
               
               Definition P_id_A__U32 (x11:Z) (x12:Z) := 0.
               
               Definition P_id_A__U82 (x11:Z) (x12:Z) (x13:Z) := 0.
               
               Definition P_id_A__ISNATKIND (x11:Z) := 0.
               
               Definition P_id_A__U94 (x11:Z) (x12:Z) := 0.
               
               Definition P_id_A__U43 (x11:Z) (x12:Z) (x13:Z) := 0.
               
               Definition P_id_A__U52 (x11:Z) := 0.
               
               Definition P_id_A__U21 (x11:Z) (x12:Z) := 0.
               
               Definition P_id_A__U92 (x11:Z) (x12:Z) (x13:Z) := 0.
               
               Definition P_id_A__U41 (x11:Z) (x12:Z) (x13:Z) := 0.
               
               Definition P_id_A__U84 (x11:Z) (x12:Z) (x13:Z) := 0.
               
               Definition P_id_A__ISNAT (x11:Z) := 0.
               
               Definition P_id_A__U12 (x11:Z) (x12:Z) := 0.
               
               Definition P_id_MARK (x11:Z) := 2* x11.
               
               Definition P_id_A__U45 (x11:Z) (x12:Z) := 0.
               
               Lemma P_id_A__U46_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_A__U46 x12 <= P_id_A__U46 x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_0].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_A__ISNATILISTKIND_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_A__ISNATILISTKIND x12 <= P_id_A__ISNATILISTKIND x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_0].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_A__U61_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_A__U61 x12 <= P_id_A__U61 x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_0].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_A__U85_monotonic :
                forall x12 x14 x13 x11, 
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_A__U85 x12 x14 <= P_id_A__U85 x11 x13.
               Proof.
                 intros x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_A__U31_monotonic :
                forall x12 x14 x13 x11, 
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_A__U31 x12 x14 <= P_id_A__U31 x11 x13.
               Proof.
                 intros x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_A__U81_monotonic :
                forall x16 x12 x14 x13 x11 x15, 
                 (0 <= x16)/\ (x16 <= x15) ->
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_A__U81 x12 x14 x16 <= P_id_A__U81 x11 x13 x15.
               Proof.
                 intros x16 x15 x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 intros [H_5 H_4].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_A__U22_monotonic :
                forall x12 x14 x13 x11, 
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_A__U22 x12 x14 <= P_id_A__U22 x11 x13.
               Proof.
                 intros x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_A__U93_monotonic :
                forall x16 x12 x14 x13 x11 x15, 
                 (0 <= x16)/\ (x16 <= x15) ->
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_A__U93 x12 x14 x16 <= P_id_A__U93 x11 x13 x15.
               Proof.
                 intros x16 x15 x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 intros [H_5 H_4].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_A__U42_monotonic :
                forall x16 x12 x14 x13 x11 x15, 
                 (0 <= x16)/\ (x16 <= x15) ->
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_A__U42 x12 x14 x16 <= P_id_A__U42 x11 x13 x15.
               Proof.
                 intros x16 x15 x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 intros [H_5 H_4].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_A__U51_monotonic :
                forall x12 x14 x13 x11, 
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_A__U51 x12 x14 <= P_id_A__U51 x11 x13.
               Proof.
                 intros x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_A__ISNATLIST_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_A__ISNATLIST x12 <= P_id_A__ISNATLIST x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_0].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_A__U91_monotonic :
                forall x16 x12 x14 x13 x11 x15, 
                 (0 <= x16)/\ (x16 <= x15) ->
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_A__U91 x12 x14 x16 <= P_id_A__U91 x11 x13 x15.
               Proof.
                 intros x16 x15 x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 intros [H_5 H_4].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_A__U33_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_A__U33 x12 <= P_id_A__U33 x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_0].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_A__U83_monotonic :
                forall x16 x12 x14 x13 x11 x15, 
                 (0 <= x16)/\ (x16 <= x15) ->
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_A__U83 x12 x14 x16 <= P_id_A__U83 x11 x13 x15.
               Proof.
                 intros x16 x15 x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 intros [H_5 H_4].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_A__U23_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_A__U23 x12 <= P_id_A__U23 x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_0].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_A__U11_monotonic :
                forall x12 x14 x13 x11, 
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_A__U11 x12 x14 <= P_id_A__U11 x11 x13.
               Proof.
                 intros x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_A__LENGTH_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_A__LENGTH x12 <= P_id_A__LENGTH x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_0].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_A__U44_monotonic :
                forall x16 x12 x14 x13 x11 x15, 
                 (0 <= x16)/\ (x16 <= x15) ->
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_A__U44 x12 x14 x16 <= P_id_A__U44 x11 x13 x15.
               Proof.
                 intros x16 x15 x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 intros [H_5 H_4].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_A__ISNATILIST_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_A__ISNATILIST x12 <= P_id_A__ISNATILIST x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_0].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_A__U13_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_A__U13 x12 <= P_id_A__U13 x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_0].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_A__U71_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_A__U71 x12 <= P_id_A__U71 x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_0].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_A__U86_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_A__U86 x12 <= P_id_A__U86 x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_0].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_A__U32_monotonic :
                forall x12 x14 x13 x11, 
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_A__U32 x12 x14 <= P_id_A__U32 x11 x13.
               Proof.
                 intros x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_A__U82_monotonic :
                forall x16 x12 x14 x13 x11 x15, 
                 (0 <= x16)/\ (x16 <= x15) ->
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_A__U82 x12 x14 x16 <= P_id_A__U82 x11 x13 x15.
               Proof.
                 intros x16 x15 x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 intros [H_5 H_4].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_A__ISNATKIND_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_A__ISNATKIND x12 <= P_id_A__ISNATKIND x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_0].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_A__U94_monotonic :
                forall x12 x14 x13 x11, 
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_A__U94 x12 x14 <= P_id_A__U94 x11 x13.
               Proof.
                 intros x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_A__U43_monotonic :
                forall x16 x12 x14 x13 x11 x15, 
                 (0 <= x16)/\ (x16 <= x15) ->
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_A__U43 x12 x14 x16 <= P_id_A__U43 x11 x13 x15.
               Proof.
                 intros x16 x15 x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 intros [H_5 H_4].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_A__U52_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_A__U52 x12 <= P_id_A__U52 x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_0].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_A__U21_monotonic :
                forall x12 x14 x13 x11, 
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_A__U21 x12 x14 <= P_id_A__U21 x11 x13.
               Proof.
                 intros x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_A__U92_monotonic :
                forall x16 x12 x14 x13 x11 x15, 
                 (0 <= x16)/\ (x16 <= x15) ->
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_A__U92 x12 x14 x16 <= P_id_A__U92 x11 x13 x15.
               Proof.
                 intros x16 x15 x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 intros [H_5 H_4].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_A__U41_monotonic :
                forall x16 x12 x14 x13 x11 x15, 
                 (0 <= x16)/\ (x16 <= x15) ->
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_A__U41 x12 x14 x16 <= P_id_A__U41 x11 x13 x15.
               Proof.
                 intros x16 x15 x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 intros [H_5 H_4].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_A__U84_monotonic :
                forall x16 x12 x14 x13 x11 x15, 
                 (0 <= x16)/\ (x16 <= x15) ->
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_A__U84 x12 x14 x16 <= P_id_A__U84 x11 x13 x15.
               Proof.
                 intros x16 x15 x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 intros [H_5 H_4].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_A__ISNAT_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_A__ISNAT x12 <= P_id_A__ISNAT x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_0].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_A__U12_monotonic :
                forall x12 x14 x13 x11, 
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_A__U12 x12 x14 <= P_id_A__U12 x11 x13.
               Proof.
                 intros x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_MARK_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->P_id_MARK x12 <= P_id_MARK x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_0].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_A__U45_monotonic :
                forall x12 x14 x13 x11, 
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_A__U45 x12 x14 <= P_id_A__U45 x11 x13.
               Proof.
                 intros x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Definition marked_measure  := 
                 InterpZ.marked_measure 0 P_id_a__zeros P_id_U51 P_id_a__U84 
                  P_id_a__U32 P_id_isNatList P_id_a__U13 P_id_U85 
                  P_id_a__length P_id_a__isNatIList P_id_U33 P_id_a__U11 
                  P_id_U81 P_id_a__U92 P_id_a__U43 P_id_U23 
                  P_id_a__isNatKind P_id_U93 P_id_U11 P_id_a__U71 P_id_U44 
                  P_id_0 P_id_U61 P_id_a__U86 P_id_a__U41 P_id_U22 
                  P_id_a__U21 P_id_U91 P_id_length P_id_a__U52 P_id_U42 
                  P_id_a__U12 P_id_U83 P_id_a__U94 P_id_a__U45 P_id_U31 
                  P_id_a__isNat P_id_isNatIListKind P_id_a__U82 P_id_U46 
                  P_id_cons P_id_U52 P_id_a__U85 P_id_a__U33 P_id_U21 
                  P_id_a__isNatList P_id_U86 P_id_mark P_id_a__U51 P_id_U41 
                  P_id_tt P_id_U82 P_id_a__U93 P_id_a__U44 P_id_isNat 
                  P_id_a__U23 P_id_U94 P_id_U12 P_id_a__U81 P_id_U45 
                  P_id_zeros P_id_U71 P_id_a__U91 P_id_a__U42 P_id_isNatKind 
                  P_id_a__U22 P_id_U92 P_id_nil P_id_a__U61 P_id_U43 
                  P_id_a__isNatIListKind P_id_U84 P_id_s P_id_a__U46 
                  P_id_U32 P_id_a__U31 P_id_U13 P_id_a__U83 P_id_isNatIList 
                  P_id_A__U46 P_id_A__ISNATILISTKIND P_id_A__U61 P_id_A__U85 
                  P_id_A__U31 P_id_A__U81 P_id_A__U22 P_id_A__U93 
                  P_id_A__U42 P_id_A__U51 P_id_A__ISNATLIST P_id_A__ZEROS 
                  P_id_A__U91 P_id_A__U33 P_id_A__U83 P_id_A__U23 
                  P_id_A__U11 P_id_A__LENGTH P_id_A__U44 P_id_A__ISNATILIST 
                  P_id_A__U13 P_id_A__U71 P_id_A__U86 P_id_A__U32 
                  P_id_A__U82 P_id_A__ISNATKIND P_id_A__U94 P_id_A__U43 
                  P_id_A__U52 P_id_A__U21 P_id_A__U92 P_id_A__U41 
                  P_id_A__U84 P_id_A__ISNAT P_id_A__U12 P_id_MARK P_id_A__U45
                 .
               
               Lemma marked_measure_equation :
                forall t, 
                 marked_measure t = match t with
                                      | (algebra.Alg.Term 
                                         algebra.F.id_a__U46 (x11::nil)) =>
                                       P_id_A__U46 (measure x11)
                                      | (algebra.Alg.Term 
                                         algebra.F.id_a__isNatIListKind 
                                         (x11::nil)) =>
                                       P_id_A__ISNATILISTKIND (measure x11)
                                      | (algebra.Alg.Term 
                                         algebra.F.id_a__U61 (x11::nil)) =>
                                       P_id_A__U61 (measure x11)
                                      | (algebra.Alg.Term 
                                         algebra.F.id_a__U85 (x12::x11::nil)) =>
                                       P_id_A__U85 (measure x12) 
                                        (measure x11)
                                      | (algebra.Alg.Term 
                                         algebra.F.id_a__U31 (x12::x11::nil)) =>
                                       P_id_A__U31 (measure x12) 
                                        (measure x11)
                                      | (algebra.Alg.Term 
                                         algebra.F.id_a__U81 (x13::x12::
                                         x11::nil)) =>
                                       P_id_A__U81 (measure x13) 
                                        (measure x12) (measure x11)
                                      | (algebra.Alg.Term 
                                         algebra.F.id_a__U22 (x12::x11::nil)) =>
                                       P_id_A__U22 (measure x12) 
                                        (measure x11)
                                      | (algebra.Alg.Term 
                                         algebra.F.id_a__U93 (x13::x12::
                                         x11::nil)) =>
                                       P_id_A__U93 (measure x13) 
                                        (measure x12) (measure x11)
                                      | (algebra.Alg.Term 
                                         algebra.F.id_a__U42 (x13::x12::
                                         x11::nil)) =>
                                       P_id_A__U42 (measure x13) 
                                        (measure x12) (measure x11)
                                      | (algebra.Alg.Term 
                                         algebra.F.id_a__U51 (x12::x11::nil)) =>
                                       P_id_A__U51 (measure x12) 
                                        (measure x11)
                                      | (algebra.Alg.Term 
                                         algebra.F.id_a__isNatList 
                                         (x11::nil)) =>
                                       P_id_A__ISNATLIST (measure x11)
                                      | (algebra.Alg.Term 
                                         algebra.F.id_a__zeros nil) =>
                                       P_id_A__ZEROS 
                                      | (algebra.Alg.Term 
                                         algebra.F.id_a__U91 (x13::x12::
                                         x11::nil)) =>
                                       P_id_A__U91 (measure x13) 
                                        (measure x12) (measure x11)
                                      | (algebra.Alg.Term 
                                         algebra.F.id_a__U33 (x11::nil)) =>
                                       P_id_A__U33 (measure x11)
                                      | (algebra.Alg.Term 
                                         algebra.F.id_a__U83 (x13::x12::
                                         x11::nil)) =>
                                       P_id_A__U83 (measure x13) 
                                        (measure x12) (measure x11)
                                      | (algebra.Alg.Term 
                                         algebra.F.id_a__U23 (x11::nil)) =>
                                       P_id_A__U23 (measure x11)
                                      | (algebra.Alg.Term 
                                         algebra.F.id_a__U11 (x12::x11::nil)) =>
                                       P_id_A__U11 (measure x12) 
                                        (measure x11)
                                      | (algebra.Alg.Term 
                                         algebra.F.id_a__length (x11::nil)) =>
                                       P_id_A__LENGTH (measure x11)
                                      | (algebra.Alg.Term 
                                         algebra.F.id_a__U44 (x13::x12::
                                         x11::nil)) =>
                                       P_id_A__U44 (measure x13) 
                                        (measure x12) (measure x11)
                                      | (algebra.Alg.Term 
                                         algebra.F.id_a__isNatIList 
                                         (x11::nil)) =>
                                       P_id_A__ISNATILIST (measure x11)
                                      | (algebra.Alg.Term 
                                         algebra.F.id_a__U13 (x11::nil)) =>
                                       P_id_A__U13 (measure x11)
                                      | (algebra.Alg.Term 
                                         algebra.F.id_a__U71 (x11::nil)) =>
                                       P_id_A__U71 (measure x11)
                                      | (algebra.Alg.Term 
                                         algebra.F.id_a__U86 (x11::nil)) =>
                                       P_id_A__U86 (measure x11)
                                      | (algebra.Alg.Term 
                                         algebra.F.id_a__U32 (x12::x11::nil)) =>
                                       P_id_A__U32 (measure x12) 
                                        (measure x11)
                                      | (algebra.Alg.Term 
                                         algebra.F.id_a__U82 (x13::x12::
                                         x11::nil)) =>
                                       P_id_A__U82 (measure x13) 
                                        (measure x12) (measure x11)
                                      | (algebra.Alg.Term 
                                         algebra.F.id_a__isNatKind 
                                         (x11::nil)) =>
                                       P_id_A__ISNATKIND (measure x11)
                                      | (algebra.Alg.Term 
                                         algebra.F.id_a__U94 (x12::x11::nil)) =>
                                       P_id_A__U94 (measure x12) 
                                        (measure x11)
                                      | (algebra.Alg.Term 
                                         algebra.F.id_a__U43 (x13::x12::
                                         x11::nil)) =>
                                       P_id_A__U43 (measure x13) 
                                        (measure x12) (measure x11)
                                      | (algebra.Alg.Term 
                                         algebra.F.id_a__U52 (x11::nil)) =>
                                       P_id_A__U52 (measure x11)
                                      | (algebra.Alg.Term 
                                         algebra.F.id_a__U21 (x12::x11::nil)) =>
                                       P_id_A__U21 (measure x12) 
                                        (measure x11)
                                      | (algebra.Alg.Term 
                                         algebra.F.id_a__U92 (x13::x12::
                                         x11::nil)) =>
                                       P_id_A__U92 (measure x13) 
                                        (measure x12) (measure x11)
                                      | (algebra.Alg.Term 
                                         algebra.F.id_a__U41 (x13::x12::
                                         x11::nil)) =>
                                       P_id_A__U41 (measure x13) 
                                        (measure x12) (measure x11)
                                      | (algebra.Alg.Term 
                                         algebra.F.id_a__U84 (x13::x12::
                                         x11::nil)) =>
                                       P_id_A__U84 (measure x13) 
                                        (measure x12) (measure x11)
                                      | (algebra.Alg.Term 
                                         algebra.F.id_a__isNat (x11::nil)) =>
                                       P_id_A__ISNAT (measure x11)
                                      | (algebra.Alg.Term 
                                         algebra.F.id_a__U12 (x12::x11::nil)) =>
                                       P_id_A__U12 (measure x12) 
                                        (measure x11)
                                      | (algebra.Alg.Term algebra.F.id_mark 
                                         (x11::nil)) =>
                                       P_id_MARK (measure x11)
                                      | (algebra.Alg.Term 
                                         algebra.F.id_a__U45 (x12::x11::nil)) =>
                                       P_id_A__U45 (measure x12) 
                                        (measure x11)
                                      | _ => measure t
                                      end.
               Proof.
                 reflexivity .
               Qed.
               
               Lemma marked_measure_star_monotonic :
                forall f l1 l2, 
                 (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step 
                                                                   R_xml_0_deep_rew.R_xml_0_rules)
                                                                  ) l1 
                   l2) ->
                  marked_measure (algebra.Alg.Term f l1) <= marked_measure 
                                                             (algebra.Alg.Term 
                                                               f l2).
               Proof.
                 unfold marked_measure in *.
                 apply InterpZ.marked_measure_star_monotonic.
                 intros ;apply P_id_U51_monotonic;assumption.
                 intros ;apply P_id_a__U84_monotonic;assumption.
                 intros ;apply P_id_a__U32_monotonic;assumption.
                 intros ;apply P_id_isNatList_monotonic;assumption.
                 intros ;apply P_id_a__U13_monotonic;assumption.
                 intros ;apply P_id_U85_monotonic;assumption.
                 intros ;apply P_id_a__length_monotonic;assumption.
                 intros ;apply P_id_a__isNatIList_monotonic;assumption.
                 intros ;apply P_id_U33_monotonic;assumption.
                 intros ;apply P_id_a__U11_monotonic;assumption.
                 intros ;apply P_id_U81_monotonic;assumption.
                 intros ;apply P_id_a__U92_monotonic;assumption.
                 intros ;apply P_id_a__U43_monotonic;assumption.
                 intros ;apply P_id_U23_monotonic;assumption.
                 intros ;apply P_id_a__isNatKind_monotonic;assumption.
                 intros ;apply P_id_U93_monotonic;assumption.
                 intros ;apply P_id_U11_monotonic;assumption.
                 intros ;apply P_id_a__U71_monotonic;assumption.
                 intros ;apply P_id_U44_monotonic;assumption.
                 intros ;apply P_id_U61_monotonic;assumption.
                 intros ;apply P_id_a__U86_monotonic;assumption.
                 intros ;apply P_id_a__U41_monotonic;assumption.
                 intros ;apply P_id_U22_monotonic;assumption.
                 intros ;apply P_id_a__U21_monotonic;assumption.
                 intros ;apply P_id_U91_monotonic;assumption.
                 intros ;apply P_id_length_monotonic;assumption.
                 intros ;apply P_id_a__U52_monotonic;assumption.
                 intros ;apply P_id_U42_monotonic;assumption.
                 intros ;apply P_id_a__U12_monotonic;assumption.
                 intros ;apply P_id_U83_monotonic;assumption.
                 intros ;apply P_id_a__U94_monotonic;assumption.
                 intros ;apply P_id_a__U45_monotonic;assumption.
                 intros ;apply P_id_U31_monotonic;assumption.
                 intros ;apply P_id_a__isNat_monotonic;assumption.
                 intros ;apply P_id_isNatIListKind_monotonic;assumption.
                 intros ;apply P_id_a__U82_monotonic;assumption.
                 intros ;apply P_id_U46_monotonic;assumption.
                 intros ;apply P_id_cons_monotonic;assumption.
                 intros ;apply P_id_U52_monotonic;assumption.
                 intros ;apply P_id_a__U85_monotonic;assumption.
                 intros ;apply P_id_a__U33_monotonic;assumption.
                 intros ;apply P_id_U21_monotonic;assumption.
                 intros ;apply P_id_a__isNatList_monotonic;assumption.
                 intros ;apply P_id_U86_monotonic;assumption.
                 intros ;apply P_id_mark_monotonic;assumption.
                 intros ;apply P_id_a__U51_monotonic;assumption.
                 intros ;apply P_id_U41_monotonic;assumption.
                 intros ;apply P_id_U82_monotonic;assumption.
                 intros ;apply P_id_a__U93_monotonic;assumption.
                 intros ;apply P_id_a__U44_monotonic;assumption.
                 intros ;apply P_id_isNat_monotonic;assumption.
                 intros ;apply P_id_a__U23_monotonic;assumption.
                 intros ;apply P_id_U94_monotonic;assumption.
                 intros ;apply P_id_U12_monotonic;assumption.
                 intros ;apply P_id_a__U81_monotonic;assumption.
                 intros ;apply P_id_U45_monotonic;assumption.
                 intros ;apply P_id_U71_monotonic;assumption.
                 intros ;apply P_id_a__U91_monotonic;assumption.
                 intros ;apply P_id_a__U42_monotonic;assumption.
                 intros ;apply P_id_isNatKind_monotonic;assumption.
                 intros ;apply P_id_a__U22_monotonic;assumption.
                 intros ;apply P_id_U92_monotonic;assumption.
                 intros ;apply P_id_a__U61_monotonic;assumption.
                 intros ;apply P_id_U43_monotonic;assumption.
                 intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
                 intros ;apply P_id_U84_monotonic;assumption.
                 intros ;apply P_id_s_monotonic;assumption.
                 intros ;apply P_id_a__U46_monotonic;assumption.
                 intros ;apply P_id_U32_monotonic;assumption.
                 intros ;apply P_id_a__U31_monotonic;assumption.
                 intros ;apply P_id_U13_monotonic;assumption.
                 intros ;apply P_id_a__U83_monotonic;assumption.
                 intros ;apply P_id_isNatIList_monotonic;assumption.
                 intros ;apply P_id_a__zeros_bounded;assumption.
                 intros ;apply P_id_U51_bounded;assumption.
                 intros ;apply P_id_a__U84_bounded;assumption.
                 intros ;apply P_id_a__U32_bounded;assumption.
                 intros ;apply P_id_isNatList_bounded;assumption.
                 intros ;apply P_id_a__U13_bounded;assumption.
                 intros ;apply P_id_U85_bounded;assumption.
                 intros ;apply P_id_a__length_bounded;assumption.
                 intros ;apply P_id_a__isNatIList_bounded;assumption.
                 intros ;apply P_id_U33_bounded;assumption.
                 intros ;apply P_id_a__U11_bounded;assumption.
                 intros ;apply P_id_U81_bounded;assumption.
                 intros ;apply P_id_a__U92_bounded;assumption.
                 intros ;apply P_id_a__U43_bounded;assumption.
                 intros ;apply P_id_U23_bounded;assumption.
                 intros ;apply P_id_a__isNatKind_bounded;assumption.
                 intros ;apply P_id_U93_bounded;assumption.
                 intros ;apply P_id_U11_bounded;assumption.
                 intros ;apply P_id_a__U71_bounded;assumption.
                 intros ;apply P_id_U44_bounded;assumption.
                 intros ;apply P_id_0_bounded;assumption.
                 intros ;apply P_id_U61_bounded;assumption.
                 intros ;apply P_id_a__U86_bounded;assumption.
                 intros ;apply P_id_a__U41_bounded;assumption.
                 intros ;apply P_id_U22_bounded;assumption.
                 intros ;apply P_id_a__U21_bounded;assumption.
                 intros ;apply P_id_U91_bounded;assumption.
                 intros ;apply P_id_length_bounded;assumption.
                 intros ;apply P_id_a__U52_bounded;assumption.
                 intros ;apply P_id_U42_bounded;assumption.
                 intros ;apply P_id_a__U12_bounded;assumption.
                 intros ;apply P_id_U83_bounded;assumption.
                 intros ;apply P_id_a__U94_bounded;assumption.
                 intros ;apply P_id_a__U45_bounded;assumption.
                 intros ;apply P_id_U31_bounded;assumption.
                 intros ;apply P_id_a__isNat_bounded;assumption.
                 intros ;apply P_id_isNatIListKind_bounded;assumption.
                 intros ;apply P_id_a__U82_bounded;assumption.
                 intros ;apply P_id_U46_bounded;assumption.
                 intros ;apply P_id_cons_bounded;assumption.
                 intros ;apply P_id_U52_bounded;assumption.
                 intros ;apply P_id_a__U85_bounded;assumption.
                 intros ;apply P_id_a__U33_bounded;assumption.
                 intros ;apply P_id_U21_bounded;assumption.
                 intros ;apply P_id_a__isNatList_bounded;assumption.
                 intros ;apply P_id_U86_bounded;assumption.
                 intros ;apply P_id_mark_bounded;assumption.
                 intros ;apply P_id_a__U51_bounded;assumption.
                 intros ;apply P_id_U41_bounded;assumption.
                 intros ;apply P_id_tt_bounded;assumption.
                 intros ;apply P_id_U82_bounded;assumption.
                 intros ;apply P_id_a__U93_bounded;assumption.
                 intros ;apply P_id_a__U44_bounded;assumption.
                 intros ;apply P_id_isNat_bounded;assumption.
                 intros ;apply P_id_a__U23_bounded;assumption.
                 intros ;apply P_id_U94_bounded;assumption.
                 intros ;apply P_id_U12_bounded;assumption.
                 intros ;apply P_id_a__U81_bounded;assumption.
                 intros ;apply P_id_U45_bounded;assumption.
                 intros ;apply P_id_zeros_bounded;assumption.
                 intros ;apply P_id_U71_bounded;assumption.
                 intros ;apply P_id_a__U91_bounded;assumption.
                 intros ;apply P_id_a__U42_bounded;assumption.
                 intros ;apply P_id_isNatKind_bounded;assumption.
                 intros ;apply P_id_a__U22_bounded;assumption.
                 intros ;apply P_id_U92_bounded;assumption.
                 intros ;apply P_id_nil_bounded;assumption.
                 intros ;apply P_id_a__U61_bounded;assumption.
                 intros ;apply P_id_U43_bounded;assumption.
                 intros ;apply P_id_a__isNatIListKind_bounded;assumption.
                 intros ;apply P_id_U84_bounded;assumption.
                 intros ;apply P_id_s_bounded;assumption.
                 intros ;apply P_id_a__U46_bounded;assumption.
                 intros ;apply P_id_U32_bounded;assumption.
                 intros ;apply P_id_a__U31_bounded;assumption.
                 intros ;apply P_id_U13_bounded;assumption.
                 intros ;apply P_id_a__U83_bounded;assumption.
                 intros ;apply P_id_isNatIList_bounded;assumption.
                 apply rules_monotonic.
                 intros ;apply P_id_A__U46_monotonic;assumption.
                 intros ;apply P_id_A__ISNATILISTKIND_monotonic;assumption.
                 intros ;apply P_id_A__U61_monotonic;assumption.
                 intros ;apply P_id_A__U85_monotonic;assumption.
                 intros ;apply P_id_A__U31_monotonic;assumption.
                 intros ;apply P_id_A__U81_monotonic;assumption.
                 intros ;apply P_id_A__U22_monotonic;assumption.
                 intros ;apply P_id_A__U93_monotonic;assumption.
                 intros ;apply P_id_A__U42_monotonic;assumption.
                 intros ;apply P_id_A__U51_monotonic;assumption.
                 intros ;apply P_id_A__ISNATLIST_monotonic;assumption.
                 intros ;apply P_id_A__U91_monotonic;assumption.
                 intros ;apply P_id_A__U33_monotonic;assumption.
                 intros ;apply P_id_A__U83_monotonic;assumption.
                 intros ;apply P_id_A__U23_monotonic;assumption.
                 intros ;apply P_id_A__U11_monotonic;assumption.
                 intros ;apply P_id_A__LENGTH_monotonic;assumption.
                 intros ;apply P_id_A__U44_monotonic;assumption.
                 intros ;apply P_id_A__ISNATILIST_monotonic;assumption.
                 intros ;apply P_id_A__U13_monotonic;assumption.
                 intros ;apply P_id_A__U71_monotonic;assumption.
                 intros ;apply P_id_A__U86_monotonic;assumption.
                 intros ;apply P_id_A__U32_monotonic;assumption.
                 intros ;apply P_id_A__U82_monotonic;assumption.
                 intros ;apply P_id_A__ISNATKIND_monotonic;assumption.
                 intros ;apply P_id_A__U94_monotonic;assumption.
                 intros ;apply P_id_A__U43_monotonic;assumption.
                 intros ;apply P_id_A__U52_monotonic;assumption.
                 intros ;apply P_id_A__U21_monotonic;assumption.
                 intros ;apply P_id_A__U92_monotonic;assumption.
                 intros ;apply P_id_A__U41_monotonic;assumption.
                 intros ;apply P_id_A__U84_monotonic;assumption.
                 intros ;apply P_id_A__ISNAT_monotonic;assumption.
                 intros ;apply P_id_A__U12_monotonic;assumption.
                 intros ;apply P_id_MARK_monotonic;assumption.
                 intros ;apply P_id_A__U45_monotonic;assumption.
               Qed.
               
               Ltac rewrite_and_unfold  :=
                do 2 (rewrite marked_measure_equation);
                 repeat (
                 match goal with
                   |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
                    rewrite (measure_equation (algebra.Alg.Term f t))
                   | H:context [measure (algebra.Alg.Term ?f ?t)] |- 
                   _ =>
                    rewrite (measure_equation (algebra.Alg.Term f t)) in H|-
                   end
                 ).
               
               Definition lt a b := 
                 (Zwf.Zwf 0) (marked_measure a) (marked_measure b).
               
               Definition le a b := marked_measure a <= marked_measure b.
               
               Lemma lt_le_compat :
                forall a b c, (lt a b) ->(le b c) ->lt a c.
               Proof.
                 unfold lt, le in *.
                 intros a b c.
                 apply (interp.le_lt_compat_right (interp.o_Z 0)).
               Qed.
               
               Lemma wf_lt : well_founded lt.
               Proof.
                 unfold lt in *.
                 apply Inverse_Image.wf_inverse_image with  (B:=Z).
                 apply Zwf.Zwf_well_founded.
               Qed.
               
               Lemma DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_strict_in_lt :
                Relation_Definitions.inclusion _ 
                 DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_strict 
                 lt.
               Proof.
                 unfold Relation_Definitions.inclusion, lt in *.
                 
                 intros a b H;destruct H;
                  match goal with
                    |  |- (Zwf.Zwf 0) _ 
                           (marked_measure (algebra.Alg.Term ?f ?l)) =>
                     let l'' := algebra.Alg_ext.find_replacement l  in 
                      ((apply (interp.le_lt_compat_right (interp.o_Z 0)) with
                         (marked_measure (algebra.Alg.Term f l''));[idtac|
                        apply marked_measure_star_monotonic;
                         repeat (
                         apply algebra.EQT_ext.one_step_list_refl_trans_clos);
                         (assumption)||(constructor 1)]))
                    end
                  ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_in_le :
                Relation_Definitions.inclusion _ 
                 DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large 
                 le.
               Proof.
                 unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *.
                 
                 intros a b H;destruct H;
                  match goal with
                    |  |- _ <= marked_measure (algebra.Alg.Term ?f ?l) =>
                     let l'' := algebra.Alg_ext.find_replacement l  in 
                      ((apply (interp.le_trans (interp.o_Z 0)) with
                         (marked_measure (algebra.Alg.Term f l''));[idtac|
                        apply marked_measure_star_monotonic;
                         repeat (
                         apply algebra.EQT_ext.one_step_list_refl_trans_clos);
                         (assumption)||(constructor 1)]))
                    end
                  ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
                  cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                   intuition;
                   (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Definition wf_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large  := 
                 WF_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large.wf
                 .
               
               
               Lemma wf :
                well_founded WF_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large.DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large
                 .
               Proof.
                 intros x.
                 apply (well_founded_ind wf_lt).
                 clear x.
                 intros x.
                 pattern x.
                 
                 apply (@Acc_ind _ 
                         DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large)
                        .
                 clear x.
                 intros x _ IHx IHx'.
                 constructor.
                 intros y H.
                 
                 destruct H;
                  (apply IHx';
                    apply DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_strict_in_lt;
                    econstructor eassumption)||
                  ((apply IHx;[econstructor eassumption|
                    intros y' Hlt;apply IHx';
                     apply lt_le_compat with (1:=Hlt) ;
                     apply DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_in_le;
                     econstructor eassumption])).
                 
                 apply wf_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_large.
               Qed.
              End WF_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large.
              
              Open Scope Z_scope.
              
              Import ring_extention.
              
              Notation Local "a <= b" := (Zle a b).
              
              Notation Local "a < b" := (Zlt a b).
              
              Definition P_id_a__zeros  := 2.
              
              Definition P_id_U51 (x11:Z) (x12:Z) := 1* x11.
              
              Definition P_id_a__U84 (x11:Z) (x12:Z) (x13:Z) := 0.
              
              Definition P_id_a__U32 (x11:Z) (x12:Z) := 0.
              
              Definition P_id_isNatList (x11:Z) := 0.
              
              Definition P_id_a__U13 (x11:Z) := 0.
              
              Definition P_id_U85 (x11:Z) (x12:Z) := 0.
              
              Definition P_id_a__length (x11:Z) := 0.
              
              Definition P_id_a__isNatIList (x11:Z) := 3.
              
              Definition P_id_U33 (x11:Z) := 0.
              
              Definition P_id_a__U11 (x11:Z) (x12:Z) := 1.
              
              Definition P_id_U81 (x11:Z) (x12:Z) (x13:Z) := 0.
              
              Definition P_id_a__U92 (x11:Z) (x12:Z) (x13:Z) := 0.
              
              Definition P_id_a__U43 (x11:Z) (x12:Z) (x13:Z) := 0.
              
              Definition P_id_U23 (x11:Z) := 1* x11.
              
              Definition P_id_a__isNatKind (x11:Z) := 0.
              
              Definition P_id_U93 (x11:Z) (x12:Z) (x13:Z) := 0.
              
              Definition P_id_U11 (x11:Z) (x12:Z) := 1.
              
              Definition P_id_a__U71 (x11:Z) := 2* x11.
              
              Definition P_id_U44 (x11:Z) (x12:Z) (x13:Z) := 0.
              
              Definition P_id_0  := 0.
              
              Definition P_id_U61 (x11:Z) := 2* x11.
              
              Definition P_id_a__U86 (x11:Z) := 1* x11.
              
              Definition P_id_a__U41 (x11:Z) (x12:Z) (x13:Z) := 0.
              
              Definition P_id_U22 (x11:Z) (x12:Z) := 1 + 1* x11.
              
              Definition P_id_a__U21 (x11:Z) (x12:Z) := 1 + 2* x11.
              
              Definition P_id_U91 (x11:Z) (x12:Z) (x13:Z) := 0.
              
              Definition P_id_length (x11:Z) := 0.
              
              Definition P_id_a__U52 (x11:Z) := 2* x11.
              
              Definition P_id_U42 (x11:Z) (x12:Z) (x13:Z) := 0.
              
              Definition P_id_a__U12 (x11:Z) (x12:Z) := 0.
              
              Definition P_id_U83 (x11:Z) (x12:Z) (x13:Z) := 0.
              
              Definition P_id_a__U94 (x11:Z) (x12:Z) := 0.
              
              Definition P_id_a__U45 (x11:Z) (x12:Z) := 0.
              
              Definition P_id_U31 (x11:Z) (x12:Z) := 0.
              
              Definition P_id_a__isNat (x11:Z) := 1.
              
              Definition P_id_isNatIListKind (x11:Z) := 0.
              
              Definition P_id_a__U82 (x11:Z) (x12:Z) (x13:Z) := 0.
              
              Definition P_id_U46 (x11:Z) := 0.
              
              Definition P_id_cons (x11:Z) (x12:Z) := 0.
              
              Definition P_id_U52 (x11:Z) := 2* x11.
              
              Definition P_id_a__U85 (x11:Z) (x12:Z) := 0.
              
              Definition P_id_a__U33 (x11:Z) := 0.
              
              Definition P_id_U21 (x11:Z) (x12:Z) := 1 + 2* x11.
              
              Definition P_id_a__isNatList (x11:Z) := 0.
              
              Definition P_id_U86 (x11:Z) := 1* x11.
              
              Definition P_id_mark (x11:Z) := 1* x11.
              
              Definition P_id_a__U51 (x11:Z) (x12:Z) := 1* x11.
              
              Definition P_id_U41 (x11:Z) (x12:Z) (x13:Z) := 0.
              
              Definition P_id_tt  := 0.
              
              Definition P_id_U82 (x11:Z) (x12:Z) (x13:Z) := 0.
              
              Definition P_id_a__U93 (x11:Z) (x12:Z) (x13:Z) := 0.
              
              Definition P_id_a__U44 (x11:Z) (x12:Z) (x13:Z) := 0.
              
              Definition P_id_isNat (x11:Z) := 1.
              
              Definition P_id_a__U23 (x11:Z) := 1* x11.
              
              Definition P_id_U94 (x11:Z) (x12:Z) := 0.
              
              Definition P_id_U12 (x11:Z) (x12:Z) := 0.
              
              Definition P_id_a__U81 (x11:Z) (x12:Z) (x13:Z) := 0.
              
              Definition P_id_U45 (x11:Z) (x12:Z) := 0.
              
              Definition P_id_zeros  := 2.
              
              Definition P_id_U71 (x11:Z) := 2* x11.
              
              Definition P_id_a__U91 (x11:Z) (x12:Z) (x13:Z) := 0.
              
              Definition P_id_a__U42 (x11:Z) (x12:Z) (x13:Z) := 0.
              
              Definition P_id_isNatKind (x11:Z) := 0.
              
              Definition P_id_a__U22 (x11:Z) (x12:Z) := 1 + 1* x11.
              
              Definition P_id_U92 (x11:Z) (x12:Z) (x13:Z) := 0.
              
              Definition P_id_nil  := 1.
              
              Definition P_id_a__U61 (x11:Z) := 2* x11.
              
              Definition P_id_U43 (x11:Z) (x12:Z) (x13:Z) := 0.
              
              Definition P_id_a__isNatIListKind (x11:Z) := 0.
              
              Definition P_id_U84 (x11:Z) (x12:Z) (x13:Z) := 0.
              
              Definition P_id_s (x11:Z) := 1* x11.
              
              Definition P_id_a__U46 (x11:Z) := 0.
              
              Definition P_id_U32 (x11:Z) (x12:Z) := 0.
              
              Definition P_id_a__U31 (x11:Z) (x12:Z) := 0.
              
              Definition P_id_U13 (x11:Z) := 0.
              
              Definition P_id_a__U83 (x11:Z) (x12:Z) (x13:Z) := 0.
              
              Definition P_id_isNatIList (x11:Z) := 3.
              
              Lemma P_id_U51_monotonic :
               forall x12 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_U51 x12 x14 <= P_id_U51 x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U84_monotonic :
               forall x16 x12 x14 x13 x11 x15, 
                (0 <= x16)/\ (x16 <= x15) ->
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_a__U84 x12 x14 x16 <= P_id_a__U84 x11 x13 x15.
              Proof.
                intros x16 x15 x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                intros [H_5 H_4].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U32_monotonic :
               forall x12 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_a__U32 x12 x14 <= P_id_a__U32 x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_isNatList_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_isNatList x12 <= P_id_isNatList x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U13_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_a__U13 x12 <= P_id_a__U13 x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U85_monotonic :
               forall x12 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_U85 x12 x14 <= P_id_U85 x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__length_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_a__length x12 <= P_id_a__length x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__isNatIList_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_a__isNatIList x12 <= P_id_a__isNatIList x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U33_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->P_id_U33 x12 <= P_id_U33 x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U11_monotonic :
               forall x12 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_a__U11 x12 x14 <= P_id_a__U11 x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U81_monotonic :
               forall x16 x12 x14 x13 x11 x15, 
                (0 <= x16)/\ (x16 <= x15) ->
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_U81 x12 x14 x16 <= P_id_U81 x11 x13 x15.
              Proof.
                intros x16 x15 x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                intros [H_5 H_4].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U92_monotonic :
               forall x16 x12 x14 x13 x11 x15, 
                (0 <= x16)/\ (x16 <= x15) ->
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_a__U92 x12 x14 x16 <= P_id_a__U92 x11 x13 x15.
              Proof.
                intros x16 x15 x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                intros [H_5 H_4].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U43_monotonic :
               forall x16 x12 x14 x13 x11 x15, 
                (0 <= x16)/\ (x16 <= x15) ->
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_a__U43 x12 x14 x16 <= P_id_a__U43 x11 x13 x15.
              Proof.
                intros x16 x15 x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                intros [H_5 H_4].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U23_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->P_id_U23 x12 <= P_id_U23 x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__isNatKind_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_a__isNatKind x12 <= P_id_a__isNatKind x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U93_monotonic :
               forall x16 x12 x14 x13 x11 x15, 
                (0 <= x16)/\ (x16 <= x15) ->
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_U93 x12 x14 x16 <= P_id_U93 x11 x13 x15.
              Proof.
                intros x16 x15 x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                intros [H_5 H_4].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U11_monotonic :
               forall x12 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_U11 x12 x14 <= P_id_U11 x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U71_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_a__U71 x12 <= P_id_a__U71 x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U44_monotonic :
               forall x16 x12 x14 x13 x11 x15, 
                (0 <= x16)/\ (x16 <= x15) ->
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_U44 x12 x14 x16 <= P_id_U44 x11 x13 x15.
              Proof.
                intros x16 x15 x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                intros [H_5 H_4].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U61_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->P_id_U61 x12 <= P_id_U61 x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U86_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_a__U86 x12 <= P_id_a__U86 x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U41_monotonic :
               forall x16 x12 x14 x13 x11 x15, 
                (0 <= x16)/\ (x16 <= x15) ->
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_a__U41 x12 x14 x16 <= P_id_a__U41 x11 x13 x15.
              Proof.
                intros x16 x15 x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                intros [H_5 H_4].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U22_monotonic :
               forall x12 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_U22 x12 x14 <= P_id_U22 x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U21_monotonic :
               forall x12 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_a__U21 x12 x14 <= P_id_a__U21 x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U91_monotonic :
               forall x16 x12 x14 x13 x11 x15, 
                (0 <= x16)/\ (x16 <= x15) ->
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_U91 x12 x14 x16 <= P_id_U91 x11 x13 x15.
              Proof.
                intros x16 x15 x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                intros [H_5 H_4].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_length_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_length x12 <= P_id_length x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U52_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_a__U52 x12 <= P_id_a__U52 x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U42_monotonic :
               forall x16 x12 x14 x13 x11 x15, 
                (0 <= x16)/\ (x16 <= x15) ->
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_U42 x12 x14 x16 <= P_id_U42 x11 x13 x15.
              Proof.
                intros x16 x15 x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                intros [H_5 H_4].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U12_monotonic :
               forall x12 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_a__U12 x12 x14 <= P_id_a__U12 x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U83_monotonic :
               forall x16 x12 x14 x13 x11 x15, 
                (0 <= x16)/\ (x16 <= x15) ->
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_U83 x12 x14 x16 <= P_id_U83 x11 x13 x15.
              Proof.
                intros x16 x15 x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                intros [H_5 H_4].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U94_monotonic :
               forall x12 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_a__U94 x12 x14 <= P_id_a__U94 x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U45_monotonic :
               forall x12 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_a__U45 x12 x14 <= P_id_a__U45 x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U31_monotonic :
               forall x12 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_U31 x12 x14 <= P_id_U31 x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__isNat_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_a__isNat x12 <= P_id_a__isNat x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_isNatIListKind_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_isNatIListKind x12 <= P_id_isNatIListKind x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U82_monotonic :
               forall x16 x12 x14 x13 x11 x15, 
                (0 <= x16)/\ (x16 <= x15) ->
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_a__U82 x12 x14 x16 <= P_id_a__U82 x11 x13 x15.
              Proof.
                intros x16 x15 x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                intros [H_5 H_4].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U46_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->P_id_U46 x12 <= P_id_U46 x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_cons_monotonic :
               forall x12 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_cons x12 x14 <= P_id_cons x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U52_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->P_id_U52 x12 <= P_id_U52 x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U85_monotonic :
               forall x12 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_a__U85 x12 x14 <= P_id_a__U85 x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U33_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_a__U33 x12 <= P_id_a__U33 x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U21_monotonic :
               forall x12 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_U21 x12 x14 <= P_id_U21 x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__isNatList_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_a__isNatList x12 <= P_id_a__isNatList x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U86_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->P_id_U86 x12 <= P_id_U86 x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_mark_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->P_id_mark x12 <= P_id_mark x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U51_monotonic :
               forall x12 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_a__U51 x12 x14 <= P_id_a__U51 x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U41_monotonic :
               forall x16 x12 x14 x13 x11 x15, 
                (0 <= x16)/\ (x16 <= x15) ->
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_U41 x12 x14 x16 <= P_id_U41 x11 x13 x15.
              Proof.
                intros x16 x15 x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                intros [H_5 H_4].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U82_monotonic :
               forall x16 x12 x14 x13 x11 x15, 
                (0 <= x16)/\ (x16 <= x15) ->
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_U82 x12 x14 x16 <= P_id_U82 x11 x13 x15.
              Proof.
                intros x16 x15 x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                intros [H_5 H_4].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U93_monotonic :
               forall x16 x12 x14 x13 x11 x15, 
                (0 <= x16)/\ (x16 <= x15) ->
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_a__U93 x12 x14 x16 <= P_id_a__U93 x11 x13 x15.
              Proof.
                intros x16 x15 x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                intros [H_5 H_4].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U44_monotonic :
               forall x16 x12 x14 x13 x11 x15, 
                (0 <= x16)/\ (x16 <= x15) ->
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_a__U44 x12 x14 x16 <= P_id_a__U44 x11 x13 x15.
              Proof.
                intros x16 x15 x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                intros [H_5 H_4].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_isNat_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->P_id_isNat x12 <= P_id_isNat x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U23_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_a__U23 x12 <= P_id_a__U23 x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U94_monotonic :
               forall x12 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_U94 x12 x14 <= P_id_U94 x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U12_monotonic :
               forall x12 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_U12 x12 x14 <= P_id_U12 x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U81_monotonic :
               forall x16 x12 x14 x13 x11 x15, 
                (0 <= x16)/\ (x16 <= x15) ->
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_a__U81 x12 x14 x16 <= P_id_a__U81 x11 x13 x15.
              Proof.
                intros x16 x15 x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                intros [H_5 H_4].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U45_monotonic :
               forall x12 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_U45 x12 x14 <= P_id_U45 x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U71_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->P_id_U71 x12 <= P_id_U71 x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U91_monotonic :
               forall x16 x12 x14 x13 x11 x15, 
                (0 <= x16)/\ (x16 <= x15) ->
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_a__U91 x12 x14 x16 <= P_id_a__U91 x11 x13 x15.
              Proof.
                intros x16 x15 x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                intros [H_5 H_4].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U42_monotonic :
               forall x16 x12 x14 x13 x11 x15, 
                (0 <= x16)/\ (x16 <= x15) ->
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_a__U42 x12 x14 x16 <= P_id_a__U42 x11 x13 x15.
              Proof.
                intros x16 x15 x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                intros [H_5 H_4].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_isNatKind_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_isNatKind x12 <= P_id_isNatKind x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U22_monotonic :
               forall x12 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_a__U22 x12 x14 <= P_id_a__U22 x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U92_monotonic :
               forall x16 x12 x14 x13 x11 x15, 
                (0 <= x16)/\ (x16 <= x15) ->
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_U92 x12 x14 x16 <= P_id_U92 x11 x13 x15.
              Proof.
                intros x16 x15 x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                intros [H_5 H_4].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U61_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_a__U61 x12 <= P_id_a__U61 x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U43_monotonic :
               forall x16 x12 x14 x13 x11 x15, 
                (0 <= x16)/\ (x16 <= x15) ->
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_U43 x12 x14 x16 <= P_id_U43 x11 x13 x15.
              Proof.
                intros x16 x15 x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                intros [H_5 H_4].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__isNatIListKind_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_a__isNatIListKind x12 <= P_id_a__isNatIListKind x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U84_monotonic :
               forall x16 x12 x14 x13 x11 x15, 
                (0 <= x16)/\ (x16 <= x15) ->
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_U84 x12 x14 x16 <= P_id_U84 x11 x13 x15.
              Proof.
                intros x16 x15 x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                intros [H_5 H_4].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_s_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->P_id_s x12 <= P_id_s x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U46_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_a__U46 x12 <= P_id_a__U46 x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U32_monotonic :
               forall x12 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_U32 x12 x14 <= P_id_U32 x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U31_monotonic :
               forall x12 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_a__U31 x12 x14 <= P_id_a__U31 x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U13_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->P_id_U13 x12 <= P_id_U13 x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U83_monotonic :
               forall x16 x12 x14 x13 x11 x15, 
                (0 <= x16)/\ (x16 <= x15) ->
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_a__U83 x12 x14 x16 <= P_id_a__U83 x11 x13 x15.
              Proof.
                intros x16 x15 x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                intros [H_5 H_4].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_isNatIList_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_isNatIList x12 <= P_id_isNatIList x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__zeros_bounded : 0 <= P_id_a__zeros .
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U51_bounded :
               forall x12 x11, 
                (0 <= x11) ->(0 <= x12) ->0 <= P_id_U51 x12 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U84_bounded :
               forall x12 x13 x11, 
                (0 <= x11) ->
                 (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U84 x13 x12 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U32_bounded :
               forall x12 x11, 
                (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U32 x12 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_isNatList_bounded :
               forall x11, (0 <= x11) ->0 <= P_id_isNatList x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U13_bounded :
               forall x11, (0 <= x11) ->0 <= P_id_a__U13 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U85_bounded :
               forall x12 x11, 
                (0 <= x11) ->(0 <= x12) ->0 <= P_id_U85 x12 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__length_bounded :
               forall x11, (0 <= x11) ->0 <= P_id_a__length x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__isNatIList_bounded :
               forall x11, (0 <= x11) ->0 <= P_id_a__isNatIList x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U33_bounded :
               forall x11, (0 <= x11) ->0 <= P_id_U33 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U11_bounded :
               forall x12 x11, 
                (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U11 x12 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U81_bounded :
               forall x12 x13 x11, 
                (0 <= x11) ->
                 (0 <= x12) ->(0 <= x13) ->0 <= P_id_U81 x13 x12 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U92_bounded :
               forall x12 x13 x11, 
                (0 <= x11) ->
                 (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U92 x13 x12 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U43_bounded :
               forall x12 x13 x11, 
                (0 <= x11) ->
                 (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U43 x13 x12 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U23_bounded :
               forall x11, (0 <= x11) ->0 <= P_id_U23 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__isNatKind_bounded :
               forall x11, (0 <= x11) ->0 <= P_id_a__isNatKind x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U93_bounded :
               forall x12 x13 x11, 
                (0 <= x11) ->
                 (0 <= x12) ->(0 <= x13) ->0 <= P_id_U93 x13 x12 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U11_bounded :
               forall x12 x11, 
                (0 <= x11) ->(0 <= x12) ->0 <= P_id_U11 x12 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U71_bounded :
               forall x11, (0 <= x11) ->0 <= P_id_a__U71 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U44_bounded :
               forall x12 x13 x11, 
                (0 <= x11) ->
                 (0 <= x12) ->(0 <= x13) ->0 <= P_id_U44 x13 x12 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_0_bounded : 0 <= P_id_0 .
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U61_bounded :
               forall x11, (0 <= x11) ->0 <= P_id_U61 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U86_bounded :
               forall x11, (0 <= x11) ->0 <= P_id_a__U86 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U41_bounded :
               forall x12 x13 x11, 
                (0 <= x11) ->
                 (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U41 x13 x12 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U22_bounded :
               forall x12 x11, 
                (0 <= x11) ->(0 <= x12) ->0 <= P_id_U22 x12 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U21_bounded :
               forall x12 x11, 
                (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U21 x12 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U91_bounded :
               forall x12 x13 x11, 
                (0 <= x11) ->
                 (0 <= x12) ->(0 <= x13) ->0 <= P_id_U91 x13 x12 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_length_bounded :
               forall x11, (0 <= x11) ->0 <= P_id_length x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U52_bounded :
               forall x11, (0 <= x11) ->0 <= P_id_a__U52 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U42_bounded :
               forall x12 x13 x11, 
                (0 <= x11) ->
                 (0 <= x12) ->(0 <= x13) ->0 <= P_id_U42 x13 x12 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U12_bounded :
               forall x12 x11, 
                (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U12 x12 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U83_bounded :
               forall x12 x13 x11, 
                (0 <= x11) ->
                 (0 <= x12) ->(0 <= x13) ->0 <= P_id_U83 x13 x12 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U94_bounded :
               forall x12 x11, 
                (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U94 x12 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U45_bounded :
               forall x12 x11, 
                (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U45 x12 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U31_bounded :
               forall x12 x11, 
                (0 <= x11) ->(0 <= x12) ->0 <= P_id_U31 x12 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__isNat_bounded :
               forall x11, (0 <= x11) ->0 <= P_id_a__isNat x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_isNatIListKind_bounded :
               forall x11, (0 <= x11) ->0 <= P_id_isNatIListKind x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U82_bounded :
               forall x12 x13 x11, 
                (0 <= x11) ->
                 (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U82 x13 x12 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U46_bounded :
               forall x11, (0 <= x11) ->0 <= P_id_U46 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_cons_bounded :
               forall x12 x11, 
                (0 <= x11) ->(0 <= x12) ->0 <= P_id_cons x12 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U52_bounded :
               forall x11, (0 <= x11) ->0 <= P_id_U52 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U85_bounded :
               forall x12 x11, 
                (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U85 x12 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U33_bounded :
               forall x11, (0 <= x11) ->0 <= P_id_a__U33 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U21_bounded :
               forall x12 x11, 
                (0 <= x11) ->(0 <= x12) ->0 <= P_id_U21 x12 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__isNatList_bounded :
               forall x11, (0 <= x11) ->0 <= P_id_a__isNatList x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U86_bounded :
               forall x11, (0 <= x11) ->0 <= P_id_U86 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_mark_bounded :
               forall x11, (0 <= x11) ->0 <= P_id_mark x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U51_bounded :
               forall x12 x11, 
                (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U51 x12 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U41_bounded :
               forall x12 x13 x11, 
                (0 <= x11) ->
                 (0 <= x12) ->(0 <= x13) ->0 <= P_id_U41 x13 x12 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_tt_bounded : 0 <= P_id_tt .
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U82_bounded :
               forall x12 x13 x11, 
                (0 <= x11) ->
                 (0 <= x12) ->(0 <= x13) ->0 <= P_id_U82 x13 x12 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U93_bounded :
               forall x12 x13 x11, 
                (0 <= x11) ->
                 (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U93 x13 x12 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U44_bounded :
               forall x12 x13 x11, 
                (0 <= x11) ->
                 (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U44 x13 x12 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_isNat_bounded :
               forall x11, (0 <= x11) ->0 <= P_id_isNat x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U23_bounded :
               forall x11, (0 <= x11) ->0 <= P_id_a__U23 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U94_bounded :
               forall x12 x11, 
                (0 <= x11) ->(0 <= x12) ->0 <= P_id_U94 x12 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U12_bounded :
               forall x12 x11, 
                (0 <= x11) ->(0 <= x12) ->0 <= P_id_U12 x12 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U81_bounded :
               forall x12 x13 x11, 
                (0 <= x11) ->
                 (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U81 x13 x12 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U45_bounded :
               forall x12 x11, 
                (0 <= x11) ->(0 <= x12) ->0 <= P_id_U45 x12 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_zeros_bounded : 0 <= P_id_zeros .
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U71_bounded :
               forall x11, (0 <= x11) ->0 <= P_id_U71 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U91_bounded :
               forall x12 x13 x11, 
                (0 <= x11) ->
                 (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U91 x13 x12 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U42_bounded :
               forall x12 x13 x11, 
                (0 <= x11) ->
                 (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U42 x13 x12 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_isNatKind_bounded :
               forall x11, (0 <= x11) ->0 <= P_id_isNatKind x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U22_bounded :
               forall x12 x11, 
                (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U22 x12 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U92_bounded :
               forall x12 x13 x11, 
                (0 <= x11) ->
                 (0 <= x12) ->(0 <= x13) ->0 <= P_id_U92 x13 x12 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_nil_bounded : 0 <= P_id_nil .
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U61_bounded :
               forall x11, (0 <= x11) ->0 <= P_id_a__U61 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U43_bounded :
               forall x12 x13 x11, 
                (0 <= x11) ->
                 (0 <= x12) ->(0 <= x13) ->0 <= P_id_U43 x13 x12 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__isNatIListKind_bounded :
               forall x11, (0 <= x11) ->0 <= P_id_a__isNatIListKind x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U84_bounded :
               forall x12 x13 x11, 
                (0 <= x11) ->
                 (0 <= x12) ->(0 <= x13) ->0 <= P_id_U84 x13 x12 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_s_bounded :
               forall x11, (0 <= x11) ->0 <= P_id_s x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U46_bounded :
               forall x11, (0 <= x11) ->0 <= P_id_a__U46 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U32_bounded :
               forall x12 x11, 
                (0 <= x11) ->(0 <= x12) ->0 <= P_id_U32 x12 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U31_bounded :
               forall x12 x11, 
                (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U31 x12 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U13_bounded :
               forall x11, (0 <= x11) ->0 <= P_id_U13 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_a__U83_bounded :
               forall x12 x13 x11, 
                (0 <= x11) ->
                 (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U83 x13 x12 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_isNatIList_bounded :
               forall x11, (0 <= x11) ->0 <= P_id_isNatIList x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Definition measure  := 
                InterpZ.measure 0 P_id_a__zeros P_id_U51 P_id_a__U84 
                 P_id_a__U32 P_id_isNatList P_id_a__U13 P_id_U85 
                 P_id_a__length P_id_a__isNatIList P_id_U33 P_id_a__U11 
                 P_id_U81 P_id_a__U92 P_id_a__U43 P_id_U23 P_id_a__isNatKind 
                 P_id_U93 P_id_U11 P_id_a__U71 P_id_U44 P_id_0 P_id_U61 
                 P_id_a__U86 P_id_a__U41 P_id_U22 P_id_a__U21 P_id_U91 
                 P_id_length P_id_a__U52 P_id_U42 P_id_a__U12 P_id_U83 
                 P_id_a__U94 P_id_a__U45 P_id_U31 P_id_a__isNat 
                 P_id_isNatIListKind P_id_a__U82 P_id_U46 P_id_cons P_id_U52 
                 P_id_a__U85 P_id_a__U33 P_id_U21 P_id_a__isNatList P_id_U86 
                 P_id_mark P_id_a__U51 P_id_U41 P_id_tt P_id_U82 P_id_a__U93 
                 P_id_a__U44 P_id_isNat P_id_a__U23 P_id_U94 P_id_U12 
                 P_id_a__U81 P_id_U45 P_id_zeros P_id_U71 P_id_a__U91 
                 P_id_a__U42 P_id_isNatKind P_id_a__U22 P_id_U92 P_id_nil 
                 P_id_a__U61 P_id_U43 P_id_a__isNatIListKind P_id_U84 
                 P_id_s P_id_a__U46 P_id_U32 P_id_a__U31 P_id_U13 
                 P_id_a__U83 P_id_isNatIList.
              
              Lemma measure_equation :
               forall t, 
                measure t = match t with
                              | (algebra.Alg.Term algebra.F.id_a__zeros nil) =>
                               P_id_a__zeros 
                              | (algebra.Alg.Term algebra.F.id_U51 (x12::
                                 x11::nil)) =>
                               P_id_U51 (measure x12) (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U84 (x13::
                                 x12::x11::nil)) =>
                               P_id_a__U84 (measure x13) (measure x12) 
                                (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U32 (x12::
                                 x11::nil)) =>
                               P_id_a__U32 (measure x12) (measure x11)
                              | (algebra.Alg.Term algebra.F.id_isNatList 
                                 (x11::nil)) =>
                               P_id_isNatList (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U13 
                                 (x11::nil)) =>
                               P_id_a__U13 (measure x11)
                              | (algebra.Alg.Term algebra.F.id_U85 (x12::
                                 x11::nil)) =>
                               P_id_U85 (measure x12) (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__length 
                                 (x11::nil)) =>
                               P_id_a__length (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__isNatIList 
                                 (x11::nil)) =>
                               P_id_a__isNatIList (measure x11)
                              | (algebra.Alg.Term algebra.F.id_U33 
                                 (x11::nil)) =>
                               P_id_U33 (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U11 (x12::
                                 x11::nil)) =>
                               P_id_a__U11 (measure x12) (measure x11)
                              | (algebra.Alg.Term algebra.F.id_U81 (x13::
                                 x12::x11::nil)) =>
                               P_id_U81 (measure x13) (measure x12) 
                                (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U92 (x13::
                                 x12::x11::nil)) =>
                               P_id_a__U92 (measure x13) (measure x12) 
                                (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U43 (x13::
                                 x12::x11::nil)) =>
                               P_id_a__U43 (measure x13) (measure x12) 
                                (measure x11)
                              | (algebra.Alg.Term algebra.F.id_U23 
                                 (x11::nil)) =>
                               P_id_U23 (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__isNatKind 
                                 (x11::nil)) =>
                               P_id_a__isNatKind (measure x11)
                              | (algebra.Alg.Term algebra.F.id_U93 (x13::
                                 x12::x11::nil)) =>
                               P_id_U93 (measure x13) (measure x12) 
                                (measure x11)
                              | (algebra.Alg.Term algebra.F.id_U11 (x12::
                                 x11::nil)) =>
                               P_id_U11 (measure x12) (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U71 
                                 (x11::nil)) =>
                               P_id_a__U71 (measure x11)
                              | (algebra.Alg.Term algebra.F.id_U44 (x13::
                                 x12::x11::nil)) =>
                               P_id_U44 (measure x13) (measure x12) 
                                (measure x11)
                              | (algebra.Alg.Term algebra.F.id_0 nil) =>
                               P_id_0 
                              | (algebra.Alg.Term algebra.F.id_U61 
                                 (x11::nil)) =>
                               P_id_U61 (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U86 
                                 (x11::nil)) =>
                               P_id_a__U86 (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U41 (x13::
                                 x12::x11::nil)) =>
                               P_id_a__U41 (measure x13) (measure x12) 
                                (measure x11)
                              | (algebra.Alg.Term algebra.F.id_U22 (x12::
                                 x11::nil)) =>
                               P_id_U22 (measure x12) (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U21 (x12::
                                 x11::nil)) =>
                               P_id_a__U21 (measure x12) (measure x11)
                              | (algebra.Alg.Term algebra.F.id_U91 (x13::
                                 x12::x11::nil)) =>
                               P_id_U91 (measure x13) (measure x12) 
                                (measure x11)
                              | (algebra.Alg.Term algebra.F.id_length 
                                 (x11::nil)) =>
                               P_id_length (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U52 
                                 (x11::nil)) =>
                               P_id_a__U52 (measure x11)
                              | (algebra.Alg.Term algebra.F.id_U42 (x13::
                                 x12::x11::nil)) =>
                               P_id_U42 (measure x13) (measure x12) 
                                (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U12 (x12::
                                 x11::nil)) =>
                               P_id_a__U12 (measure x12) (measure x11)
                              | (algebra.Alg.Term algebra.F.id_U83 (x13::
                                 x12::x11::nil)) =>
                               P_id_U83 (measure x13) (measure x12) 
                                (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U94 (x12::
                                 x11::nil)) =>
                               P_id_a__U94 (measure x12) (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U45 (x12::
                                 x11::nil)) =>
                               P_id_a__U45 (measure x12) (measure x11)
                              | (algebra.Alg.Term algebra.F.id_U31 (x12::
                                 x11::nil)) =>
                               P_id_U31 (measure x12) (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__isNat 
                                 (x11::nil)) =>
                               P_id_a__isNat (measure x11)
                              | (algebra.Alg.Term 
                                 algebra.F.id_isNatIListKind (x11::nil)) =>
                               P_id_isNatIListKind (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U82 (x13::
                                 x12::x11::nil)) =>
                               P_id_a__U82 (measure x13) (measure x12) 
                                (measure x11)
                              | (algebra.Alg.Term algebra.F.id_U46 
                                 (x11::nil)) =>
                               P_id_U46 (measure x11)
                              | (algebra.Alg.Term algebra.F.id_cons (x12::
                                 x11::nil)) =>
                               P_id_cons (measure x12) (measure x11)
                              | (algebra.Alg.Term algebra.F.id_U52 
                                 (x11::nil)) =>
                               P_id_U52 (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U85 (x12::
                                 x11::nil)) =>
                               P_id_a__U85 (measure x12) (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U33 
                                 (x11::nil)) =>
                               P_id_a__U33 (measure x11)
                              | (algebra.Alg.Term algebra.F.id_U21 (x12::
                                 x11::nil)) =>
                               P_id_U21 (measure x12) (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__isNatList 
                                 (x11::nil)) =>
                               P_id_a__isNatList (measure x11)
                              | (algebra.Alg.Term algebra.F.id_U86 
                                 (x11::nil)) =>
                               P_id_U86 (measure x11)
                              | (algebra.Alg.Term algebra.F.id_mark 
                                 (x11::nil)) =>
                               P_id_mark (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U51 (x12::
                                 x11::nil)) =>
                               P_id_a__U51 (measure x12) (measure x11)
                              | (algebra.Alg.Term algebra.F.id_U41 (x13::
                                 x12::x11::nil)) =>
                               P_id_U41 (measure x13) (measure x12) 
                                (measure x11)
                              | (algebra.Alg.Term algebra.F.id_tt nil) =>
                               P_id_tt 
                              | (algebra.Alg.Term algebra.F.id_U82 (x13::
                                 x12::x11::nil)) =>
                               P_id_U82 (measure x13) (measure x12) 
                                (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U93 (x13::
                                 x12::x11::nil)) =>
                               P_id_a__U93 (measure x13) (measure x12) 
                                (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U44 (x13::
                                 x12::x11::nil)) =>
                               P_id_a__U44 (measure x13) (measure x12) 
                                (measure x11)
                              | (algebra.Alg.Term algebra.F.id_isNat 
                                 (x11::nil)) =>
                               P_id_isNat (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U23 
                                 (x11::nil)) =>
                               P_id_a__U23 (measure x11)
                              | (algebra.Alg.Term algebra.F.id_U94 (x12::
                                 x11::nil)) =>
                               P_id_U94 (measure x12) (measure x11)
                              | (algebra.Alg.Term algebra.F.id_U12 (x12::
                                 x11::nil)) =>
                               P_id_U12 (measure x12) (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U81 (x13::
                                 x12::x11::nil)) =>
                               P_id_a__U81 (measure x13) (measure x12) 
                                (measure x11)
                              | (algebra.Alg.Term algebra.F.id_U45 (x12::
                                 x11::nil)) =>
                               P_id_U45 (measure x12) (measure x11)
                              | (algebra.Alg.Term algebra.F.id_zeros nil) =>
                               P_id_zeros 
                              | (algebra.Alg.Term algebra.F.id_U71 
                                 (x11::nil)) =>
                               P_id_U71 (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U91 (x13::
                                 x12::x11::nil)) =>
                               P_id_a__U91 (measure x13) (measure x12) 
                                (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U42 (x13::
                                 x12::x11::nil)) =>
                               P_id_a__U42 (measure x13) (measure x12) 
                                (measure x11)
                              | (algebra.Alg.Term algebra.F.id_isNatKind 
                                 (x11::nil)) =>
                               P_id_isNatKind (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U22 (x12::
                                 x11::nil)) =>
                               P_id_a__U22 (measure x12) (measure x11)
                              | (algebra.Alg.Term algebra.F.id_U92 (x13::
                                 x12::x11::nil)) =>
                               P_id_U92 (measure x13) (measure x12) 
                                (measure x11)
                              | (algebra.Alg.Term algebra.F.id_nil nil) =>
                               P_id_nil 
                              | (algebra.Alg.Term algebra.F.id_a__U61 
                                 (x11::nil)) =>
                               P_id_a__U61 (measure x11)
                              | (algebra.Alg.Term algebra.F.id_U43 (x13::
                                 x12::x11::nil)) =>
                               P_id_U43 (measure x13) (measure x12) 
                                (measure x11)
                              | (algebra.Alg.Term 
                                 algebra.F.id_a__isNatIListKind (x11::nil)) =>
                               P_id_a__isNatIListKind (measure x11)
                              | (algebra.Alg.Term algebra.F.id_U84 (x13::
                                 x12::x11::nil)) =>
                               P_id_U84 (measure x13) (measure x12) 
                                (measure x11)
                              | (algebra.Alg.Term algebra.F.id_s (x11::nil)) =>
                               P_id_s (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U46 
                                 (x11::nil)) =>
                               P_id_a__U46 (measure x11)
                              | (algebra.Alg.Term algebra.F.id_U32 (x12::
                                 x11::nil)) =>
                               P_id_U32 (measure x12) (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U31 (x12::
                                 x11::nil)) =>
                               P_id_a__U31 (measure x12) (measure x11)
                              | (algebra.Alg.Term algebra.F.id_U13 
                                 (x11::nil)) =>
                               P_id_U13 (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U83 (x13::
                                 x12::x11::nil)) =>
                               P_id_a__U83 (measure x13) (measure x12) 
                                (measure x11)
                              | (algebra.Alg.Term algebra.F.id_isNatIList 
                                 (x11::nil)) =>
                               P_id_isNatIList (measure x11)
                              | _ => 0
                              end.
              Proof.
                intros t;case t;intros ;apply refl_equal.
              Qed.
              
              Lemma measure_bounded : forall t, 0 <= measure t.
              Proof.
                unfold measure in |-*.
                
                apply InterpZ.measure_bounded;
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Ltac generate_pos_hyp  :=
               match goal with
                 | H:context [measure ?x] |- _ =>
                  let v := fresh "v" in 
                   (let H := fresh "h" in 
                     (set (H:=measure_bounded x) in *;
                       set (v:=measure x) in *;clearbody H;clearbody v))
                 |  |- context [measure ?x] =>
                  let v := fresh "v" in 
                   (let H := fresh "h" in 
                     (set (H:=measure_bounded x) in *;
                       set (v:=measure x) in *;clearbody H;clearbody v))
                 end
               .
              
              Lemma rules_monotonic :
               forall l r, 
                (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) ->
                 measure r <= measure l.
              Proof.
                intros l r H.
                fold measure in |-*.
                
                inversion H;clear H;subst;inversion H0;clear H0;subst;
                 simpl algebra.EQT.T.apply_subst in |-*;
                 repeat (
                 match goal with
                   |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
                    rewrite (measure_equation (algebra.Alg.Term f t))
                   end
                 );repeat (generate_pos_hyp );
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma measure_star_monotonic :
               forall l r, 
                (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                           r l) ->measure r <= measure l.
              Proof.
                unfold measure in *.
                apply InterpZ.measure_star_monotonic.
                intros ;apply P_id_U51_monotonic;assumption.
                intros ;apply P_id_a__U84_monotonic;assumption.
                intros ;apply P_id_a__U32_monotonic;assumption.
                intros ;apply P_id_isNatList_monotonic;assumption.
                intros ;apply P_id_a__U13_monotonic;assumption.
                intros ;apply P_id_U85_monotonic;assumption.
                intros ;apply P_id_a__length_monotonic;assumption.
                intros ;apply P_id_a__isNatIList_monotonic;assumption.
                intros ;apply P_id_U33_monotonic;assumption.
                intros ;apply P_id_a__U11_monotonic;assumption.
                intros ;apply P_id_U81_monotonic;assumption.
                intros ;apply P_id_a__U92_monotonic;assumption.
                intros ;apply P_id_a__U43_monotonic;assumption.
                intros ;apply P_id_U23_monotonic;assumption.
                intros ;apply P_id_a__isNatKind_monotonic;assumption.
                intros ;apply P_id_U93_monotonic;assumption.
                intros ;apply P_id_U11_monotonic;assumption.
                intros ;apply P_id_a__U71_monotonic;assumption.
                intros ;apply P_id_U44_monotonic;assumption.
                intros ;apply P_id_U61_monotonic;assumption.
                intros ;apply P_id_a__U86_monotonic;assumption.
                intros ;apply P_id_a__U41_monotonic;assumption.
                intros ;apply P_id_U22_monotonic;assumption.
                intros ;apply P_id_a__U21_monotonic;assumption.
                intros ;apply P_id_U91_monotonic;assumption.
                intros ;apply P_id_length_monotonic;assumption.
                intros ;apply P_id_a__U52_monotonic;assumption.
                intros ;apply P_id_U42_monotonic;assumption.
                intros ;apply P_id_a__U12_monotonic;assumption.
                intros ;apply P_id_U83_monotonic;assumption.
                intros ;apply P_id_a__U94_monotonic;assumption.
                intros ;apply P_id_a__U45_monotonic;assumption.
                intros ;apply P_id_U31_monotonic;assumption.
                intros ;apply P_id_a__isNat_monotonic;assumption.
                intros ;apply P_id_isNatIListKind_monotonic;assumption.
                intros ;apply P_id_a__U82_monotonic;assumption.
                intros ;apply P_id_U46_monotonic;assumption.
                intros ;apply P_id_cons_monotonic;assumption.
                intros ;apply P_id_U52_monotonic;assumption.
                intros ;apply P_id_a__U85_monotonic;assumption.
                intros ;apply P_id_a__U33_monotonic;assumption.
                intros ;apply P_id_U21_monotonic;assumption.
                intros ;apply P_id_a__isNatList_monotonic;assumption.
                intros ;apply P_id_U86_monotonic;assumption.
                intros ;apply P_id_mark_monotonic;assumption.
                intros ;apply P_id_a__U51_monotonic;assumption.
                intros ;apply P_id_U41_monotonic;assumption.
                intros ;apply P_id_U82_monotonic;assumption.
                intros ;apply P_id_a__U93_monotonic;assumption.
                intros ;apply P_id_a__U44_monotonic;assumption.
                intros ;apply P_id_isNat_monotonic;assumption.
                intros ;apply P_id_a__U23_monotonic;assumption.
                intros ;apply P_id_U94_monotonic;assumption.
                intros ;apply P_id_U12_monotonic;assumption.
                intros ;apply P_id_a__U81_monotonic;assumption.
                intros ;apply P_id_U45_monotonic;assumption.
                intros ;apply P_id_U71_monotonic;assumption.
                intros ;apply P_id_a__U91_monotonic;assumption.
                intros ;apply P_id_a__U42_monotonic;assumption.
                intros ;apply P_id_isNatKind_monotonic;assumption.
                intros ;apply P_id_a__U22_monotonic;assumption.
                intros ;apply P_id_U92_monotonic;assumption.
                intros ;apply P_id_a__U61_monotonic;assumption.
                intros ;apply P_id_U43_monotonic;assumption.
                intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
                intros ;apply P_id_U84_monotonic;assumption.
                intros ;apply P_id_s_monotonic;assumption.
                intros ;apply P_id_a__U46_monotonic;assumption.
                intros ;apply P_id_U32_monotonic;assumption.
                intros ;apply P_id_a__U31_monotonic;assumption.
                intros ;apply P_id_U13_monotonic;assumption.
                intros ;apply P_id_a__U83_monotonic;assumption.
                intros ;apply P_id_isNatIList_monotonic;assumption.
                intros ;apply P_id_a__zeros_bounded;assumption.
                intros ;apply P_id_U51_bounded;assumption.
                intros ;apply P_id_a__U84_bounded;assumption.
                intros ;apply P_id_a__U32_bounded;assumption.
                intros ;apply P_id_isNatList_bounded;assumption.
                intros ;apply P_id_a__U13_bounded;assumption.
                intros ;apply P_id_U85_bounded;assumption.
                intros ;apply P_id_a__length_bounded;assumption.
                intros ;apply P_id_a__isNatIList_bounded;assumption.
                intros ;apply P_id_U33_bounded;assumption.
                intros ;apply P_id_a__U11_bounded;assumption.
                intros ;apply P_id_U81_bounded;assumption.
                intros ;apply P_id_a__U92_bounded;assumption.
                intros ;apply P_id_a__U43_bounded;assumption.
                intros ;apply P_id_U23_bounded;assumption.
                intros ;apply P_id_a__isNatKind_bounded;assumption.
                intros ;apply P_id_U93_bounded;assumption.
                intros ;apply P_id_U11_bounded;assumption.
                intros ;apply P_id_a__U71_bounded;assumption.
                intros ;apply P_id_U44_bounded;assumption.
                intros ;apply P_id_0_bounded;assumption.
                intros ;apply P_id_U61_bounded;assumption.
                intros ;apply P_id_a__U86_bounded;assumption.
                intros ;apply P_id_a__U41_bounded;assumption.
                intros ;apply P_id_U22_bounded;assumption.
                intros ;apply P_id_a__U21_bounded;assumption.
                intros ;apply P_id_U91_bounded;assumption.
                intros ;apply P_id_length_bounded;assumption.
                intros ;apply P_id_a__U52_bounded;assumption.
                intros ;apply P_id_U42_bounded;assumption.
                intros ;apply P_id_a__U12_bounded;assumption.
                intros ;apply P_id_U83_bounded;assumption.
                intros ;apply P_id_a__U94_bounded;assumption.
                intros ;apply P_id_a__U45_bounded;assumption.
                intros ;apply P_id_U31_bounded;assumption.
                intros ;apply P_id_a__isNat_bounded;assumption.
                intros ;apply P_id_isNatIListKind_bounded;assumption.
                intros ;apply P_id_a__U82_bounded;assumption.
                intros ;apply P_id_U46_bounded;assumption.
                intros ;apply P_id_cons_bounded;assumption.
                intros ;apply P_id_U52_bounded;assumption.
                intros ;apply P_id_a__U85_bounded;assumption.
                intros ;apply P_id_a__U33_bounded;assumption.
                intros ;apply P_id_U21_bounded;assumption.
                intros ;apply P_id_a__isNatList_bounded;assumption.
                intros ;apply P_id_U86_bounded;assumption.
                intros ;apply P_id_mark_bounded;assumption.
                intros ;apply P_id_a__U51_bounded;assumption.
                intros ;apply P_id_U41_bounded;assumption.
                intros ;apply P_id_tt_bounded;assumption.
                intros ;apply P_id_U82_bounded;assumption.
                intros ;apply P_id_a__U93_bounded;assumption.
                intros ;apply P_id_a__U44_bounded;assumption.
                intros ;apply P_id_isNat_bounded;assumption.
                intros ;apply P_id_a__U23_bounded;assumption.
                intros ;apply P_id_U94_bounded;assumption.
                intros ;apply P_id_U12_bounded;assumption.
                intros ;apply P_id_a__U81_bounded;assumption.
                intros ;apply P_id_U45_bounded;assumption.
                intros ;apply P_id_zeros_bounded;assumption.
                intros ;apply P_id_U71_bounded;assumption.
                intros ;apply P_id_a__U91_bounded;assumption.
                intros ;apply P_id_a__U42_bounded;assumption.
                intros ;apply P_id_isNatKind_bounded;assumption.
                intros ;apply P_id_a__U22_bounded;assumption.
                intros ;apply P_id_U92_bounded;assumption.
                intros ;apply P_id_nil_bounded;assumption.
                intros ;apply P_id_a__U61_bounded;assumption.
                intros ;apply P_id_U43_bounded;assumption.
                intros ;apply P_id_a__isNatIListKind_bounded;assumption.
                intros ;apply P_id_U84_bounded;assumption.
                intros ;apply P_id_s_bounded;assumption.
                intros ;apply P_id_a__U46_bounded;assumption.
                intros ;apply P_id_U32_bounded;assumption.
                intros ;apply P_id_a__U31_bounded;assumption.
                intros ;apply P_id_U13_bounded;assumption.
                intros ;apply P_id_a__U83_bounded;assumption.
                intros ;apply P_id_isNatIList_bounded;assumption.
                apply rules_monotonic.
              Qed.
              
              Definition P_id_A__U46 (x11:Z) := 0.
              
              Definition P_id_A__ISNATILISTKIND (x11:Z) := 0.
              
              Definition P_id_A__U61 (x11:Z) := 0.
              
              Definition P_id_A__U85 (x11:Z) (x12:Z) := 0.
              
              Definition P_id_A__U31 (x11:Z) (x12:Z) := 0.
              
              Definition P_id_A__U81 (x11:Z) (x12:Z) (x13:Z) := 0.
              
              Definition P_id_A__U22 (x11:Z) (x12:Z) := 0.
              
              Definition P_id_A__U93 (x11:Z) (x12:Z) (x13:Z) := 0.
              
              Definition P_id_A__U42 (x11:Z) (x12:Z) (x13:Z) := 0.
              
              Definition P_id_A__U51 (x11:Z) (x12:Z) := 0.
              
              Definition P_id_A__ISNATLIST (x11:Z) := 0.
              
              Definition P_id_A__ZEROS  := 0.
              
              Definition P_id_A__U91 (x11:Z) (x12:Z) (x13:Z) := 0.
              
              Definition P_id_A__U33 (x11:Z) := 0.
              
              Definition P_id_A__U83 (x11:Z) (x12:Z) (x13:Z) := 0.
              
              Definition P_id_A__U23 (x11:Z) := 0.
              
              Definition P_id_A__U11 (x11:Z) (x12:Z) := 0.
              
              Definition P_id_A__LENGTH (x11:Z) := 0.
              
              Definition P_id_A__U44 (x11:Z) (x12:Z) (x13:Z) := 0.
              
              Definition P_id_A__ISNATILIST (x11:Z) := 0.
              
              Definition P_id_A__U13 (x11:Z) := 0.
              
              Definition P_id_A__U71 (x11:Z) := 0.
              
              Definition P_id_A__U86 (x11:Z) := 0.
              
              Definition P_id_A__U32 (x11:Z) (x12:Z) := 0.
              
              Definition P_id_A__U82 (x11:Z) (x12:Z) (x13:Z) := 0.
              
              Definition P_id_A__ISNATKIND (x11:Z) := 0.
              
              Definition P_id_A__U94 (x11:Z) (x12:Z) := 0.
              
              Definition P_id_A__U43 (x11:Z) (x12:Z) (x13:Z) := 0.
              
              Definition P_id_A__U52 (x11:Z) := 0.
              
              Definition P_id_A__U21 (x11:Z) (x12:Z) := 0.
              
              Definition P_id_A__U92 (x11:Z) (x12:Z) (x13:Z) := 0.
              
              Definition P_id_A__U41 (x11:Z) (x12:Z) (x13:Z) := 0.
              
              Definition P_id_A__U84 (x11:Z) (x12:Z) (x13:Z) := 0.
              
              Definition P_id_A__ISNAT (x11:Z) := 0.
              
              Definition P_id_A__U12 (x11:Z) (x12:Z) := 0.
              
              Definition P_id_MARK (x11:Z) := 2* x11.
              
              Definition P_id_A__U45 (x11:Z) (x12:Z) := 0.
              
              Lemma P_id_A__U46_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_A__U46 x12 <= P_id_A__U46 x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_A__ISNATILISTKIND_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_A__ISNATILISTKIND x12 <= P_id_A__ISNATILISTKIND x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_A__U61_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_A__U61 x12 <= P_id_A__U61 x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_A__U85_monotonic :
               forall x12 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_A__U85 x12 x14 <= P_id_A__U85 x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_A__U31_monotonic :
               forall x12 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_A__U31 x12 x14 <= P_id_A__U31 x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_A__U81_monotonic :
               forall x16 x12 x14 x13 x11 x15, 
                (0 <= x16)/\ (x16 <= x15) ->
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_A__U81 x12 x14 x16 <= P_id_A__U81 x11 x13 x15.
              Proof.
                intros x16 x15 x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                intros [H_5 H_4].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_A__U22_monotonic :
               forall x12 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_A__U22 x12 x14 <= P_id_A__U22 x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_A__U93_monotonic :
               forall x16 x12 x14 x13 x11 x15, 
                (0 <= x16)/\ (x16 <= x15) ->
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_A__U93 x12 x14 x16 <= P_id_A__U93 x11 x13 x15.
              Proof.
                intros x16 x15 x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                intros [H_5 H_4].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_A__U42_monotonic :
               forall x16 x12 x14 x13 x11 x15, 
                (0 <= x16)/\ (x16 <= x15) ->
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_A__U42 x12 x14 x16 <= P_id_A__U42 x11 x13 x15.
              Proof.
                intros x16 x15 x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                intros [H_5 H_4].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_A__U51_monotonic :
               forall x12 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_A__U51 x12 x14 <= P_id_A__U51 x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_A__ISNATLIST_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_A__ISNATLIST x12 <= P_id_A__ISNATLIST x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_A__U91_monotonic :
               forall x16 x12 x14 x13 x11 x15, 
                (0 <= x16)/\ (x16 <= x15) ->
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_A__U91 x12 x14 x16 <= P_id_A__U91 x11 x13 x15.
              Proof.
                intros x16 x15 x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                intros [H_5 H_4].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_A__U33_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_A__U33 x12 <= P_id_A__U33 x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_A__U83_monotonic :
               forall x16 x12 x14 x13 x11 x15, 
                (0 <= x16)/\ (x16 <= x15) ->
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_A__U83 x12 x14 x16 <= P_id_A__U83 x11 x13 x15.
              Proof.
                intros x16 x15 x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                intros [H_5 H_4].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_A__U23_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_A__U23 x12 <= P_id_A__U23 x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_A__U11_monotonic :
               forall x12 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_A__U11 x12 x14 <= P_id_A__U11 x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_A__LENGTH_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_A__LENGTH x12 <= P_id_A__LENGTH x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_A__U44_monotonic :
               forall x16 x12 x14 x13 x11 x15, 
                (0 <= x16)/\ (x16 <= x15) ->
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_A__U44 x12 x14 x16 <= P_id_A__U44 x11 x13 x15.
              Proof.
                intros x16 x15 x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                intros [H_5 H_4].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_A__ISNATILIST_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_A__ISNATILIST x12 <= P_id_A__ISNATILIST x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_A__U13_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_A__U13 x12 <= P_id_A__U13 x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_A__U71_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_A__U71 x12 <= P_id_A__U71 x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_A__U86_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_A__U86 x12 <= P_id_A__U86 x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_A__U32_monotonic :
               forall x12 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_A__U32 x12 x14 <= P_id_A__U32 x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_A__U82_monotonic :
               forall x16 x12 x14 x13 x11 x15, 
                (0 <= x16)/\ (x16 <= x15) ->
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_A__U82 x12 x14 x16 <= P_id_A__U82 x11 x13 x15.
              Proof.
                intros x16 x15 x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                intros [H_5 H_4].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_A__ISNATKIND_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_A__ISNATKIND x12 <= P_id_A__ISNATKIND x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_A__U94_monotonic :
               forall x12 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_A__U94 x12 x14 <= P_id_A__U94 x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_A__U43_monotonic :
               forall x16 x12 x14 x13 x11 x15, 
                (0 <= x16)/\ (x16 <= x15) ->
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_A__U43 x12 x14 x16 <= P_id_A__U43 x11 x13 x15.
              Proof.
                intros x16 x15 x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                intros [H_5 H_4].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_A__U52_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_A__U52 x12 <= P_id_A__U52 x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_A__U21_monotonic :
               forall x12 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_A__U21 x12 x14 <= P_id_A__U21 x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_A__U92_monotonic :
               forall x16 x12 x14 x13 x11 x15, 
                (0 <= x16)/\ (x16 <= x15) ->
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_A__U92 x12 x14 x16 <= P_id_A__U92 x11 x13 x15.
              Proof.
                intros x16 x15 x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                intros [H_5 H_4].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_A__U41_monotonic :
               forall x16 x12 x14 x13 x11 x15, 
                (0 <= x16)/\ (x16 <= x15) ->
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_A__U41 x12 x14 x16 <= P_id_A__U41 x11 x13 x15.
              Proof.
                intros x16 x15 x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                intros [H_5 H_4].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_A__U84_monotonic :
               forall x16 x12 x14 x13 x11 x15, 
                (0 <= x16)/\ (x16 <= x15) ->
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_A__U84 x12 x14 x16 <= P_id_A__U84 x11 x13 x15.
              Proof.
                intros x16 x15 x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                intros [H_5 H_4].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_A__ISNAT_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_A__ISNAT x12 <= P_id_A__ISNAT x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_A__U12_monotonic :
               forall x12 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_A__U12 x12 x14 <= P_id_A__U12 x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_MARK_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->P_id_MARK x12 <= P_id_MARK x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_A__U45_monotonic :
               forall x12 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_A__U45 x12 x14 <= P_id_A__U45 x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Definition marked_measure  := 
                InterpZ.marked_measure 0 P_id_a__zeros P_id_U51 P_id_a__U84 
                 P_id_a__U32 P_id_isNatList P_id_a__U13 P_id_U85 
                 P_id_a__length P_id_a__isNatIList P_id_U33 P_id_a__U11 
                 P_id_U81 P_id_a__U92 P_id_a__U43 P_id_U23 P_id_a__isNatKind 
                 P_id_U93 P_id_U11 P_id_a__U71 P_id_U44 P_id_0 P_id_U61 
                 P_id_a__U86 P_id_a__U41 P_id_U22 P_id_a__U21 P_id_U91 
                 P_id_length P_id_a__U52 P_id_U42 P_id_a__U12 P_id_U83 
                 P_id_a__U94 P_id_a__U45 P_id_U31 P_id_a__isNat 
                 P_id_isNatIListKind P_id_a__U82 P_id_U46 P_id_cons P_id_U52 
                 P_id_a__U85 P_id_a__U33 P_id_U21 P_id_a__isNatList P_id_U86 
                 P_id_mark P_id_a__U51 P_id_U41 P_id_tt P_id_U82 P_id_a__U93 
                 P_id_a__U44 P_id_isNat P_id_a__U23 P_id_U94 P_id_U12 
                 P_id_a__U81 P_id_U45 P_id_zeros P_id_U71 P_id_a__U91 
                 P_id_a__U42 P_id_isNatKind P_id_a__U22 P_id_U92 P_id_nil 
                 P_id_a__U61 P_id_U43 P_id_a__isNatIListKind P_id_U84 
                 P_id_s P_id_a__U46 P_id_U32 P_id_a__U31 P_id_U13 
                 P_id_a__U83 P_id_isNatIList P_id_A__U46 
                 P_id_A__ISNATILISTKIND P_id_A__U61 P_id_A__U85 P_id_A__U31 
                 P_id_A__U81 P_id_A__U22 P_id_A__U93 P_id_A__U42 P_id_A__U51 
                 P_id_A__ISNATLIST P_id_A__ZEROS P_id_A__U91 P_id_A__U33 
                 P_id_A__U83 P_id_A__U23 P_id_A__U11 P_id_A__LENGTH 
                 P_id_A__U44 P_id_A__ISNATILIST P_id_A__U13 P_id_A__U71 
                 P_id_A__U86 P_id_A__U32 P_id_A__U82 P_id_A__ISNATKIND 
                 P_id_A__U94 P_id_A__U43 P_id_A__U52 P_id_A__U21 P_id_A__U92 
                 P_id_A__U41 P_id_A__U84 P_id_A__ISNAT P_id_A__U12 P_id_MARK 
                 P_id_A__U45.
              
              Lemma marked_measure_equation :
               forall t, 
                marked_measure t = match t with
                                     | (algebra.Alg.Term algebra.F.id_a__U46 
                                        (x11::nil)) =>
                                      P_id_A__U46 (measure x11)
                                     | (algebra.Alg.Term 
                                        algebra.F.id_a__isNatIListKind 
                                        (x11::nil)) =>
                                      P_id_A__ISNATILISTKIND (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_a__U61 
                                        (x11::nil)) =>
                                      P_id_A__U61 (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_a__U85 
                                        (x12::x11::nil)) =>
                                      P_id_A__U85 (measure x12) (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_a__U31 
                                        (x12::x11::nil)) =>
                                      P_id_A__U31 (measure x12) (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_a__U81 
                                        (x13::x12::x11::nil)) =>
                                      P_id_A__U81 (measure x13) 
                                       (measure x12) (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_a__U22 
                                        (x12::x11::nil)) =>
                                      P_id_A__U22 (measure x12) (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_a__U93 
                                        (x13::x12::x11::nil)) =>
                                      P_id_A__U93 (measure x13) 
                                       (measure x12) (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_a__U42 
                                        (x13::x12::x11::nil)) =>
                                      P_id_A__U42 (measure x13) 
                                       (measure x12) (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_a__U51 
                                        (x12::x11::nil)) =>
                                      P_id_A__U51 (measure x12) (measure x11)
                                     | (algebra.Alg.Term 
                                        algebra.F.id_a__isNatList (x11::nil)) =>
                                      P_id_A__ISNATLIST (measure x11)
                                     | (algebra.Alg.Term 
                                        algebra.F.id_a__zeros nil) =>
                                      P_id_A__ZEROS 
                                     | (algebra.Alg.Term algebra.F.id_a__U91 
                                        (x13::x12::x11::nil)) =>
                                      P_id_A__U91 (measure x13) 
                                       (measure x12) (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_a__U33 
                                        (x11::nil)) =>
                                      P_id_A__U33 (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_a__U83 
                                        (x13::x12::x11::nil)) =>
                                      P_id_A__U83 (measure x13) 
                                       (measure x12) (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_a__U23 
                                        (x11::nil)) =>
                                      P_id_A__U23 (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_a__U11 
                                        (x12::x11::nil)) =>
                                      P_id_A__U11 (measure x12) (measure x11)
                                     | (algebra.Alg.Term 
                                        algebra.F.id_a__length (x11::nil)) =>
                                      P_id_A__LENGTH (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_a__U44 
                                        (x13::x12::x11::nil)) =>
                                      P_id_A__U44 (measure x13) 
                                       (measure x12) (measure x11)
                                     | (algebra.Alg.Term 
                                        algebra.F.id_a__isNatIList 
                                        (x11::nil)) =>
                                      P_id_A__ISNATILIST (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_a__U13 
                                        (x11::nil)) =>
                                      P_id_A__U13 (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_a__U71 
                                        (x11::nil)) =>
                                      P_id_A__U71 (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_a__U86 
                                        (x11::nil)) =>
                                      P_id_A__U86 (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_a__U32 
                                        (x12::x11::nil)) =>
                                      P_id_A__U32 (measure x12) (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_a__U82 
                                        (x13::x12::x11::nil)) =>
                                      P_id_A__U82 (measure x13) 
                                       (measure x12) (measure x11)
                                     | (algebra.Alg.Term 
                                        algebra.F.id_a__isNatKind (x11::nil)) =>
                                      P_id_A__ISNATKIND (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_a__U94 
                                        (x12::x11::nil)) =>
                                      P_id_A__U94 (measure x12) (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_a__U43 
                                        (x13::x12::x11::nil)) =>
                                      P_id_A__U43 (measure x13) 
                                       (measure x12) (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_a__U52 
                                        (x11::nil)) =>
                                      P_id_A__U52 (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_a__U21 
                                        (x12::x11::nil)) =>
                                      P_id_A__U21 (measure x12) (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_a__U92 
                                        (x13::x12::x11::nil)) =>
                                      P_id_A__U92 (measure x13) 
                                       (measure x12) (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_a__U41 
                                        (x13::x12::x11::nil)) =>
                                      P_id_A__U41 (measure x13) 
                                       (measure x12) (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_a__U84 
                                        (x13::x12::x11::nil)) =>
                                      P_id_A__U84 (measure x13) 
                                       (measure x12) (measure x11)
                                     | (algebra.Alg.Term 
                                        algebra.F.id_a__isNat (x11::nil)) =>
                                      P_id_A__ISNAT (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_a__U12 
                                        (x12::x11::nil)) =>
                                      P_id_A__U12 (measure x12) (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_mark 
                                        (x11::nil)) =>
                                      P_id_MARK (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_a__U45 
                                        (x12::x11::nil)) =>
                                      P_id_A__U45 (measure x12) (measure x11)
                                     | _ => measure t
                                     end.
              Proof.
                reflexivity .
              Qed.
              
              Lemma marked_measure_star_monotonic :
               forall f l1 l2, 
                (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step 
                                                                  R_xml_0_deep_rew.R_xml_0_rules)
                                                                 ) l1 
                  l2) ->
                 marked_measure (algebra.Alg.Term f l1) <= marked_measure 
                                                            (algebra.Alg.Term 
                                                              f l2).
              Proof.
                unfold marked_measure in *.
                apply InterpZ.marked_measure_star_monotonic.
                intros ;apply P_id_U51_monotonic;assumption.
                intros ;apply P_id_a__U84_monotonic;assumption.
                intros ;apply P_id_a__U32_monotonic;assumption.
                intros ;apply P_id_isNatList_monotonic;assumption.
                intros ;apply P_id_a__U13_monotonic;assumption.
                intros ;apply P_id_U85_monotonic;assumption.
                intros ;apply P_id_a__length_monotonic;assumption.
                intros ;apply P_id_a__isNatIList_monotonic;assumption.
                intros ;apply P_id_U33_monotonic;assumption.
                intros ;apply P_id_a__U11_monotonic;assumption.
                intros ;apply P_id_U81_monotonic;assumption.
                intros ;apply P_id_a__U92_monotonic;assumption.
                intros ;apply P_id_a__U43_monotonic;assumption.
                intros ;apply P_id_U23_monotonic;assumption.
                intros ;apply P_id_a__isNatKind_monotonic;assumption.
                intros ;apply P_id_U93_monotonic;assumption.
                intros ;apply P_id_U11_monotonic;assumption.
                intros ;apply P_id_a__U71_monotonic;assumption.
                intros ;apply P_id_U44_monotonic;assumption.
                intros ;apply P_id_U61_monotonic;assumption.
                intros ;apply P_id_a__U86_monotonic;assumption.
                intros ;apply P_id_a__U41_monotonic;assumption.
                intros ;apply P_id_U22_monotonic;assumption.
                intros ;apply P_id_a__U21_monotonic;assumption.
                intros ;apply P_id_U91_monotonic;assumption.
                intros ;apply P_id_length_monotonic;assumption.
                intros ;apply P_id_a__U52_monotonic;assumption.
                intros ;apply P_id_U42_monotonic;assumption.
                intros ;apply P_id_a__U12_monotonic;assumption.
                intros ;apply P_id_U83_monotonic;assumption.
                intros ;apply P_id_a__U94_monotonic;assumption.
                intros ;apply P_id_a__U45_monotonic;assumption.
                intros ;apply P_id_U31_monotonic;assumption.
                intros ;apply P_id_a__isNat_monotonic;assumption.
                intros ;apply P_id_isNatIListKind_monotonic;assumption.
                intros ;apply P_id_a__U82_monotonic;assumption.
                intros ;apply P_id_U46_monotonic;assumption.
                intros ;apply P_id_cons_monotonic;assumption.
                intros ;apply P_id_U52_monotonic;assumption.
                intros ;apply P_id_a__U85_monotonic;assumption.
                intros ;apply P_id_a__U33_monotonic;assumption.
                intros ;apply P_id_U21_monotonic;assumption.
                intros ;apply P_id_a__isNatList_monotonic;assumption.
                intros ;apply P_id_U86_monotonic;assumption.
                intros ;apply P_id_mark_monotonic;assumption.
                intros ;apply P_id_a__U51_monotonic;assumption.
                intros ;apply P_id_U41_monotonic;assumption.
                intros ;apply P_id_U82_monotonic;assumption.
                intros ;apply P_id_a__U93_monotonic;assumption.
                intros ;apply P_id_a__U44_monotonic;assumption.
                intros ;apply P_id_isNat_monotonic;assumption.
                intros ;apply P_id_a__U23_monotonic;assumption.
                intros ;apply P_id_U94_monotonic;assumption.
                intros ;apply P_id_U12_monotonic;assumption.
                intros ;apply P_id_a__U81_monotonic;assumption.
                intros ;apply P_id_U45_monotonic;assumption.
                intros ;apply P_id_U71_monotonic;assumption.
                intros ;apply P_id_a__U91_monotonic;assumption.
                intros ;apply P_id_a__U42_monotonic;assumption.
                intros ;apply P_id_isNatKind_monotonic;assumption.
                intros ;apply P_id_a__U22_monotonic;assumption.
                intros ;apply P_id_U92_monotonic;assumption.
                intros ;apply P_id_a__U61_monotonic;assumption.
                intros ;apply P_id_U43_monotonic;assumption.
                intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
                intros ;apply P_id_U84_monotonic;assumption.
                intros ;apply P_id_s_monotonic;assumption.
                intros ;apply P_id_a__U46_monotonic;assumption.
                intros ;apply P_id_U32_monotonic;assumption.
                intros ;apply P_id_a__U31_monotonic;assumption.
                intros ;apply P_id_U13_monotonic;assumption.
                intros ;apply P_id_a__U83_monotonic;assumption.
                intros ;apply P_id_isNatIList_monotonic;assumption.
                intros ;apply P_id_a__zeros_bounded;assumption.
                intros ;apply P_id_U51_bounded;assumption.
                intros ;apply P_id_a__U84_bounded;assumption.
                intros ;apply P_id_a__U32_bounded;assumption.
                intros ;apply P_id_isNatList_bounded;assumption.
                intros ;apply P_id_a__U13_bounded;assumption.
                intros ;apply P_id_U85_bounded;assumption.
                intros ;apply P_id_a__length_bounded;assumption.
                intros ;apply P_id_a__isNatIList_bounded;assumption.
                intros ;apply P_id_U33_bounded;assumption.
                intros ;apply P_id_a__U11_bounded;assumption.
                intros ;apply P_id_U81_bounded;assumption.
                intros ;apply P_id_a__U92_bounded;assumption.
                intros ;apply P_id_a__U43_bounded;assumption.
                intros ;apply P_id_U23_bounded;assumption.
                intros ;apply P_id_a__isNatKind_bounded;assumption.
                intros ;apply P_id_U93_bounded;assumption.
                intros ;apply P_id_U11_bounded;assumption.
                intros ;apply P_id_a__U71_bounded;assumption.
                intros ;apply P_id_U44_bounded;assumption.
                intros ;apply P_id_0_bounded;assumption.
                intros ;apply P_id_U61_bounded;assumption.
                intros ;apply P_id_a__U86_bounded;assumption.
                intros ;apply P_id_a__U41_bounded;assumption.
                intros ;apply P_id_U22_bounded;assumption.
                intros ;apply P_id_a__U21_bounded;assumption.
                intros ;apply P_id_U91_bounded;assumption.
                intros ;apply P_id_length_bounded;assumption.
                intros ;apply P_id_a__U52_bounded;assumption.
                intros ;apply P_id_U42_bounded;assumption.
                intros ;apply P_id_a__U12_bounded;assumption.
                intros ;apply P_id_U83_bounded;assumption.
                intros ;apply P_id_a__U94_bounded;assumption.
                intros ;apply P_id_a__U45_bounded;assumption.
                intros ;apply P_id_U31_bounded;assumption.
                intros ;apply P_id_a__isNat_bounded;assumption.
                intros ;apply P_id_isNatIListKind_bounded;assumption.
                intros ;apply P_id_a__U82_bounded;assumption.
                intros ;apply P_id_U46_bounded;assumption.
                intros ;apply P_id_cons_bounded;assumption.
                intros ;apply P_id_U52_bounded;assumption.
                intros ;apply P_id_a__U85_bounded;assumption.
                intros ;apply P_id_a__U33_bounded;assumption.
                intros ;apply P_id_U21_bounded;assumption.
                intros ;apply P_id_a__isNatList_bounded;assumption.
                intros ;apply P_id_U86_bounded;assumption.
                intros ;apply P_id_mark_bounded;assumption.
                intros ;apply P_id_a__U51_bounded;assumption.
                intros ;apply P_id_U41_bounded;assumption.
                intros ;apply P_id_tt_bounded;assumption.
                intros ;apply P_id_U82_bounded;assumption.
                intros ;apply P_id_a__U93_bounded;assumption.
                intros ;apply P_id_a__U44_bounded;assumption.
                intros ;apply P_id_isNat_bounded;assumption.
                intros ;apply P_id_a__U23_bounded;assumption.
                intros ;apply P_id_U94_bounded;assumption.
                intros ;apply P_id_U12_bounded;assumption.
                intros ;apply P_id_a__U81_bounded;assumption.
                intros ;apply P_id_U45_bounded;assumption.
                intros ;apply P_id_zeros_bounded;assumption.
                intros ;apply P_id_U71_bounded;assumption.
                intros ;apply P_id_a__U91_bounded;assumption.
                intros ;apply P_id_a__U42_bounded;assumption.
                intros ;apply P_id_isNatKind_bounded;assumption.
                intros ;apply P_id_a__U22_bounded;assumption.
                intros ;apply P_id_U92_bounded;assumption.
                intros ;apply P_id_nil_bounded;assumption.
                intros ;apply P_id_a__U61_bounded;assumption.
                intros ;apply P_id_U43_bounded;assumption.
                intros ;apply P_id_a__isNatIListKind_bounded;assumption.
                intros ;apply P_id_U84_bounded;assumption.
                intros ;apply P_id_s_bounded;assumption.
                intros ;apply P_id_a__U46_bounded;assumption.
                intros ;apply P_id_U32_bounded;assumption.
                intros ;apply P_id_a__U31_bounded;assumption.
                intros ;apply P_id_U13_bounded;assumption.
                intros ;apply P_id_a__U83_bounded;assumption.
                intros ;apply P_id_isNatIList_bounded;assumption.
                apply rules_monotonic.
                intros ;apply P_id_A__U46_monotonic;assumption.
                intros ;apply P_id_A__ISNATILISTKIND_monotonic;assumption.
                intros ;apply P_id_A__U61_monotonic;assumption.
                intros ;apply P_id_A__U85_monotonic;assumption.
                intros ;apply P_id_A__U31_monotonic;assumption.
                intros ;apply P_id_A__U81_monotonic;assumption.
                intros ;apply P_id_A__U22_monotonic;assumption.
                intros ;apply P_id_A__U93_monotonic;assumption.
                intros ;apply P_id_A__U42_monotonic;assumption.
                intros ;apply P_id_A__U51_monotonic;assumption.
                intros ;apply P_id_A__ISNATLIST_monotonic;assumption.
                intros ;apply P_id_A__U91_monotonic;assumption.
                intros ;apply P_id_A__U33_monotonic;assumption.
                intros ;apply P_id_A__U83_monotonic;assumption.
                intros ;apply P_id_A__U23_monotonic;assumption.
                intros ;apply P_id_A__U11_monotonic;assumption.
                intros ;apply P_id_A__LENGTH_monotonic;assumption.
                intros ;apply P_id_A__U44_monotonic;assumption.
                intros ;apply P_id_A__ISNATILIST_monotonic;assumption.
                intros ;apply P_id_A__U13_monotonic;assumption.
                intros ;apply P_id_A__U71_monotonic;assumption.
                intros ;apply P_id_A__U86_monotonic;assumption.
                intros ;apply P_id_A__U32_monotonic;assumption.
                intros ;apply P_id_A__U82_monotonic;assumption.
                intros ;apply P_id_A__ISNATKIND_monotonic;assumption.
                intros ;apply P_id_A__U94_monotonic;assumption.
                intros ;apply P_id_A__U43_monotonic;assumption.
                intros ;apply P_id_A__U52_monotonic;assumption.
                intros ;apply P_id_A__U21_monotonic;assumption.
                intros ;apply P_id_A__U92_monotonic;assumption.
                intros ;apply P_id_A__U41_monotonic;assumption.
                intros ;apply P_id_A__U84_monotonic;assumption.
                intros ;apply P_id_A__ISNAT_monotonic;assumption.
                intros ;apply P_id_A__U12_monotonic;assumption.
                intros ;apply P_id_MARK_monotonic;assumption.
                intros ;apply P_id_A__U45_monotonic;assumption.
              Qed.
              
              Ltac rewrite_and_unfold  :=
               do 2 (rewrite marked_measure_equation);
                repeat (
                match goal with
                  |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
                   rewrite (measure_equation (algebra.Alg.Term f t))
                  | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ =>
                   rewrite (measure_equation (algebra.Alg.Term f t)) in H|-
                  end
                ).
              
              Definition lt a b := 
                (Zwf.Zwf 0) (marked_measure a) (marked_measure b).
              
              Definition le a b := marked_measure a <= marked_measure b.
              
              Lemma lt_le_compat :
               forall a b c, (lt a b) ->(le b c) ->lt a c.
              Proof.
                unfold lt, le in *.
                intros a b c.
                apply (interp.le_lt_compat_right (interp.o_Z 0)).
              Qed.
              
              Lemma wf_lt : well_founded lt.
              Proof.
                unfold lt in *.
                apply Inverse_Image.wf_inverse_image with  (B:=Z).
                apply Zwf.Zwf_well_founded.
              Qed.
              
              Lemma DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_strict_in_lt :
               Relation_Definitions.inclusion _ 
                DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_strict 
                lt.
              Proof.
                unfold Relation_Definitions.inclusion, lt in *.
                
                intros a b H;destruct H;
                 match goal with
                   |  |- (Zwf.Zwf 0) _ 
                          (marked_measure (algebra.Alg.Term ?f ?l)) =>
                    let l'' := algebra.Alg_ext.find_replacement l  in 
                     ((apply (interp.le_lt_compat_right (interp.o_Z 0)) with
                        (marked_measure (algebra.Alg.Term f l''));[idtac|
                       apply marked_measure_star_monotonic;
                        repeat (
                        apply algebra.EQT_ext.one_step_list_refl_trans_clos);
                        (assumption)||(constructor 1)]))
                   end
                 ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_in_le :
               Relation_Definitions.inclusion _ 
                DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large 
                le.
              Proof.
                unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *.
                
                intros a b H;destruct H;
                 match goal with
                   |  |- _ <= marked_measure (algebra.Alg.Term ?f ?l) =>
                    let l'' := algebra.Alg_ext.find_replacement l  in 
                     ((apply (interp.le_trans (interp.o_Z 0)) with
                        (marked_measure (algebra.Alg.Term f l''));[idtac|
                       apply marked_measure_star_monotonic;
                        repeat (
                        apply algebra.EQT_ext.one_step_list_refl_trans_clos);
                        (assumption)||(constructor 1)]))
                   end
                 ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Definition wf_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large  := 
                WF_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large.wf
                .
              
              
              Lemma wf :
               well_founded WF_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large.DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large
                .
              Proof.
                intros x.
                apply (well_founded_ind wf_lt).
                clear x.
                intros x.
                pattern x.
                
                apply (@Acc_ind _ 
                        DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large)
                       .
                clear x.
                intros x _ IHx IHx'.
                constructor.
                intros y H.
                
                destruct H;
                 (apply IHx';
                   apply DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_strict_in_lt;
                   econstructor eassumption)||
                 ((apply IHx;[econstructor eassumption|
                   intros y' Hlt;apply IHx';
                    apply lt_le_compat with (1:=Hlt) ;
                    apply DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large_in_le;
                    econstructor eassumption])).
                
                apply wf_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_large.
              Qed.
             End WF_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large.
             
             Open Scope Z_scope.
             
             Import ring_extention.
             
             Notation Local "a <= b" := (Zle a b).
             
             Notation Local "a < b" := (Zlt a b).
             
             Definition P_id_a__zeros  := 2.
             
             Definition P_id_U51 (x11:Z) (x12:Z) := 2* x11.
             
             Definition P_id_a__U84 (x11:Z) (x12:Z) (x13:Z) := 1 + 1* x11.
             
             Definition P_id_a__U32 (x11:Z) (x12:Z) := 3 + 1* x11.
             
             Definition P_id_isNatList (x11:Z) := 1.
             
             Definition P_id_a__U13 (x11:Z) := 0.
             
             Definition P_id_U85 (x11:Z) (x12:Z) := 1 + 2* x11.
             
             Definition P_id_a__length (x11:Z) := 0.
             
             Definition P_id_a__isNatIList (x11:Z) := 3.
             
             Definition P_id_U33 (x11:Z) := 0.
             
             Definition P_id_a__U11 (x11:Z) (x12:Z) := 0.
             
             Definition P_id_U81 (x11:Z) (x12:Z) (x13:Z) := 1 + 2* x11.
             
             Definition P_id_a__U92 (x11:Z) (x12:Z) (x13:Z) := 0.
             
             Definition P_id_a__U43 (x11:Z) (x12:Z) (x13:Z) := 2.
             
             Definition P_id_U23 (x11:Z) := 1* x11.
             
             Definition P_id_a__isNatKind (x11:Z) := 0.
             
             Definition P_id_U93 (x11:Z) (x12:Z) (x13:Z) := 0.
             
             Definition P_id_U11 (x11:Z) (x12:Z) := 0.
             
             Definition P_id_a__U71 (x11:Z) := 1* x11.
             
             Definition P_id_U44 (x11:Z) (x12:Z) (x13:Z) := 1.
             
             Definition P_id_0  := 0.
             
             Definition P_id_U61 (x11:Z) := 1* x11.
             
             Definition P_id_a__U86 (x11:Z) := 1* x11.
             
             Definition P_id_a__U41 (x11:Z) (x12:Z) (x13:Z) := 2.
             
             Definition P_id_U22 (x11:Z) (x12:Z) := 2* x11.
             
             Definition P_id_a__U21 (x11:Z) (x12:Z) := 2* x11.
             
             Definition P_id_U91 (x11:Z) (x12:Z) (x13:Z) := 0.
             
             Definition P_id_length (x11:Z) := 0.
             
             Definition P_id_a__U52 (x11:Z) := 1* x11.
             
             Definition P_id_U42 (x11:Z) (x12:Z) (x13:Z) := 2.
             
             Definition P_id_a__U12 (x11:Z) (x12:Z) := 2* x11.
             
             Definition P_id_U83 (x11:Z) (x12:Z) (x13:Z) := 1 + 1* x11.
             
             Definition P_id_a__U94 (x11:Z) (x12:Z) := 0.
             
             Definition P_id_a__U45 (x11:Z) (x12:Z) := 1.
             
             Definition P_id_U31 (x11:Z) (x12:Z) := 2 + 3* x11.
             
             Definition P_id_a__isNat (x11:Z) := 0.
             
             Definition P_id_isNatIListKind (x11:Z) := 0.
             
             Definition P_id_a__U82 (x11:Z) (x12:Z) (x13:Z) := 1 + 2* x11.
             
             Definition P_id_U46 (x11:Z) := 1.
             
             Definition P_id_cons (x11:Z) (x12:Z) := 0.
             
             Definition P_id_U52 (x11:Z) := 1* x11.
             
             Definition P_id_a__U85 (x11:Z) (x12:Z) := 1 + 2* x11.
             
             Definition P_id_a__U33 (x11:Z) := 0.
             
             Definition P_id_U21 (x11:Z) (x12:Z) := 2* x11.
             
             Definition P_id_a__isNatList (x11:Z) := 1.
             
             Definition P_id_U86 (x11:Z) := 1* x11.
             
             Definition P_id_mark (x11:Z) := 2* x11.
             
             Definition P_id_a__U51 (x11:Z) (x12:Z) := 2* x11.
             
             Definition P_id_U41 (x11:Z) (x12:Z) (x13:Z) := 1.
             
             Definition P_id_tt  := 0.
             
             Definition P_id_U82 (x11:Z) (x12:Z) (x13:Z) := 1 + 2* x11.
             
             Definition P_id_a__U93 (x11:Z) (x12:Z) (x13:Z) := 0.
             
             Definition P_id_a__U44 (x11:Z) (x12:Z) (x13:Z) := 1.
             
             Definition P_id_isNat (x11:Z) := 0.
             
             Definition P_id_a__U23 (x11:Z) := 1* x11.
             
             Definition P_id_U94 (x11:Z) (x12:Z) := 0.
             
             Definition P_id_U12 (x11:Z) (x12:Z) := 2* x11.
             
             Definition P_id_a__U81 (x11:Z) (x12:Z) (x13:Z) := 1 + 2* x11.
             
             Definition P_id_U45 (x11:Z) (x12:Z) := 1.
             
             Definition P_id_zeros  := 1.
             
             Definition P_id_U71 (x11:Z) := 1* x11.
             
             Definition P_id_a__U91 (x11:Z) (x12:Z) (x13:Z) := 0.
             
             Definition P_id_a__U42 (x11:Z) (x12:Z) (x13:Z) := 2.
             
             Definition P_id_isNatKind (x11:Z) := 0.
             
             Definition P_id_a__U22 (x11:Z) (x12:Z) := 2* x11.
             
             Definition P_id_U92 (x11:Z) (x12:Z) (x13:Z) := 0.
             
             Definition P_id_nil  := 2.
             
             Definition P_id_a__U61 (x11:Z) := 1* x11.
             
             Definition P_id_U43 (x11:Z) (x12:Z) (x13:Z) := 1.
             
             Definition P_id_a__isNatIListKind (x11:Z) := 0.
             
             Definition P_id_U84 (x11:Z) (x12:Z) (x13:Z) := 1 + 1* x11.
             
             Definition P_id_s (x11:Z) := 2* x11.
             
             Definition P_id_a__U46 (x11:Z) := 1.
             
             Definition P_id_U32 (x11:Z) (x12:Z) := 2 + 1* x11.
             
             Definition P_id_a__U31 (x11:Z) (x12:Z) := 3 + 3* x11.
             
             Definition P_id_U13 (x11:Z) := 0.
             
             Definition P_id_a__U83 (x11:Z) (x12:Z) (x13:Z) := 1 + 1* x11.
             
             Definition P_id_isNatIList (x11:Z) := 2.
             
             Lemma P_id_U51_monotonic :
              forall x12 x14 x13 x11, 
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_U51 x12 x14 <= P_id_U51 x11 x13.
             Proof.
               intros x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U84_monotonic :
              forall x16 x12 x14 x13 x11 x15, 
               (0 <= x16)/\ (x16 <= x15) ->
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_a__U84 x12 x14 x16 <= P_id_a__U84 x11 x13 x15.
             Proof.
               intros x16 x15 x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               intros [H_5 H_4].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U32_monotonic :
              forall x12 x14 x13 x11, 
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_a__U32 x12 x14 <= P_id_a__U32 x11 x13.
             Proof.
               intros x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_isNatList_monotonic :
              forall x12 x11, 
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_isNatList x12 <= P_id_isNatList x11.
             Proof.
               intros x12 x11.
               intros [H_1 H_0].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U13_monotonic :
              forall x12 x11, 
               (0 <= x12)/\ (x12 <= x11) ->P_id_a__U13 x12 <= P_id_a__U13 x11.
             Proof.
               intros x12 x11.
               intros [H_1 H_0].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U85_monotonic :
              forall x12 x14 x13 x11, 
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_U85 x12 x14 <= P_id_U85 x11 x13.
             Proof.
               intros x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__length_monotonic :
              forall x12 x11, 
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_a__length x12 <= P_id_a__length x11.
             Proof.
               intros x12 x11.
               intros [H_1 H_0].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__isNatIList_monotonic :
              forall x12 x11, 
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_a__isNatIList x12 <= P_id_a__isNatIList x11.
             Proof.
               intros x12 x11.
               intros [H_1 H_0].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U33_monotonic :
              forall x12 x11, 
               (0 <= x12)/\ (x12 <= x11) ->P_id_U33 x12 <= P_id_U33 x11.
             Proof.
               intros x12 x11.
               intros [H_1 H_0].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U11_monotonic :
              forall x12 x14 x13 x11, 
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_a__U11 x12 x14 <= P_id_a__U11 x11 x13.
             Proof.
               intros x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U81_monotonic :
              forall x16 x12 x14 x13 x11 x15, 
               (0 <= x16)/\ (x16 <= x15) ->
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_U81 x12 x14 x16 <= P_id_U81 x11 x13 x15.
             Proof.
               intros x16 x15 x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               intros [H_5 H_4].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U92_monotonic :
              forall x16 x12 x14 x13 x11 x15, 
               (0 <= x16)/\ (x16 <= x15) ->
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_a__U92 x12 x14 x16 <= P_id_a__U92 x11 x13 x15.
             Proof.
               intros x16 x15 x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               intros [H_5 H_4].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U43_monotonic :
              forall x16 x12 x14 x13 x11 x15, 
               (0 <= x16)/\ (x16 <= x15) ->
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_a__U43 x12 x14 x16 <= P_id_a__U43 x11 x13 x15.
             Proof.
               intros x16 x15 x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               intros [H_5 H_4].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U23_monotonic :
              forall x12 x11, 
               (0 <= x12)/\ (x12 <= x11) ->P_id_U23 x12 <= P_id_U23 x11.
             Proof.
               intros x12 x11.
               intros [H_1 H_0].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__isNatKind_monotonic :
              forall x12 x11, 
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_a__isNatKind x12 <= P_id_a__isNatKind x11.
             Proof.
               intros x12 x11.
               intros [H_1 H_0].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U93_monotonic :
              forall x16 x12 x14 x13 x11 x15, 
               (0 <= x16)/\ (x16 <= x15) ->
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_U93 x12 x14 x16 <= P_id_U93 x11 x13 x15.
             Proof.
               intros x16 x15 x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               intros [H_5 H_4].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U11_monotonic :
              forall x12 x14 x13 x11, 
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_U11 x12 x14 <= P_id_U11 x11 x13.
             Proof.
               intros x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U71_monotonic :
              forall x12 x11, 
               (0 <= x12)/\ (x12 <= x11) ->P_id_a__U71 x12 <= P_id_a__U71 x11.
             Proof.
               intros x12 x11.
               intros [H_1 H_0].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U44_monotonic :
              forall x16 x12 x14 x13 x11 x15, 
               (0 <= x16)/\ (x16 <= x15) ->
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_U44 x12 x14 x16 <= P_id_U44 x11 x13 x15.
             Proof.
               intros x16 x15 x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               intros [H_5 H_4].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U61_monotonic :
              forall x12 x11, 
               (0 <= x12)/\ (x12 <= x11) ->P_id_U61 x12 <= P_id_U61 x11.
             Proof.
               intros x12 x11.
               intros [H_1 H_0].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U86_monotonic :
              forall x12 x11, 
               (0 <= x12)/\ (x12 <= x11) ->P_id_a__U86 x12 <= P_id_a__U86 x11.
             Proof.
               intros x12 x11.
               intros [H_1 H_0].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U41_monotonic :
              forall x16 x12 x14 x13 x11 x15, 
               (0 <= x16)/\ (x16 <= x15) ->
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_a__U41 x12 x14 x16 <= P_id_a__U41 x11 x13 x15.
             Proof.
               intros x16 x15 x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               intros [H_5 H_4].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U22_monotonic :
              forall x12 x14 x13 x11, 
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_U22 x12 x14 <= P_id_U22 x11 x13.
             Proof.
               intros x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U21_monotonic :
              forall x12 x14 x13 x11, 
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_a__U21 x12 x14 <= P_id_a__U21 x11 x13.
             Proof.
               intros x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U91_monotonic :
              forall x16 x12 x14 x13 x11 x15, 
               (0 <= x16)/\ (x16 <= x15) ->
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_U91 x12 x14 x16 <= P_id_U91 x11 x13 x15.
             Proof.
               intros x16 x15 x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               intros [H_5 H_4].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_length_monotonic :
              forall x12 x11, 
               (0 <= x12)/\ (x12 <= x11) ->P_id_length x12 <= P_id_length x11.
             Proof.
               intros x12 x11.
               intros [H_1 H_0].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U52_monotonic :
              forall x12 x11, 
               (0 <= x12)/\ (x12 <= x11) ->P_id_a__U52 x12 <= P_id_a__U52 x11.
             Proof.
               intros x12 x11.
               intros [H_1 H_0].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U42_monotonic :
              forall x16 x12 x14 x13 x11 x15, 
               (0 <= x16)/\ (x16 <= x15) ->
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_U42 x12 x14 x16 <= P_id_U42 x11 x13 x15.
             Proof.
               intros x16 x15 x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               intros [H_5 H_4].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U12_monotonic :
              forall x12 x14 x13 x11, 
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_a__U12 x12 x14 <= P_id_a__U12 x11 x13.
             Proof.
               intros x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U83_monotonic :
              forall x16 x12 x14 x13 x11 x15, 
               (0 <= x16)/\ (x16 <= x15) ->
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_U83 x12 x14 x16 <= P_id_U83 x11 x13 x15.
             Proof.
               intros x16 x15 x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               intros [H_5 H_4].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U94_monotonic :
              forall x12 x14 x13 x11, 
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_a__U94 x12 x14 <= P_id_a__U94 x11 x13.
             Proof.
               intros x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U45_monotonic :
              forall x12 x14 x13 x11, 
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_a__U45 x12 x14 <= P_id_a__U45 x11 x13.
             Proof.
               intros x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U31_monotonic :
              forall x12 x14 x13 x11, 
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_U31 x12 x14 <= P_id_U31 x11 x13.
             Proof.
               intros x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__isNat_monotonic :
              forall x12 x11, 
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_a__isNat x12 <= P_id_a__isNat x11.
             Proof.
               intros x12 x11.
               intros [H_1 H_0].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_isNatIListKind_monotonic :
              forall x12 x11, 
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_isNatIListKind x12 <= P_id_isNatIListKind x11.
             Proof.
               intros x12 x11.
               intros [H_1 H_0].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U82_monotonic :
              forall x16 x12 x14 x13 x11 x15, 
               (0 <= x16)/\ (x16 <= x15) ->
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_a__U82 x12 x14 x16 <= P_id_a__U82 x11 x13 x15.
             Proof.
               intros x16 x15 x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               intros [H_5 H_4].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U46_monotonic :
              forall x12 x11, 
               (0 <= x12)/\ (x12 <= x11) ->P_id_U46 x12 <= P_id_U46 x11.
             Proof.
               intros x12 x11.
               intros [H_1 H_0].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_cons_monotonic :
              forall x12 x14 x13 x11, 
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_cons x12 x14 <= P_id_cons x11 x13.
             Proof.
               intros x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U52_monotonic :
              forall x12 x11, 
               (0 <= x12)/\ (x12 <= x11) ->P_id_U52 x12 <= P_id_U52 x11.
             Proof.
               intros x12 x11.
               intros [H_1 H_0].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U85_monotonic :
              forall x12 x14 x13 x11, 
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_a__U85 x12 x14 <= P_id_a__U85 x11 x13.
             Proof.
               intros x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U33_monotonic :
              forall x12 x11, 
               (0 <= x12)/\ (x12 <= x11) ->P_id_a__U33 x12 <= P_id_a__U33 x11.
             Proof.
               intros x12 x11.
               intros [H_1 H_0].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U21_monotonic :
              forall x12 x14 x13 x11, 
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_U21 x12 x14 <= P_id_U21 x11 x13.
             Proof.
               intros x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__isNatList_monotonic :
              forall x12 x11, 
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_a__isNatList x12 <= P_id_a__isNatList x11.
             Proof.
               intros x12 x11.
               intros [H_1 H_0].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U86_monotonic :
              forall x12 x11, 
               (0 <= x12)/\ (x12 <= x11) ->P_id_U86 x12 <= P_id_U86 x11.
             Proof.
               intros x12 x11.
               intros [H_1 H_0].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_mark_monotonic :
              forall x12 x11, 
               (0 <= x12)/\ (x12 <= x11) ->P_id_mark x12 <= P_id_mark x11.
             Proof.
               intros x12 x11.
               intros [H_1 H_0].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U51_monotonic :
              forall x12 x14 x13 x11, 
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_a__U51 x12 x14 <= P_id_a__U51 x11 x13.
             Proof.
               intros x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U41_monotonic :
              forall x16 x12 x14 x13 x11 x15, 
               (0 <= x16)/\ (x16 <= x15) ->
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_U41 x12 x14 x16 <= P_id_U41 x11 x13 x15.
             Proof.
               intros x16 x15 x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               intros [H_5 H_4].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U82_monotonic :
              forall x16 x12 x14 x13 x11 x15, 
               (0 <= x16)/\ (x16 <= x15) ->
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_U82 x12 x14 x16 <= P_id_U82 x11 x13 x15.
             Proof.
               intros x16 x15 x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               intros [H_5 H_4].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U93_monotonic :
              forall x16 x12 x14 x13 x11 x15, 
               (0 <= x16)/\ (x16 <= x15) ->
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_a__U93 x12 x14 x16 <= P_id_a__U93 x11 x13 x15.
             Proof.
               intros x16 x15 x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               intros [H_5 H_4].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U44_monotonic :
              forall x16 x12 x14 x13 x11 x15, 
               (0 <= x16)/\ (x16 <= x15) ->
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_a__U44 x12 x14 x16 <= P_id_a__U44 x11 x13 x15.
             Proof.
               intros x16 x15 x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               intros [H_5 H_4].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_isNat_monotonic :
              forall x12 x11, 
               (0 <= x12)/\ (x12 <= x11) ->P_id_isNat x12 <= P_id_isNat x11.
             Proof.
               intros x12 x11.
               intros [H_1 H_0].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U23_monotonic :
              forall x12 x11, 
               (0 <= x12)/\ (x12 <= x11) ->P_id_a__U23 x12 <= P_id_a__U23 x11.
             Proof.
               intros x12 x11.
               intros [H_1 H_0].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U94_monotonic :
              forall x12 x14 x13 x11, 
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_U94 x12 x14 <= P_id_U94 x11 x13.
             Proof.
               intros x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U12_monotonic :
              forall x12 x14 x13 x11, 
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_U12 x12 x14 <= P_id_U12 x11 x13.
             Proof.
               intros x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U81_monotonic :
              forall x16 x12 x14 x13 x11 x15, 
               (0 <= x16)/\ (x16 <= x15) ->
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_a__U81 x12 x14 x16 <= P_id_a__U81 x11 x13 x15.
             Proof.
               intros x16 x15 x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               intros [H_5 H_4].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U45_monotonic :
              forall x12 x14 x13 x11, 
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_U45 x12 x14 <= P_id_U45 x11 x13.
             Proof.
               intros x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U71_monotonic :
              forall x12 x11, 
               (0 <= x12)/\ (x12 <= x11) ->P_id_U71 x12 <= P_id_U71 x11.
             Proof.
               intros x12 x11.
               intros [H_1 H_0].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U91_monotonic :
              forall x16 x12 x14 x13 x11 x15, 
               (0 <= x16)/\ (x16 <= x15) ->
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_a__U91 x12 x14 x16 <= P_id_a__U91 x11 x13 x15.
             Proof.
               intros x16 x15 x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               intros [H_5 H_4].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U42_monotonic :
              forall x16 x12 x14 x13 x11 x15, 
               (0 <= x16)/\ (x16 <= x15) ->
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_a__U42 x12 x14 x16 <= P_id_a__U42 x11 x13 x15.
             Proof.
               intros x16 x15 x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               intros [H_5 H_4].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_isNatKind_monotonic :
              forall x12 x11, 
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_isNatKind x12 <= P_id_isNatKind x11.
             Proof.
               intros x12 x11.
               intros [H_1 H_0].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U22_monotonic :
              forall x12 x14 x13 x11, 
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_a__U22 x12 x14 <= P_id_a__U22 x11 x13.
             Proof.
               intros x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U92_monotonic :
              forall x16 x12 x14 x13 x11 x15, 
               (0 <= x16)/\ (x16 <= x15) ->
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_U92 x12 x14 x16 <= P_id_U92 x11 x13 x15.
             Proof.
               intros x16 x15 x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               intros [H_5 H_4].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U61_monotonic :
              forall x12 x11, 
               (0 <= x12)/\ (x12 <= x11) ->P_id_a__U61 x12 <= P_id_a__U61 x11.
             Proof.
               intros x12 x11.
               intros [H_1 H_0].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U43_monotonic :
              forall x16 x12 x14 x13 x11 x15, 
               (0 <= x16)/\ (x16 <= x15) ->
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_U43 x12 x14 x16 <= P_id_U43 x11 x13 x15.
             Proof.
               intros x16 x15 x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               intros [H_5 H_4].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__isNatIListKind_monotonic :
              forall x12 x11, 
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_a__isNatIListKind x12 <= P_id_a__isNatIListKind x11.
             Proof.
               intros x12 x11.
               intros [H_1 H_0].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U84_monotonic :
              forall x16 x12 x14 x13 x11 x15, 
               (0 <= x16)/\ (x16 <= x15) ->
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_U84 x12 x14 x16 <= P_id_U84 x11 x13 x15.
             Proof.
               intros x16 x15 x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               intros [H_5 H_4].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_s_monotonic :
              forall x12 x11, 
               (0 <= x12)/\ (x12 <= x11) ->P_id_s x12 <= P_id_s x11.
             Proof.
               intros x12 x11.
               intros [H_1 H_0].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U46_monotonic :
              forall x12 x11, 
               (0 <= x12)/\ (x12 <= x11) ->P_id_a__U46 x12 <= P_id_a__U46 x11.
             Proof.
               intros x12 x11.
               intros [H_1 H_0].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U32_monotonic :
              forall x12 x14 x13 x11, 
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_U32 x12 x14 <= P_id_U32 x11 x13.
             Proof.
               intros x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U31_monotonic :
              forall x12 x14 x13 x11, 
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_a__U31 x12 x14 <= P_id_a__U31 x11 x13.
             Proof.
               intros x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U13_monotonic :
              forall x12 x11, 
               (0 <= x12)/\ (x12 <= x11) ->P_id_U13 x12 <= P_id_U13 x11.
             Proof.
               intros x12 x11.
               intros [H_1 H_0].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U83_monotonic :
              forall x16 x12 x14 x13 x11 x15, 
               (0 <= x16)/\ (x16 <= x15) ->
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_a__U83 x12 x14 x16 <= P_id_a__U83 x11 x13 x15.
             Proof.
               intros x16 x15 x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               intros [H_5 H_4].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_isNatIList_monotonic :
              forall x12 x11, 
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_isNatIList x12 <= P_id_isNatIList x11.
             Proof.
               intros x12 x11.
               intros [H_1 H_0].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__zeros_bounded : 0 <= P_id_a__zeros .
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U51_bounded :
              forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U51 x12 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U84_bounded :
              forall x12 x13 x11, 
               (0 <= x11) ->
                (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U84 x13 x12 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U32_bounded :
              forall x12 x11, 
               (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U32 x12 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_isNatList_bounded :
              forall x11, (0 <= x11) ->0 <= P_id_isNatList x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U13_bounded :
              forall x11, (0 <= x11) ->0 <= P_id_a__U13 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U85_bounded :
              forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U85 x12 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__length_bounded :
              forall x11, (0 <= x11) ->0 <= P_id_a__length x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__isNatIList_bounded :
              forall x11, (0 <= x11) ->0 <= P_id_a__isNatIList x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U33_bounded :
              forall x11, (0 <= x11) ->0 <= P_id_U33 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U11_bounded :
              forall x12 x11, 
               (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U11 x12 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U81_bounded :
              forall x12 x13 x11, 
               (0 <= x11) ->
                (0 <= x12) ->(0 <= x13) ->0 <= P_id_U81 x13 x12 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U92_bounded :
              forall x12 x13 x11, 
               (0 <= x11) ->
                (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U92 x13 x12 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U43_bounded :
              forall x12 x13 x11, 
               (0 <= x11) ->
                (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U43 x13 x12 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U23_bounded :
              forall x11, (0 <= x11) ->0 <= P_id_U23 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__isNatKind_bounded :
              forall x11, (0 <= x11) ->0 <= P_id_a__isNatKind x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U93_bounded :
              forall x12 x13 x11, 
               (0 <= x11) ->
                (0 <= x12) ->(0 <= x13) ->0 <= P_id_U93 x13 x12 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U11_bounded :
              forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U11 x12 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U71_bounded :
              forall x11, (0 <= x11) ->0 <= P_id_a__U71 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U44_bounded :
              forall x12 x13 x11, 
               (0 <= x11) ->
                (0 <= x12) ->(0 <= x13) ->0 <= P_id_U44 x13 x12 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_0_bounded : 0 <= P_id_0 .
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U61_bounded :
              forall x11, (0 <= x11) ->0 <= P_id_U61 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U86_bounded :
              forall x11, (0 <= x11) ->0 <= P_id_a__U86 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U41_bounded :
              forall x12 x13 x11, 
               (0 <= x11) ->
                (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U41 x13 x12 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U22_bounded :
              forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U22 x12 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U21_bounded :
              forall x12 x11, 
               (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U21 x12 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U91_bounded :
              forall x12 x13 x11, 
               (0 <= x11) ->
                (0 <= x12) ->(0 <= x13) ->0 <= P_id_U91 x13 x12 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_length_bounded :
              forall x11, (0 <= x11) ->0 <= P_id_length x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U52_bounded :
              forall x11, (0 <= x11) ->0 <= P_id_a__U52 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U42_bounded :
              forall x12 x13 x11, 
               (0 <= x11) ->
                (0 <= x12) ->(0 <= x13) ->0 <= P_id_U42 x13 x12 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U12_bounded :
              forall x12 x11, 
               (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U12 x12 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U83_bounded :
              forall x12 x13 x11, 
               (0 <= x11) ->
                (0 <= x12) ->(0 <= x13) ->0 <= P_id_U83 x13 x12 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U94_bounded :
              forall x12 x11, 
               (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U94 x12 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U45_bounded :
              forall x12 x11, 
               (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U45 x12 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U31_bounded :
              forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U31 x12 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__isNat_bounded :
              forall x11, (0 <= x11) ->0 <= P_id_a__isNat x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_isNatIListKind_bounded :
              forall x11, (0 <= x11) ->0 <= P_id_isNatIListKind x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U82_bounded :
              forall x12 x13 x11, 
               (0 <= x11) ->
                (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U82 x13 x12 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U46_bounded :
              forall x11, (0 <= x11) ->0 <= P_id_U46 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_cons_bounded :
              forall x12 x11, 
               (0 <= x11) ->(0 <= x12) ->0 <= P_id_cons x12 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U52_bounded :
              forall x11, (0 <= x11) ->0 <= P_id_U52 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U85_bounded :
              forall x12 x11, 
               (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U85 x12 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U33_bounded :
              forall x11, (0 <= x11) ->0 <= P_id_a__U33 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U21_bounded :
              forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U21 x12 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__isNatList_bounded :
              forall x11, (0 <= x11) ->0 <= P_id_a__isNatList x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U86_bounded :
              forall x11, (0 <= x11) ->0 <= P_id_U86 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_mark_bounded :
              forall x11, (0 <= x11) ->0 <= P_id_mark x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U51_bounded :
              forall x12 x11, 
               (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U51 x12 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U41_bounded :
              forall x12 x13 x11, 
               (0 <= x11) ->
                (0 <= x12) ->(0 <= x13) ->0 <= P_id_U41 x13 x12 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_tt_bounded : 0 <= P_id_tt .
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U82_bounded :
              forall x12 x13 x11, 
               (0 <= x11) ->
                (0 <= x12) ->(0 <= x13) ->0 <= P_id_U82 x13 x12 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U93_bounded :
              forall x12 x13 x11, 
               (0 <= x11) ->
                (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U93 x13 x12 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U44_bounded :
              forall x12 x13 x11, 
               (0 <= x11) ->
                (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U44 x13 x12 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_isNat_bounded :
              forall x11, (0 <= x11) ->0 <= P_id_isNat x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U23_bounded :
              forall x11, (0 <= x11) ->0 <= P_id_a__U23 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U94_bounded :
              forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U94 x12 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U12_bounded :
              forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U12 x12 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U81_bounded :
              forall x12 x13 x11, 
               (0 <= x11) ->
                (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U81 x13 x12 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U45_bounded :
              forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U45 x12 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_zeros_bounded : 0 <= P_id_zeros .
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U71_bounded :
              forall x11, (0 <= x11) ->0 <= P_id_U71 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U91_bounded :
              forall x12 x13 x11, 
               (0 <= x11) ->
                (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U91 x13 x12 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U42_bounded :
              forall x12 x13 x11, 
               (0 <= x11) ->
                (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U42 x13 x12 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_isNatKind_bounded :
              forall x11, (0 <= x11) ->0 <= P_id_isNatKind x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U22_bounded :
              forall x12 x11, 
               (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U22 x12 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U92_bounded :
              forall x12 x13 x11, 
               (0 <= x11) ->
                (0 <= x12) ->(0 <= x13) ->0 <= P_id_U92 x13 x12 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_nil_bounded : 0 <= P_id_nil .
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U61_bounded :
              forall x11, (0 <= x11) ->0 <= P_id_a__U61 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U43_bounded :
              forall x12 x13 x11, 
               (0 <= x11) ->
                (0 <= x12) ->(0 <= x13) ->0 <= P_id_U43 x13 x12 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__isNatIListKind_bounded :
              forall x11, (0 <= x11) ->0 <= P_id_a__isNatIListKind x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U84_bounded :
              forall x12 x13 x11, 
               (0 <= x11) ->
                (0 <= x12) ->(0 <= x13) ->0 <= P_id_U84 x13 x12 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_s_bounded : forall x11, (0 <= x11) ->0 <= P_id_s x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U46_bounded :
              forall x11, (0 <= x11) ->0 <= P_id_a__U46 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U32_bounded :
              forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U32 x12 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U31_bounded :
              forall x12 x11, 
               (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U31 x12 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_U13_bounded :
              forall x11, (0 <= x11) ->0 <= P_id_U13 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_a__U83_bounded :
              forall x12 x13 x11, 
               (0 <= x11) ->
                (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U83 x13 x12 x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_isNatIList_bounded :
              forall x11, (0 <= x11) ->0 <= P_id_isNatIList x11.
             Proof.
               intros .
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Definition measure  := 
               InterpZ.measure 0 P_id_a__zeros P_id_U51 P_id_a__U84 
                P_id_a__U32 P_id_isNatList P_id_a__U13 P_id_U85 
                P_id_a__length P_id_a__isNatIList P_id_U33 P_id_a__U11 
                P_id_U81 P_id_a__U92 P_id_a__U43 P_id_U23 P_id_a__isNatKind 
                P_id_U93 P_id_U11 P_id_a__U71 P_id_U44 P_id_0 P_id_U61 
                P_id_a__U86 P_id_a__U41 P_id_U22 P_id_a__U21 P_id_U91 
                P_id_length P_id_a__U52 P_id_U42 P_id_a__U12 P_id_U83 
                P_id_a__U94 P_id_a__U45 P_id_U31 P_id_a__isNat 
                P_id_isNatIListKind P_id_a__U82 P_id_U46 P_id_cons P_id_U52 
                P_id_a__U85 P_id_a__U33 P_id_U21 P_id_a__isNatList P_id_U86 
                P_id_mark P_id_a__U51 P_id_U41 P_id_tt P_id_U82 P_id_a__U93 
                P_id_a__U44 P_id_isNat P_id_a__U23 P_id_U94 P_id_U12 
                P_id_a__U81 P_id_U45 P_id_zeros P_id_U71 P_id_a__U91 
                P_id_a__U42 P_id_isNatKind P_id_a__U22 P_id_U92 P_id_nil 
                P_id_a__U61 P_id_U43 P_id_a__isNatIListKind P_id_U84 
                P_id_s P_id_a__U46 P_id_U32 P_id_a__U31 P_id_U13 P_id_a__U83 
                P_id_isNatIList.
             
             Lemma measure_equation :
              forall t, 
               measure t = match t with
                             | (algebra.Alg.Term algebra.F.id_a__zeros nil) =>
                              P_id_a__zeros 
                             | (algebra.Alg.Term algebra.F.id_U51 (x12::
                                x11::nil)) =>
                              P_id_U51 (measure x12) (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U84 (x13::
                                x12::x11::nil)) =>
                              P_id_a__U84 (measure x13) (measure x12) 
                               (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U32 (x12::
                                x11::nil)) =>
                              P_id_a__U32 (measure x12) (measure x11)
                             | (algebra.Alg.Term algebra.F.id_isNatList 
                                (x11::nil)) =>
                              P_id_isNatList (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U13 
                                (x11::nil)) =>
                              P_id_a__U13 (measure x11)
                             | (algebra.Alg.Term algebra.F.id_U85 (x12::
                                x11::nil)) =>
                              P_id_U85 (measure x12) (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__length 
                                (x11::nil)) =>
                              P_id_a__length (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__isNatIList 
                                (x11::nil)) =>
                              P_id_a__isNatIList (measure x11)
                             | (algebra.Alg.Term algebra.F.id_U33 (x11::nil)) =>
                              P_id_U33 (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U11 (x12::
                                x11::nil)) =>
                              P_id_a__U11 (measure x12) (measure x11)
                             | (algebra.Alg.Term algebra.F.id_U81 (x13::x12::
                                x11::nil)) =>
                              P_id_U81 (measure x13) (measure x12) 
                               (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U92 (x13::
                                x12::x11::nil)) =>
                              P_id_a__U92 (measure x13) (measure x12) 
                               (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U43 (x13::
                                x12::x11::nil)) =>
                              P_id_a__U43 (measure x13) (measure x12) 
                               (measure x11)
                             | (algebra.Alg.Term algebra.F.id_U23 (x11::nil)) =>
                              P_id_U23 (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__isNatKind 
                                (x11::nil)) =>
                              P_id_a__isNatKind (measure x11)
                             | (algebra.Alg.Term algebra.F.id_U93 (x13::x12::
                                x11::nil)) =>
                              P_id_U93 (measure x13) (measure x12) 
                               (measure x11)
                             | (algebra.Alg.Term algebra.F.id_U11 (x12::
                                x11::nil)) =>
                              P_id_U11 (measure x12) (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U71 
                                (x11::nil)) =>
                              P_id_a__U71 (measure x11)
                             | (algebra.Alg.Term algebra.F.id_U44 (x13::x12::
                                x11::nil)) =>
                              P_id_U44 (measure x13) (measure x12) 
                               (measure x11)
                             | (algebra.Alg.Term algebra.F.id_0 nil) =>
                              P_id_0 
                             | (algebra.Alg.Term algebra.F.id_U61 (x11::nil)) =>
                              P_id_U61 (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U86 
                                (x11::nil)) =>
                              P_id_a__U86 (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U41 (x13::
                                x12::x11::nil)) =>
                              P_id_a__U41 (measure x13) (measure x12) 
                               (measure x11)
                             | (algebra.Alg.Term algebra.F.id_U22 (x12::
                                x11::nil)) =>
                              P_id_U22 (measure x12) (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U21 (x12::
                                x11::nil)) =>
                              P_id_a__U21 (measure x12) (measure x11)
                             | (algebra.Alg.Term algebra.F.id_U91 (x13::x12::
                                x11::nil)) =>
                              P_id_U91 (measure x13) (measure x12) 
                               (measure x11)
                             | (algebra.Alg.Term algebra.F.id_length 
                                (x11::nil)) =>
                              P_id_length (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U52 
                                (x11::nil)) =>
                              P_id_a__U52 (measure x11)
                             | (algebra.Alg.Term algebra.F.id_U42 (x13::x12::
                                x11::nil)) =>
                              P_id_U42 (measure x13) (measure x12) 
                               (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U12 (x12::
                                x11::nil)) =>
                              P_id_a__U12 (measure x12) (measure x11)
                             | (algebra.Alg.Term algebra.F.id_U83 (x13::x12::
                                x11::nil)) =>
                              P_id_U83 (measure x13) (measure x12) 
                               (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U94 (x12::
                                x11::nil)) =>
                              P_id_a__U94 (measure x12) (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U45 (x12::
                                x11::nil)) =>
                              P_id_a__U45 (measure x12) (measure x11)
                             | (algebra.Alg.Term algebra.F.id_U31 (x12::
                                x11::nil)) =>
                              P_id_U31 (measure x12) (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__isNat 
                                (x11::nil)) =>
                              P_id_a__isNat (measure x11)
                             | (algebra.Alg.Term algebra.F.id_isNatIListKind 
                                (x11::nil)) =>
                              P_id_isNatIListKind (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U82 (x13::
                                x12::x11::nil)) =>
                              P_id_a__U82 (measure x13) (measure x12) 
                               (measure x11)
                             | (algebra.Alg.Term algebra.F.id_U46 (x11::nil)) =>
                              P_id_U46 (measure x11)
                             | (algebra.Alg.Term algebra.F.id_cons (x12::
                                x11::nil)) =>
                              P_id_cons (measure x12) (measure x11)
                             | (algebra.Alg.Term algebra.F.id_U52 (x11::nil)) =>
                              P_id_U52 (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U85 (x12::
                                x11::nil)) =>
                              P_id_a__U85 (measure x12) (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U33 
                                (x11::nil)) =>
                              P_id_a__U33 (measure x11)
                             | (algebra.Alg.Term algebra.F.id_U21 (x12::
                                x11::nil)) =>
                              P_id_U21 (measure x12) (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__isNatList 
                                (x11::nil)) =>
                              P_id_a__isNatList (measure x11)
                             | (algebra.Alg.Term algebra.F.id_U86 (x11::nil)) =>
                              P_id_U86 (measure x11)
                             | (algebra.Alg.Term algebra.F.id_mark 
                                (x11::nil)) =>
                              P_id_mark (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U51 (x12::
                                x11::nil)) =>
                              P_id_a__U51 (measure x12) (measure x11)
                             | (algebra.Alg.Term algebra.F.id_U41 (x13::x12::
                                x11::nil)) =>
                              P_id_U41 (measure x13) (measure x12) 
                               (measure x11)
                             | (algebra.Alg.Term algebra.F.id_tt nil) =>
                              P_id_tt 
                             | (algebra.Alg.Term algebra.F.id_U82 (x13::x12::
                                x11::nil)) =>
                              P_id_U82 (measure x13) (measure x12) 
                               (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U93 (x13::
                                x12::x11::nil)) =>
                              P_id_a__U93 (measure x13) (measure x12) 
                               (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U44 (x13::
                                x12::x11::nil)) =>
                              P_id_a__U44 (measure x13) (measure x12) 
                               (measure x11)
                             | (algebra.Alg.Term algebra.F.id_isNat 
                                (x11::nil)) =>
                              P_id_isNat (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U23 
                                (x11::nil)) =>
                              P_id_a__U23 (measure x11)
                             | (algebra.Alg.Term algebra.F.id_U94 (x12::
                                x11::nil)) =>
                              P_id_U94 (measure x12) (measure x11)
                             | (algebra.Alg.Term algebra.F.id_U12 (x12::
                                x11::nil)) =>
                              P_id_U12 (measure x12) (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U81 (x13::
                                x12::x11::nil)) =>
                              P_id_a__U81 (measure x13) (measure x12) 
                               (measure x11)
                             | (algebra.Alg.Term algebra.F.id_U45 (x12::
                                x11::nil)) =>
                              P_id_U45 (measure x12) (measure x11)
                             | (algebra.Alg.Term algebra.F.id_zeros nil) =>
                              P_id_zeros 
                             | (algebra.Alg.Term algebra.F.id_U71 (x11::nil)) =>
                              P_id_U71 (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U91 (x13::
                                x12::x11::nil)) =>
                              P_id_a__U91 (measure x13) (measure x12) 
                               (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U42 (x13::
                                x12::x11::nil)) =>
                              P_id_a__U42 (measure x13) (measure x12) 
                               (measure x11)
                             | (algebra.Alg.Term algebra.F.id_isNatKind 
                                (x11::nil)) =>
                              P_id_isNatKind (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U22 (x12::
                                x11::nil)) =>
                              P_id_a__U22 (measure x12) (measure x11)
                             | (algebra.Alg.Term algebra.F.id_U92 (x13::x12::
                                x11::nil)) =>
                              P_id_U92 (measure x13) (measure x12) 
                               (measure x11)
                             | (algebra.Alg.Term algebra.F.id_nil nil) =>
                              P_id_nil 
                             | (algebra.Alg.Term algebra.F.id_a__U61 
                                (x11::nil)) =>
                              P_id_a__U61 (measure x11)
                             | (algebra.Alg.Term algebra.F.id_U43 (x13::x12::
                                x11::nil)) =>
                              P_id_U43 (measure x13) (measure x12) 
                               (measure x11)
                             | (algebra.Alg.Term 
                                algebra.F.id_a__isNatIListKind (x11::nil)) =>
                              P_id_a__isNatIListKind (measure x11)
                             | (algebra.Alg.Term algebra.F.id_U84 (x13::x12::
                                x11::nil)) =>
                              P_id_U84 (measure x13) (measure x12) 
                               (measure x11)
                             | (algebra.Alg.Term algebra.F.id_s (x11::nil)) =>
                              P_id_s (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U46 
                                (x11::nil)) =>
                              P_id_a__U46 (measure x11)
                             | (algebra.Alg.Term algebra.F.id_U32 (x12::
                                x11::nil)) =>
                              P_id_U32 (measure x12) (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U31 (x12::
                                x11::nil)) =>
                              P_id_a__U31 (measure x12) (measure x11)
                             | (algebra.Alg.Term algebra.F.id_U13 (x11::nil)) =>
                              P_id_U13 (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U83 (x13::
                                x12::x11::nil)) =>
                              P_id_a__U83 (measure x13) (measure x12) 
                               (measure x11)
                             | (algebra.Alg.Term algebra.F.id_isNatIList 
                                (x11::nil)) =>
                              P_id_isNatIList (measure x11)
                             | _ => 0
                             end.
             Proof.
               intros t;case t;intros ;apply refl_equal.
             Qed.
             
             Lemma measure_bounded : forall t, 0 <= measure t.
             Proof.
               unfold measure in |-*.
               
               apply InterpZ.measure_bounded;
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Ltac generate_pos_hyp  :=
              match goal with
                | H:context [measure ?x] |- _ =>
                 let v := fresh "v" in 
                  (let H := fresh "h" in 
                    (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
                      clearbody H;clearbody v))
                |  |- context [measure ?x] =>
                 let v := fresh "v" in 
                  (let H := fresh "h" in 
                    (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
                      clearbody H;clearbody v))
                end
              .
             
             Lemma rules_monotonic :
              forall l r, 
               (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) ->
                measure r <= measure l.
             Proof.
               intros l r H.
               fold measure in |-*.
               
               inversion H;clear H;subst;inversion H0;clear H0;subst;
                simpl algebra.EQT.T.apply_subst in |-*;
                repeat (
                match goal with
                  |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
                   rewrite (measure_equation (algebra.Alg.Term f t))
                  end
                );repeat (generate_pos_hyp );
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma measure_star_monotonic :
              forall l r, 
               (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                          r l) ->measure r <= measure l.
             Proof.
               unfold measure in *.
               apply InterpZ.measure_star_monotonic.
               intros ;apply P_id_U51_monotonic;assumption.
               intros ;apply P_id_a__U84_monotonic;assumption.
               intros ;apply P_id_a__U32_monotonic;assumption.
               intros ;apply P_id_isNatList_monotonic;assumption.
               intros ;apply P_id_a__U13_monotonic;assumption.
               intros ;apply P_id_U85_monotonic;assumption.
               intros ;apply P_id_a__length_monotonic;assumption.
               intros ;apply P_id_a__isNatIList_monotonic;assumption.
               intros ;apply P_id_U33_monotonic;assumption.
               intros ;apply P_id_a__U11_monotonic;assumption.
               intros ;apply P_id_U81_monotonic;assumption.
               intros ;apply P_id_a__U92_monotonic;assumption.
               intros ;apply P_id_a__U43_monotonic;assumption.
               intros ;apply P_id_U23_monotonic;assumption.
               intros ;apply P_id_a__isNatKind_monotonic;assumption.
               intros ;apply P_id_U93_monotonic;assumption.
               intros ;apply P_id_U11_monotonic;assumption.
               intros ;apply P_id_a__U71_monotonic;assumption.
               intros ;apply P_id_U44_monotonic;assumption.
               intros ;apply P_id_U61_monotonic;assumption.
               intros ;apply P_id_a__U86_monotonic;assumption.
               intros ;apply P_id_a__U41_monotonic;assumption.
               intros ;apply P_id_U22_monotonic;assumption.
               intros ;apply P_id_a__U21_monotonic;assumption.
               intros ;apply P_id_U91_monotonic;assumption.
               intros ;apply P_id_length_monotonic;assumption.
               intros ;apply P_id_a__U52_monotonic;assumption.
               intros ;apply P_id_U42_monotonic;assumption.
               intros ;apply P_id_a__U12_monotonic;assumption.
               intros ;apply P_id_U83_monotonic;assumption.
               intros ;apply P_id_a__U94_monotonic;assumption.
               intros ;apply P_id_a__U45_monotonic;assumption.
               intros ;apply P_id_U31_monotonic;assumption.
               intros ;apply P_id_a__isNat_monotonic;assumption.
               intros ;apply P_id_isNatIListKind_monotonic;assumption.
               intros ;apply P_id_a__U82_monotonic;assumption.
               intros ;apply P_id_U46_monotonic;assumption.
               intros ;apply P_id_cons_monotonic;assumption.
               intros ;apply P_id_U52_monotonic;assumption.
               intros ;apply P_id_a__U85_monotonic;assumption.
               intros ;apply P_id_a__U33_monotonic;assumption.
               intros ;apply P_id_U21_monotonic;assumption.
               intros ;apply P_id_a__isNatList_monotonic;assumption.
               intros ;apply P_id_U86_monotonic;assumption.
               intros ;apply P_id_mark_monotonic;assumption.
               intros ;apply P_id_a__U51_monotonic;assumption.
               intros ;apply P_id_U41_monotonic;assumption.
               intros ;apply P_id_U82_monotonic;assumption.
               intros ;apply P_id_a__U93_monotonic;assumption.
               intros ;apply P_id_a__U44_monotonic;assumption.
               intros ;apply P_id_isNat_monotonic;assumption.
               intros ;apply P_id_a__U23_monotonic;assumption.
               intros ;apply P_id_U94_monotonic;assumption.
               intros ;apply P_id_U12_monotonic;assumption.
               intros ;apply P_id_a__U81_monotonic;assumption.
               intros ;apply P_id_U45_monotonic;assumption.
               intros ;apply P_id_U71_monotonic;assumption.
               intros ;apply P_id_a__U91_monotonic;assumption.
               intros ;apply P_id_a__U42_monotonic;assumption.
               intros ;apply P_id_isNatKind_monotonic;assumption.
               intros ;apply P_id_a__U22_monotonic;assumption.
               intros ;apply P_id_U92_monotonic;assumption.
               intros ;apply P_id_a__U61_monotonic;assumption.
               intros ;apply P_id_U43_monotonic;assumption.
               intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
               intros ;apply P_id_U84_monotonic;assumption.
               intros ;apply P_id_s_monotonic;assumption.
               intros ;apply P_id_a__U46_monotonic;assumption.
               intros ;apply P_id_U32_monotonic;assumption.
               intros ;apply P_id_a__U31_monotonic;assumption.
               intros ;apply P_id_U13_monotonic;assumption.
               intros ;apply P_id_a__U83_monotonic;assumption.
               intros ;apply P_id_isNatIList_monotonic;assumption.
               intros ;apply P_id_a__zeros_bounded;assumption.
               intros ;apply P_id_U51_bounded;assumption.
               intros ;apply P_id_a__U84_bounded;assumption.
               intros ;apply P_id_a__U32_bounded;assumption.
               intros ;apply P_id_isNatList_bounded;assumption.
               intros ;apply P_id_a__U13_bounded;assumption.
               intros ;apply P_id_U85_bounded;assumption.
               intros ;apply P_id_a__length_bounded;assumption.
               intros ;apply P_id_a__isNatIList_bounded;assumption.
               intros ;apply P_id_U33_bounded;assumption.
               intros ;apply P_id_a__U11_bounded;assumption.
               intros ;apply P_id_U81_bounded;assumption.
               intros ;apply P_id_a__U92_bounded;assumption.
               intros ;apply P_id_a__U43_bounded;assumption.
               intros ;apply P_id_U23_bounded;assumption.
               intros ;apply P_id_a__isNatKind_bounded;assumption.
               intros ;apply P_id_U93_bounded;assumption.
               intros ;apply P_id_U11_bounded;assumption.
               intros ;apply P_id_a__U71_bounded;assumption.
               intros ;apply P_id_U44_bounded;assumption.
               intros ;apply P_id_0_bounded;assumption.
               intros ;apply P_id_U61_bounded;assumption.
               intros ;apply P_id_a__U86_bounded;assumption.
               intros ;apply P_id_a__U41_bounded;assumption.
               intros ;apply P_id_U22_bounded;assumption.
               intros ;apply P_id_a__U21_bounded;assumption.
               intros ;apply P_id_U91_bounded;assumption.
               intros ;apply P_id_length_bounded;assumption.
               intros ;apply P_id_a__U52_bounded;assumption.
               intros ;apply P_id_U42_bounded;assumption.
               intros ;apply P_id_a__U12_bounded;assumption.
               intros ;apply P_id_U83_bounded;assumption.
               intros ;apply P_id_a__U94_bounded;assumption.
               intros ;apply P_id_a__U45_bounded;assumption.
               intros ;apply P_id_U31_bounded;assumption.
               intros ;apply P_id_a__isNat_bounded;assumption.
               intros ;apply P_id_isNatIListKind_bounded;assumption.
               intros ;apply P_id_a__U82_bounded;assumption.
               intros ;apply P_id_U46_bounded;assumption.
               intros ;apply P_id_cons_bounded;assumption.
               intros ;apply P_id_U52_bounded;assumption.
               intros ;apply P_id_a__U85_bounded;assumption.
               intros ;apply P_id_a__U33_bounded;assumption.
               intros ;apply P_id_U21_bounded;assumption.
               intros ;apply P_id_a__isNatList_bounded;assumption.
               intros ;apply P_id_U86_bounded;assumption.
               intros ;apply P_id_mark_bounded;assumption.
               intros ;apply P_id_a__U51_bounded;assumption.
               intros ;apply P_id_U41_bounded;assumption.
               intros ;apply P_id_tt_bounded;assumption.
               intros ;apply P_id_U82_bounded;assumption.
               intros ;apply P_id_a__U93_bounded;assumption.
               intros ;apply P_id_a__U44_bounded;assumption.
               intros ;apply P_id_isNat_bounded;assumption.
               intros ;apply P_id_a__U23_bounded;assumption.
               intros ;apply P_id_U94_bounded;assumption.
               intros ;apply P_id_U12_bounded;assumption.
               intros ;apply P_id_a__U81_bounded;assumption.
               intros ;apply P_id_U45_bounded;assumption.
               intros ;apply P_id_zeros_bounded;assumption.
               intros ;apply P_id_U71_bounded;assumption.
               intros ;apply P_id_a__U91_bounded;assumption.
               intros ;apply P_id_a__U42_bounded;assumption.
               intros ;apply P_id_isNatKind_bounded;assumption.
               intros ;apply P_id_a__U22_bounded;assumption.
               intros ;apply P_id_U92_bounded;assumption.
               intros ;apply P_id_nil_bounded;assumption.
               intros ;apply P_id_a__U61_bounded;assumption.
               intros ;apply P_id_U43_bounded;assumption.
               intros ;apply P_id_a__isNatIListKind_bounded;assumption.
               intros ;apply P_id_U84_bounded;assumption.
               intros ;apply P_id_s_bounded;assumption.
               intros ;apply P_id_a__U46_bounded;assumption.
               intros ;apply P_id_U32_bounded;assumption.
               intros ;apply P_id_a__U31_bounded;assumption.
               intros ;apply P_id_U13_bounded;assumption.
               intros ;apply P_id_a__U83_bounded;assumption.
               intros ;apply P_id_isNatIList_bounded;assumption.
               apply rules_monotonic.
             Qed.
             
             Definition P_id_A__U46 (x11:Z) := 0.
             
             Definition P_id_A__ISNATILISTKIND (x11:Z) := 0.
             
             Definition P_id_A__U61 (x11:Z) := 0.
             
             Definition P_id_A__U85 (x11:Z) (x12:Z) := 0.
             
             Definition P_id_A__U31 (x11:Z) (x12:Z) := 0.
             
             Definition P_id_A__U81 (x11:Z) (x12:Z) (x13:Z) := 0.
             
             Definition P_id_A__U22 (x11:Z) (x12:Z) := 0.
             
             Definition P_id_A__U93 (x11:Z) (x12:Z) (x13:Z) := 0.
             
             Definition P_id_A__U42 (x11:Z) (x12:Z) (x13:Z) := 0.
             
             Definition P_id_A__U51 (x11:Z) (x12:Z) := 0.
             
             Definition P_id_A__ISNATLIST (x11:Z) := 0.
             
             Definition P_id_A__ZEROS  := 0.
             
             Definition P_id_A__U91 (x11:Z) (x12:Z) (x13:Z) := 0.
             
             Definition P_id_A__U33 (x11:Z) := 0.
             
             Definition P_id_A__U83 (x11:Z) (x12:Z) (x13:Z) := 0.
             
             Definition P_id_A__U23 (x11:Z) := 0.
             
             Definition P_id_A__U11 (x11:Z) (x12:Z) := 0.
             
             Definition P_id_A__LENGTH (x11:Z) := 0.
             
             Definition P_id_A__U44 (x11:Z) (x12:Z) (x13:Z) := 0.
             
             Definition P_id_A__ISNATILIST (x11:Z) := 0.
             
             Definition P_id_A__U13 (x11:Z) := 0.
             
             Definition P_id_A__U71 (x11:Z) := 0.
             
             Definition P_id_A__U86 (x11:Z) := 0.
             
             Definition P_id_A__U32 (x11:Z) (x12:Z) := 0.
             
             Definition P_id_A__U82 (x11:Z) (x12:Z) (x13:Z) := 0.
             
             Definition P_id_A__ISNATKIND (x11:Z) := 0.
             
             Definition P_id_A__U94 (x11:Z) (x12:Z) := 0.
             
             Definition P_id_A__U43 (x11:Z) (x12:Z) (x13:Z) := 0.
             
             Definition P_id_A__U52 (x11:Z) := 0.
             
             Definition P_id_A__U21 (x11:Z) (x12:Z) := 0.
             
             Definition P_id_A__U92 (x11:Z) (x12:Z) (x13:Z) := 0.
             
             Definition P_id_A__U41 (x11:Z) (x12:Z) (x13:Z) := 0.
             
             Definition P_id_A__U84 (x11:Z) (x12:Z) (x13:Z) := 0.
             
             Definition P_id_A__ISNAT (x11:Z) := 0.
             
             Definition P_id_A__U12 (x11:Z) (x12:Z) := 0.
             
             Definition P_id_MARK (x11:Z) := 2* x11.
             
             Definition P_id_A__U45 (x11:Z) (x12:Z) := 0.
             
             Lemma P_id_A__U46_monotonic :
              forall x12 x11, 
               (0 <= x12)/\ (x12 <= x11) ->P_id_A__U46 x12 <= P_id_A__U46 x11.
             Proof.
               intros x12 x11.
               intros [H_1 H_0].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_A__ISNATILISTKIND_monotonic :
              forall x12 x11, 
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_A__ISNATILISTKIND x12 <= P_id_A__ISNATILISTKIND x11.
             Proof.
               intros x12 x11.
               intros [H_1 H_0].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_A__U61_monotonic :
              forall x12 x11, 
               (0 <= x12)/\ (x12 <= x11) ->P_id_A__U61 x12 <= P_id_A__U61 x11.
             Proof.
               intros x12 x11.
               intros [H_1 H_0].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_A__U85_monotonic :
              forall x12 x14 x13 x11, 
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_A__U85 x12 x14 <= P_id_A__U85 x11 x13.
             Proof.
               intros x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_A__U31_monotonic :
              forall x12 x14 x13 x11, 
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_A__U31 x12 x14 <= P_id_A__U31 x11 x13.
             Proof.
               intros x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_A__U81_monotonic :
              forall x16 x12 x14 x13 x11 x15, 
               (0 <= x16)/\ (x16 <= x15) ->
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_A__U81 x12 x14 x16 <= P_id_A__U81 x11 x13 x15.
             Proof.
               intros x16 x15 x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               intros [H_5 H_4].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_A__U22_monotonic :
              forall x12 x14 x13 x11, 
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_A__U22 x12 x14 <= P_id_A__U22 x11 x13.
             Proof.
               intros x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_A__U93_monotonic :
              forall x16 x12 x14 x13 x11 x15, 
               (0 <= x16)/\ (x16 <= x15) ->
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_A__U93 x12 x14 x16 <= P_id_A__U93 x11 x13 x15.
             Proof.
               intros x16 x15 x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               intros [H_5 H_4].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_A__U42_monotonic :
              forall x16 x12 x14 x13 x11 x15, 
               (0 <= x16)/\ (x16 <= x15) ->
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_A__U42 x12 x14 x16 <= P_id_A__U42 x11 x13 x15.
             Proof.
               intros x16 x15 x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               intros [H_5 H_4].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_A__U51_monotonic :
              forall x12 x14 x13 x11, 
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_A__U51 x12 x14 <= P_id_A__U51 x11 x13.
             Proof.
               intros x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_A__ISNATLIST_monotonic :
              forall x12 x11, 
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_A__ISNATLIST x12 <= P_id_A__ISNATLIST x11.
             Proof.
               intros x12 x11.
               intros [H_1 H_0].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_A__U91_monotonic :
              forall x16 x12 x14 x13 x11 x15, 
               (0 <= x16)/\ (x16 <= x15) ->
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_A__U91 x12 x14 x16 <= P_id_A__U91 x11 x13 x15.
             Proof.
               intros x16 x15 x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               intros [H_5 H_4].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_A__U33_monotonic :
              forall x12 x11, 
               (0 <= x12)/\ (x12 <= x11) ->P_id_A__U33 x12 <= P_id_A__U33 x11.
             Proof.
               intros x12 x11.
               intros [H_1 H_0].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_A__U83_monotonic :
              forall x16 x12 x14 x13 x11 x15, 
               (0 <= x16)/\ (x16 <= x15) ->
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_A__U83 x12 x14 x16 <= P_id_A__U83 x11 x13 x15.
             Proof.
               intros x16 x15 x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               intros [H_5 H_4].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_A__U23_monotonic :
              forall x12 x11, 
               (0 <= x12)/\ (x12 <= x11) ->P_id_A__U23 x12 <= P_id_A__U23 x11.
             Proof.
               intros x12 x11.
               intros [H_1 H_0].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_A__U11_monotonic :
              forall x12 x14 x13 x11, 
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_A__U11 x12 x14 <= P_id_A__U11 x11 x13.
             Proof.
               intros x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_A__LENGTH_monotonic :
              forall x12 x11, 
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_A__LENGTH x12 <= P_id_A__LENGTH x11.
             Proof.
               intros x12 x11.
               intros [H_1 H_0].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_A__U44_monotonic :
              forall x16 x12 x14 x13 x11 x15, 
               (0 <= x16)/\ (x16 <= x15) ->
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_A__U44 x12 x14 x16 <= P_id_A__U44 x11 x13 x15.
             Proof.
               intros x16 x15 x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               intros [H_5 H_4].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_A__ISNATILIST_monotonic :
              forall x12 x11, 
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_A__ISNATILIST x12 <= P_id_A__ISNATILIST x11.
             Proof.
               intros x12 x11.
               intros [H_1 H_0].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_A__U13_monotonic :
              forall x12 x11, 
               (0 <= x12)/\ (x12 <= x11) ->P_id_A__U13 x12 <= P_id_A__U13 x11.
             Proof.
               intros x12 x11.
               intros [H_1 H_0].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_A__U71_monotonic :
              forall x12 x11, 
               (0 <= x12)/\ (x12 <= x11) ->P_id_A__U71 x12 <= P_id_A__U71 x11.
             Proof.
               intros x12 x11.
               intros [H_1 H_0].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_A__U86_monotonic :
              forall x12 x11, 
               (0 <= x12)/\ (x12 <= x11) ->P_id_A__U86 x12 <= P_id_A__U86 x11.
             Proof.
               intros x12 x11.
               intros [H_1 H_0].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_A__U32_monotonic :
              forall x12 x14 x13 x11, 
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_A__U32 x12 x14 <= P_id_A__U32 x11 x13.
             Proof.
               intros x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_A__U82_monotonic :
              forall x16 x12 x14 x13 x11 x15, 
               (0 <= x16)/\ (x16 <= x15) ->
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_A__U82 x12 x14 x16 <= P_id_A__U82 x11 x13 x15.
             Proof.
               intros x16 x15 x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               intros [H_5 H_4].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_A__ISNATKIND_monotonic :
              forall x12 x11, 
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_A__ISNATKIND x12 <= P_id_A__ISNATKIND x11.
             Proof.
               intros x12 x11.
               intros [H_1 H_0].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_A__U94_monotonic :
              forall x12 x14 x13 x11, 
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_A__U94 x12 x14 <= P_id_A__U94 x11 x13.
             Proof.
               intros x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_A__U43_monotonic :
              forall x16 x12 x14 x13 x11 x15, 
               (0 <= x16)/\ (x16 <= x15) ->
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_A__U43 x12 x14 x16 <= P_id_A__U43 x11 x13 x15.
             Proof.
               intros x16 x15 x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               intros [H_5 H_4].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_A__U52_monotonic :
              forall x12 x11, 
               (0 <= x12)/\ (x12 <= x11) ->P_id_A__U52 x12 <= P_id_A__U52 x11.
             Proof.
               intros x12 x11.
               intros [H_1 H_0].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_A__U21_monotonic :
              forall x12 x14 x13 x11, 
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_A__U21 x12 x14 <= P_id_A__U21 x11 x13.
             Proof.
               intros x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_A__U92_monotonic :
              forall x16 x12 x14 x13 x11 x15, 
               (0 <= x16)/\ (x16 <= x15) ->
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_A__U92 x12 x14 x16 <= P_id_A__U92 x11 x13 x15.
             Proof.
               intros x16 x15 x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               intros [H_5 H_4].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_A__U41_monotonic :
              forall x16 x12 x14 x13 x11 x15, 
               (0 <= x16)/\ (x16 <= x15) ->
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_A__U41 x12 x14 x16 <= P_id_A__U41 x11 x13 x15.
             Proof.
               intros x16 x15 x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               intros [H_5 H_4].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_A__U84_monotonic :
              forall x16 x12 x14 x13 x11 x15, 
               (0 <= x16)/\ (x16 <= x15) ->
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_A__U84 x12 x14 x16 <= P_id_A__U84 x11 x13 x15.
             Proof.
               intros x16 x15 x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               intros [H_5 H_4].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_A__ISNAT_monotonic :
              forall x12 x11, 
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_A__ISNAT x12 <= P_id_A__ISNAT x11.
             Proof.
               intros x12 x11.
               intros [H_1 H_0].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_A__U12_monotonic :
              forall x12 x14 x13 x11, 
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_A__U12 x12 x14 <= P_id_A__U12 x11 x13.
             Proof.
               intros x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_MARK_monotonic :
              forall x12 x11, 
               (0 <= x12)/\ (x12 <= x11) ->P_id_MARK x12 <= P_id_MARK x11.
             Proof.
               intros x12 x11.
               intros [H_1 H_0].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma P_id_A__U45_monotonic :
              forall x12 x14 x13 x11, 
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_A__U45 x12 x14 <= P_id_A__U45 x11 x13.
             Proof.
               intros x14 x13 x12 x11.
               intros [H_1 H_0].
               intros [H_3 H_2].
               
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Definition marked_measure  := 
               InterpZ.marked_measure 0 P_id_a__zeros P_id_U51 P_id_a__U84 
                P_id_a__U32 P_id_isNatList P_id_a__U13 P_id_U85 
                P_id_a__length P_id_a__isNatIList P_id_U33 P_id_a__U11 
                P_id_U81 P_id_a__U92 P_id_a__U43 P_id_U23 P_id_a__isNatKind 
                P_id_U93 P_id_U11 P_id_a__U71 P_id_U44 P_id_0 P_id_U61 
                P_id_a__U86 P_id_a__U41 P_id_U22 P_id_a__U21 P_id_U91 
                P_id_length P_id_a__U52 P_id_U42 P_id_a__U12 P_id_U83 
                P_id_a__U94 P_id_a__U45 P_id_U31 P_id_a__isNat 
                P_id_isNatIListKind P_id_a__U82 P_id_U46 P_id_cons P_id_U52 
                P_id_a__U85 P_id_a__U33 P_id_U21 P_id_a__isNatList P_id_U86 
                P_id_mark P_id_a__U51 P_id_U41 P_id_tt P_id_U82 P_id_a__U93 
                P_id_a__U44 P_id_isNat P_id_a__U23 P_id_U94 P_id_U12 
                P_id_a__U81 P_id_U45 P_id_zeros P_id_U71 P_id_a__U91 
                P_id_a__U42 P_id_isNatKind P_id_a__U22 P_id_U92 P_id_nil 
                P_id_a__U61 P_id_U43 P_id_a__isNatIListKind P_id_U84 
                P_id_s P_id_a__U46 P_id_U32 P_id_a__U31 P_id_U13 P_id_a__U83 
                P_id_isNatIList P_id_A__U46 P_id_A__ISNATILISTKIND 
                P_id_A__U61 P_id_A__U85 P_id_A__U31 P_id_A__U81 P_id_A__U22 
                P_id_A__U93 P_id_A__U42 P_id_A__U51 P_id_A__ISNATLIST 
                P_id_A__ZEROS P_id_A__U91 P_id_A__U33 P_id_A__U83 
                P_id_A__U23 P_id_A__U11 P_id_A__LENGTH P_id_A__U44 
                P_id_A__ISNATILIST P_id_A__U13 P_id_A__U71 P_id_A__U86 
                P_id_A__U32 P_id_A__U82 P_id_A__ISNATKIND P_id_A__U94 
                P_id_A__U43 P_id_A__U52 P_id_A__U21 P_id_A__U92 P_id_A__U41 
                P_id_A__U84 P_id_A__ISNAT P_id_A__U12 P_id_MARK P_id_A__U45.
             
             Lemma marked_measure_equation :
              forall t, 
               marked_measure t = match t with
                                    | (algebra.Alg.Term algebra.F.id_a__U46 
                                       (x11::nil)) =>
                                     P_id_A__U46 (measure x11)
                                    | (algebra.Alg.Term 
                                       algebra.F.id_a__isNatIListKind 
                                       (x11::nil)) =>
                                     P_id_A__ISNATILISTKIND (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U61 
                                       (x11::nil)) =>
                                     P_id_A__U61 (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U85 
                                       (x12::x11::nil)) =>
                                     P_id_A__U85 (measure x12) (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U31 
                                       (x12::x11::nil)) =>
                                     P_id_A__U31 (measure x12) (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U81 
                                       (x13::x12::x11::nil)) =>
                                     P_id_A__U81 (measure x13) (measure x12) 
                                      (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U22 
                                       (x12::x11::nil)) =>
                                     P_id_A__U22 (measure x12) (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U93 
                                       (x13::x12::x11::nil)) =>
                                     P_id_A__U93 (measure x13) (measure x12) 
                                      (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U42 
                                       (x13::x12::x11::nil)) =>
                                     P_id_A__U42 (measure x13) (measure x12) 
                                      (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U51 
                                       (x12::x11::nil)) =>
                                     P_id_A__U51 (measure x12) (measure x11)
                                    | (algebra.Alg.Term 
                                       algebra.F.id_a__isNatList (x11::nil)) =>
                                     P_id_A__ISNATLIST (measure x11)
                                    | (algebra.Alg.Term 
                                       algebra.F.id_a__zeros nil) =>
                                     P_id_A__ZEROS 
                                    | (algebra.Alg.Term algebra.F.id_a__U91 
                                       (x13::x12::x11::nil)) =>
                                     P_id_A__U91 (measure x13) (measure x12) 
                                      (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U33 
                                       (x11::nil)) =>
                                     P_id_A__U33 (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U83 
                                       (x13::x12::x11::nil)) =>
                                     P_id_A__U83 (measure x13) (measure x12) 
                                      (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U23 
                                       (x11::nil)) =>
                                     P_id_A__U23 (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U11 
                                       (x12::x11::nil)) =>
                                     P_id_A__U11 (measure x12) (measure x11)
                                    | (algebra.Alg.Term 
                                       algebra.F.id_a__length (x11::nil)) =>
                                     P_id_A__LENGTH (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U44 
                                       (x13::x12::x11::nil)) =>
                                     P_id_A__U44 (measure x13) (measure x12) 
                                      (measure x11)
                                    | (algebra.Alg.Term 
                                       algebra.F.id_a__isNatIList (x11::nil)) =>
                                     P_id_A__ISNATILIST (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U13 
                                       (x11::nil)) =>
                                     P_id_A__U13 (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U71 
                                       (x11::nil)) =>
                                     P_id_A__U71 (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U86 
                                       (x11::nil)) =>
                                     P_id_A__U86 (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U32 
                                       (x12::x11::nil)) =>
                                     P_id_A__U32 (measure x12) (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U82 
                                       (x13::x12::x11::nil)) =>
                                     P_id_A__U82 (measure x13) (measure x12) 
                                      (measure x11)
                                    | (algebra.Alg.Term 
                                       algebra.F.id_a__isNatKind (x11::nil)) =>
                                     P_id_A__ISNATKIND (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U94 
                                       (x12::x11::nil)) =>
                                     P_id_A__U94 (measure x12) (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U43 
                                       (x13::x12::x11::nil)) =>
                                     P_id_A__U43 (measure x13) (measure x12) 
                                      (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U52 
                                       (x11::nil)) =>
                                     P_id_A__U52 (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U21 
                                       (x12::x11::nil)) =>
                                     P_id_A__U21 (measure x12) (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U92 
                                       (x13::x12::x11::nil)) =>
                                     P_id_A__U92 (measure x13) (measure x12) 
                                      (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U41 
                                       (x13::x12::x11::nil)) =>
                                     P_id_A__U41 (measure x13) (measure x12) 
                                      (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U84 
                                       (x13::x12::x11::nil)) =>
                                     P_id_A__U84 (measure x13) (measure x12) 
                                      (measure x11)
                                    | (algebra.Alg.Term 
                                       algebra.F.id_a__isNat (x11::nil)) =>
                                     P_id_A__ISNAT (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U12 
                                       (x12::x11::nil)) =>
                                     P_id_A__U12 (measure x12) (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_mark 
                                       (x11::nil)) =>
                                     P_id_MARK (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U45 
                                       (x12::x11::nil)) =>
                                     P_id_A__U45 (measure x12) (measure x11)
                                    | _ => measure t
                                    end.
             Proof.
               reflexivity .
             Qed.
             
             Lemma marked_measure_star_monotonic :
              forall f l1 l2, 
               (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step 
                                                                 R_xml_0_deep_rew.R_xml_0_rules)
                                                                ) l1 
                 l2) ->
                marked_measure (algebra.Alg.Term f l1) <= marked_measure 
                                                           (algebra.Alg.Term 
                                                             f l2).
             Proof.
               unfold marked_measure in *.
               apply InterpZ.marked_measure_star_monotonic.
               intros ;apply P_id_U51_monotonic;assumption.
               intros ;apply P_id_a__U84_monotonic;assumption.
               intros ;apply P_id_a__U32_monotonic;assumption.
               intros ;apply P_id_isNatList_monotonic;assumption.
               intros ;apply P_id_a__U13_monotonic;assumption.
               intros ;apply P_id_U85_monotonic;assumption.
               intros ;apply P_id_a__length_monotonic;assumption.
               intros ;apply P_id_a__isNatIList_monotonic;assumption.
               intros ;apply P_id_U33_monotonic;assumption.
               intros ;apply P_id_a__U11_monotonic;assumption.
               intros ;apply P_id_U81_monotonic;assumption.
               intros ;apply P_id_a__U92_monotonic;assumption.
               intros ;apply P_id_a__U43_monotonic;assumption.
               intros ;apply P_id_U23_monotonic;assumption.
               intros ;apply P_id_a__isNatKind_monotonic;assumption.
               intros ;apply P_id_U93_monotonic;assumption.
               intros ;apply P_id_U11_monotonic;assumption.
               intros ;apply P_id_a__U71_monotonic;assumption.
               intros ;apply P_id_U44_monotonic;assumption.
               intros ;apply P_id_U61_monotonic;assumption.
               intros ;apply P_id_a__U86_monotonic;assumption.
               intros ;apply P_id_a__U41_monotonic;assumption.
               intros ;apply P_id_U22_monotonic;assumption.
               intros ;apply P_id_a__U21_monotonic;assumption.
               intros ;apply P_id_U91_monotonic;assumption.
               intros ;apply P_id_length_monotonic;assumption.
               intros ;apply P_id_a__U52_monotonic;assumption.
               intros ;apply P_id_U42_monotonic;assumption.
               intros ;apply P_id_a__U12_monotonic;assumption.
               intros ;apply P_id_U83_monotonic;assumption.
               intros ;apply P_id_a__U94_monotonic;assumption.
               intros ;apply P_id_a__U45_monotonic;assumption.
               intros ;apply P_id_U31_monotonic;assumption.
               intros ;apply P_id_a__isNat_monotonic;assumption.
               intros ;apply P_id_isNatIListKind_monotonic;assumption.
               intros ;apply P_id_a__U82_monotonic;assumption.
               intros ;apply P_id_U46_monotonic;assumption.
               intros ;apply P_id_cons_monotonic;assumption.
               intros ;apply P_id_U52_monotonic;assumption.
               intros ;apply P_id_a__U85_monotonic;assumption.
               intros ;apply P_id_a__U33_monotonic;assumption.
               intros ;apply P_id_U21_monotonic;assumption.
               intros ;apply P_id_a__isNatList_monotonic;assumption.
               intros ;apply P_id_U86_monotonic;assumption.
               intros ;apply P_id_mark_monotonic;assumption.
               intros ;apply P_id_a__U51_monotonic;assumption.
               intros ;apply P_id_U41_monotonic;assumption.
               intros ;apply P_id_U82_monotonic;assumption.
               intros ;apply P_id_a__U93_monotonic;assumption.
               intros ;apply P_id_a__U44_monotonic;assumption.
               intros ;apply P_id_isNat_monotonic;assumption.
               intros ;apply P_id_a__U23_monotonic;assumption.
               intros ;apply P_id_U94_monotonic;assumption.
               intros ;apply P_id_U12_monotonic;assumption.
               intros ;apply P_id_a__U81_monotonic;assumption.
               intros ;apply P_id_U45_monotonic;assumption.
               intros ;apply P_id_U71_monotonic;assumption.
               intros ;apply P_id_a__U91_monotonic;assumption.
               intros ;apply P_id_a__U42_monotonic;assumption.
               intros ;apply P_id_isNatKind_monotonic;assumption.
               intros ;apply P_id_a__U22_monotonic;assumption.
               intros ;apply P_id_U92_monotonic;assumption.
               intros ;apply P_id_a__U61_monotonic;assumption.
               intros ;apply P_id_U43_monotonic;assumption.
               intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
               intros ;apply P_id_U84_monotonic;assumption.
               intros ;apply P_id_s_monotonic;assumption.
               intros ;apply P_id_a__U46_monotonic;assumption.
               intros ;apply P_id_U32_monotonic;assumption.
               intros ;apply P_id_a__U31_monotonic;assumption.
               intros ;apply P_id_U13_monotonic;assumption.
               intros ;apply P_id_a__U83_monotonic;assumption.
               intros ;apply P_id_isNatIList_monotonic;assumption.
               intros ;apply P_id_a__zeros_bounded;assumption.
               intros ;apply P_id_U51_bounded;assumption.
               intros ;apply P_id_a__U84_bounded;assumption.
               intros ;apply P_id_a__U32_bounded;assumption.
               intros ;apply P_id_isNatList_bounded;assumption.
               intros ;apply P_id_a__U13_bounded;assumption.
               intros ;apply P_id_U85_bounded;assumption.
               intros ;apply P_id_a__length_bounded;assumption.
               intros ;apply P_id_a__isNatIList_bounded;assumption.
               intros ;apply P_id_U33_bounded;assumption.
               intros ;apply P_id_a__U11_bounded;assumption.
               intros ;apply P_id_U81_bounded;assumption.
               intros ;apply P_id_a__U92_bounded;assumption.
               intros ;apply P_id_a__U43_bounded;assumption.
               intros ;apply P_id_U23_bounded;assumption.
               intros ;apply P_id_a__isNatKind_bounded;assumption.
               intros ;apply P_id_U93_bounded;assumption.
               intros ;apply P_id_U11_bounded;assumption.
               intros ;apply P_id_a__U71_bounded;assumption.
               intros ;apply P_id_U44_bounded;assumption.
               intros ;apply P_id_0_bounded;assumption.
               intros ;apply P_id_U61_bounded;assumption.
               intros ;apply P_id_a__U86_bounded;assumption.
               intros ;apply P_id_a__U41_bounded;assumption.
               intros ;apply P_id_U22_bounded;assumption.
               intros ;apply P_id_a__U21_bounded;assumption.
               intros ;apply P_id_U91_bounded;assumption.
               intros ;apply P_id_length_bounded;assumption.
               intros ;apply P_id_a__U52_bounded;assumption.
               intros ;apply P_id_U42_bounded;assumption.
               intros ;apply P_id_a__U12_bounded;assumption.
               intros ;apply P_id_U83_bounded;assumption.
               intros ;apply P_id_a__U94_bounded;assumption.
               intros ;apply P_id_a__U45_bounded;assumption.
               intros ;apply P_id_U31_bounded;assumption.
               intros ;apply P_id_a__isNat_bounded;assumption.
               intros ;apply P_id_isNatIListKind_bounded;assumption.
               intros ;apply P_id_a__U82_bounded;assumption.
               intros ;apply P_id_U46_bounded;assumption.
               intros ;apply P_id_cons_bounded;assumption.
               intros ;apply P_id_U52_bounded;assumption.
               intros ;apply P_id_a__U85_bounded;assumption.
               intros ;apply P_id_a__U33_bounded;assumption.
               intros ;apply P_id_U21_bounded;assumption.
               intros ;apply P_id_a__isNatList_bounded;assumption.
               intros ;apply P_id_U86_bounded;assumption.
               intros ;apply P_id_mark_bounded;assumption.
               intros ;apply P_id_a__U51_bounded;assumption.
               intros ;apply P_id_U41_bounded;assumption.
               intros ;apply P_id_tt_bounded;assumption.
               intros ;apply P_id_U82_bounded;assumption.
               intros ;apply P_id_a__U93_bounded;assumption.
               intros ;apply P_id_a__U44_bounded;assumption.
               intros ;apply P_id_isNat_bounded;assumption.
               intros ;apply P_id_a__U23_bounded;assumption.
               intros ;apply P_id_U94_bounded;assumption.
               intros ;apply P_id_U12_bounded;assumption.
               intros ;apply P_id_a__U81_bounded;assumption.
               intros ;apply P_id_U45_bounded;assumption.
               intros ;apply P_id_zeros_bounded;assumption.
               intros ;apply P_id_U71_bounded;assumption.
               intros ;apply P_id_a__U91_bounded;assumption.
               intros ;apply P_id_a__U42_bounded;assumption.
               intros ;apply P_id_isNatKind_bounded;assumption.
               intros ;apply P_id_a__U22_bounded;assumption.
               intros ;apply P_id_U92_bounded;assumption.
               intros ;apply P_id_nil_bounded;assumption.
               intros ;apply P_id_a__U61_bounded;assumption.
               intros ;apply P_id_U43_bounded;assumption.
               intros ;apply P_id_a__isNatIListKind_bounded;assumption.
               intros ;apply P_id_U84_bounded;assumption.
               intros ;apply P_id_s_bounded;assumption.
               intros ;apply P_id_a__U46_bounded;assumption.
               intros ;apply P_id_U32_bounded;assumption.
               intros ;apply P_id_a__U31_bounded;assumption.
               intros ;apply P_id_U13_bounded;assumption.
               intros ;apply P_id_a__U83_bounded;assumption.
               intros ;apply P_id_isNatIList_bounded;assumption.
               apply rules_monotonic.
               intros ;apply P_id_A__U46_monotonic;assumption.
               intros ;apply P_id_A__ISNATILISTKIND_monotonic;assumption.
               intros ;apply P_id_A__U61_monotonic;assumption.
               intros ;apply P_id_A__U85_monotonic;assumption.
               intros ;apply P_id_A__U31_monotonic;assumption.
               intros ;apply P_id_A__U81_monotonic;assumption.
               intros ;apply P_id_A__U22_monotonic;assumption.
               intros ;apply P_id_A__U93_monotonic;assumption.
               intros ;apply P_id_A__U42_monotonic;assumption.
               intros ;apply P_id_A__U51_monotonic;assumption.
               intros ;apply P_id_A__ISNATLIST_monotonic;assumption.
               intros ;apply P_id_A__U91_monotonic;assumption.
               intros ;apply P_id_A__U33_monotonic;assumption.
               intros ;apply P_id_A__U83_monotonic;assumption.
               intros ;apply P_id_A__U23_monotonic;assumption.
               intros ;apply P_id_A__U11_monotonic;assumption.
               intros ;apply P_id_A__LENGTH_monotonic;assumption.
               intros ;apply P_id_A__U44_monotonic;assumption.
               intros ;apply P_id_A__ISNATILIST_monotonic;assumption.
               intros ;apply P_id_A__U13_monotonic;assumption.
               intros ;apply P_id_A__U71_monotonic;assumption.
               intros ;apply P_id_A__U86_monotonic;assumption.
               intros ;apply P_id_A__U32_monotonic;assumption.
               intros ;apply P_id_A__U82_monotonic;assumption.
               intros ;apply P_id_A__ISNATKIND_monotonic;assumption.
               intros ;apply P_id_A__U94_monotonic;assumption.
               intros ;apply P_id_A__U43_monotonic;assumption.
               intros ;apply P_id_A__U52_monotonic;assumption.
               intros ;apply P_id_A__U21_monotonic;assumption.
               intros ;apply P_id_A__U92_monotonic;assumption.
               intros ;apply P_id_A__U41_monotonic;assumption.
               intros ;apply P_id_A__U84_monotonic;assumption.
               intros ;apply P_id_A__ISNAT_monotonic;assumption.
               intros ;apply P_id_A__U12_monotonic;assumption.
               intros ;apply P_id_MARK_monotonic;assumption.
               intros ;apply P_id_A__U45_monotonic;assumption.
             Qed.
             
             Ltac rewrite_and_unfold  :=
              do 2 (rewrite marked_measure_equation);
               repeat (
               match goal with
                 |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
                  rewrite (measure_equation (algebra.Alg.Term f t))
                 | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ =>
                  rewrite (measure_equation (algebra.Alg.Term f t)) in H|-
                 end
               ).
             
             Definition lt a b := 
               (Zwf.Zwf 0) (marked_measure a) (marked_measure b).
             
             Definition le a b := marked_measure a <= marked_measure b.
             
             Lemma lt_le_compat : forall a b c, (lt a b) ->(le b c) ->lt a c.
             Proof.
               unfold lt, le in *.
               intros a b c.
               apply (interp.le_lt_compat_right (interp.o_Z 0)).
             Qed.
             
             Lemma wf_lt : well_founded lt.
             Proof.
               unfold lt in *.
               apply Inverse_Image.wf_inverse_image with  (B:=Z).
               apply Zwf.Zwf_well_founded.
             Qed.
             
             Lemma DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_strict_in_lt :
              Relation_Definitions.inclusion _ 
               DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_strict 
               lt.
             Proof.
               unfold Relation_Definitions.inclusion, lt in *.
               
               intros a b H;destruct H;
                match goal with
                  |  |- (Zwf.Zwf 0) _ 
                         (marked_measure (algebra.Alg.Term ?f ?l)) =>
                   let l'' := algebra.Alg_ext.find_replacement l  in 
                    ((apply (interp.le_lt_compat_right (interp.o_Z 0)) with
                       (marked_measure (algebra.Alg.Term f l''));[idtac|
                      apply marked_measure_star_monotonic;
                       repeat (
                       apply algebra.EQT_ext.one_step_list_refl_trans_clos);
                       (assumption)||(constructor 1)]))
                  end
                ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Lemma DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_in_le :
              Relation_Definitions.inclusion _ 
               DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large 
               le.
             Proof.
               unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *.
               
               intros a b H;destruct H;
                match goal with
                  |  |- _ <= marked_measure (algebra.Alg.Term ?f ?l) =>
                   let l'' := algebra.Alg_ext.find_replacement l  in 
                    ((apply (interp.le_trans (interp.o_Z 0)) with
                       (marked_measure (algebra.Alg.Term f l''));[idtac|
                      apply marked_measure_star_monotonic;
                       repeat (
                       apply algebra.EQT_ext.one_step_list_refl_trans_clos);
                       (assumption)||(constructor 1)]))
                  end
                ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
             Qed.
             
             Definition wf_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large  := 
               WF_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large.wf
               .
             
             
             Lemma wf :
              well_founded WF_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large.DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large
               .
             Proof.
               intros x.
               apply (well_founded_ind wf_lt).
               clear x.
               intros x.
               pattern x.
               
               apply (@Acc_ind _ 
                       DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large)
                      .
               clear x.
               intros x _ IHx IHx'.
               constructor.
               intros y H.
               
               destruct H;
                (apply IHx';
                  apply DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_strict_in_lt;
                  econstructor eassumption)||
                ((apply IHx;[econstructor eassumption|
                  intros y' Hlt;apply IHx';apply lt_le_compat with (1:=Hlt) ;
                   apply DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large_in_le;
                   econstructor eassumption])).
               
               apply wf_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_large.
             Qed.
            End WF_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large.
            
            Open Scope Z_scope.
            
            Import ring_extention.
            
            Notation Local "a <= b" := (Zle a b).
            
            Notation Local "a < b" := (Zlt a b).
            
            Definition P_id_a__zeros  := 0.
            
            Definition P_id_U51 (x11:Z) (x12:Z) := 1* x11.
            
            Definition P_id_a__U84 (x11:Z) (x12:Z) (x13:Z) := 
              1* x11 + 1* x12 + 1* x13.
            
            Definition P_id_a__U32 (x11:Z) (x12:Z) := 0.
            
            Definition P_id_isNatList (x11:Z) := 1* x11.
            
            Definition P_id_a__U13 (x11:Z) := 1 + 1* x11.
            
            Definition P_id_U85 (x11:Z) (x12:Z) := 1* x11 + 1* x12.
            
            Definition P_id_a__length (x11:Z) := 1 + 1* x11.
            
            Definition P_id_a__isNatIList (x11:Z) := 2.
            
            Definition P_id_U33 (x11:Z) := 0.
            
            Definition P_id_a__U11 (x11:Z) (x12:Z) := 1 + 1* x12.
            
            Definition P_id_U81 (x11:Z) (x12:Z) (x13:Z) := 
              1* x11 + 1* x12 + 1* x13.
            
            Definition P_id_a__U92 (x11:Z) (x12:Z) (x13:Z) := 1 + 1* x12.
            
            Definition P_id_a__U43 (x11:Z) (x12:Z) (x13:Z) := 2.
            
            Definition P_id_U23 (x11:Z) := 1* x11.
            
            Definition P_id_a__isNatKind (x11:Z) := 0.
            
            Definition P_id_U93 (x11:Z) (x12:Z) (x13:Z) := 1 + 1* x12.
            
            Definition P_id_U11 (x11:Z) (x12:Z) := 1 + 1* x12.
            
            Definition P_id_a__U71 (x11:Z) := 1* x11.
            
            Definition P_id_U44 (x11:Z) (x12:Z) (x13:Z) := 0.
            
            Definition P_id_0  := 0.
            
            Definition P_id_U61 (x11:Z) := 1* x11.
            
            Definition P_id_a__U86 (x11:Z) := 1* x11.
            
            Definition P_id_a__U41 (x11:Z) (x12:Z) (x13:Z) := 2.
            
            Definition P_id_U22 (x11:Z) (x12:Z) := 1* x11 + 1* x12.
            
            Definition P_id_a__U21 (x11:Z) (x12:Z) := 1* x11 + 1* x12.
            
            Definition P_id_U91 (x11:Z) (x12:Z) (x13:Z) := 1 + 1* x12.
            
            Definition P_id_length (x11:Z) := 1 + 1* x11.
            
            Definition P_id_a__U52 (x11:Z) := 1* x11.
            
            Definition P_id_U42 (x11:Z) (x12:Z) (x13:Z) := 2 + 2* x11.
            
            Definition P_id_a__U12 (x11:Z) (x12:Z) := 1 + 1* x12.
            
            Definition P_id_U83 (x11:Z) (x12:Z) (x13:Z) := 
              1* x11 + 1* x12 + 1* x13.
            
            Definition P_id_a__U94 (x11:Z) (x12:Z) := 1 + 1* x12.
            
            Definition P_id_a__U45 (x11:Z) (x12:Z) := 0.
            
            Definition P_id_U31 (x11:Z) (x12:Z) := 0.
            
            Definition P_id_a__isNat (x11:Z) := 1* x11.
            
            Definition P_id_isNatIListKind (x11:Z) := 0.
            
            Definition P_id_a__U82 (x11:Z) (x12:Z) (x13:Z) := 
              1* x11 + 1* x12 + 1* x13.
            
            Definition P_id_U46 (x11:Z) := 0.
            
            Definition P_id_cons (x11:Z) (x12:Z) := 2* x11 + 2* x12.
            
            Definition P_id_U52 (x11:Z) := 1* x11.
            
            Definition P_id_a__U85 (x11:Z) (x12:Z) := 1* x11 + 1* x12.
            
            Definition P_id_a__U33 (x11:Z) := 0.
            
            Definition P_id_U21 (x11:Z) (x12:Z) := 1* x11 + 1* x12.
            
            Definition P_id_a__isNatList (x11:Z) := 1* x11.
            
            Definition P_id_U86 (x11:Z) := 1* x11.
            
            Definition P_id_mark (x11:Z) := 1* x11.
            
            Definition P_id_a__U51 (x11:Z) (x12:Z) := 1* x11.
            
            Definition P_id_U41 (x11:Z) (x12:Z) (x13:Z) := 2.
            
            Definition P_id_tt  := 0.
            
            Definition P_id_U82 (x11:Z) (x12:Z) (x13:Z) := 
              1* x11 + 1* x12 + 1* x13.
            
            Definition P_id_a__U93 (x11:Z) (x12:Z) (x13:Z) := 1 + 1* x12.
            
            Definition P_id_a__U44 (x11:Z) (x12:Z) (x13:Z) := 0.
            
            Definition P_id_isNat (x11:Z) := 1* x11.
            
            Definition P_id_a__U23 (x11:Z) := 1* x11.
            
            Definition P_id_U94 (x11:Z) (x12:Z) := 1 + 1* x12.
            
            Definition P_id_U12 (x11:Z) (x12:Z) := 1 + 1* x12.
            
            Definition P_id_a__U81 (x11:Z) (x12:Z) (x13:Z) := 
              1* x11 + 1* x12 + 1* x13.
            
            Definition P_id_U45 (x11:Z) (x12:Z) := 0.
            
            Definition P_id_zeros  := 0.
            
            Definition P_id_U71 (x11:Z) := 1* x11.
            
            Definition P_id_a__U91 (x11:Z) (x12:Z) (x13:Z) := 1 + 1* x12.
            
            Definition P_id_a__U42 (x11:Z) (x12:Z) (x13:Z) := 2 + 2* x11.
            
            Definition P_id_isNatKind (x11:Z) := 0.
            
            Definition P_id_a__U22 (x11:Z) (x12:Z) := 1* x11 + 1* x12.
            
            Definition P_id_U92 (x11:Z) (x12:Z) (x13:Z) := 1 + 1* x12.
            
            Definition P_id_nil  := 0.
            
            Definition P_id_a__U61 (x11:Z) := 1* x11.
            
            Definition P_id_U43 (x11:Z) (x12:Z) (x13:Z) := 2.
            
            Definition P_id_a__isNatIListKind (x11:Z) := 0.
            
            Definition P_id_U84 (x11:Z) (x12:Z) (x13:Z) := 
              1* x11 + 1* x12 + 1* x13.
            
            Definition P_id_s (x11:Z) := 1* x11.
            
            Definition P_id_a__U46 (x11:Z) := 0.
            
            Definition P_id_U32 (x11:Z) (x12:Z) := 0.
            
            Definition P_id_a__U31 (x11:Z) (x12:Z) := 0.
            
            Definition P_id_U13 (x11:Z) := 1 + 1* x11.
            
            Definition P_id_a__U83 (x11:Z) (x12:Z) (x13:Z) := 
              1* x11 + 1* x12 + 1* x13.
            
            Definition P_id_isNatIList (x11:Z) := 2.
            
            Lemma P_id_U51_monotonic :
             forall x12 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_U51 x12 x14 <= P_id_U51 x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U84_monotonic :
             forall x16 x12 x14 x13 x11 x15, 
              (0 <= x16)/\ (x16 <= x15) ->
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_a__U84 x12 x14 x16 <= P_id_a__U84 x11 x13 x15.
            Proof.
              intros x16 x15 x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              intros [H_5 H_4].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U32_monotonic :
             forall x12 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_a__U32 x12 x14 <= P_id_a__U32 x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_isNatList_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_isNatList x12 <= P_id_isNatList x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U13_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_a__U13 x12 <= P_id_a__U13 x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U85_monotonic :
             forall x12 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_U85 x12 x14 <= P_id_U85 x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__length_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_a__length x12 <= P_id_a__length x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__isNatIList_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_a__isNatIList x12 <= P_id_a__isNatIList x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U33_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_U33 x12 <= P_id_U33 x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U11_monotonic :
             forall x12 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_a__U11 x12 x14 <= P_id_a__U11 x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U81_monotonic :
             forall x16 x12 x14 x13 x11 x15, 
              (0 <= x16)/\ (x16 <= x15) ->
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_U81 x12 x14 x16 <= P_id_U81 x11 x13 x15.
            Proof.
              intros x16 x15 x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              intros [H_5 H_4].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U92_monotonic :
             forall x16 x12 x14 x13 x11 x15, 
              (0 <= x16)/\ (x16 <= x15) ->
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_a__U92 x12 x14 x16 <= P_id_a__U92 x11 x13 x15.
            Proof.
              intros x16 x15 x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              intros [H_5 H_4].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U43_monotonic :
             forall x16 x12 x14 x13 x11 x15, 
              (0 <= x16)/\ (x16 <= x15) ->
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_a__U43 x12 x14 x16 <= P_id_a__U43 x11 x13 x15.
            Proof.
              intros x16 x15 x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              intros [H_5 H_4].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U23_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_U23 x12 <= P_id_U23 x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__isNatKind_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_a__isNatKind x12 <= P_id_a__isNatKind x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U93_monotonic :
             forall x16 x12 x14 x13 x11 x15, 
              (0 <= x16)/\ (x16 <= x15) ->
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_U93 x12 x14 x16 <= P_id_U93 x11 x13 x15.
            Proof.
              intros x16 x15 x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              intros [H_5 H_4].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U11_monotonic :
             forall x12 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_U11 x12 x14 <= P_id_U11 x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U71_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_a__U71 x12 <= P_id_a__U71 x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U44_monotonic :
             forall x16 x12 x14 x13 x11 x15, 
              (0 <= x16)/\ (x16 <= x15) ->
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_U44 x12 x14 x16 <= P_id_U44 x11 x13 x15.
            Proof.
              intros x16 x15 x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              intros [H_5 H_4].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U61_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_U61 x12 <= P_id_U61 x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U86_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_a__U86 x12 <= P_id_a__U86 x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U41_monotonic :
             forall x16 x12 x14 x13 x11 x15, 
              (0 <= x16)/\ (x16 <= x15) ->
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_a__U41 x12 x14 x16 <= P_id_a__U41 x11 x13 x15.
            Proof.
              intros x16 x15 x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              intros [H_5 H_4].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U22_monotonic :
             forall x12 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_U22 x12 x14 <= P_id_U22 x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U21_monotonic :
             forall x12 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_a__U21 x12 x14 <= P_id_a__U21 x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U91_monotonic :
             forall x16 x12 x14 x13 x11 x15, 
              (0 <= x16)/\ (x16 <= x15) ->
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_U91 x12 x14 x16 <= P_id_U91 x11 x13 x15.
            Proof.
              intros x16 x15 x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              intros [H_5 H_4].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_length_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_length x12 <= P_id_length x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U52_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_a__U52 x12 <= P_id_a__U52 x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U42_monotonic :
             forall x16 x12 x14 x13 x11 x15, 
              (0 <= x16)/\ (x16 <= x15) ->
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_U42 x12 x14 x16 <= P_id_U42 x11 x13 x15.
            Proof.
              intros x16 x15 x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              intros [H_5 H_4].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U12_monotonic :
             forall x12 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_a__U12 x12 x14 <= P_id_a__U12 x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U83_monotonic :
             forall x16 x12 x14 x13 x11 x15, 
              (0 <= x16)/\ (x16 <= x15) ->
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_U83 x12 x14 x16 <= P_id_U83 x11 x13 x15.
            Proof.
              intros x16 x15 x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              intros [H_5 H_4].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U94_monotonic :
             forall x12 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_a__U94 x12 x14 <= P_id_a__U94 x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U45_monotonic :
             forall x12 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_a__U45 x12 x14 <= P_id_a__U45 x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U31_monotonic :
             forall x12 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_U31 x12 x14 <= P_id_U31 x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__isNat_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_a__isNat x12 <= P_id_a__isNat x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_isNatIListKind_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_isNatIListKind x12 <= P_id_isNatIListKind x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U82_monotonic :
             forall x16 x12 x14 x13 x11 x15, 
              (0 <= x16)/\ (x16 <= x15) ->
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_a__U82 x12 x14 x16 <= P_id_a__U82 x11 x13 x15.
            Proof.
              intros x16 x15 x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              intros [H_5 H_4].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U46_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_U46 x12 <= P_id_U46 x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_cons_monotonic :
             forall x12 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_cons x12 x14 <= P_id_cons x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U52_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_U52 x12 <= P_id_U52 x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U85_monotonic :
             forall x12 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_a__U85 x12 x14 <= P_id_a__U85 x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U33_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_a__U33 x12 <= P_id_a__U33 x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U21_monotonic :
             forall x12 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_U21 x12 x14 <= P_id_U21 x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__isNatList_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_a__isNatList x12 <= P_id_a__isNatList x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U86_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_U86 x12 <= P_id_U86 x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_mark_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_mark x12 <= P_id_mark x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U51_monotonic :
             forall x12 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_a__U51 x12 x14 <= P_id_a__U51 x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U41_monotonic :
             forall x16 x12 x14 x13 x11 x15, 
              (0 <= x16)/\ (x16 <= x15) ->
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_U41 x12 x14 x16 <= P_id_U41 x11 x13 x15.
            Proof.
              intros x16 x15 x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              intros [H_5 H_4].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U82_monotonic :
             forall x16 x12 x14 x13 x11 x15, 
              (0 <= x16)/\ (x16 <= x15) ->
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_U82 x12 x14 x16 <= P_id_U82 x11 x13 x15.
            Proof.
              intros x16 x15 x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              intros [H_5 H_4].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U93_monotonic :
             forall x16 x12 x14 x13 x11 x15, 
              (0 <= x16)/\ (x16 <= x15) ->
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_a__U93 x12 x14 x16 <= P_id_a__U93 x11 x13 x15.
            Proof.
              intros x16 x15 x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              intros [H_5 H_4].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U44_monotonic :
             forall x16 x12 x14 x13 x11 x15, 
              (0 <= x16)/\ (x16 <= x15) ->
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_a__U44 x12 x14 x16 <= P_id_a__U44 x11 x13 x15.
            Proof.
              intros x16 x15 x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              intros [H_5 H_4].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_isNat_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_isNat x12 <= P_id_isNat x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U23_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_a__U23 x12 <= P_id_a__U23 x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U94_monotonic :
             forall x12 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_U94 x12 x14 <= P_id_U94 x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U12_monotonic :
             forall x12 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_U12 x12 x14 <= P_id_U12 x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U81_monotonic :
             forall x16 x12 x14 x13 x11 x15, 
              (0 <= x16)/\ (x16 <= x15) ->
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_a__U81 x12 x14 x16 <= P_id_a__U81 x11 x13 x15.
            Proof.
              intros x16 x15 x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              intros [H_5 H_4].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U45_monotonic :
             forall x12 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_U45 x12 x14 <= P_id_U45 x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U71_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_U71 x12 <= P_id_U71 x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U91_monotonic :
             forall x16 x12 x14 x13 x11 x15, 
              (0 <= x16)/\ (x16 <= x15) ->
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_a__U91 x12 x14 x16 <= P_id_a__U91 x11 x13 x15.
            Proof.
              intros x16 x15 x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              intros [H_5 H_4].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U42_monotonic :
             forall x16 x12 x14 x13 x11 x15, 
              (0 <= x16)/\ (x16 <= x15) ->
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_a__U42 x12 x14 x16 <= P_id_a__U42 x11 x13 x15.
            Proof.
              intros x16 x15 x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              intros [H_5 H_4].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_isNatKind_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_isNatKind x12 <= P_id_isNatKind x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U22_monotonic :
             forall x12 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_a__U22 x12 x14 <= P_id_a__U22 x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U92_monotonic :
             forall x16 x12 x14 x13 x11 x15, 
              (0 <= x16)/\ (x16 <= x15) ->
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_U92 x12 x14 x16 <= P_id_U92 x11 x13 x15.
            Proof.
              intros x16 x15 x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              intros [H_5 H_4].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U61_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_a__U61 x12 <= P_id_a__U61 x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U43_monotonic :
             forall x16 x12 x14 x13 x11 x15, 
              (0 <= x16)/\ (x16 <= x15) ->
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_U43 x12 x14 x16 <= P_id_U43 x11 x13 x15.
            Proof.
              intros x16 x15 x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              intros [H_5 H_4].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__isNatIListKind_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_a__isNatIListKind x12 <= P_id_a__isNatIListKind x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U84_monotonic :
             forall x16 x12 x14 x13 x11 x15, 
              (0 <= x16)/\ (x16 <= x15) ->
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_U84 x12 x14 x16 <= P_id_U84 x11 x13 x15.
            Proof.
              intros x16 x15 x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              intros [H_5 H_4].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_s_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_s x12 <= P_id_s x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U46_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_a__U46 x12 <= P_id_a__U46 x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U32_monotonic :
             forall x12 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_U32 x12 x14 <= P_id_U32 x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U31_monotonic :
             forall x12 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_a__U31 x12 x14 <= P_id_a__U31 x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U13_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_U13 x12 <= P_id_U13 x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U83_monotonic :
             forall x16 x12 x14 x13 x11 x15, 
              (0 <= x16)/\ (x16 <= x15) ->
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_a__U83 x12 x14 x16 <= P_id_a__U83 x11 x13 x15.
            Proof.
              intros x16 x15 x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              intros [H_5 H_4].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_isNatIList_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_isNatIList x12 <= P_id_isNatIList x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__zeros_bounded : 0 <= P_id_a__zeros .
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U51_bounded :
             forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U51 x12 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U84_bounded :
             forall x12 x13 x11, 
              (0 <= x11) ->
               (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U84 x13 x12 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U32_bounded :
             forall x12 x11, 
              (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U32 x12 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_isNatList_bounded :
             forall x11, (0 <= x11) ->0 <= P_id_isNatList x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U13_bounded :
             forall x11, (0 <= x11) ->0 <= P_id_a__U13 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U85_bounded :
             forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U85 x12 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__length_bounded :
             forall x11, (0 <= x11) ->0 <= P_id_a__length x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__isNatIList_bounded :
             forall x11, (0 <= x11) ->0 <= P_id_a__isNatIList x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U33_bounded :
             forall x11, (0 <= x11) ->0 <= P_id_U33 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U11_bounded :
             forall x12 x11, 
              (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U11 x12 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U81_bounded :
             forall x12 x13 x11, 
              (0 <= x11) ->
               (0 <= x12) ->(0 <= x13) ->0 <= P_id_U81 x13 x12 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U92_bounded :
             forall x12 x13 x11, 
              (0 <= x11) ->
               (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U92 x13 x12 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U43_bounded :
             forall x12 x13 x11, 
              (0 <= x11) ->
               (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U43 x13 x12 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U23_bounded :
             forall x11, (0 <= x11) ->0 <= P_id_U23 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__isNatKind_bounded :
             forall x11, (0 <= x11) ->0 <= P_id_a__isNatKind x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U93_bounded :
             forall x12 x13 x11, 
              (0 <= x11) ->
               (0 <= x12) ->(0 <= x13) ->0 <= P_id_U93 x13 x12 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U11_bounded :
             forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U11 x12 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U71_bounded :
             forall x11, (0 <= x11) ->0 <= P_id_a__U71 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U44_bounded :
             forall x12 x13 x11, 
              (0 <= x11) ->
               (0 <= x12) ->(0 <= x13) ->0 <= P_id_U44 x13 x12 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_0_bounded : 0 <= P_id_0 .
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U61_bounded :
             forall x11, (0 <= x11) ->0 <= P_id_U61 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U86_bounded :
             forall x11, (0 <= x11) ->0 <= P_id_a__U86 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U41_bounded :
             forall x12 x13 x11, 
              (0 <= x11) ->
               (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U41 x13 x12 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U22_bounded :
             forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U22 x12 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U21_bounded :
             forall x12 x11, 
              (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U21 x12 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U91_bounded :
             forall x12 x13 x11, 
              (0 <= x11) ->
               (0 <= x12) ->(0 <= x13) ->0 <= P_id_U91 x13 x12 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_length_bounded :
             forall x11, (0 <= x11) ->0 <= P_id_length x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U52_bounded :
             forall x11, (0 <= x11) ->0 <= P_id_a__U52 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U42_bounded :
             forall x12 x13 x11, 
              (0 <= x11) ->
               (0 <= x12) ->(0 <= x13) ->0 <= P_id_U42 x13 x12 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U12_bounded :
             forall x12 x11, 
              (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U12 x12 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U83_bounded :
             forall x12 x13 x11, 
              (0 <= x11) ->
               (0 <= x12) ->(0 <= x13) ->0 <= P_id_U83 x13 x12 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U94_bounded :
             forall x12 x11, 
              (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U94 x12 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U45_bounded :
             forall x12 x11, 
              (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U45 x12 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U31_bounded :
             forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U31 x12 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__isNat_bounded :
             forall x11, (0 <= x11) ->0 <= P_id_a__isNat x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_isNatIListKind_bounded :
             forall x11, (0 <= x11) ->0 <= P_id_isNatIListKind x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U82_bounded :
             forall x12 x13 x11, 
              (0 <= x11) ->
               (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U82 x13 x12 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U46_bounded :
             forall x11, (0 <= x11) ->0 <= P_id_U46 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_cons_bounded :
             forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_cons x12 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U52_bounded :
             forall x11, (0 <= x11) ->0 <= P_id_U52 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U85_bounded :
             forall x12 x11, 
              (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U85 x12 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U33_bounded :
             forall x11, (0 <= x11) ->0 <= P_id_a__U33 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U21_bounded :
             forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U21 x12 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__isNatList_bounded :
             forall x11, (0 <= x11) ->0 <= P_id_a__isNatList x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U86_bounded :
             forall x11, (0 <= x11) ->0 <= P_id_U86 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_mark_bounded :
             forall x11, (0 <= x11) ->0 <= P_id_mark x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U51_bounded :
             forall x12 x11, 
              (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U51 x12 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U41_bounded :
             forall x12 x13 x11, 
              (0 <= x11) ->
               (0 <= x12) ->(0 <= x13) ->0 <= P_id_U41 x13 x12 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_tt_bounded : 0 <= P_id_tt .
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U82_bounded :
             forall x12 x13 x11, 
              (0 <= x11) ->
               (0 <= x12) ->(0 <= x13) ->0 <= P_id_U82 x13 x12 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U93_bounded :
             forall x12 x13 x11, 
              (0 <= x11) ->
               (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U93 x13 x12 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U44_bounded :
             forall x12 x13 x11, 
              (0 <= x11) ->
               (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U44 x13 x12 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_isNat_bounded :
             forall x11, (0 <= x11) ->0 <= P_id_isNat x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U23_bounded :
             forall x11, (0 <= x11) ->0 <= P_id_a__U23 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U94_bounded :
             forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U94 x12 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U12_bounded :
             forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U12 x12 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U81_bounded :
             forall x12 x13 x11, 
              (0 <= x11) ->
               (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U81 x13 x12 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U45_bounded :
             forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U45 x12 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_zeros_bounded : 0 <= P_id_zeros .
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U71_bounded :
             forall x11, (0 <= x11) ->0 <= P_id_U71 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U91_bounded :
             forall x12 x13 x11, 
              (0 <= x11) ->
               (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U91 x13 x12 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U42_bounded :
             forall x12 x13 x11, 
              (0 <= x11) ->
               (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U42 x13 x12 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_isNatKind_bounded :
             forall x11, (0 <= x11) ->0 <= P_id_isNatKind x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U22_bounded :
             forall x12 x11, 
              (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U22 x12 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U92_bounded :
             forall x12 x13 x11, 
              (0 <= x11) ->
               (0 <= x12) ->(0 <= x13) ->0 <= P_id_U92 x13 x12 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_nil_bounded : 0 <= P_id_nil .
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U61_bounded :
             forall x11, (0 <= x11) ->0 <= P_id_a__U61 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U43_bounded :
             forall x12 x13 x11, 
              (0 <= x11) ->
               (0 <= x12) ->(0 <= x13) ->0 <= P_id_U43 x13 x12 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__isNatIListKind_bounded :
             forall x11, (0 <= x11) ->0 <= P_id_a__isNatIListKind x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U84_bounded :
             forall x12 x13 x11, 
              (0 <= x11) ->
               (0 <= x12) ->(0 <= x13) ->0 <= P_id_U84 x13 x12 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_s_bounded : forall x11, (0 <= x11) ->0 <= P_id_s x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U46_bounded :
             forall x11, (0 <= x11) ->0 <= P_id_a__U46 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U32_bounded :
             forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U32 x12 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U31_bounded :
             forall x12 x11, 
              (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U31 x12 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U13_bounded :
             forall x11, (0 <= x11) ->0 <= P_id_U13 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_a__U83_bounded :
             forall x12 x13 x11, 
              (0 <= x11) ->
               (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U83 x13 x12 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_isNatIList_bounded :
             forall x11, (0 <= x11) ->0 <= P_id_isNatIList x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Definition measure  := 
              InterpZ.measure 0 P_id_a__zeros P_id_U51 P_id_a__U84 
               P_id_a__U32 P_id_isNatList P_id_a__U13 P_id_U85 
               P_id_a__length P_id_a__isNatIList P_id_U33 P_id_a__U11 
               P_id_U81 P_id_a__U92 P_id_a__U43 P_id_U23 P_id_a__isNatKind 
               P_id_U93 P_id_U11 P_id_a__U71 P_id_U44 P_id_0 P_id_U61 
               P_id_a__U86 P_id_a__U41 P_id_U22 P_id_a__U21 P_id_U91 
               P_id_length P_id_a__U52 P_id_U42 P_id_a__U12 P_id_U83 
               P_id_a__U94 P_id_a__U45 P_id_U31 P_id_a__isNat 
               P_id_isNatIListKind P_id_a__U82 P_id_U46 P_id_cons P_id_U52 
               P_id_a__U85 P_id_a__U33 P_id_U21 P_id_a__isNatList P_id_U86 
               P_id_mark P_id_a__U51 P_id_U41 P_id_tt P_id_U82 P_id_a__U93 
               P_id_a__U44 P_id_isNat P_id_a__U23 P_id_U94 P_id_U12 
               P_id_a__U81 P_id_U45 P_id_zeros P_id_U71 P_id_a__U91 
               P_id_a__U42 P_id_isNatKind P_id_a__U22 P_id_U92 P_id_nil 
               P_id_a__U61 P_id_U43 P_id_a__isNatIListKind P_id_U84 P_id_s 
               P_id_a__U46 P_id_U32 P_id_a__U31 P_id_U13 P_id_a__U83 
               P_id_isNatIList.
            
            Lemma measure_equation :
             forall t, 
              measure t = match t with
                            | (algebra.Alg.Term algebra.F.id_a__zeros nil) =>
                             P_id_a__zeros 
                            | (algebra.Alg.Term algebra.F.id_U51 (x12::
                               x11::nil)) =>
                             P_id_U51 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U84 (x13::
                               x12::x11::nil)) =>
                             P_id_a__U84 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U32 (x12::
                               x11::nil)) =>
                             P_id_a__U32 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_isNatList 
                               (x11::nil)) =>
                             P_id_isNatList (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U13 
                               (x11::nil)) =>
                             P_id_a__U13 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_U85 (x12::
                               x11::nil)) =>
                             P_id_U85 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__length 
                               (x11::nil)) =>
                             P_id_a__length (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__isNatIList 
                               (x11::nil)) =>
                             P_id_a__isNatIList (measure x11)
                            | (algebra.Alg.Term algebra.F.id_U33 (x11::nil)) =>
                             P_id_U33 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U11 (x12::
                               x11::nil)) =>
                             P_id_a__U11 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_U81 (x13::x12::
                               x11::nil)) =>
                             P_id_U81 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U92 (x13::
                               x12::x11::nil)) =>
                             P_id_a__U92 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U43 (x13::
                               x12::x11::nil)) =>
                             P_id_a__U43 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_U23 (x11::nil)) =>
                             P_id_U23 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__isNatKind 
                               (x11::nil)) =>
                             P_id_a__isNatKind (measure x11)
                            | (algebra.Alg.Term algebra.F.id_U93 (x13::x12::
                               x11::nil)) =>
                             P_id_U93 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_U11 (x12::
                               x11::nil)) =>
                             P_id_U11 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U71 
                               (x11::nil)) =>
                             P_id_a__U71 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_U44 (x13::x12::
                               x11::nil)) =>
                             P_id_U44 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_0 nil) =>
                             P_id_0 
                            | (algebra.Alg.Term algebra.F.id_U61 (x11::nil)) =>
                             P_id_U61 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U86 
                               (x11::nil)) =>
                             P_id_a__U86 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U41 (x13::
                               x12::x11::nil)) =>
                             P_id_a__U41 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_U22 (x12::
                               x11::nil)) =>
                             P_id_U22 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U21 (x12::
                               x11::nil)) =>
                             P_id_a__U21 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_U91 (x13::x12::
                               x11::nil)) =>
                             P_id_U91 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_length 
                               (x11::nil)) =>
                             P_id_length (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U52 
                               (x11::nil)) =>
                             P_id_a__U52 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_U42 (x13::x12::
                               x11::nil)) =>
                             P_id_U42 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U12 (x12::
                               x11::nil)) =>
                             P_id_a__U12 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_U83 (x13::x12::
                               x11::nil)) =>
                             P_id_U83 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U94 (x12::
                               x11::nil)) =>
                             P_id_a__U94 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U45 (x12::
                               x11::nil)) =>
                             P_id_a__U45 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_U31 (x12::
                               x11::nil)) =>
                             P_id_U31 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__isNat 
                               (x11::nil)) =>
                             P_id_a__isNat (measure x11)
                            | (algebra.Alg.Term algebra.F.id_isNatIListKind 
                               (x11::nil)) =>
                             P_id_isNatIListKind (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U82 (x13::
                               x12::x11::nil)) =>
                             P_id_a__U82 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_U46 (x11::nil)) =>
                             P_id_U46 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_cons (x12::
                               x11::nil)) =>
                             P_id_cons (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_U52 (x11::nil)) =>
                             P_id_U52 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U85 (x12::
                               x11::nil)) =>
                             P_id_a__U85 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U33 
                               (x11::nil)) =>
                             P_id_a__U33 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_U21 (x12::
                               x11::nil)) =>
                             P_id_U21 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__isNatList 
                               (x11::nil)) =>
                             P_id_a__isNatList (measure x11)
                            | (algebra.Alg.Term algebra.F.id_U86 (x11::nil)) =>
                             P_id_U86 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_mark (x11::nil)) =>
                             P_id_mark (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U51 (x12::
                               x11::nil)) =>
                             P_id_a__U51 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_U41 (x13::x12::
                               x11::nil)) =>
                             P_id_U41 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_tt nil) =>
                             P_id_tt 
                            | (algebra.Alg.Term algebra.F.id_U82 (x13::x12::
                               x11::nil)) =>
                             P_id_U82 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U93 (x13::
                               x12::x11::nil)) =>
                             P_id_a__U93 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U44 (x13::
                               x12::x11::nil)) =>
                             P_id_a__U44 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_isNat 
                               (x11::nil)) =>
                             P_id_isNat (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U23 
                               (x11::nil)) =>
                             P_id_a__U23 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_U94 (x12::
                               x11::nil)) =>
                             P_id_U94 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_U12 (x12::
                               x11::nil)) =>
                             P_id_U12 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U81 (x13::
                               x12::x11::nil)) =>
                             P_id_a__U81 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_U45 (x12::
                               x11::nil)) =>
                             P_id_U45 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_zeros nil) =>
                             P_id_zeros 
                            | (algebra.Alg.Term algebra.F.id_U71 (x11::nil)) =>
                             P_id_U71 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U91 (x13::
                               x12::x11::nil)) =>
                             P_id_a__U91 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U42 (x13::
                               x12::x11::nil)) =>
                             P_id_a__U42 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_isNatKind 
                               (x11::nil)) =>
                             P_id_isNatKind (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U22 (x12::
                               x11::nil)) =>
                             P_id_a__U22 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_U92 (x13::x12::
                               x11::nil)) =>
                             P_id_U92 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_nil nil) =>
                             P_id_nil 
                            | (algebra.Alg.Term algebra.F.id_a__U61 
                               (x11::nil)) =>
                             P_id_a__U61 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_U43 (x13::x12::
                               x11::nil)) =>
                             P_id_U43 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term 
                               algebra.F.id_a__isNatIListKind (x11::nil)) =>
                             P_id_a__isNatIListKind (measure x11)
                            | (algebra.Alg.Term algebra.F.id_U84 (x13::x12::
                               x11::nil)) =>
                             P_id_U84 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_s (x11::nil)) =>
                             P_id_s (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U46 
                               (x11::nil)) =>
                             P_id_a__U46 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_U32 (x12::
                               x11::nil)) =>
                             P_id_U32 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U31 (x12::
                               x11::nil)) =>
                             P_id_a__U31 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_U13 (x11::nil)) =>
                             P_id_U13 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U83 (x13::
                               x12::x11::nil)) =>
                             P_id_a__U83 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_isNatIList 
                               (x11::nil)) =>
                             P_id_isNatIList (measure x11)
                            | _ => 0
                            end.
            Proof.
              intros t;case t;intros ;apply refl_equal.
            Qed.
            
            Lemma measure_bounded : forall t, 0 <= measure t.
            Proof.
              unfold measure in |-*.
              
              apply InterpZ.measure_bounded;
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Ltac generate_pos_hyp  :=
             match goal with
               | H:context [measure ?x] |- _ =>
                let v := fresh "v" in 
                 (let H := fresh "h" in 
                   (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
                     clearbody H;clearbody v))
               |  |- context [measure ?x] =>
                let v := fresh "v" in 
                 (let H := fresh "h" in 
                   (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
                     clearbody H;clearbody v))
               end
             .
            
            Lemma rules_monotonic :
             forall l r, 
              (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) ->
               measure r <= measure l.
            Proof.
              intros l r H.
              fold measure in |-*.
              
              inversion H;clear H;subst;inversion H0;clear H0;subst;
               simpl algebra.EQT.T.apply_subst in |-*;
               repeat (
               match goal with
                 |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
                  rewrite (measure_equation (algebra.Alg.Term f t))
                 end
               );repeat (generate_pos_hyp );
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma measure_star_monotonic :
             forall l r, 
              (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                         r l) ->measure r <= measure l.
            Proof.
              unfold measure in *.
              apply InterpZ.measure_star_monotonic.
              intros ;apply P_id_U51_monotonic;assumption.
              intros ;apply P_id_a__U84_monotonic;assumption.
              intros ;apply P_id_a__U32_monotonic;assumption.
              intros ;apply P_id_isNatList_monotonic;assumption.
              intros ;apply P_id_a__U13_monotonic;assumption.
              intros ;apply P_id_U85_monotonic;assumption.
              intros ;apply P_id_a__length_monotonic;assumption.
              intros ;apply P_id_a__isNatIList_monotonic;assumption.
              intros ;apply P_id_U33_monotonic;assumption.
              intros ;apply P_id_a__U11_monotonic;assumption.
              intros ;apply P_id_U81_monotonic;assumption.
              intros ;apply P_id_a__U92_monotonic;assumption.
              intros ;apply P_id_a__U43_monotonic;assumption.
              intros ;apply P_id_U23_monotonic;assumption.
              intros ;apply P_id_a__isNatKind_monotonic;assumption.
              intros ;apply P_id_U93_monotonic;assumption.
              intros ;apply P_id_U11_monotonic;assumption.
              intros ;apply P_id_a__U71_monotonic;assumption.
              intros ;apply P_id_U44_monotonic;assumption.
              intros ;apply P_id_U61_monotonic;assumption.
              intros ;apply P_id_a__U86_monotonic;assumption.
              intros ;apply P_id_a__U41_monotonic;assumption.
              intros ;apply P_id_U22_monotonic;assumption.
              intros ;apply P_id_a__U21_monotonic;assumption.
              intros ;apply P_id_U91_monotonic;assumption.
              intros ;apply P_id_length_monotonic;assumption.
              intros ;apply P_id_a__U52_monotonic;assumption.
              intros ;apply P_id_U42_monotonic;assumption.
              intros ;apply P_id_a__U12_monotonic;assumption.
              intros ;apply P_id_U83_monotonic;assumption.
              intros ;apply P_id_a__U94_monotonic;assumption.
              intros ;apply P_id_a__U45_monotonic;assumption.
              intros ;apply P_id_U31_monotonic;assumption.
              intros ;apply P_id_a__isNat_monotonic;assumption.
              intros ;apply P_id_isNatIListKind_monotonic;assumption.
              intros ;apply P_id_a__U82_monotonic;assumption.
              intros ;apply P_id_U46_monotonic;assumption.
              intros ;apply P_id_cons_monotonic;assumption.
              intros ;apply P_id_U52_monotonic;assumption.
              intros ;apply P_id_a__U85_monotonic;assumption.
              intros ;apply P_id_a__U33_monotonic;assumption.
              intros ;apply P_id_U21_monotonic;assumption.
              intros ;apply P_id_a__isNatList_monotonic;assumption.
              intros ;apply P_id_U86_monotonic;assumption.
              intros ;apply P_id_mark_monotonic;assumption.
              intros ;apply P_id_a__U51_monotonic;assumption.
              intros ;apply P_id_U41_monotonic;assumption.
              intros ;apply P_id_U82_monotonic;assumption.
              intros ;apply P_id_a__U93_monotonic;assumption.
              intros ;apply P_id_a__U44_monotonic;assumption.
              intros ;apply P_id_isNat_monotonic;assumption.
              intros ;apply P_id_a__U23_monotonic;assumption.
              intros ;apply P_id_U94_monotonic;assumption.
              intros ;apply P_id_U12_monotonic;assumption.
              intros ;apply P_id_a__U81_monotonic;assumption.
              intros ;apply P_id_U45_monotonic;assumption.
              intros ;apply P_id_U71_monotonic;assumption.
              intros ;apply P_id_a__U91_monotonic;assumption.
              intros ;apply P_id_a__U42_monotonic;assumption.
              intros ;apply P_id_isNatKind_monotonic;assumption.
              intros ;apply P_id_a__U22_monotonic;assumption.
              intros ;apply P_id_U92_monotonic;assumption.
              intros ;apply P_id_a__U61_monotonic;assumption.
              intros ;apply P_id_U43_monotonic;assumption.
              intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
              intros ;apply P_id_U84_monotonic;assumption.
              intros ;apply P_id_s_monotonic;assumption.
              intros ;apply P_id_a__U46_monotonic;assumption.
              intros ;apply P_id_U32_monotonic;assumption.
              intros ;apply P_id_a__U31_monotonic;assumption.
              intros ;apply P_id_U13_monotonic;assumption.
              intros ;apply P_id_a__U83_monotonic;assumption.
              intros ;apply P_id_isNatIList_monotonic;assumption.
              intros ;apply P_id_a__zeros_bounded;assumption.
              intros ;apply P_id_U51_bounded;assumption.
              intros ;apply P_id_a__U84_bounded;assumption.
              intros ;apply P_id_a__U32_bounded;assumption.
              intros ;apply P_id_isNatList_bounded;assumption.
              intros ;apply P_id_a__U13_bounded;assumption.
              intros ;apply P_id_U85_bounded;assumption.
              intros ;apply P_id_a__length_bounded;assumption.
              intros ;apply P_id_a__isNatIList_bounded;assumption.
              intros ;apply P_id_U33_bounded;assumption.
              intros ;apply P_id_a__U11_bounded;assumption.
              intros ;apply P_id_U81_bounded;assumption.
              intros ;apply P_id_a__U92_bounded;assumption.
              intros ;apply P_id_a__U43_bounded;assumption.
              intros ;apply P_id_U23_bounded;assumption.
              intros ;apply P_id_a__isNatKind_bounded;assumption.
              intros ;apply P_id_U93_bounded;assumption.
              intros ;apply P_id_U11_bounded;assumption.
              intros ;apply P_id_a__U71_bounded;assumption.
              intros ;apply P_id_U44_bounded;assumption.
              intros ;apply P_id_0_bounded;assumption.
              intros ;apply P_id_U61_bounded;assumption.
              intros ;apply P_id_a__U86_bounded;assumption.
              intros ;apply P_id_a__U41_bounded;assumption.
              intros ;apply P_id_U22_bounded;assumption.
              intros ;apply P_id_a__U21_bounded;assumption.
              intros ;apply P_id_U91_bounded;assumption.
              intros ;apply P_id_length_bounded;assumption.
              intros ;apply P_id_a__U52_bounded;assumption.
              intros ;apply P_id_U42_bounded;assumption.
              intros ;apply P_id_a__U12_bounded;assumption.
              intros ;apply P_id_U83_bounded;assumption.
              intros ;apply P_id_a__U94_bounded;assumption.
              intros ;apply P_id_a__U45_bounded;assumption.
              intros ;apply P_id_U31_bounded;assumption.
              intros ;apply P_id_a__isNat_bounded;assumption.
              intros ;apply P_id_isNatIListKind_bounded;assumption.
              intros ;apply P_id_a__U82_bounded;assumption.
              intros ;apply P_id_U46_bounded;assumption.
              intros ;apply P_id_cons_bounded;assumption.
              intros ;apply P_id_U52_bounded;assumption.
              intros ;apply P_id_a__U85_bounded;assumption.
              intros ;apply P_id_a__U33_bounded;assumption.
              intros ;apply P_id_U21_bounded;assumption.
              intros ;apply P_id_a__isNatList_bounded;assumption.
              intros ;apply P_id_U86_bounded;assumption.
              intros ;apply P_id_mark_bounded;assumption.
              intros ;apply P_id_a__U51_bounded;assumption.
              intros ;apply P_id_U41_bounded;assumption.
              intros ;apply P_id_tt_bounded;assumption.
              intros ;apply P_id_U82_bounded;assumption.
              intros ;apply P_id_a__U93_bounded;assumption.
              intros ;apply P_id_a__U44_bounded;assumption.
              intros ;apply P_id_isNat_bounded;assumption.
              intros ;apply P_id_a__U23_bounded;assumption.
              intros ;apply P_id_U94_bounded;assumption.
              intros ;apply P_id_U12_bounded;assumption.
              intros ;apply P_id_a__U81_bounded;assumption.
              intros ;apply P_id_U45_bounded;assumption.
              intros ;apply P_id_zeros_bounded;assumption.
              intros ;apply P_id_U71_bounded;assumption.
              intros ;apply P_id_a__U91_bounded;assumption.
              intros ;apply P_id_a__U42_bounded;assumption.
              intros ;apply P_id_isNatKind_bounded;assumption.
              intros ;apply P_id_a__U22_bounded;assumption.
              intros ;apply P_id_U92_bounded;assumption.
              intros ;apply P_id_nil_bounded;assumption.
              intros ;apply P_id_a__U61_bounded;assumption.
              intros ;apply P_id_U43_bounded;assumption.
              intros ;apply P_id_a__isNatIListKind_bounded;assumption.
              intros ;apply P_id_U84_bounded;assumption.
              intros ;apply P_id_s_bounded;assumption.
              intros ;apply P_id_a__U46_bounded;assumption.
              intros ;apply P_id_U32_bounded;assumption.
              intros ;apply P_id_a__U31_bounded;assumption.
              intros ;apply P_id_U13_bounded;assumption.
              intros ;apply P_id_a__U83_bounded;assumption.
              intros ;apply P_id_isNatIList_bounded;assumption.
              apply rules_monotonic.
            Qed.
            
            Definition P_id_A__U46 (x11:Z) := 0.
            
            Definition P_id_A__ISNATILISTKIND (x11:Z) := 0.
            
            Definition P_id_A__U61 (x11:Z) := 0.
            
            Definition P_id_A__U85 (x11:Z) (x12:Z) := 0.
            
            Definition P_id_A__U31 (x11:Z) (x12:Z) := 0.
            
            Definition P_id_A__U81 (x11:Z) (x12:Z) (x13:Z) := 0.
            
            Definition P_id_A__U22 (x11:Z) (x12:Z) := 0.
            
            Definition P_id_A__U93 (x11:Z) (x12:Z) (x13:Z) := 0.
            
            Definition P_id_A__U42 (x11:Z) (x12:Z) (x13:Z) := 0.
            
            Definition P_id_A__U51 (x11:Z) (x12:Z) := 0.
            
            Definition P_id_A__ISNATLIST (x11:Z) := 0.
            
            Definition P_id_A__ZEROS  := 0.
            
            Definition P_id_A__U91 (x11:Z) (x12:Z) (x13:Z) := 0.
            
            Definition P_id_A__U33 (x11:Z) := 0.
            
            Definition P_id_A__U83 (x11:Z) (x12:Z) (x13:Z) := 0.
            
            Definition P_id_A__U23 (x11:Z) := 0.
            
            Definition P_id_A__U11 (x11:Z) (x12:Z) := 0.
            
            Definition P_id_A__LENGTH (x11:Z) := 0.
            
            Definition P_id_A__U44 (x11:Z) (x12:Z) (x13:Z) := 0.
            
            Definition P_id_A__ISNATILIST (x11:Z) := 0.
            
            Definition P_id_A__U13 (x11:Z) := 0.
            
            Definition P_id_A__U71 (x11:Z) := 0.
            
            Definition P_id_A__U86 (x11:Z) := 0.
            
            Definition P_id_A__U32 (x11:Z) (x12:Z) := 0.
            
            Definition P_id_A__U82 (x11:Z) (x12:Z) (x13:Z) := 0.
            
            Definition P_id_A__ISNATKIND (x11:Z) := 0.
            
            Definition P_id_A__U94 (x11:Z) (x12:Z) := 0.
            
            Definition P_id_A__U43 (x11:Z) (x12:Z) (x13:Z) := 0.
            
            Definition P_id_A__U52 (x11:Z) := 0.
            
            Definition P_id_A__U21 (x11:Z) (x12:Z) := 0.
            
            Definition P_id_A__U92 (x11:Z) (x12:Z) (x13:Z) := 0.
            
            Definition P_id_A__U41 (x11:Z) (x12:Z) (x13:Z) := 0.
            
            Definition P_id_A__U84 (x11:Z) (x12:Z) (x13:Z) := 0.
            
            Definition P_id_A__ISNAT (x11:Z) := 0.
            
            Definition P_id_A__U12 (x11:Z) (x12:Z) := 0.
            
            Definition P_id_MARK (x11:Z) := 1* x11.
            
            Definition P_id_A__U45 (x11:Z) (x12:Z) := 0.
            
            Lemma P_id_A__U46_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_A__U46 x12 <= P_id_A__U46 x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_A__ISNATILISTKIND_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_A__ISNATILISTKIND x12 <= P_id_A__ISNATILISTKIND x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_A__U61_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_A__U61 x12 <= P_id_A__U61 x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_A__U85_monotonic :
             forall x12 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_A__U85 x12 x14 <= P_id_A__U85 x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_A__U31_monotonic :
             forall x12 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_A__U31 x12 x14 <= P_id_A__U31 x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_A__U81_monotonic :
             forall x16 x12 x14 x13 x11 x15, 
              (0 <= x16)/\ (x16 <= x15) ->
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_A__U81 x12 x14 x16 <= P_id_A__U81 x11 x13 x15.
            Proof.
              intros x16 x15 x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              intros [H_5 H_4].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_A__U22_monotonic :
             forall x12 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_A__U22 x12 x14 <= P_id_A__U22 x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_A__U93_monotonic :
             forall x16 x12 x14 x13 x11 x15, 
              (0 <= x16)/\ (x16 <= x15) ->
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_A__U93 x12 x14 x16 <= P_id_A__U93 x11 x13 x15.
            Proof.
              intros x16 x15 x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              intros [H_5 H_4].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_A__U42_monotonic :
             forall x16 x12 x14 x13 x11 x15, 
              (0 <= x16)/\ (x16 <= x15) ->
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_A__U42 x12 x14 x16 <= P_id_A__U42 x11 x13 x15.
            Proof.
              intros x16 x15 x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              intros [H_5 H_4].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_A__U51_monotonic :
             forall x12 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_A__U51 x12 x14 <= P_id_A__U51 x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_A__ISNATLIST_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_A__ISNATLIST x12 <= P_id_A__ISNATLIST x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_A__U91_monotonic :
             forall x16 x12 x14 x13 x11 x15, 
              (0 <= x16)/\ (x16 <= x15) ->
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_A__U91 x12 x14 x16 <= P_id_A__U91 x11 x13 x15.
            Proof.
              intros x16 x15 x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              intros [H_5 H_4].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_A__U33_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_A__U33 x12 <= P_id_A__U33 x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_A__U83_monotonic :
             forall x16 x12 x14 x13 x11 x15, 
              (0 <= x16)/\ (x16 <= x15) ->
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_A__U83 x12 x14 x16 <= P_id_A__U83 x11 x13 x15.
            Proof.
              intros x16 x15 x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              intros [H_5 H_4].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_A__U23_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_A__U23 x12 <= P_id_A__U23 x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_A__U11_monotonic :
             forall x12 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_A__U11 x12 x14 <= P_id_A__U11 x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_A__LENGTH_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_A__LENGTH x12 <= P_id_A__LENGTH x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_A__U44_monotonic :
             forall x16 x12 x14 x13 x11 x15, 
              (0 <= x16)/\ (x16 <= x15) ->
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_A__U44 x12 x14 x16 <= P_id_A__U44 x11 x13 x15.
            Proof.
              intros x16 x15 x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              intros [H_5 H_4].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_A__ISNATILIST_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_A__ISNATILIST x12 <= P_id_A__ISNATILIST x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_A__U13_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_A__U13 x12 <= P_id_A__U13 x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_A__U71_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_A__U71 x12 <= P_id_A__U71 x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_A__U86_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_A__U86 x12 <= P_id_A__U86 x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_A__U32_monotonic :
             forall x12 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_A__U32 x12 x14 <= P_id_A__U32 x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_A__U82_monotonic :
             forall x16 x12 x14 x13 x11 x15, 
              (0 <= x16)/\ (x16 <= x15) ->
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_A__U82 x12 x14 x16 <= P_id_A__U82 x11 x13 x15.
            Proof.
              intros x16 x15 x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              intros [H_5 H_4].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_A__ISNATKIND_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_A__ISNATKIND x12 <= P_id_A__ISNATKIND x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_A__U94_monotonic :
             forall x12 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_A__U94 x12 x14 <= P_id_A__U94 x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_A__U43_monotonic :
             forall x16 x12 x14 x13 x11 x15, 
              (0 <= x16)/\ (x16 <= x15) ->
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_A__U43 x12 x14 x16 <= P_id_A__U43 x11 x13 x15.
            Proof.
              intros x16 x15 x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              intros [H_5 H_4].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_A__U52_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_A__U52 x12 <= P_id_A__U52 x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_A__U21_monotonic :
             forall x12 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_A__U21 x12 x14 <= P_id_A__U21 x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_A__U92_monotonic :
             forall x16 x12 x14 x13 x11 x15, 
              (0 <= x16)/\ (x16 <= x15) ->
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_A__U92 x12 x14 x16 <= P_id_A__U92 x11 x13 x15.
            Proof.
              intros x16 x15 x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              intros [H_5 H_4].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_A__U41_monotonic :
             forall x16 x12 x14 x13 x11 x15, 
              (0 <= x16)/\ (x16 <= x15) ->
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_A__U41 x12 x14 x16 <= P_id_A__U41 x11 x13 x15.
            Proof.
              intros x16 x15 x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              intros [H_5 H_4].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_A__U84_monotonic :
             forall x16 x12 x14 x13 x11 x15, 
              (0 <= x16)/\ (x16 <= x15) ->
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_A__U84 x12 x14 x16 <= P_id_A__U84 x11 x13 x15.
            Proof.
              intros x16 x15 x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              intros [H_5 H_4].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_A__ISNAT_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_A__ISNAT x12 <= P_id_A__ISNAT x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_A__U12_monotonic :
             forall x12 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_A__U12 x12 x14 <= P_id_A__U12 x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_MARK_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_MARK x12 <= P_id_MARK x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_A__U45_monotonic :
             forall x12 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_A__U45 x12 x14 <= P_id_A__U45 x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Definition marked_measure  := 
              InterpZ.marked_measure 0 P_id_a__zeros P_id_U51 P_id_a__U84 
               P_id_a__U32 P_id_isNatList P_id_a__U13 P_id_U85 
               P_id_a__length P_id_a__isNatIList P_id_U33 P_id_a__U11 
               P_id_U81 P_id_a__U92 P_id_a__U43 P_id_U23 P_id_a__isNatKind 
               P_id_U93 P_id_U11 P_id_a__U71 P_id_U44 P_id_0 P_id_U61 
               P_id_a__U86 P_id_a__U41 P_id_U22 P_id_a__U21 P_id_U91 
               P_id_length P_id_a__U52 P_id_U42 P_id_a__U12 P_id_U83 
               P_id_a__U94 P_id_a__U45 P_id_U31 P_id_a__isNat 
               P_id_isNatIListKind P_id_a__U82 P_id_U46 P_id_cons P_id_U52 
               P_id_a__U85 P_id_a__U33 P_id_U21 P_id_a__isNatList P_id_U86 
               P_id_mark P_id_a__U51 P_id_U41 P_id_tt P_id_U82 P_id_a__U93 
               P_id_a__U44 P_id_isNat P_id_a__U23 P_id_U94 P_id_U12 
               P_id_a__U81 P_id_U45 P_id_zeros P_id_U71 P_id_a__U91 
               P_id_a__U42 P_id_isNatKind P_id_a__U22 P_id_U92 P_id_nil 
               P_id_a__U61 P_id_U43 P_id_a__isNatIListKind P_id_U84 P_id_s 
               P_id_a__U46 P_id_U32 P_id_a__U31 P_id_U13 P_id_a__U83 
               P_id_isNatIList P_id_A__U46 P_id_A__ISNATILISTKIND 
               P_id_A__U61 P_id_A__U85 P_id_A__U31 P_id_A__U81 P_id_A__U22 
               P_id_A__U93 P_id_A__U42 P_id_A__U51 P_id_A__ISNATLIST 
               P_id_A__ZEROS P_id_A__U91 P_id_A__U33 P_id_A__U83 P_id_A__U23 
               P_id_A__U11 P_id_A__LENGTH P_id_A__U44 P_id_A__ISNATILIST 
               P_id_A__U13 P_id_A__U71 P_id_A__U86 P_id_A__U32 P_id_A__U82 
               P_id_A__ISNATKIND P_id_A__U94 P_id_A__U43 P_id_A__U52 
               P_id_A__U21 P_id_A__U92 P_id_A__U41 P_id_A__U84 P_id_A__ISNAT 
               P_id_A__U12 P_id_MARK P_id_A__U45.
            
            Lemma marked_measure_equation :
             forall t, 
              marked_measure t = match t with
                                   | (algebra.Alg.Term algebra.F.id_a__U46 
                                      (x11::nil)) =>
                                    P_id_A__U46 (measure x11)
                                   | (algebra.Alg.Term 
                                      algebra.F.id_a__isNatIListKind 
                                      (x11::nil)) =>
                                    P_id_A__ISNATILISTKIND (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U61 
                                      (x11::nil)) =>
                                    P_id_A__U61 (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U85 
                                      (x12::x11::nil)) =>
                                    P_id_A__U85 (measure x12) (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U31 
                                      (x12::x11::nil)) =>
                                    P_id_A__U31 (measure x12) (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U81 
                                      (x13::x12::x11::nil)) =>
                                    P_id_A__U81 (measure x13) (measure x12) 
                                     (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U22 
                                      (x12::x11::nil)) =>
                                    P_id_A__U22 (measure x12) (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U93 
                                      (x13::x12::x11::nil)) =>
                                    P_id_A__U93 (measure x13) (measure x12) 
                                     (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U42 
                                      (x13::x12::x11::nil)) =>
                                    P_id_A__U42 (measure x13) (measure x12) 
                                     (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U51 
                                      (x12::x11::nil)) =>
                                    P_id_A__U51 (measure x12) (measure x11)
                                   | (algebra.Alg.Term 
                                      algebra.F.id_a__isNatList (x11::nil)) =>
                                    P_id_A__ISNATLIST (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__zeros 
                                      nil) =>
                                    P_id_A__ZEROS 
                                   | (algebra.Alg.Term algebra.F.id_a__U91 
                                      (x13::x12::x11::nil)) =>
                                    P_id_A__U91 (measure x13) (measure x12) 
                                     (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U33 
                                      (x11::nil)) =>
                                    P_id_A__U33 (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U83 
                                      (x13::x12::x11::nil)) =>
                                    P_id_A__U83 (measure x13) (measure x12) 
                                     (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U23 
                                      (x11::nil)) =>
                                    P_id_A__U23 (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U11 
                                      (x12::x11::nil)) =>
                                    P_id_A__U11 (measure x12) (measure x11)
                                   | (algebra.Alg.Term 
                                      algebra.F.id_a__length (x11::nil)) =>
                                    P_id_A__LENGTH (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U44 
                                      (x13::x12::x11::nil)) =>
                                    P_id_A__U44 (measure x13) (measure x12) 
                                     (measure x11)
                                   | (algebra.Alg.Term 
                                      algebra.F.id_a__isNatIList (x11::nil)) =>
                                    P_id_A__ISNATILIST (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U13 
                                      (x11::nil)) =>
                                    P_id_A__U13 (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U71 
                                      (x11::nil)) =>
                                    P_id_A__U71 (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U86 
                                      (x11::nil)) =>
                                    P_id_A__U86 (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U32 
                                      (x12::x11::nil)) =>
                                    P_id_A__U32 (measure x12) (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U82 
                                      (x13::x12::x11::nil)) =>
                                    P_id_A__U82 (measure x13) (measure x12) 
                                     (measure x11)
                                   | (algebra.Alg.Term 
                                      algebra.F.id_a__isNatKind (x11::nil)) =>
                                    P_id_A__ISNATKIND (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U94 
                                      (x12::x11::nil)) =>
                                    P_id_A__U94 (measure x12) (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U43 
                                      (x13::x12::x11::nil)) =>
                                    P_id_A__U43 (measure x13) (measure x12) 
                                     (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U52 
                                      (x11::nil)) =>
                                    P_id_A__U52 (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U21 
                                      (x12::x11::nil)) =>
                                    P_id_A__U21 (measure x12) (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U92 
                                      (x13::x12::x11::nil)) =>
                                    P_id_A__U92 (measure x13) (measure x12) 
                                     (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U41 
                                      (x13::x12::x11::nil)) =>
                                    P_id_A__U41 (measure x13) (measure x12) 
                                     (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U84 
                                      (x13::x12::x11::nil)) =>
                                    P_id_A__U84 (measure x13) (measure x12) 
                                     (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__isNat 
                                      (x11::nil)) =>
                                    P_id_A__ISNAT (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U12 
                                      (x12::x11::nil)) =>
                                    P_id_A__U12 (measure x12) (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_mark 
                                      (x11::nil)) =>
                                    P_id_MARK (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U45 
                                      (x12::x11::nil)) =>
                                    P_id_A__U45 (measure x12) (measure x11)
                                   | _ => measure t
                                   end.
            Proof.
              reflexivity .
            Qed.
            
            Lemma marked_measure_star_monotonic :
             forall f l1 l2, 
              (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step 
                                                                R_xml_0_deep_rew.R_xml_0_rules)
                                                               ) l1 l2) ->
               marked_measure (algebra.Alg.Term f l1) <= marked_measure 
                                                          (algebra.Alg.Term 
                                                            f l2).
            Proof.
              unfold marked_measure in *.
              apply InterpZ.marked_measure_star_monotonic.
              intros ;apply P_id_U51_monotonic;assumption.
              intros ;apply P_id_a__U84_monotonic;assumption.
              intros ;apply P_id_a__U32_monotonic;assumption.
              intros ;apply P_id_isNatList_monotonic;assumption.
              intros ;apply P_id_a__U13_monotonic;assumption.
              intros ;apply P_id_U85_monotonic;assumption.
              intros ;apply P_id_a__length_monotonic;assumption.
              intros ;apply P_id_a__isNatIList_monotonic;assumption.
              intros ;apply P_id_U33_monotonic;assumption.
              intros ;apply P_id_a__U11_monotonic;assumption.
              intros ;apply P_id_U81_monotonic;assumption.
              intros ;apply P_id_a__U92_monotonic;assumption.
              intros ;apply P_id_a__U43_monotonic;assumption.
              intros ;apply P_id_U23_monotonic;assumption.
              intros ;apply P_id_a__isNatKind_monotonic;assumption.
              intros ;apply P_id_U93_monotonic;assumption.
              intros ;apply P_id_U11_monotonic;assumption.
              intros ;apply P_id_a__U71_monotonic;assumption.
              intros ;apply P_id_U44_monotonic;assumption.
              intros ;apply P_id_U61_monotonic;assumption.
              intros ;apply P_id_a__U86_monotonic;assumption.
              intros ;apply P_id_a__U41_monotonic;assumption.
              intros ;apply P_id_U22_monotonic;assumption.
              intros ;apply P_id_a__U21_monotonic;assumption.
              intros ;apply P_id_U91_monotonic;assumption.
              intros ;apply P_id_length_monotonic;assumption.
              intros ;apply P_id_a__U52_monotonic;assumption.
              intros ;apply P_id_U42_monotonic;assumption.
              intros ;apply P_id_a__U12_monotonic;assumption.
              intros ;apply P_id_U83_monotonic;assumption.
              intros ;apply P_id_a__U94_monotonic;assumption.
              intros ;apply P_id_a__U45_monotonic;assumption.
              intros ;apply P_id_U31_monotonic;assumption.
              intros ;apply P_id_a__isNat_monotonic;assumption.
              intros ;apply P_id_isNatIListKind_monotonic;assumption.
              intros ;apply P_id_a__U82_monotonic;assumption.
              intros ;apply P_id_U46_monotonic;assumption.
              intros ;apply P_id_cons_monotonic;assumption.
              intros ;apply P_id_U52_monotonic;assumption.
              intros ;apply P_id_a__U85_monotonic;assumption.
              intros ;apply P_id_a__U33_monotonic;assumption.
              intros ;apply P_id_U21_monotonic;assumption.
              intros ;apply P_id_a__isNatList_monotonic;assumption.
              intros ;apply P_id_U86_monotonic;assumption.
              intros ;apply P_id_mark_monotonic;assumption.
              intros ;apply P_id_a__U51_monotonic;assumption.
              intros ;apply P_id_U41_monotonic;assumption.
              intros ;apply P_id_U82_monotonic;assumption.
              intros ;apply P_id_a__U93_monotonic;assumption.
              intros ;apply P_id_a__U44_monotonic;assumption.
              intros ;apply P_id_isNat_monotonic;assumption.
              intros ;apply P_id_a__U23_monotonic;assumption.
              intros ;apply P_id_U94_monotonic;assumption.
              intros ;apply P_id_U12_monotonic;assumption.
              intros ;apply P_id_a__U81_monotonic;assumption.
              intros ;apply P_id_U45_monotonic;assumption.
              intros ;apply P_id_U71_monotonic;assumption.
              intros ;apply P_id_a__U91_monotonic;assumption.
              intros ;apply P_id_a__U42_monotonic;assumption.
              intros ;apply P_id_isNatKind_monotonic;assumption.
              intros ;apply P_id_a__U22_monotonic;assumption.
              intros ;apply P_id_U92_monotonic;assumption.
              intros ;apply P_id_a__U61_monotonic;assumption.
              intros ;apply P_id_U43_monotonic;assumption.
              intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
              intros ;apply P_id_U84_monotonic;assumption.
              intros ;apply P_id_s_monotonic;assumption.
              intros ;apply P_id_a__U46_monotonic;assumption.
              intros ;apply P_id_U32_monotonic;assumption.
              intros ;apply P_id_a__U31_monotonic;assumption.
              intros ;apply P_id_U13_monotonic;assumption.
              intros ;apply P_id_a__U83_monotonic;assumption.
              intros ;apply P_id_isNatIList_monotonic;assumption.
              intros ;apply P_id_a__zeros_bounded;assumption.
              intros ;apply P_id_U51_bounded;assumption.
              intros ;apply P_id_a__U84_bounded;assumption.
              intros ;apply P_id_a__U32_bounded;assumption.
              intros ;apply P_id_isNatList_bounded;assumption.
              intros ;apply P_id_a__U13_bounded;assumption.
              intros ;apply P_id_U85_bounded;assumption.
              intros ;apply P_id_a__length_bounded;assumption.
              intros ;apply P_id_a__isNatIList_bounded;assumption.
              intros ;apply P_id_U33_bounded;assumption.
              intros ;apply P_id_a__U11_bounded;assumption.
              intros ;apply P_id_U81_bounded;assumption.
              intros ;apply P_id_a__U92_bounded;assumption.
              intros ;apply P_id_a__U43_bounded;assumption.
              intros ;apply P_id_U23_bounded;assumption.
              intros ;apply P_id_a__isNatKind_bounded;assumption.
              intros ;apply P_id_U93_bounded;assumption.
              intros ;apply P_id_U11_bounded;assumption.
              intros ;apply P_id_a__U71_bounded;assumption.
              intros ;apply P_id_U44_bounded;assumption.
              intros ;apply P_id_0_bounded;assumption.
              intros ;apply P_id_U61_bounded;assumption.
              intros ;apply P_id_a__U86_bounded;assumption.
              intros ;apply P_id_a__U41_bounded;assumption.
              intros ;apply P_id_U22_bounded;assumption.
              intros ;apply P_id_a__U21_bounded;assumption.
              intros ;apply P_id_U91_bounded;assumption.
              intros ;apply P_id_length_bounded;assumption.
              intros ;apply P_id_a__U52_bounded;assumption.
              intros ;apply P_id_U42_bounded;assumption.
              intros ;apply P_id_a__U12_bounded;assumption.
              intros ;apply P_id_U83_bounded;assumption.
              intros ;apply P_id_a__U94_bounded;assumption.
              intros ;apply P_id_a__U45_bounded;assumption.
              intros ;apply P_id_U31_bounded;assumption.
              intros ;apply P_id_a__isNat_bounded;assumption.
              intros ;apply P_id_isNatIListKind_bounded;assumption.
              intros ;apply P_id_a__U82_bounded;assumption.
              intros ;apply P_id_U46_bounded;assumption.
              intros ;apply P_id_cons_bounded;assumption.
              intros ;apply P_id_U52_bounded;assumption.
              intros ;apply P_id_a__U85_bounded;assumption.
              intros ;apply P_id_a__U33_bounded;assumption.
              intros ;apply P_id_U21_bounded;assumption.
              intros ;apply P_id_a__isNatList_bounded;assumption.
              intros ;apply P_id_U86_bounded;assumption.
              intros ;apply P_id_mark_bounded;assumption.
              intros ;apply P_id_a__U51_bounded;assumption.
              intros ;apply P_id_U41_bounded;assumption.
              intros ;apply P_id_tt_bounded;assumption.
              intros ;apply P_id_U82_bounded;assumption.
              intros ;apply P_id_a__U93_bounded;assumption.
              intros ;apply P_id_a__U44_bounded;assumption.
              intros ;apply P_id_isNat_bounded;assumption.
              intros ;apply P_id_a__U23_bounded;assumption.
              intros ;apply P_id_U94_bounded;assumption.
              intros ;apply P_id_U12_bounded;assumption.
              intros ;apply P_id_a__U81_bounded;assumption.
              intros ;apply P_id_U45_bounded;assumption.
              intros ;apply P_id_zeros_bounded;assumption.
              intros ;apply P_id_U71_bounded;assumption.
              intros ;apply P_id_a__U91_bounded;assumption.
              intros ;apply P_id_a__U42_bounded;assumption.
              intros ;apply P_id_isNatKind_bounded;assumption.
              intros ;apply P_id_a__U22_bounded;assumption.
              intros ;apply P_id_U92_bounded;assumption.
              intros ;apply P_id_nil_bounded;assumption.
              intros ;apply P_id_a__U61_bounded;assumption.
              intros ;apply P_id_U43_bounded;assumption.
              intros ;apply P_id_a__isNatIListKind_bounded;assumption.
              intros ;apply P_id_U84_bounded;assumption.
              intros ;apply P_id_s_bounded;assumption.
              intros ;apply P_id_a__U46_bounded;assumption.
              intros ;apply P_id_U32_bounded;assumption.
              intros ;apply P_id_a__U31_bounded;assumption.
              intros ;apply P_id_U13_bounded;assumption.
              intros ;apply P_id_a__U83_bounded;assumption.
              intros ;apply P_id_isNatIList_bounded;assumption.
              apply rules_monotonic.
              intros ;apply P_id_A__U46_monotonic;assumption.
              intros ;apply P_id_A__ISNATILISTKIND_monotonic;assumption.
              intros ;apply P_id_A__U61_monotonic;assumption.
              intros ;apply P_id_A__U85_monotonic;assumption.
              intros ;apply P_id_A__U31_monotonic;assumption.
              intros ;apply P_id_A__U81_monotonic;assumption.
              intros ;apply P_id_A__U22_monotonic;assumption.
              intros ;apply P_id_A__U93_monotonic;assumption.
              intros ;apply P_id_A__U42_monotonic;assumption.
              intros ;apply P_id_A__U51_monotonic;assumption.
              intros ;apply P_id_A__ISNATLIST_monotonic;assumption.
              intros ;apply P_id_A__U91_monotonic;assumption.
              intros ;apply P_id_A__U33_monotonic;assumption.
              intros ;apply P_id_A__U83_monotonic;assumption.
              intros ;apply P_id_A__U23_monotonic;assumption.
              intros ;apply P_id_A__U11_monotonic;assumption.
              intros ;apply P_id_A__LENGTH_monotonic;assumption.
              intros ;apply P_id_A__U44_monotonic;assumption.
              intros ;apply P_id_A__ISNATILIST_monotonic;assumption.
              intros ;apply P_id_A__U13_monotonic;assumption.
              intros ;apply P_id_A__U71_monotonic;assumption.
              intros ;apply P_id_A__U86_monotonic;assumption.
              intros ;apply P_id_A__U32_monotonic;assumption.
              intros ;apply P_id_A__U82_monotonic;assumption.
              intros ;apply P_id_A__ISNATKIND_monotonic;assumption.
              intros ;apply P_id_A__U94_monotonic;assumption.
              intros ;apply P_id_A__U43_monotonic;assumption.
              intros ;apply P_id_A__U52_monotonic;assumption.
              intros ;apply P_id_A__U21_monotonic;assumption.
              intros ;apply P_id_A__U92_monotonic;assumption.
              intros ;apply P_id_A__U41_monotonic;assumption.
              intros ;apply P_id_A__U84_monotonic;assumption.
              intros ;apply P_id_A__ISNAT_monotonic;assumption.
              intros ;apply P_id_A__U12_monotonic;assumption.
              intros ;apply P_id_MARK_monotonic;assumption.
              intros ;apply P_id_A__U45_monotonic;assumption.
            Qed.
            
            Ltac rewrite_and_unfold  :=
             do 2 (rewrite marked_measure_equation);
              repeat (
              match goal with
                |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
                 rewrite (measure_equation (algebra.Alg.Term f t))
                | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ =>
                 rewrite (measure_equation (algebra.Alg.Term f t)) in H|-
                end
              ).
            
            Definition lt a b := 
              (Zwf.Zwf 0) (marked_measure a) (marked_measure b).
            
            Definition le a b := marked_measure a <= marked_measure b.
            
            Lemma lt_le_compat : forall a b c, (lt a b) ->(le b c) ->lt a c.
            Proof.
              unfold lt, le in *.
              intros a b c.
              apply (interp.le_lt_compat_right (interp.o_Z 0)).
            Qed.
            
            Lemma wf_lt : well_founded lt.
            Proof.
              unfold lt in *.
              apply Inverse_Image.wf_inverse_image with  (B:=Z).
              apply Zwf.Zwf_well_founded.
            Qed.
            
            Lemma DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_strict_in_lt :
             Relation_Definitions.inclusion _ 
              DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_strict 
              lt.
            Proof.
              unfold Relation_Definitions.inclusion, lt in *.
              
              intros a b H;destruct H;
               match goal with
                 |  |- (Zwf.Zwf 0) _ 
                        (marked_measure (algebra.Alg.Term ?f ?l)) =>
                  let l'' := algebra.Alg_ext.find_replacement l  in 
                   ((apply (interp.le_lt_compat_right (interp.o_Z 0)) with
                      (marked_measure (algebra.Alg.Term f l''));[idtac|
                     apply marked_measure_star_monotonic;
                      repeat (
                      apply algebra.EQT_ext.one_step_list_refl_trans_clos);
                      (assumption)||(constructor 1)]))
                 end
               ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_in_le :
             Relation_Definitions.inclusion _ 
              DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large 
              le.
            Proof.
              unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *.
              
              intros a b H;destruct H;
               match goal with
                 |  |- _ <= marked_measure (algebra.Alg.Term ?f ?l) =>
                  let l'' := algebra.Alg_ext.find_replacement l  in 
                   ((apply (interp.le_trans (interp.o_Z 0)) with
                      (marked_measure (algebra.Alg.Term f l''));[idtac|
                     apply marked_measure_star_monotonic;
                      repeat (
                      apply algebra.EQT_ext.one_step_list_refl_trans_clos);
                      (assumption)||(constructor 1)]))
                 end
               ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
               cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Definition wf_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large  := 
              WF_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large.wf
              .
            
            
            Lemma wf :
             well_founded WF_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large.DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large
              .
            Proof.
              intros x.
              apply (well_founded_ind wf_lt).
              clear x.
              intros x.
              pattern x.
              
              apply (@Acc_ind _ 
                      DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large)
                     .
              clear x.
              intros x _ IHx IHx'.
              constructor.
              intros y H.
              
              destruct H;
               (apply IHx';
                 apply DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_strict_in_lt;
                 econstructor eassumption)||
               ((apply IHx;[econstructor eassumption|
                 intros y' Hlt;apply IHx';apply lt_le_compat with (1:=Hlt) ;
                  apply DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large_in_le;
                  econstructor eassumption])).
              
              apply wf_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_large.
            Qed.
           End WF_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large.
           
           Open Scope Z_scope.
           
           Import ring_extention.
           
           Notation Local "a <= b" := (Zle a b).
           
           Notation Local "a < b" := (Zlt a b).
           
           Definition P_id_a__zeros  := 0.
           
           Definition P_id_U51 (x11:Z) (x12:Z) := 1* x11.
           
           Definition P_id_a__U84 (x11:Z) (x12:Z) (x13:Z) := 
             1* x11 + 2* x12 + 2* x13.
           
           Definition P_id_a__U32 (x11:Z) (x12:Z) := 0.
           
           Definition P_id_isNatList (x11:Z) := 2* x11.
           
           Definition P_id_a__U13 (x11:Z) := 1* x11.
           
           Definition P_id_U85 (x11:Z) (x12:Z) := 1* x11 + 2* x12.
           
           Definition P_id_a__length (x11:Z) := 2 + 2* x11.
           
           Definition P_id_a__isNatIList (x11:Z) := 0.
           
           Definition P_id_U33 (x11:Z) := 0.
           
           Definition P_id_a__U11 (x11:Z) (x12:Z) := 1 + 2* x12.
           
           Definition P_id_U81 (x11:Z) (x12:Z) (x13:Z) := 
             1* x11 + 2* x12 + 2* x13.
           
           Definition P_id_a__U92 (x11:Z) (x12:Z) (x13:Z) := 2 + 2* x12.
           
           Definition P_id_a__U43 (x11:Z) (x12:Z) (x13:Z) := 0.
           
           Definition P_id_U23 (x11:Z) := 1* x11.
           
           Definition P_id_a__isNatKind (x11:Z) := 0.
           
           Definition P_id_U93 (x11:Z) (x12:Z) (x13:Z) := 2 + 2* x12.
           
           Definition P_id_U11 (x11:Z) (x12:Z) := 1 + 2* x12.
           
           Definition P_id_a__U71 (x11:Z) := 1* x11.
           
           Definition P_id_U44 (x11:Z) (x12:Z) (x13:Z) := 0.
           
           Definition P_id_0  := 0.
           
           Definition P_id_U61 (x11:Z) := 2* x11.
           
           Definition P_id_a__U86 (x11:Z) := 1* x11.
           
           Definition P_id_a__U41 (x11:Z) (x12:Z) (x13:Z) := 0.
           
           Definition P_id_U22 (x11:Z) (x12:Z) := 1* x11 + 2* x12.
           
           Definition P_id_a__U21 (x11:Z) (x12:Z) := 1* x11 + 2* x12.
           
           Definition P_id_U91 (x11:Z) (x12:Z) (x13:Z) := 2 + 2* x12 + 2* x13
             .
           
           Definition P_id_length (x11:Z) := 2 + 2* x11.
           
           Definition P_id_a__U52 (x11:Z) := 1* x11.
           
           Definition P_id_U42 (x11:Z) (x12:Z) (x13:Z) := 0.
           
           Definition P_id_a__U12 (x11:Z) (x12:Z) := 1 + 1* x11 + 2* x12.
           
           Definition P_id_U83 (x11:Z) (x12:Z) (x13:Z) := 
             1* x11 + 2* x12 + 2* x13.
           
           Definition P_id_a__U94 (x11:Z) (x12:Z) := 2 + 2* x12.
           
           Definition P_id_a__U45 (x11:Z) (x12:Z) := 0.
           
           Definition P_id_U31 (x11:Z) (x12:Z) := 0.
           
           Definition P_id_a__isNat (x11:Z) := 2* x11.
           
           Definition P_id_isNatIListKind (x11:Z) := 0.
           
           Definition P_id_a__U82 (x11:Z) (x12:Z) (x13:Z) := 
             1* x11 + 2* x12 + 2* x13.
           
           Definition P_id_U46 (x11:Z) := 0.
           
           Definition P_id_cons (x11:Z) (x12:Z) := 2* x11 + 2* x12.
           
           Definition P_id_U52 (x11:Z) := 1* x11.
           
           Definition P_id_a__U85 (x11:Z) (x12:Z) := 1* x11 + 2* x12.
           
           Definition P_id_a__U33 (x11:Z) := 0.
           
           Definition P_id_U21 (x11:Z) (x12:Z) := 1* x11 + 2* x12.
           
           Definition P_id_a__isNatList (x11:Z) := 2* x11.
           
           Definition P_id_U86 (x11:Z) := 1* x11.
           
           Definition P_id_mark (x11:Z) := 1* x11.
           
           Definition P_id_a__U51 (x11:Z) (x12:Z) := 1* x11.
           
           Definition P_id_U41 (x11:Z) (x12:Z) (x13:Z) := 0.
           
           Definition P_id_tt  := 0.
           
           Definition P_id_U82 (x11:Z) (x12:Z) (x13:Z) := 
             1* x11 + 2* x12 + 2* x13.
           
           Definition P_id_a__U93 (x11:Z) (x12:Z) (x13:Z) := 2 + 2* x12.
           
           Definition P_id_a__U44 (x11:Z) (x12:Z) (x13:Z) := 0.
           
           Definition P_id_isNat (x11:Z) := 2* x11.
           
           Definition P_id_a__U23 (x11:Z) := 1* x11.
           
           Definition P_id_U94 (x11:Z) (x12:Z) := 2 + 2* x12.
           
           Definition P_id_U12 (x11:Z) (x12:Z) := 1 + 1* x11 + 2* x12.
           
           Definition P_id_a__U81 (x11:Z) (x12:Z) (x13:Z) := 
             1* x11 + 2* x12 + 2* x13.
           
           Definition P_id_U45 (x11:Z) (x12:Z) := 0.
           
           Definition P_id_zeros  := 0.
           
           Definition P_id_U71 (x11:Z) := 1* x11.
           
           Definition P_id_a__U91 (x11:Z) (x12:Z) (x13:Z) := 
             2 + 2* x12 + 2* x13.
           
           Definition P_id_a__U42 (x11:Z) (x12:Z) (x13:Z) := 0.
           
           Definition P_id_isNatKind (x11:Z) := 0.
           
           Definition P_id_a__U22 (x11:Z) (x12:Z) := 1* x11 + 2* x12.
           
           Definition P_id_U92 (x11:Z) (x12:Z) (x13:Z) := 2 + 2* x12.
           
           Definition P_id_nil  := 0.
           
           Definition P_id_a__U61 (x11:Z) := 2* x11.
           
           Definition P_id_U43 (x11:Z) (x12:Z) (x13:Z) := 0.
           
           Definition P_id_a__isNatIListKind (x11:Z) := 0.
           
           Definition P_id_U84 (x11:Z) (x12:Z) (x13:Z) := 
             1* x11 + 2* x12 + 2* x13.
           
           Definition P_id_s (x11:Z) := 1* x11.
           
           Definition P_id_a__U46 (x11:Z) := 0.
           
           Definition P_id_U32 (x11:Z) (x12:Z) := 0.
           
           Definition P_id_a__U31 (x11:Z) (x12:Z) := 0.
           
           Definition P_id_U13 (x11:Z) := 1* x11.
           
           Definition P_id_a__U83 (x11:Z) (x12:Z) (x13:Z) := 
             1* x11 + 2* x12 + 2* x13.
           
           Definition P_id_isNatIList (x11:Z) := 0.
           
           Lemma P_id_U51_monotonic :
            forall x12 x14 x13 x11, 
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_U51 x12 x14 <= P_id_U51 x11 x13.
           Proof.
             intros x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U84_monotonic :
            forall x16 x12 x14 x13 x11 x15, 
             (0 <= x16)/\ (x16 <= x15) ->
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_a__U84 x12 x14 x16 <= P_id_a__U84 x11 x13 x15.
           Proof.
             intros x16 x15 x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             intros [H_5 H_4].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U32_monotonic :
            forall x12 x14 x13 x11, 
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_a__U32 x12 x14 <= P_id_a__U32 x11 x13.
           Proof.
             intros x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_isNatList_monotonic :
            forall x12 x11, 
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_isNatList x12 <= P_id_isNatList x11.
           Proof.
             intros x12 x11.
             intros [H_1 H_0].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U13_monotonic :
            forall x12 x11, 
             (0 <= x12)/\ (x12 <= x11) ->P_id_a__U13 x12 <= P_id_a__U13 x11.
           Proof.
             intros x12 x11.
             intros [H_1 H_0].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U85_monotonic :
            forall x12 x14 x13 x11, 
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_U85 x12 x14 <= P_id_U85 x11 x13.
           Proof.
             intros x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__length_monotonic :
            forall x12 x11, 
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_a__length x12 <= P_id_a__length x11.
           Proof.
             intros x12 x11.
             intros [H_1 H_0].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__isNatIList_monotonic :
            forall x12 x11, 
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_a__isNatIList x12 <= P_id_a__isNatIList x11.
           Proof.
             intros x12 x11.
             intros [H_1 H_0].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U33_monotonic :
            forall x12 x11, 
             (0 <= x12)/\ (x12 <= x11) ->P_id_U33 x12 <= P_id_U33 x11.
           Proof.
             intros x12 x11.
             intros [H_1 H_0].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U11_monotonic :
            forall x12 x14 x13 x11, 
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_a__U11 x12 x14 <= P_id_a__U11 x11 x13.
           Proof.
             intros x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U81_monotonic :
            forall x16 x12 x14 x13 x11 x15, 
             (0 <= x16)/\ (x16 <= x15) ->
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_U81 x12 x14 x16 <= P_id_U81 x11 x13 x15.
           Proof.
             intros x16 x15 x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             intros [H_5 H_4].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U92_monotonic :
            forall x16 x12 x14 x13 x11 x15, 
             (0 <= x16)/\ (x16 <= x15) ->
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_a__U92 x12 x14 x16 <= P_id_a__U92 x11 x13 x15.
           Proof.
             intros x16 x15 x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             intros [H_5 H_4].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U43_monotonic :
            forall x16 x12 x14 x13 x11 x15, 
             (0 <= x16)/\ (x16 <= x15) ->
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_a__U43 x12 x14 x16 <= P_id_a__U43 x11 x13 x15.
           Proof.
             intros x16 x15 x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             intros [H_5 H_4].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U23_monotonic :
            forall x12 x11, 
             (0 <= x12)/\ (x12 <= x11) ->P_id_U23 x12 <= P_id_U23 x11.
           Proof.
             intros x12 x11.
             intros [H_1 H_0].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__isNatKind_monotonic :
            forall x12 x11, 
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_a__isNatKind x12 <= P_id_a__isNatKind x11.
           Proof.
             intros x12 x11.
             intros [H_1 H_0].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U93_monotonic :
            forall x16 x12 x14 x13 x11 x15, 
             (0 <= x16)/\ (x16 <= x15) ->
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_U93 x12 x14 x16 <= P_id_U93 x11 x13 x15.
           Proof.
             intros x16 x15 x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             intros [H_5 H_4].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U11_monotonic :
            forall x12 x14 x13 x11, 
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_U11 x12 x14 <= P_id_U11 x11 x13.
           Proof.
             intros x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U71_monotonic :
            forall x12 x11, 
             (0 <= x12)/\ (x12 <= x11) ->P_id_a__U71 x12 <= P_id_a__U71 x11.
           Proof.
             intros x12 x11.
             intros [H_1 H_0].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U44_monotonic :
            forall x16 x12 x14 x13 x11 x15, 
             (0 <= x16)/\ (x16 <= x15) ->
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_U44 x12 x14 x16 <= P_id_U44 x11 x13 x15.
           Proof.
             intros x16 x15 x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             intros [H_5 H_4].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U61_monotonic :
            forall x12 x11, 
             (0 <= x12)/\ (x12 <= x11) ->P_id_U61 x12 <= P_id_U61 x11.
           Proof.
             intros x12 x11.
             intros [H_1 H_0].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U86_monotonic :
            forall x12 x11, 
             (0 <= x12)/\ (x12 <= x11) ->P_id_a__U86 x12 <= P_id_a__U86 x11.
           Proof.
             intros x12 x11.
             intros [H_1 H_0].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U41_monotonic :
            forall x16 x12 x14 x13 x11 x15, 
             (0 <= x16)/\ (x16 <= x15) ->
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_a__U41 x12 x14 x16 <= P_id_a__U41 x11 x13 x15.
           Proof.
             intros x16 x15 x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             intros [H_5 H_4].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U22_monotonic :
            forall x12 x14 x13 x11, 
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_U22 x12 x14 <= P_id_U22 x11 x13.
           Proof.
             intros x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U21_monotonic :
            forall x12 x14 x13 x11, 
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_a__U21 x12 x14 <= P_id_a__U21 x11 x13.
           Proof.
             intros x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U91_monotonic :
            forall x16 x12 x14 x13 x11 x15, 
             (0 <= x16)/\ (x16 <= x15) ->
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_U91 x12 x14 x16 <= P_id_U91 x11 x13 x15.
           Proof.
             intros x16 x15 x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             intros [H_5 H_4].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_length_monotonic :
            forall x12 x11, 
             (0 <= x12)/\ (x12 <= x11) ->P_id_length x12 <= P_id_length x11.
           Proof.
             intros x12 x11.
             intros [H_1 H_0].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U52_monotonic :
            forall x12 x11, 
             (0 <= x12)/\ (x12 <= x11) ->P_id_a__U52 x12 <= P_id_a__U52 x11.
           Proof.
             intros x12 x11.
             intros [H_1 H_0].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U42_monotonic :
            forall x16 x12 x14 x13 x11 x15, 
             (0 <= x16)/\ (x16 <= x15) ->
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_U42 x12 x14 x16 <= P_id_U42 x11 x13 x15.
           Proof.
             intros x16 x15 x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             intros [H_5 H_4].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U12_monotonic :
            forall x12 x14 x13 x11, 
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_a__U12 x12 x14 <= P_id_a__U12 x11 x13.
           Proof.
             intros x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U83_monotonic :
            forall x16 x12 x14 x13 x11 x15, 
             (0 <= x16)/\ (x16 <= x15) ->
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_U83 x12 x14 x16 <= P_id_U83 x11 x13 x15.
           Proof.
             intros x16 x15 x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             intros [H_5 H_4].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U94_monotonic :
            forall x12 x14 x13 x11, 
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_a__U94 x12 x14 <= P_id_a__U94 x11 x13.
           Proof.
             intros x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U45_monotonic :
            forall x12 x14 x13 x11, 
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_a__U45 x12 x14 <= P_id_a__U45 x11 x13.
           Proof.
             intros x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U31_monotonic :
            forall x12 x14 x13 x11, 
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_U31 x12 x14 <= P_id_U31 x11 x13.
           Proof.
             intros x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__isNat_monotonic :
            forall x12 x11, 
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_a__isNat x12 <= P_id_a__isNat x11.
           Proof.
             intros x12 x11.
             intros [H_1 H_0].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_isNatIListKind_monotonic :
            forall x12 x11, 
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_isNatIListKind x12 <= P_id_isNatIListKind x11.
           Proof.
             intros x12 x11.
             intros [H_1 H_0].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U82_monotonic :
            forall x16 x12 x14 x13 x11 x15, 
             (0 <= x16)/\ (x16 <= x15) ->
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_a__U82 x12 x14 x16 <= P_id_a__U82 x11 x13 x15.
           Proof.
             intros x16 x15 x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             intros [H_5 H_4].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U46_monotonic :
            forall x12 x11, 
             (0 <= x12)/\ (x12 <= x11) ->P_id_U46 x12 <= P_id_U46 x11.
           Proof.
             intros x12 x11.
             intros [H_1 H_0].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_cons_monotonic :
            forall x12 x14 x13 x11, 
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_cons x12 x14 <= P_id_cons x11 x13.
           Proof.
             intros x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U52_monotonic :
            forall x12 x11, 
             (0 <= x12)/\ (x12 <= x11) ->P_id_U52 x12 <= P_id_U52 x11.
           Proof.
             intros x12 x11.
             intros [H_1 H_0].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U85_monotonic :
            forall x12 x14 x13 x11, 
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_a__U85 x12 x14 <= P_id_a__U85 x11 x13.
           Proof.
             intros x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U33_monotonic :
            forall x12 x11, 
             (0 <= x12)/\ (x12 <= x11) ->P_id_a__U33 x12 <= P_id_a__U33 x11.
           Proof.
             intros x12 x11.
             intros [H_1 H_0].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U21_monotonic :
            forall x12 x14 x13 x11, 
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_U21 x12 x14 <= P_id_U21 x11 x13.
           Proof.
             intros x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__isNatList_monotonic :
            forall x12 x11, 
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_a__isNatList x12 <= P_id_a__isNatList x11.
           Proof.
             intros x12 x11.
             intros [H_1 H_0].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U86_monotonic :
            forall x12 x11, 
             (0 <= x12)/\ (x12 <= x11) ->P_id_U86 x12 <= P_id_U86 x11.
           Proof.
             intros x12 x11.
             intros [H_1 H_0].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_mark_monotonic :
            forall x12 x11, 
             (0 <= x12)/\ (x12 <= x11) ->P_id_mark x12 <= P_id_mark x11.
           Proof.
             intros x12 x11.
             intros [H_1 H_0].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U51_monotonic :
            forall x12 x14 x13 x11, 
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_a__U51 x12 x14 <= P_id_a__U51 x11 x13.
           Proof.
             intros x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U41_monotonic :
            forall x16 x12 x14 x13 x11 x15, 
             (0 <= x16)/\ (x16 <= x15) ->
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_U41 x12 x14 x16 <= P_id_U41 x11 x13 x15.
           Proof.
             intros x16 x15 x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             intros [H_5 H_4].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U82_monotonic :
            forall x16 x12 x14 x13 x11 x15, 
             (0 <= x16)/\ (x16 <= x15) ->
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_U82 x12 x14 x16 <= P_id_U82 x11 x13 x15.
           Proof.
             intros x16 x15 x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             intros [H_5 H_4].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U93_monotonic :
            forall x16 x12 x14 x13 x11 x15, 
             (0 <= x16)/\ (x16 <= x15) ->
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_a__U93 x12 x14 x16 <= P_id_a__U93 x11 x13 x15.
           Proof.
             intros x16 x15 x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             intros [H_5 H_4].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U44_monotonic :
            forall x16 x12 x14 x13 x11 x15, 
             (0 <= x16)/\ (x16 <= x15) ->
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_a__U44 x12 x14 x16 <= P_id_a__U44 x11 x13 x15.
           Proof.
             intros x16 x15 x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             intros [H_5 H_4].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_isNat_monotonic :
            forall x12 x11, 
             (0 <= x12)/\ (x12 <= x11) ->P_id_isNat x12 <= P_id_isNat x11.
           Proof.
             intros x12 x11.
             intros [H_1 H_0].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U23_monotonic :
            forall x12 x11, 
             (0 <= x12)/\ (x12 <= x11) ->P_id_a__U23 x12 <= P_id_a__U23 x11.
           Proof.
             intros x12 x11.
             intros [H_1 H_0].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U94_monotonic :
            forall x12 x14 x13 x11, 
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_U94 x12 x14 <= P_id_U94 x11 x13.
           Proof.
             intros x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U12_monotonic :
            forall x12 x14 x13 x11, 
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_U12 x12 x14 <= P_id_U12 x11 x13.
           Proof.
             intros x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U81_monotonic :
            forall x16 x12 x14 x13 x11 x15, 
             (0 <= x16)/\ (x16 <= x15) ->
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_a__U81 x12 x14 x16 <= P_id_a__U81 x11 x13 x15.
           Proof.
             intros x16 x15 x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             intros [H_5 H_4].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U45_monotonic :
            forall x12 x14 x13 x11, 
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_U45 x12 x14 <= P_id_U45 x11 x13.
           Proof.
             intros x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U71_monotonic :
            forall x12 x11, 
             (0 <= x12)/\ (x12 <= x11) ->P_id_U71 x12 <= P_id_U71 x11.
           Proof.
             intros x12 x11.
             intros [H_1 H_0].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U91_monotonic :
            forall x16 x12 x14 x13 x11 x15, 
             (0 <= x16)/\ (x16 <= x15) ->
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_a__U91 x12 x14 x16 <= P_id_a__U91 x11 x13 x15.
           Proof.
             intros x16 x15 x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             intros [H_5 H_4].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U42_monotonic :
            forall x16 x12 x14 x13 x11 x15, 
             (0 <= x16)/\ (x16 <= x15) ->
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_a__U42 x12 x14 x16 <= P_id_a__U42 x11 x13 x15.
           Proof.
             intros x16 x15 x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             intros [H_5 H_4].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_isNatKind_monotonic :
            forall x12 x11, 
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_isNatKind x12 <= P_id_isNatKind x11.
           Proof.
             intros x12 x11.
             intros [H_1 H_0].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U22_monotonic :
            forall x12 x14 x13 x11, 
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_a__U22 x12 x14 <= P_id_a__U22 x11 x13.
           Proof.
             intros x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U92_monotonic :
            forall x16 x12 x14 x13 x11 x15, 
             (0 <= x16)/\ (x16 <= x15) ->
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_U92 x12 x14 x16 <= P_id_U92 x11 x13 x15.
           Proof.
             intros x16 x15 x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             intros [H_5 H_4].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U61_monotonic :
            forall x12 x11, 
             (0 <= x12)/\ (x12 <= x11) ->P_id_a__U61 x12 <= P_id_a__U61 x11.
           Proof.
             intros x12 x11.
             intros [H_1 H_0].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U43_monotonic :
            forall x16 x12 x14 x13 x11 x15, 
             (0 <= x16)/\ (x16 <= x15) ->
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_U43 x12 x14 x16 <= P_id_U43 x11 x13 x15.
           Proof.
             intros x16 x15 x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             intros [H_5 H_4].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__isNatIListKind_monotonic :
            forall x12 x11, 
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_a__isNatIListKind x12 <= P_id_a__isNatIListKind x11.
           Proof.
             intros x12 x11.
             intros [H_1 H_0].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U84_monotonic :
            forall x16 x12 x14 x13 x11 x15, 
             (0 <= x16)/\ (x16 <= x15) ->
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_U84 x12 x14 x16 <= P_id_U84 x11 x13 x15.
           Proof.
             intros x16 x15 x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             intros [H_5 H_4].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_s_monotonic :
            forall x12 x11, 
             (0 <= x12)/\ (x12 <= x11) ->P_id_s x12 <= P_id_s x11.
           Proof.
             intros x12 x11.
             intros [H_1 H_0].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U46_monotonic :
            forall x12 x11, 
             (0 <= x12)/\ (x12 <= x11) ->P_id_a__U46 x12 <= P_id_a__U46 x11.
           Proof.
             intros x12 x11.
             intros [H_1 H_0].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U32_monotonic :
            forall x12 x14 x13 x11, 
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_U32 x12 x14 <= P_id_U32 x11 x13.
           Proof.
             intros x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U31_monotonic :
            forall x12 x14 x13 x11, 
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_a__U31 x12 x14 <= P_id_a__U31 x11 x13.
           Proof.
             intros x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U13_monotonic :
            forall x12 x11, 
             (0 <= x12)/\ (x12 <= x11) ->P_id_U13 x12 <= P_id_U13 x11.
           Proof.
             intros x12 x11.
             intros [H_1 H_0].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U83_monotonic :
            forall x16 x12 x14 x13 x11 x15, 
             (0 <= x16)/\ (x16 <= x15) ->
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_a__U83 x12 x14 x16 <= P_id_a__U83 x11 x13 x15.
           Proof.
             intros x16 x15 x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             intros [H_5 H_4].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_isNatIList_monotonic :
            forall x12 x11, 
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_isNatIList x12 <= P_id_isNatIList x11.
           Proof.
             intros x12 x11.
             intros [H_1 H_0].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__zeros_bounded : 0 <= P_id_a__zeros .
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U51_bounded :
            forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U51 x12 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U84_bounded :
            forall x12 x13 x11, 
             (0 <= x11) ->
              (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U84 x13 x12 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U32_bounded :
            forall x12 x11, 
             (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U32 x12 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_isNatList_bounded :
            forall x11, (0 <= x11) ->0 <= P_id_isNatList x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U13_bounded :
            forall x11, (0 <= x11) ->0 <= P_id_a__U13 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U85_bounded :
            forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U85 x12 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__length_bounded :
            forall x11, (0 <= x11) ->0 <= P_id_a__length x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__isNatIList_bounded :
            forall x11, (0 <= x11) ->0 <= P_id_a__isNatIList x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U33_bounded :
            forall x11, (0 <= x11) ->0 <= P_id_U33 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U11_bounded :
            forall x12 x11, 
             (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U11 x12 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U81_bounded :
            forall x12 x13 x11, 
             (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U81 x13 x12 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U92_bounded :
            forall x12 x13 x11, 
             (0 <= x11) ->
              (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U92 x13 x12 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U43_bounded :
            forall x12 x13 x11, 
             (0 <= x11) ->
              (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U43 x13 x12 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U23_bounded :
            forall x11, (0 <= x11) ->0 <= P_id_U23 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__isNatKind_bounded :
            forall x11, (0 <= x11) ->0 <= P_id_a__isNatKind x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U93_bounded :
            forall x12 x13 x11, 
             (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U93 x13 x12 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U11_bounded :
            forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U11 x12 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U71_bounded :
            forall x11, (0 <= x11) ->0 <= P_id_a__U71 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U44_bounded :
            forall x12 x13 x11, 
             (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U44 x13 x12 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_0_bounded : 0 <= P_id_0 .
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U61_bounded :
            forall x11, (0 <= x11) ->0 <= P_id_U61 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U86_bounded :
            forall x11, (0 <= x11) ->0 <= P_id_a__U86 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U41_bounded :
            forall x12 x13 x11, 
             (0 <= x11) ->
              (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U41 x13 x12 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U22_bounded :
            forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U22 x12 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U21_bounded :
            forall x12 x11, 
             (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U21 x12 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U91_bounded :
            forall x12 x13 x11, 
             (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U91 x13 x12 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_length_bounded :
            forall x11, (0 <= x11) ->0 <= P_id_length x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U52_bounded :
            forall x11, (0 <= x11) ->0 <= P_id_a__U52 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U42_bounded :
            forall x12 x13 x11, 
             (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U42 x13 x12 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U12_bounded :
            forall x12 x11, 
             (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U12 x12 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U83_bounded :
            forall x12 x13 x11, 
             (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U83 x13 x12 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U94_bounded :
            forall x12 x11, 
             (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U94 x12 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U45_bounded :
            forall x12 x11, 
             (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U45 x12 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U31_bounded :
            forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U31 x12 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__isNat_bounded :
            forall x11, (0 <= x11) ->0 <= P_id_a__isNat x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_isNatIListKind_bounded :
            forall x11, (0 <= x11) ->0 <= P_id_isNatIListKind x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U82_bounded :
            forall x12 x13 x11, 
             (0 <= x11) ->
              (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U82 x13 x12 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U46_bounded :
            forall x11, (0 <= x11) ->0 <= P_id_U46 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_cons_bounded :
            forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_cons x12 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U52_bounded :
            forall x11, (0 <= x11) ->0 <= P_id_U52 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U85_bounded :
            forall x12 x11, 
             (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U85 x12 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U33_bounded :
            forall x11, (0 <= x11) ->0 <= P_id_a__U33 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U21_bounded :
            forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U21 x12 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__isNatList_bounded :
            forall x11, (0 <= x11) ->0 <= P_id_a__isNatList x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U86_bounded :
            forall x11, (0 <= x11) ->0 <= P_id_U86 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_mark_bounded :
            forall x11, (0 <= x11) ->0 <= P_id_mark x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U51_bounded :
            forall x12 x11, 
             (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U51 x12 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U41_bounded :
            forall x12 x13 x11, 
             (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U41 x13 x12 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_tt_bounded : 0 <= P_id_tt .
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U82_bounded :
            forall x12 x13 x11, 
             (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U82 x13 x12 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U93_bounded :
            forall x12 x13 x11, 
             (0 <= x11) ->
              (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U93 x13 x12 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U44_bounded :
            forall x12 x13 x11, 
             (0 <= x11) ->
              (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U44 x13 x12 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_isNat_bounded :
            forall x11, (0 <= x11) ->0 <= P_id_isNat x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U23_bounded :
            forall x11, (0 <= x11) ->0 <= P_id_a__U23 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U94_bounded :
            forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U94 x12 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U12_bounded :
            forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U12 x12 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U81_bounded :
            forall x12 x13 x11, 
             (0 <= x11) ->
              (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U81 x13 x12 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U45_bounded :
            forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U45 x12 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_zeros_bounded : 0 <= P_id_zeros .
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U71_bounded :
            forall x11, (0 <= x11) ->0 <= P_id_U71 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U91_bounded :
            forall x12 x13 x11, 
             (0 <= x11) ->
              (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U91 x13 x12 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U42_bounded :
            forall x12 x13 x11, 
             (0 <= x11) ->
              (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U42 x13 x12 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_isNatKind_bounded :
            forall x11, (0 <= x11) ->0 <= P_id_isNatKind x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U22_bounded :
            forall x12 x11, 
             (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U22 x12 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U92_bounded :
            forall x12 x13 x11, 
             (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U92 x13 x12 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_nil_bounded : 0 <= P_id_nil .
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U61_bounded :
            forall x11, (0 <= x11) ->0 <= P_id_a__U61 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U43_bounded :
            forall x12 x13 x11, 
             (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U43 x13 x12 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__isNatIListKind_bounded :
            forall x11, (0 <= x11) ->0 <= P_id_a__isNatIListKind x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U84_bounded :
            forall x12 x13 x11, 
             (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U84 x13 x12 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_s_bounded : forall x11, (0 <= x11) ->0 <= P_id_s x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U46_bounded :
            forall x11, (0 <= x11) ->0 <= P_id_a__U46 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U32_bounded :
            forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U32 x12 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U31_bounded :
            forall x12 x11, 
             (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U31 x12 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_U13_bounded :
            forall x11, (0 <= x11) ->0 <= P_id_U13 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_a__U83_bounded :
            forall x12 x13 x11, 
             (0 <= x11) ->
              (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U83 x13 x12 x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_isNatIList_bounded :
            forall x11, (0 <= x11) ->0 <= P_id_isNatIList x11.
           Proof.
             intros .
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Definition measure  := 
             InterpZ.measure 0 P_id_a__zeros P_id_U51 P_id_a__U84 
              P_id_a__U32 P_id_isNatList P_id_a__U13 P_id_U85 P_id_a__length 
              P_id_a__isNatIList P_id_U33 P_id_a__U11 P_id_U81 P_id_a__U92 
              P_id_a__U43 P_id_U23 P_id_a__isNatKind P_id_U93 P_id_U11 
              P_id_a__U71 P_id_U44 P_id_0 P_id_U61 P_id_a__U86 P_id_a__U41 
              P_id_U22 P_id_a__U21 P_id_U91 P_id_length P_id_a__U52 P_id_U42 
              P_id_a__U12 P_id_U83 P_id_a__U94 P_id_a__U45 P_id_U31 
              P_id_a__isNat P_id_isNatIListKind P_id_a__U82 P_id_U46 
              P_id_cons P_id_U52 P_id_a__U85 P_id_a__U33 P_id_U21 
              P_id_a__isNatList P_id_U86 P_id_mark P_id_a__U51 P_id_U41 
              P_id_tt P_id_U82 P_id_a__U93 P_id_a__U44 P_id_isNat 
              P_id_a__U23 P_id_U94 P_id_U12 P_id_a__U81 P_id_U45 P_id_zeros 
              P_id_U71 P_id_a__U91 P_id_a__U42 P_id_isNatKind P_id_a__U22 
              P_id_U92 P_id_nil P_id_a__U61 P_id_U43 P_id_a__isNatIListKind 
              P_id_U84 P_id_s P_id_a__U46 P_id_U32 P_id_a__U31 P_id_U13 
              P_id_a__U83 P_id_isNatIList.
           
           Lemma measure_equation :
            forall t, 
             measure t = match t with
                           | (algebra.Alg.Term algebra.F.id_a__zeros nil) =>
                            P_id_a__zeros 
                           | (algebra.Alg.Term algebra.F.id_U51 (x12::
                              x11::nil)) =>
                            P_id_U51 (measure x12) (measure x11)
                           | (algebra.Alg.Term algebra.F.id_a__U84 (x13::
                              x12::x11::nil)) =>
                            P_id_a__U84 (measure x13) (measure x12) 
                             (measure x11)
                           | (algebra.Alg.Term algebra.F.id_a__U32 (x12::
                              x11::nil)) =>
                            P_id_a__U32 (measure x12) (measure x11)
                           | (algebra.Alg.Term algebra.F.id_isNatList 
                              (x11::nil)) =>
                            P_id_isNatList (measure x11)
                           | (algebra.Alg.Term algebra.F.id_a__U13 
                              (x11::nil)) =>
                            P_id_a__U13 (measure x11)
                           | (algebra.Alg.Term algebra.F.id_U85 (x12::
                              x11::nil)) =>
                            P_id_U85 (measure x12) (measure x11)
                           | (algebra.Alg.Term algebra.F.id_a__length 
                              (x11::nil)) =>
                            P_id_a__length (measure x11)
                           | (algebra.Alg.Term algebra.F.id_a__isNatIList 
                              (x11::nil)) =>
                            P_id_a__isNatIList (measure x11)
                           | (algebra.Alg.Term algebra.F.id_U33 (x11::nil)) =>
                            P_id_U33 (measure x11)
                           | (algebra.Alg.Term algebra.F.id_a__U11 (x12::
                              x11::nil)) =>
                            P_id_a__U11 (measure x12) (measure x11)
                           | (algebra.Alg.Term algebra.F.id_U81 (x13::x12::
                              x11::nil)) =>
                            P_id_U81 (measure x13) (measure x12) 
                             (measure x11)
                           | (algebra.Alg.Term algebra.F.id_a__U92 (x13::
                              x12::x11::nil)) =>
                            P_id_a__U92 (measure x13) (measure x12) 
                             (measure x11)
                           | (algebra.Alg.Term algebra.F.id_a__U43 (x13::
                              x12::x11::nil)) =>
                            P_id_a__U43 (measure x13) (measure x12) 
                             (measure x11)
                           | (algebra.Alg.Term algebra.F.id_U23 (x11::nil)) =>
                            P_id_U23 (measure x11)
                           | (algebra.Alg.Term algebra.F.id_a__isNatKind 
                              (x11::nil)) =>
                            P_id_a__isNatKind (measure x11)
                           | (algebra.Alg.Term algebra.F.id_U93 (x13::x12::
                              x11::nil)) =>
                            P_id_U93 (measure x13) (measure x12) 
                             (measure x11)
                           | (algebra.Alg.Term algebra.F.id_U11 (x12::
                              x11::nil)) =>
                            P_id_U11 (measure x12) (measure x11)
                           | (algebra.Alg.Term algebra.F.id_a__U71 
                              (x11::nil)) =>
                            P_id_a__U71 (measure x11)
                           | (algebra.Alg.Term algebra.F.id_U44 (x13::x12::
                              x11::nil)) =>
                            P_id_U44 (measure x13) (measure x12) 
                             (measure x11)
                           | (algebra.Alg.Term algebra.F.id_0 nil) => 
                           P_id_0 
                           | (algebra.Alg.Term algebra.F.id_U61 (x11::nil)) =>
                            P_id_U61 (measure x11)
                           | (algebra.Alg.Term algebra.F.id_a__U86 
                              (x11::nil)) =>
                            P_id_a__U86 (measure x11)
                           | (algebra.Alg.Term algebra.F.id_a__U41 (x13::
                              x12::x11::nil)) =>
                            P_id_a__U41 (measure x13) (measure x12) 
                             (measure x11)
                           | (algebra.Alg.Term algebra.F.id_U22 (x12::
                              x11::nil)) =>
                            P_id_U22 (measure x12) (measure x11)
                           | (algebra.Alg.Term algebra.F.id_a__U21 (x12::
                              x11::nil)) =>
                            P_id_a__U21 (measure x12) (measure x11)
                           | (algebra.Alg.Term algebra.F.id_U91 (x13::x12::
                              x11::nil)) =>
                            P_id_U91 (measure x13) (measure x12) 
                             (measure x11)
                           | (algebra.Alg.Term algebra.F.id_length 
                              (x11::nil)) =>
                            P_id_length (measure x11)
                           | (algebra.Alg.Term algebra.F.id_a__U52 
                              (x11::nil)) =>
                            P_id_a__U52 (measure x11)
                           | (algebra.Alg.Term algebra.F.id_U42 (x13::x12::
                              x11::nil)) =>
                            P_id_U42 (measure x13) (measure x12) 
                             (measure x11)
                           | (algebra.Alg.Term algebra.F.id_a__U12 (x12::
                              x11::nil)) =>
                            P_id_a__U12 (measure x12) (measure x11)
                           | (algebra.Alg.Term algebra.F.id_U83 (x13::x12::
                              x11::nil)) =>
                            P_id_U83 (measure x13) (measure x12) 
                             (measure x11)
                           | (algebra.Alg.Term algebra.F.id_a__U94 (x12::
                              x11::nil)) =>
                            P_id_a__U94 (measure x12) (measure x11)
                           | (algebra.Alg.Term algebra.F.id_a__U45 (x12::
                              x11::nil)) =>
                            P_id_a__U45 (measure x12) (measure x11)
                           | (algebra.Alg.Term algebra.F.id_U31 (x12::
                              x11::nil)) =>
                            P_id_U31 (measure x12) (measure x11)
                           | (algebra.Alg.Term algebra.F.id_a__isNat 
                              (x11::nil)) =>
                            P_id_a__isNat (measure x11)
                           | (algebra.Alg.Term algebra.F.id_isNatIListKind 
                              (x11::nil)) =>
                            P_id_isNatIListKind (measure x11)
                           | (algebra.Alg.Term algebra.F.id_a__U82 (x13::
                              x12::x11::nil)) =>
                            P_id_a__U82 (measure x13) (measure x12) 
                             (measure x11)
                           | (algebra.Alg.Term algebra.F.id_U46 (x11::nil)) =>
                            P_id_U46 (measure x11)
                           | (algebra.Alg.Term algebra.F.id_cons (x12::
                              x11::nil)) =>
                            P_id_cons (measure x12) (measure x11)
                           | (algebra.Alg.Term algebra.F.id_U52 (x11::nil)) =>
                            P_id_U52 (measure x11)
                           | (algebra.Alg.Term algebra.F.id_a__U85 (x12::
                              x11::nil)) =>
                            P_id_a__U85 (measure x12) (measure x11)
                           | (algebra.Alg.Term algebra.F.id_a__U33 
                              (x11::nil)) =>
                            P_id_a__U33 (measure x11)
                           | (algebra.Alg.Term algebra.F.id_U21 (x12::
                              x11::nil)) =>
                            P_id_U21 (measure x12) (measure x11)
                           | (algebra.Alg.Term algebra.F.id_a__isNatList 
                              (x11::nil)) =>
                            P_id_a__isNatList (measure x11)
                           | (algebra.Alg.Term algebra.F.id_U86 (x11::nil)) =>
                            P_id_U86 (measure x11)
                           | (algebra.Alg.Term algebra.F.id_mark (x11::nil)) =>
                            P_id_mark (measure x11)
                           | (algebra.Alg.Term algebra.F.id_a__U51 (x12::
                              x11::nil)) =>
                            P_id_a__U51 (measure x12) (measure x11)
                           | (algebra.Alg.Term algebra.F.id_U41 (x13::x12::
                              x11::nil)) =>
                            P_id_U41 (measure x13) (measure x12) 
                             (measure x11)
                           | (algebra.Alg.Term algebra.F.id_tt nil) =>
                            P_id_tt 
                           | (algebra.Alg.Term algebra.F.id_U82 (x13::x12::
                              x11::nil)) =>
                            P_id_U82 (measure x13) (measure x12) 
                             (measure x11)
                           | (algebra.Alg.Term algebra.F.id_a__U93 (x13::
                              x12::x11::nil)) =>
                            P_id_a__U93 (measure x13) (measure x12) 
                             (measure x11)
                           | (algebra.Alg.Term algebra.F.id_a__U44 (x13::
                              x12::x11::nil)) =>
                            P_id_a__U44 (measure x13) (measure x12) 
                             (measure x11)
                           | (algebra.Alg.Term algebra.F.id_isNat (x11::nil)) =>
                            P_id_isNat (measure x11)
                           | (algebra.Alg.Term algebra.F.id_a__U23 
                              (x11::nil)) =>
                            P_id_a__U23 (measure x11)
                           | (algebra.Alg.Term algebra.F.id_U94 (x12::
                              x11::nil)) =>
                            P_id_U94 (measure x12) (measure x11)
                           | (algebra.Alg.Term algebra.F.id_U12 (x12::
                              x11::nil)) =>
                            P_id_U12 (measure x12) (measure x11)
                           | (algebra.Alg.Term algebra.F.id_a__U81 (x13::
                              x12::x11::nil)) =>
                            P_id_a__U81 (measure x13) (measure x12) 
                             (measure x11)
                           | (algebra.Alg.Term algebra.F.id_U45 (x12::
                              x11::nil)) =>
                            P_id_U45 (measure x12) (measure x11)
                           | (algebra.Alg.Term algebra.F.id_zeros nil) =>
                            P_id_zeros 
                           | (algebra.Alg.Term algebra.F.id_U71 (x11::nil)) =>
                            P_id_U71 (measure x11)
                           | (algebra.Alg.Term algebra.F.id_a__U91 (x13::
                              x12::x11::nil)) =>
                            P_id_a__U91 (measure x13) (measure x12) 
                             (measure x11)
                           | (algebra.Alg.Term algebra.F.id_a__U42 (x13::
                              x12::x11::nil)) =>
                            P_id_a__U42 (measure x13) (measure x12) 
                             (measure x11)
                           | (algebra.Alg.Term algebra.F.id_isNatKind 
                              (x11::nil)) =>
                            P_id_isNatKind (measure x11)
                           | (algebra.Alg.Term algebra.F.id_a__U22 (x12::
                              x11::nil)) =>
                            P_id_a__U22 (measure x12) (measure x11)
                           | (algebra.Alg.Term algebra.F.id_U92 (x13::x12::
                              x11::nil)) =>
                            P_id_U92 (measure x13) (measure x12) 
                             (measure x11)
                           | (algebra.Alg.Term algebra.F.id_nil nil) =>
                            P_id_nil 
                           | (algebra.Alg.Term algebra.F.id_a__U61 
                              (x11::nil)) =>
                            P_id_a__U61 (measure x11)
                           | (algebra.Alg.Term algebra.F.id_U43 (x13::x12::
                              x11::nil)) =>
                            P_id_U43 (measure x13) (measure x12) 
                             (measure x11)
                           | (algebra.Alg.Term 
                              algebra.F.id_a__isNatIListKind (x11::nil)) =>
                            P_id_a__isNatIListKind (measure x11)
                           | (algebra.Alg.Term algebra.F.id_U84 (x13::x12::
                              x11::nil)) =>
                            P_id_U84 (measure x13) (measure x12) 
                             (measure x11)
                           | (algebra.Alg.Term algebra.F.id_s (x11::nil)) =>
                            P_id_s (measure x11)
                           | (algebra.Alg.Term algebra.F.id_a__U46 
                              (x11::nil)) =>
                            P_id_a__U46 (measure x11)
                           | (algebra.Alg.Term algebra.F.id_U32 (x12::
                              x11::nil)) =>
                            P_id_U32 (measure x12) (measure x11)
                           | (algebra.Alg.Term algebra.F.id_a__U31 (x12::
                              x11::nil)) =>
                            P_id_a__U31 (measure x12) (measure x11)
                           | (algebra.Alg.Term algebra.F.id_U13 (x11::nil)) =>
                            P_id_U13 (measure x11)
                           | (algebra.Alg.Term algebra.F.id_a__U83 (x13::
                              x12::x11::nil)) =>
                            P_id_a__U83 (measure x13) (measure x12) 
                             (measure x11)
                           | (algebra.Alg.Term algebra.F.id_isNatIList 
                              (x11::nil)) =>
                            P_id_isNatIList (measure x11)
                           | _ => 0
                           end.
           Proof.
             intros t;case t;intros ;apply refl_equal.
           Qed.
           
           Lemma measure_bounded : forall t, 0 <= measure t.
           Proof.
             unfold measure in |-*.
             
             apply InterpZ.measure_bounded;
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Ltac generate_pos_hyp  :=
            match goal with
              | H:context [measure ?x] |- _ =>
               let v := fresh "v" in 
                (let H := fresh "h" in 
                  (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
                    clearbody H;clearbody v))
              |  |- context [measure ?x] =>
               let v := fresh "v" in 
                (let H := fresh "h" in 
                  (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
                    clearbody H;clearbody v))
              end
            .
           
           Lemma rules_monotonic :
            forall l r, 
             (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) ->
              measure r <= measure l.
           Proof.
             intros l r H.
             fold measure in |-*.
             
             inversion H;clear H;subst;inversion H0;clear H0;subst;
              simpl algebra.EQT.T.apply_subst in |-*;
              repeat (
              match goal with
                |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
                 rewrite (measure_equation (algebra.Alg.Term f t))
                end
              );repeat (generate_pos_hyp );
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma measure_star_monotonic :
            forall l r, 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                        r l) ->measure r <= measure l.
           Proof.
             unfold measure in *.
             apply InterpZ.measure_star_monotonic.
             intros ;apply P_id_U51_monotonic;assumption.
             intros ;apply P_id_a__U84_monotonic;assumption.
             intros ;apply P_id_a__U32_monotonic;assumption.
             intros ;apply P_id_isNatList_monotonic;assumption.
             intros ;apply P_id_a__U13_monotonic;assumption.
             intros ;apply P_id_U85_monotonic;assumption.
             intros ;apply P_id_a__length_monotonic;assumption.
             intros ;apply P_id_a__isNatIList_monotonic;assumption.
             intros ;apply P_id_U33_monotonic;assumption.
             intros ;apply P_id_a__U11_monotonic;assumption.
             intros ;apply P_id_U81_monotonic;assumption.
             intros ;apply P_id_a__U92_monotonic;assumption.
             intros ;apply P_id_a__U43_monotonic;assumption.
             intros ;apply P_id_U23_monotonic;assumption.
             intros ;apply P_id_a__isNatKind_monotonic;assumption.
             intros ;apply P_id_U93_monotonic;assumption.
             intros ;apply P_id_U11_monotonic;assumption.
             intros ;apply P_id_a__U71_monotonic;assumption.
             intros ;apply P_id_U44_monotonic;assumption.
             intros ;apply P_id_U61_monotonic;assumption.
             intros ;apply P_id_a__U86_monotonic;assumption.
             intros ;apply P_id_a__U41_monotonic;assumption.
             intros ;apply P_id_U22_monotonic;assumption.
             intros ;apply P_id_a__U21_monotonic;assumption.
             intros ;apply P_id_U91_monotonic;assumption.
             intros ;apply P_id_length_monotonic;assumption.
             intros ;apply P_id_a__U52_monotonic;assumption.
             intros ;apply P_id_U42_monotonic;assumption.
             intros ;apply P_id_a__U12_monotonic;assumption.
             intros ;apply P_id_U83_monotonic;assumption.
             intros ;apply P_id_a__U94_monotonic;assumption.
             intros ;apply P_id_a__U45_monotonic;assumption.
             intros ;apply P_id_U31_monotonic;assumption.
             intros ;apply P_id_a__isNat_monotonic;assumption.
             intros ;apply P_id_isNatIListKind_monotonic;assumption.
             intros ;apply P_id_a__U82_monotonic;assumption.
             intros ;apply P_id_U46_monotonic;assumption.
             intros ;apply P_id_cons_monotonic;assumption.
             intros ;apply P_id_U52_monotonic;assumption.
             intros ;apply P_id_a__U85_monotonic;assumption.
             intros ;apply P_id_a__U33_monotonic;assumption.
             intros ;apply P_id_U21_monotonic;assumption.
             intros ;apply P_id_a__isNatList_monotonic;assumption.
             intros ;apply P_id_U86_monotonic;assumption.
             intros ;apply P_id_mark_monotonic;assumption.
             intros ;apply P_id_a__U51_monotonic;assumption.
             intros ;apply P_id_U41_monotonic;assumption.
             intros ;apply P_id_U82_monotonic;assumption.
             intros ;apply P_id_a__U93_monotonic;assumption.
             intros ;apply P_id_a__U44_monotonic;assumption.
             intros ;apply P_id_isNat_monotonic;assumption.
             intros ;apply P_id_a__U23_monotonic;assumption.
             intros ;apply P_id_U94_monotonic;assumption.
             intros ;apply P_id_U12_monotonic;assumption.
             intros ;apply P_id_a__U81_monotonic;assumption.
             intros ;apply P_id_U45_monotonic;assumption.
             intros ;apply P_id_U71_monotonic;assumption.
             intros ;apply P_id_a__U91_monotonic;assumption.
             intros ;apply P_id_a__U42_monotonic;assumption.
             intros ;apply P_id_isNatKind_monotonic;assumption.
             intros ;apply P_id_a__U22_monotonic;assumption.
             intros ;apply P_id_U92_monotonic;assumption.
             intros ;apply P_id_a__U61_monotonic;assumption.
             intros ;apply P_id_U43_monotonic;assumption.
             intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
             intros ;apply P_id_U84_monotonic;assumption.
             intros ;apply P_id_s_monotonic;assumption.
             intros ;apply P_id_a__U46_monotonic;assumption.
             intros ;apply P_id_U32_monotonic;assumption.
             intros ;apply P_id_a__U31_monotonic;assumption.
             intros ;apply P_id_U13_monotonic;assumption.
             intros ;apply P_id_a__U83_monotonic;assumption.
             intros ;apply P_id_isNatIList_monotonic;assumption.
             intros ;apply P_id_a__zeros_bounded;assumption.
             intros ;apply P_id_U51_bounded;assumption.
             intros ;apply P_id_a__U84_bounded;assumption.
             intros ;apply P_id_a__U32_bounded;assumption.
             intros ;apply P_id_isNatList_bounded;assumption.
             intros ;apply P_id_a__U13_bounded;assumption.
             intros ;apply P_id_U85_bounded;assumption.
             intros ;apply P_id_a__length_bounded;assumption.
             intros ;apply P_id_a__isNatIList_bounded;assumption.
             intros ;apply P_id_U33_bounded;assumption.
             intros ;apply P_id_a__U11_bounded;assumption.
             intros ;apply P_id_U81_bounded;assumption.
             intros ;apply P_id_a__U92_bounded;assumption.
             intros ;apply P_id_a__U43_bounded;assumption.
             intros ;apply P_id_U23_bounded;assumption.
             intros ;apply P_id_a__isNatKind_bounded;assumption.
             intros ;apply P_id_U93_bounded;assumption.
             intros ;apply P_id_U11_bounded;assumption.
             intros ;apply P_id_a__U71_bounded;assumption.
             intros ;apply P_id_U44_bounded;assumption.
             intros ;apply P_id_0_bounded;assumption.
             intros ;apply P_id_U61_bounded;assumption.
             intros ;apply P_id_a__U86_bounded;assumption.
             intros ;apply P_id_a__U41_bounded;assumption.
             intros ;apply P_id_U22_bounded;assumption.
             intros ;apply P_id_a__U21_bounded;assumption.
             intros ;apply P_id_U91_bounded;assumption.
             intros ;apply P_id_length_bounded;assumption.
             intros ;apply P_id_a__U52_bounded;assumption.
             intros ;apply P_id_U42_bounded;assumption.
             intros ;apply P_id_a__U12_bounded;assumption.
             intros ;apply P_id_U83_bounded;assumption.
             intros ;apply P_id_a__U94_bounded;assumption.
             intros ;apply P_id_a__U45_bounded;assumption.
             intros ;apply P_id_U31_bounded;assumption.
             intros ;apply P_id_a__isNat_bounded;assumption.
             intros ;apply P_id_isNatIListKind_bounded;assumption.
             intros ;apply P_id_a__U82_bounded;assumption.
             intros ;apply P_id_U46_bounded;assumption.
             intros ;apply P_id_cons_bounded;assumption.
             intros ;apply P_id_U52_bounded;assumption.
             intros ;apply P_id_a__U85_bounded;assumption.
             intros ;apply P_id_a__U33_bounded;assumption.
             intros ;apply P_id_U21_bounded;assumption.
             intros ;apply P_id_a__isNatList_bounded;assumption.
             intros ;apply P_id_U86_bounded;assumption.
             intros ;apply P_id_mark_bounded;assumption.
             intros ;apply P_id_a__U51_bounded;assumption.
             intros ;apply P_id_U41_bounded;assumption.
             intros ;apply P_id_tt_bounded;assumption.
             intros ;apply P_id_U82_bounded;assumption.
             intros ;apply P_id_a__U93_bounded;assumption.
             intros ;apply P_id_a__U44_bounded;assumption.
             intros ;apply P_id_isNat_bounded;assumption.
             intros ;apply P_id_a__U23_bounded;assumption.
             intros ;apply P_id_U94_bounded;assumption.
             intros ;apply P_id_U12_bounded;assumption.
             intros ;apply P_id_a__U81_bounded;assumption.
             intros ;apply P_id_U45_bounded;assumption.
             intros ;apply P_id_zeros_bounded;assumption.
             intros ;apply P_id_U71_bounded;assumption.
             intros ;apply P_id_a__U91_bounded;assumption.
             intros ;apply P_id_a__U42_bounded;assumption.
             intros ;apply P_id_isNatKind_bounded;assumption.
             intros ;apply P_id_a__U22_bounded;assumption.
             intros ;apply P_id_U92_bounded;assumption.
             intros ;apply P_id_nil_bounded;assumption.
             intros ;apply P_id_a__U61_bounded;assumption.
             intros ;apply P_id_U43_bounded;assumption.
             intros ;apply P_id_a__isNatIListKind_bounded;assumption.
             intros ;apply P_id_U84_bounded;assumption.
             intros ;apply P_id_s_bounded;assumption.
             intros ;apply P_id_a__U46_bounded;assumption.
             intros ;apply P_id_U32_bounded;assumption.
             intros ;apply P_id_a__U31_bounded;assumption.
             intros ;apply P_id_U13_bounded;assumption.
             intros ;apply P_id_a__U83_bounded;assumption.
             intros ;apply P_id_isNatIList_bounded;assumption.
             apply rules_monotonic.
           Qed.
           
           Definition P_id_A__U46 (x11:Z) := 0.
           
           Definition P_id_A__ISNATILISTKIND (x11:Z) := 0.
           
           Definition P_id_A__U61 (x11:Z) := 0.
           
           Definition P_id_A__U85 (x11:Z) (x12:Z) := 0.
           
           Definition P_id_A__U31 (x11:Z) (x12:Z) := 0.
           
           Definition P_id_A__U81 (x11:Z) (x12:Z) (x13:Z) := 0.
           
           Definition P_id_A__U22 (x11:Z) (x12:Z) := 0.
           
           Definition P_id_A__U93 (x11:Z) (x12:Z) (x13:Z) := 0.
           
           Definition P_id_A__U42 (x11:Z) (x12:Z) (x13:Z) := 0.
           
           Definition P_id_A__U51 (x11:Z) (x12:Z) := 0.
           
           Definition P_id_A__ISNATLIST (x11:Z) := 0.
           
           Definition P_id_A__ZEROS  := 0.
           
           Definition P_id_A__U91 (x11:Z) (x12:Z) (x13:Z) := 0.
           
           Definition P_id_A__U33 (x11:Z) := 0.
           
           Definition P_id_A__U83 (x11:Z) (x12:Z) (x13:Z) := 0.
           
           Definition P_id_A__U23 (x11:Z) := 0.
           
           Definition P_id_A__U11 (x11:Z) (x12:Z) := 0.
           
           Definition P_id_A__LENGTH (x11:Z) := 0.
           
           Definition P_id_A__U44 (x11:Z) (x12:Z) (x13:Z) := 0.
           
           Definition P_id_A__ISNATILIST (x11:Z) := 0.
           
           Definition P_id_A__U13 (x11:Z) := 0.
           
           Definition P_id_A__U71 (x11:Z) := 0.
           
           Definition P_id_A__U86 (x11:Z) := 0.
           
           Definition P_id_A__U32 (x11:Z) (x12:Z) := 0.
           
           Definition P_id_A__U82 (x11:Z) (x12:Z) (x13:Z) := 0.
           
           Definition P_id_A__ISNATKIND (x11:Z) := 0.
           
           Definition P_id_A__U94 (x11:Z) (x12:Z) := 0.
           
           Definition P_id_A__U43 (x11:Z) (x12:Z) (x13:Z) := 0.
           
           Definition P_id_A__U52 (x11:Z) := 0.
           
           Definition P_id_A__U21 (x11:Z) (x12:Z) := 0.
           
           Definition P_id_A__U92 (x11:Z) (x12:Z) (x13:Z) := 0.
           
           Definition P_id_A__U41 (x11:Z) (x12:Z) (x13:Z) := 0.
           
           Definition P_id_A__U84 (x11:Z) (x12:Z) (x13:Z) := 0.
           
           Definition P_id_A__ISNAT (x11:Z) := 0.
           
           Definition P_id_A__U12 (x11:Z) (x12:Z) := 0.
           
           Definition P_id_MARK (x11:Z) := 1* x11.
           
           Definition P_id_A__U45 (x11:Z) (x12:Z) := 0.
           
           Lemma P_id_A__U46_monotonic :
            forall x12 x11, 
             (0 <= x12)/\ (x12 <= x11) ->P_id_A__U46 x12 <= P_id_A__U46 x11.
           Proof.
             intros x12 x11.
             intros [H_1 H_0].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_A__ISNATILISTKIND_monotonic :
            forall x12 x11, 
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_A__ISNATILISTKIND x12 <= P_id_A__ISNATILISTKIND x11.
           Proof.
             intros x12 x11.
             intros [H_1 H_0].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_A__U61_monotonic :
            forall x12 x11, 
             (0 <= x12)/\ (x12 <= x11) ->P_id_A__U61 x12 <= P_id_A__U61 x11.
           Proof.
             intros x12 x11.
             intros [H_1 H_0].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_A__U85_monotonic :
            forall x12 x14 x13 x11, 
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_A__U85 x12 x14 <= P_id_A__U85 x11 x13.
           Proof.
             intros x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_A__U31_monotonic :
            forall x12 x14 x13 x11, 
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_A__U31 x12 x14 <= P_id_A__U31 x11 x13.
           Proof.
             intros x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_A__U81_monotonic :
            forall x16 x12 x14 x13 x11 x15, 
             (0 <= x16)/\ (x16 <= x15) ->
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_A__U81 x12 x14 x16 <= P_id_A__U81 x11 x13 x15.
           Proof.
             intros x16 x15 x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             intros [H_5 H_4].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_A__U22_monotonic :
            forall x12 x14 x13 x11, 
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_A__U22 x12 x14 <= P_id_A__U22 x11 x13.
           Proof.
             intros x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_A__U93_monotonic :
            forall x16 x12 x14 x13 x11 x15, 
             (0 <= x16)/\ (x16 <= x15) ->
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_A__U93 x12 x14 x16 <= P_id_A__U93 x11 x13 x15.
           Proof.
             intros x16 x15 x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             intros [H_5 H_4].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_A__U42_monotonic :
            forall x16 x12 x14 x13 x11 x15, 
             (0 <= x16)/\ (x16 <= x15) ->
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_A__U42 x12 x14 x16 <= P_id_A__U42 x11 x13 x15.
           Proof.
             intros x16 x15 x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             intros [H_5 H_4].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_A__U51_monotonic :
            forall x12 x14 x13 x11, 
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_A__U51 x12 x14 <= P_id_A__U51 x11 x13.
           Proof.
             intros x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_A__ISNATLIST_monotonic :
            forall x12 x11, 
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_A__ISNATLIST x12 <= P_id_A__ISNATLIST x11.
           Proof.
             intros x12 x11.
             intros [H_1 H_0].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_A__U91_monotonic :
            forall x16 x12 x14 x13 x11 x15, 
             (0 <= x16)/\ (x16 <= x15) ->
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_A__U91 x12 x14 x16 <= P_id_A__U91 x11 x13 x15.
           Proof.
             intros x16 x15 x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             intros [H_5 H_4].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_A__U33_monotonic :
            forall x12 x11, 
             (0 <= x12)/\ (x12 <= x11) ->P_id_A__U33 x12 <= P_id_A__U33 x11.
           Proof.
             intros x12 x11.
             intros [H_1 H_0].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_A__U83_monotonic :
            forall x16 x12 x14 x13 x11 x15, 
             (0 <= x16)/\ (x16 <= x15) ->
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_A__U83 x12 x14 x16 <= P_id_A__U83 x11 x13 x15.
           Proof.
             intros x16 x15 x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             intros [H_5 H_4].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_A__U23_monotonic :
            forall x12 x11, 
             (0 <= x12)/\ (x12 <= x11) ->P_id_A__U23 x12 <= P_id_A__U23 x11.
           Proof.
             intros x12 x11.
             intros [H_1 H_0].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_A__U11_monotonic :
            forall x12 x14 x13 x11, 
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_A__U11 x12 x14 <= P_id_A__U11 x11 x13.
           Proof.
             intros x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_A__LENGTH_monotonic :
            forall x12 x11, 
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_A__LENGTH x12 <= P_id_A__LENGTH x11.
           Proof.
             intros x12 x11.
             intros [H_1 H_0].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_A__U44_monotonic :
            forall x16 x12 x14 x13 x11 x15, 
             (0 <= x16)/\ (x16 <= x15) ->
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_A__U44 x12 x14 x16 <= P_id_A__U44 x11 x13 x15.
           Proof.
             intros x16 x15 x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             intros [H_5 H_4].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_A__ISNATILIST_monotonic :
            forall x12 x11, 
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_A__ISNATILIST x12 <= P_id_A__ISNATILIST x11.
           Proof.
             intros x12 x11.
             intros [H_1 H_0].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_A__U13_monotonic :
            forall x12 x11, 
             (0 <= x12)/\ (x12 <= x11) ->P_id_A__U13 x12 <= P_id_A__U13 x11.
           Proof.
             intros x12 x11.
             intros [H_1 H_0].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_A__U71_monotonic :
            forall x12 x11, 
             (0 <= x12)/\ (x12 <= x11) ->P_id_A__U71 x12 <= P_id_A__U71 x11.
           Proof.
             intros x12 x11.
             intros [H_1 H_0].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_A__U86_monotonic :
            forall x12 x11, 
             (0 <= x12)/\ (x12 <= x11) ->P_id_A__U86 x12 <= P_id_A__U86 x11.
           Proof.
             intros x12 x11.
             intros [H_1 H_0].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_A__U32_monotonic :
            forall x12 x14 x13 x11, 
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_A__U32 x12 x14 <= P_id_A__U32 x11 x13.
           Proof.
             intros x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_A__U82_monotonic :
            forall x16 x12 x14 x13 x11 x15, 
             (0 <= x16)/\ (x16 <= x15) ->
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_A__U82 x12 x14 x16 <= P_id_A__U82 x11 x13 x15.
           Proof.
             intros x16 x15 x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             intros [H_5 H_4].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_A__ISNATKIND_monotonic :
            forall x12 x11, 
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_A__ISNATKIND x12 <= P_id_A__ISNATKIND x11.
           Proof.
             intros x12 x11.
             intros [H_1 H_0].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_A__U94_monotonic :
            forall x12 x14 x13 x11, 
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_A__U94 x12 x14 <= P_id_A__U94 x11 x13.
           Proof.
             intros x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_A__U43_monotonic :
            forall x16 x12 x14 x13 x11 x15, 
             (0 <= x16)/\ (x16 <= x15) ->
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_A__U43 x12 x14 x16 <= P_id_A__U43 x11 x13 x15.
           Proof.
             intros x16 x15 x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             intros [H_5 H_4].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_A__U52_monotonic :
            forall x12 x11, 
             (0 <= x12)/\ (x12 <= x11) ->P_id_A__U52 x12 <= P_id_A__U52 x11.
           Proof.
             intros x12 x11.
             intros [H_1 H_0].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_A__U21_monotonic :
            forall x12 x14 x13 x11, 
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_A__U21 x12 x14 <= P_id_A__U21 x11 x13.
           Proof.
             intros x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_A__U92_monotonic :
            forall x16 x12 x14 x13 x11 x15, 
             (0 <= x16)/\ (x16 <= x15) ->
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_A__U92 x12 x14 x16 <= P_id_A__U92 x11 x13 x15.
           Proof.
             intros x16 x15 x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             intros [H_5 H_4].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_A__U41_monotonic :
            forall x16 x12 x14 x13 x11 x15, 
             (0 <= x16)/\ (x16 <= x15) ->
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_A__U41 x12 x14 x16 <= P_id_A__U41 x11 x13 x15.
           Proof.
             intros x16 x15 x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             intros [H_5 H_4].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_A__U84_monotonic :
            forall x16 x12 x14 x13 x11 x15, 
             (0 <= x16)/\ (x16 <= x15) ->
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_A__U84 x12 x14 x16 <= P_id_A__U84 x11 x13 x15.
           Proof.
             intros x16 x15 x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             intros [H_5 H_4].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_A__ISNAT_monotonic :
            forall x12 x11, 
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_A__ISNAT x12 <= P_id_A__ISNAT x11.
           Proof.
             intros x12 x11.
             intros [H_1 H_0].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_A__U12_monotonic :
            forall x12 x14 x13 x11, 
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_A__U12 x12 x14 <= P_id_A__U12 x11 x13.
           Proof.
             intros x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_MARK_monotonic :
            forall x12 x11, 
             (0 <= x12)/\ (x12 <= x11) ->P_id_MARK x12 <= P_id_MARK x11.
           Proof.
             intros x12 x11.
             intros [H_1 H_0].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma P_id_A__U45_monotonic :
            forall x12 x14 x13 x11, 
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_A__U45 x12 x14 <= P_id_A__U45 x11 x13.
           Proof.
             intros x14 x13 x12 x11.
             intros [H_1 H_0].
             intros [H_3 H_2].
             
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Definition marked_measure  := 
             InterpZ.marked_measure 0 P_id_a__zeros P_id_U51 P_id_a__U84 
              P_id_a__U32 P_id_isNatList P_id_a__U13 P_id_U85 P_id_a__length 
              P_id_a__isNatIList P_id_U33 P_id_a__U11 P_id_U81 P_id_a__U92 
              P_id_a__U43 P_id_U23 P_id_a__isNatKind P_id_U93 P_id_U11 
              P_id_a__U71 P_id_U44 P_id_0 P_id_U61 P_id_a__U86 P_id_a__U41 
              P_id_U22 P_id_a__U21 P_id_U91 P_id_length P_id_a__U52 P_id_U42 
              P_id_a__U12 P_id_U83 P_id_a__U94 P_id_a__U45 P_id_U31 
              P_id_a__isNat P_id_isNatIListKind P_id_a__U82 P_id_U46 
              P_id_cons P_id_U52 P_id_a__U85 P_id_a__U33 P_id_U21 
              P_id_a__isNatList P_id_U86 P_id_mark P_id_a__U51 P_id_U41 
              P_id_tt P_id_U82 P_id_a__U93 P_id_a__U44 P_id_isNat 
              P_id_a__U23 P_id_U94 P_id_U12 P_id_a__U81 P_id_U45 P_id_zeros 
              P_id_U71 P_id_a__U91 P_id_a__U42 P_id_isNatKind P_id_a__U22 
              P_id_U92 P_id_nil P_id_a__U61 P_id_U43 P_id_a__isNatIListKind 
              P_id_U84 P_id_s P_id_a__U46 P_id_U32 P_id_a__U31 P_id_U13 
              P_id_a__U83 P_id_isNatIList P_id_A__U46 P_id_A__ISNATILISTKIND 
              P_id_A__U61 P_id_A__U85 P_id_A__U31 P_id_A__U81 P_id_A__U22 
              P_id_A__U93 P_id_A__U42 P_id_A__U51 P_id_A__ISNATLIST 
              P_id_A__ZEROS P_id_A__U91 P_id_A__U33 P_id_A__U83 P_id_A__U23 
              P_id_A__U11 P_id_A__LENGTH P_id_A__U44 P_id_A__ISNATILIST 
              P_id_A__U13 P_id_A__U71 P_id_A__U86 P_id_A__U32 P_id_A__U82 
              P_id_A__ISNATKIND P_id_A__U94 P_id_A__U43 P_id_A__U52 
              P_id_A__U21 P_id_A__U92 P_id_A__U41 P_id_A__U84 P_id_A__ISNAT 
              P_id_A__U12 P_id_MARK P_id_A__U45.
           
           Lemma marked_measure_equation :
            forall t, 
             marked_measure t = match t with
                                  | (algebra.Alg.Term algebra.F.id_a__U46 
                                     (x11::nil)) =>
                                   P_id_A__U46 (measure x11)
                                  | (algebra.Alg.Term 
                                     algebra.F.id_a__isNatIListKind 
                                     (x11::nil)) =>
                                   P_id_A__ISNATILISTKIND (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U61 
                                     (x11::nil)) =>
                                   P_id_A__U61 (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U85 
                                     (x12::x11::nil)) =>
                                   P_id_A__U85 (measure x12) (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U31 
                                     (x12::x11::nil)) =>
                                   P_id_A__U31 (measure x12) (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U81 
                                     (x13::x12::x11::nil)) =>
                                   P_id_A__U81 (measure x13) (measure x12) 
                                    (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U22 
                                     (x12::x11::nil)) =>
                                   P_id_A__U22 (measure x12) (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U93 
                                     (x13::x12::x11::nil)) =>
                                   P_id_A__U93 (measure x13) (measure x12) 
                                    (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U42 
                                     (x13::x12::x11::nil)) =>
                                   P_id_A__U42 (measure x13) (measure x12) 
                                    (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U51 
                                     (x12::x11::nil)) =>
                                   P_id_A__U51 (measure x12) (measure x11)
                                  | (algebra.Alg.Term 
                                     algebra.F.id_a__isNatList (x11::nil)) =>
                                   P_id_A__ISNATLIST (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__zeros 
                                     nil) =>
                                   P_id_A__ZEROS 
                                  | (algebra.Alg.Term algebra.F.id_a__U91 
                                     (x13::x12::x11::nil)) =>
                                   P_id_A__U91 (measure x13) (measure x12) 
                                    (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U33 
                                     (x11::nil)) =>
                                   P_id_A__U33 (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U83 
                                     (x13::x12::x11::nil)) =>
                                   P_id_A__U83 (measure x13) (measure x12) 
                                    (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U23 
                                     (x11::nil)) =>
                                   P_id_A__U23 (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U11 
                                     (x12::x11::nil)) =>
                                   P_id_A__U11 (measure x12) (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__length 
                                     (x11::nil)) =>
                                   P_id_A__LENGTH (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U44 
                                     (x13::x12::x11::nil)) =>
                                   P_id_A__U44 (measure x13) (measure x12) 
                                    (measure x11)
                                  | (algebra.Alg.Term 
                                     algebra.F.id_a__isNatIList (x11::nil)) =>
                                   P_id_A__ISNATILIST (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U13 
                                     (x11::nil)) =>
                                   P_id_A__U13 (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U71 
                                     (x11::nil)) =>
                                   P_id_A__U71 (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U86 
                                     (x11::nil)) =>
                                   P_id_A__U86 (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U32 
                                     (x12::x11::nil)) =>
                                   P_id_A__U32 (measure x12) (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U82 
                                     (x13::x12::x11::nil)) =>
                                   P_id_A__U82 (measure x13) (measure x12) 
                                    (measure x11)
                                  | (algebra.Alg.Term 
                                     algebra.F.id_a__isNatKind (x11::nil)) =>
                                   P_id_A__ISNATKIND (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U94 
                                     (x12::x11::nil)) =>
                                   P_id_A__U94 (measure x12) (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U43 
                                     (x13::x12::x11::nil)) =>
                                   P_id_A__U43 (measure x13) (measure x12) 
                                    (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U52 
                                     (x11::nil)) =>
                                   P_id_A__U52 (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U21 
                                     (x12::x11::nil)) =>
                                   P_id_A__U21 (measure x12) (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U92 
                                     (x13::x12::x11::nil)) =>
                                   P_id_A__U92 (measure x13) (measure x12) 
                                    (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U41 
                                     (x13::x12::x11::nil)) =>
                                   P_id_A__U41 (measure x13) (measure x12) 
                                    (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U84 
                                     (x13::x12::x11::nil)) =>
                                   P_id_A__U84 (measure x13) (measure x12) 
                                    (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__isNat 
                                     (x11::nil)) =>
                                   P_id_A__ISNAT (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U12 
                                     (x12::x11::nil)) =>
                                   P_id_A__U12 (measure x12) (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_mark 
                                     (x11::nil)) =>
                                   P_id_MARK (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U45 
                                     (x12::x11::nil)) =>
                                   P_id_A__U45 (measure x12) (measure x11)
                                  | _ => measure t
                                  end.
           Proof.
             reflexivity .
           Qed.
           
           Lemma marked_measure_star_monotonic :
            forall f l1 l2, 
             (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step 
                                                               R_xml_0_deep_rew.R_xml_0_rules)
                                                              ) l1 l2) ->
              marked_measure (algebra.Alg.Term f l1) <= marked_measure 
                                                         (algebra.Alg.Term 
                                                           f l2).
           Proof.
             unfold marked_measure in *.
             apply InterpZ.marked_measure_star_monotonic.
             intros ;apply P_id_U51_monotonic;assumption.
             intros ;apply P_id_a__U84_monotonic;assumption.
             intros ;apply P_id_a__U32_monotonic;assumption.
             intros ;apply P_id_isNatList_monotonic;assumption.
             intros ;apply P_id_a__U13_monotonic;assumption.
             intros ;apply P_id_U85_monotonic;assumption.
             intros ;apply P_id_a__length_monotonic;assumption.
             intros ;apply P_id_a__isNatIList_monotonic;assumption.
             intros ;apply P_id_U33_monotonic;assumption.
             intros ;apply P_id_a__U11_monotonic;assumption.
             intros ;apply P_id_U81_monotonic;assumption.
             intros ;apply P_id_a__U92_monotonic;assumption.
             intros ;apply P_id_a__U43_monotonic;assumption.
             intros ;apply P_id_U23_monotonic;assumption.
             intros ;apply P_id_a__isNatKind_monotonic;assumption.
             intros ;apply P_id_U93_monotonic;assumption.
             intros ;apply P_id_U11_monotonic;assumption.
             intros ;apply P_id_a__U71_monotonic;assumption.
             intros ;apply P_id_U44_monotonic;assumption.
             intros ;apply P_id_U61_monotonic;assumption.
             intros ;apply P_id_a__U86_monotonic;assumption.
             intros ;apply P_id_a__U41_monotonic;assumption.
             intros ;apply P_id_U22_monotonic;assumption.
             intros ;apply P_id_a__U21_monotonic;assumption.
             intros ;apply P_id_U91_monotonic;assumption.
             intros ;apply P_id_length_monotonic;assumption.
             intros ;apply P_id_a__U52_monotonic;assumption.
             intros ;apply P_id_U42_monotonic;assumption.
             intros ;apply P_id_a__U12_monotonic;assumption.
             intros ;apply P_id_U83_monotonic;assumption.
             intros ;apply P_id_a__U94_monotonic;assumption.
             intros ;apply P_id_a__U45_monotonic;assumption.
             intros ;apply P_id_U31_monotonic;assumption.
             intros ;apply P_id_a__isNat_monotonic;assumption.
             intros ;apply P_id_isNatIListKind_monotonic;assumption.
             intros ;apply P_id_a__U82_monotonic;assumption.
             intros ;apply P_id_U46_monotonic;assumption.
             intros ;apply P_id_cons_monotonic;assumption.
             intros ;apply P_id_U52_monotonic;assumption.
             intros ;apply P_id_a__U85_monotonic;assumption.
             intros ;apply P_id_a__U33_monotonic;assumption.
             intros ;apply P_id_U21_monotonic;assumption.
             intros ;apply P_id_a__isNatList_monotonic;assumption.
             intros ;apply P_id_U86_monotonic;assumption.
             intros ;apply P_id_mark_monotonic;assumption.
             intros ;apply P_id_a__U51_monotonic;assumption.
             intros ;apply P_id_U41_monotonic;assumption.
             intros ;apply P_id_U82_monotonic;assumption.
             intros ;apply P_id_a__U93_monotonic;assumption.
             intros ;apply P_id_a__U44_monotonic;assumption.
             intros ;apply P_id_isNat_monotonic;assumption.
             intros ;apply P_id_a__U23_monotonic;assumption.
             intros ;apply P_id_U94_monotonic;assumption.
             intros ;apply P_id_U12_monotonic;assumption.
             intros ;apply P_id_a__U81_monotonic;assumption.
             intros ;apply P_id_U45_monotonic;assumption.
             intros ;apply P_id_U71_monotonic;assumption.
             intros ;apply P_id_a__U91_monotonic;assumption.
             intros ;apply P_id_a__U42_monotonic;assumption.
             intros ;apply P_id_isNatKind_monotonic;assumption.
             intros ;apply P_id_a__U22_monotonic;assumption.
             intros ;apply P_id_U92_monotonic;assumption.
             intros ;apply P_id_a__U61_monotonic;assumption.
             intros ;apply P_id_U43_monotonic;assumption.
             intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
             intros ;apply P_id_U84_monotonic;assumption.
             intros ;apply P_id_s_monotonic;assumption.
             intros ;apply P_id_a__U46_monotonic;assumption.
             intros ;apply P_id_U32_monotonic;assumption.
             intros ;apply P_id_a__U31_monotonic;assumption.
             intros ;apply P_id_U13_monotonic;assumption.
             intros ;apply P_id_a__U83_monotonic;assumption.
             intros ;apply P_id_isNatIList_monotonic;assumption.
             intros ;apply P_id_a__zeros_bounded;assumption.
             intros ;apply P_id_U51_bounded;assumption.
             intros ;apply P_id_a__U84_bounded;assumption.
             intros ;apply P_id_a__U32_bounded;assumption.
             intros ;apply P_id_isNatList_bounded;assumption.
             intros ;apply P_id_a__U13_bounded;assumption.
             intros ;apply P_id_U85_bounded;assumption.
             intros ;apply P_id_a__length_bounded;assumption.
             intros ;apply P_id_a__isNatIList_bounded;assumption.
             intros ;apply P_id_U33_bounded;assumption.
             intros ;apply P_id_a__U11_bounded;assumption.
             intros ;apply P_id_U81_bounded;assumption.
             intros ;apply P_id_a__U92_bounded;assumption.
             intros ;apply P_id_a__U43_bounded;assumption.
             intros ;apply P_id_U23_bounded;assumption.
             intros ;apply P_id_a__isNatKind_bounded;assumption.
             intros ;apply P_id_U93_bounded;assumption.
             intros ;apply P_id_U11_bounded;assumption.
             intros ;apply P_id_a__U71_bounded;assumption.
             intros ;apply P_id_U44_bounded;assumption.
             intros ;apply P_id_0_bounded;assumption.
             intros ;apply P_id_U61_bounded;assumption.
             intros ;apply P_id_a__U86_bounded;assumption.
             intros ;apply P_id_a__U41_bounded;assumption.
             intros ;apply P_id_U22_bounded;assumption.
             intros ;apply P_id_a__U21_bounded;assumption.
             intros ;apply P_id_U91_bounded;assumption.
             intros ;apply P_id_length_bounded;assumption.
             intros ;apply P_id_a__U52_bounded;assumption.
             intros ;apply P_id_U42_bounded;assumption.
             intros ;apply P_id_a__U12_bounded;assumption.
             intros ;apply P_id_U83_bounded;assumption.
             intros ;apply P_id_a__U94_bounded;assumption.
             intros ;apply P_id_a__U45_bounded;assumption.
             intros ;apply P_id_U31_bounded;assumption.
             intros ;apply P_id_a__isNat_bounded;assumption.
             intros ;apply P_id_isNatIListKind_bounded;assumption.
             intros ;apply P_id_a__U82_bounded;assumption.
             intros ;apply P_id_U46_bounded;assumption.
             intros ;apply P_id_cons_bounded;assumption.
             intros ;apply P_id_U52_bounded;assumption.
             intros ;apply P_id_a__U85_bounded;assumption.
             intros ;apply P_id_a__U33_bounded;assumption.
             intros ;apply P_id_U21_bounded;assumption.
             intros ;apply P_id_a__isNatList_bounded;assumption.
             intros ;apply P_id_U86_bounded;assumption.
             intros ;apply P_id_mark_bounded;assumption.
             intros ;apply P_id_a__U51_bounded;assumption.
             intros ;apply P_id_U41_bounded;assumption.
             intros ;apply P_id_tt_bounded;assumption.
             intros ;apply P_id_U82_bounded;assumption.
             intros ;apply P_id_a__U93_bounded;assumption.
             intros ;apply P_id_a__U44_bounded;assumption.
             intros ;apply P_id_isNat_bounded;assumption.
             intros ;apply P_id_a__U23_bounded;assumption.
             intros ;apply P_id_U94_bounded;assumption.
             intros ;apply P_id_U12_bounded;assumption.
             intros ;apply P_id_a__U81_bounded;assumption.
             intros ;apply P_id_U45_bounded;assumption.
             intros ;apply P_id_zeros_bounded;assumption.
             intros ;apply P_id_U71_bounded;assumption.
             intros ;apply P_id_a__U91_bounded;assumption.
             intros ;apply P_id_a__U42_bounded;assumption.
             intros ;apply P_id_isNatKind_bounded;assumption.
             intros ;apply P_id_a__U22_bounded;assumption.
             intros ;apply P_id_U92_bounded;assumption.
             intros ;apply P_id_nil_bounded;assumption.
             intros ;apply P_id_a__U61_bounded;assumption.
             intros ;apply P_id_U43_bounded;assumption.
             intros ;apply P_id_a__isNatIListKind_bounded;assumption.
             intros ;apply P_id_U84_bounded;assumption.
             intros ;apply P_id_s_bounded;assumption.
             intros ;apply P_id_a__U46_bounded;assumption.
             intros ;apply P_id_U32_bounded;assumption.
             intros ;apply P_id_a__U31_bounded;assumption.
             intros ;apply P_id_U13_bounded;assumption.
             intros ;apply P_id_a__U83_bounded;assumption.
             intros ;apply P_id_isNatIList_bounded;assumption.
             apply rules_monotonic.
             intros ;apply P_id_A__U46_monotonic;assumption.
             intros ;apply P_id_A__ISNATILISTKIND_monotonic;assumption.
             intros ;apply P_id_A__U61_monotonic;assumption.
             intros ;apply P_id_A__U85_monotonic;assumption.
             intros ;apply P_id_A__U31_monotonic;assumption.
             intros ;apply P_id_A__U81_monotonic;assumption.
             intros ;apply P_id_A__U22_monotonic;assumption.
             intros ;apply P_id_A__U93_monotonic;assumption.
             intros ;apply P_id_A__U42_monotonic;assumption.
             intros ;apply P_id_A__U51_monotonic;assumption.
             intros ;apply P_id_A__ISNATLIST_monotonic;assumption.
             intros ;apply P_id_A__U91_monotonic;assumption.
             intros ;apply P_id_A__U33_monotonic;assumption.
             intros ;apply P_id_A__U83_monotonic;assumption.
             intros ;apply P_id_A__U23_monotonic;assumption.
             intros ;apply P_id_A__U11_monotonic;assumption.
             intros ;apply P_id_A__LENGTH_monotonic;assumption.
             intros ;apply P_id_A__U44_monotonic;assumption.
             intros ;apply P_id_A__ISNATILIST_monotonic;assumption.
             intros ;apply P_id_A__U13_monotonic;assumption.
             intros ;apply P_id_A__U71_monotonic;assumption.
             intros ;apply P_id_A__U86_monotonic;assumption.
             intros ;apply P_id_A__U32_monotonic;assumption.
             intros ;apply P_id_A__U82_monotonic;assumption.
             intros ;apply P_id_A__ISNATKIND_monotonic;assumption.
             intros ;apply P_id_A__U94_monotonic;assumption.
             intros ;apply P_id_A__U43_monotonic;assumption.
             intros ;apply P_id_A__U52_monotonic;assumption.
             intros ;apply P_id_A__U21_monotonic;assumption.
             intros ;apply P_id_A__U92_monotonic;assumption.
             intros ;apply P_id_A__U41_monotonic;assumption.
             intros ;apply P_id_A__U84_monotonic;assumption.
             intros ;apply P_id_A__ISNAT_monotonic;assumption.
             intros ;apply P_id_A__U12_monotonic;assumption.
             intros ;apply P_id_MARK_monotonic;assumption.
             intros ;apply P_id_A__U45_monotonic;assumption.
           Qed.
           
           Ltac rewrite_and_unfold  :=
            do 2 (rewrite marked_measure_equation);
             repeat (
             match goal with
               |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
                rewrite (measure_equation (algebra.Alg.Term f t))
               | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ =>
                rewrite (measure_equation (algebra.Alg.Term f t)) in H|-
               end
             ).
           
           Definition lt a b := 
             (Zwf.Zwf 0) (marked_measure a) (marked_measure b).
           
           Definition le a b := marked_measure a <= marked_measure b.
           
           Lemma lt_le_compat : forall a b c, (lt a b) ->(le b c) ->lt a c.
           Proof.
             unfold lt, le in *.
             intros a b c.
             apply (interp.le_lt_compat_right (interp.o_Z 0)).
           Qed.
           
           Lemma wf_lt : well_founded lt.
           Proof.
             unfold lt in *.
             apply Inverse_Image.wf_inverse_image with  (B:=Z).
             apply Zwf.Zwf_well_founded.
           Qed.
           
           Lemma DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_strict_in_lt :
            Relation_Definitions.inclusion _ 
             DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_strict 
             lt.
           Proof.
             unfold Relation_Definitions.inclusion, lt in *.
             
             intros a b H;destruct H;
              match goal with
                |  |- (Zwf.Zwf 0) _ (marked_measure (algebra.Alg.Term ?f ?l)) =>
                 let l'' := algebra.Alg_ext.find_replacement l  in 
                  ((apply (interp.le_lt_compat_right (interp.o_Z 0)) with
                     (marked_measure (algebra.Alg.Term f l''));[idtac|
                    apply marked_measure_star_monotonic;
                     repeat (
                     apply algebra.EQT_ext.one_step_list_refl_trans_clos);
                     (assumption)||(constructor 1)]))
                end
              ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Lemma DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_in_le :
            Relation_Definitions.inclusion _ 
             DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large 
             le.
           Proof.
             unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *.
             
             intros a b H;destruct H;
              match goal with
                |  |- _ <= marked_measure (algebra.Alg.Term ?f ?l) =>
                 let l'' := algebra.Alg_ext.find_replacement l  in 
                  ((apply (interp.le_trans (interp.o_Z 0)) with
                     (marked_measure (algebra.Alg.Term f l''));[idtac|
                    apply marked_measure_star_monotonic;
                     repeat (
                     apply algebra.EQT_ext.one_step_list_refl_trans_clos);
                     (assumption)||(constructor 1)]))
                end
              ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
           Qed.
           
           Definition wf_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large  := 
             WF_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large.wf
             .
           
           
           Lemma wf :
            well_founded WF_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large.DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large
             .
           Proof.
             intros x.
             apply (well_founded_ind wf_lt).
             clear x.
             intros x.
             pattern x.
             
             apply (@Acc_ind _ 
                     DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large)
                    .
             clear x.
             intros x _ IHx IHx'.
             constructor.
             intros y H.
             
             destruct H;
              (apply IHx';
                apply DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_strict_in_lt;
                econstructor eassumption)||
              ((apply IHx;[econstructor eassumption|
                intros y' Hlt;apply IHx';apply lt_le_compat with (1:=Hlt) ;
                 apply DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large_in_le;
                 econstructor eassumption])).
             
             apply wf_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_large.
           Qed.
          End WF_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large.
          
          Open Scope Z_scope.
          
          Import ring_extention.
          
          Notation Local "a <= b" := (Zle a b).
          
          Notation Local "a < b" := (Zlt a b).
          
          Definition P_id_a__zeros  := 0.
          
          Definition P_id_U51 (x11:Z) (x12:Z) := 2* x11.
          
          Definition P_id_a__U84 (x11:Z) (x12:Z) (x13:Z) := 
            2* x11 + 1* x12 + 2* x13.
          
          Definition P_id_a__U32 (x11:Z) (x12:Z) := 0.
          
          Definition P_id_isNatList (x11:Z) := 2* x11.
          
          Definition P_id_a__U13 (x11:Z) := 1* x11.
          
          Definition P_id_U85 (x11:Z) (x12:Z) := 1* x11 + 2* x12.
          
          Definition P_id_a__length (x11:Z) := 1 + 2* x11.
          
          Definition P_id_a__isNatIList (x11:Z) := 3* x11.
          
          Definition P_id_U33 (x11:Z) := 0.
          
          Definition P_id_a__U11 (x11:Z) (x12:Z) := 1 + 2* x11 + 2* x12.
          
          Definition P_id_U81 (x11:Z) (x12:Z) (x13:Z) := 
            2* x11 + 1* x12 + 2* x13.
          
          Definition P_id_a__U92 (x11:Z) (x12:Z) (x13:Z) := 1 + 2* x12.
          
          Definition P_id_a__U43 (x11:Z) (x12:Z) (x13:Z) := 0.
          
          Definition P_id_U23 (x11:Z) := 1* x11.
          
          Definition P_id_a__isNatKind (x11:Z) := 0.
          
          Definition P_id_U93 (x11:Z) (x12:Z) (x13:Z) := 1 + 2* x12.
          
          Definition P_id_U11 (x11:Z) (x12:Z) := 1 + 2* x11 + 2* x12.
          
          Definition P_id_a__U71 (x11:Z) := 2* x11.
          
          Definition P_id_U44 (x11:Z) (x12:Z) (x13:Z) := 0.
          
          Definition P_id_0  := 0.
          
          Definition P_id_U61 (x11:Z) := 1* x11.
          
          Definition P_id_a__U86 (x11:Z) := 1* x11.
          
          Definition P_id_a__U41 (x11:Z) (x12:Z) (x13:Z) := 1* x13.
          
          Definition P_id_U22 (x11:Z) (x12:Z) := 2* x11 + 1* x12.
          
          Definition P_id_a__U21 (x11:Z) (x12:Z) := 2* x11 + 1* x12.
          
          Definition P_id_U91 (x11:Z) (x12:Z) (x13:Z) := 1 + 2* x12.
          
          Definition P_id_length (x11:Z) := 1 + 2* x11.
          
          Definition P_id_a__U52 (x11:Z) := 2* x11.
          
          Definition P_id_U42 (x11:Z) (x12:Z) (x13:Z) := 0.
          
          Definition P_id_a__U12 (x11:Z) (x12:Z) := 2* x11 + 2* x12.
          
          Definition P_id_U83 (x11:Z) (x12:Z) (x13:Z) := 
            2* x11 + 1* x12 + 2* x13.
          
          Definition P_id_a__U94 (x11:Z) (x12:Z) := 1 + 2* x12.
          
          Definition P_id_a__U45 (x11:Z) (x12:Z) := 0.
          
          Definition P_id_U31 (x11:Z) (x12:Z) := 0.
          
          Definition P_id_a__isNat (x11:Z) := 1* x11.
          
          Definition P_id_isNatIListKind (x11:Z) := 0.
          
          Definition P_id_a__U82 (x11:Z) (x12:Z) (x13:Z) := 
            2* x11 + 1* x12 + 2* x13.
          
          Definition P_id_U46 (x11:Z) := 0.
          
          Definition P_id_cons (x11:Z) (x12:Z) := 1* x11 + 1* x12.
          
          Definition P_id_U52 (x11:Z) := 2* x11.
          
          Definition P_id_a__U85 (x11:Z) (x12:Z) := 1* x11 + 2* x12.
          
          Definition P_id_a__U33 (x11:Z) := 0.
          
          Definition P_id_U21 (x11:Z) (x12:Z) := 2* x11 + 1* x12.
          
          Definition P_id_a__isNatList (x11:Z) := 2* x11.
          
          Definition P_id_U86 (x11:Z) := 1* x11.
          
          Definition P_id_mark (x11:Z) := 1* x11.
          
          Definition P_id_a__U51 (x11:Z) (x12:Z) := 2* x11.
          
          Definition P_id_U41 (x11:Z) (x12:Z) (x13:Z) := 1* x13.
          
          Definition P_id_tt  := 0.
          
          Definition P_id_U82 (x11:Z) (x12:Z) (x13:Z) := 
            2* x11 + 1* x12 + 2* x13.
          
          Definition P_id_a__U93 (x11:Z) (x12:Z) (x13:Z) := 1 + 2* x12.
          
          Definition P_id_a__U44 (x11:Z) (x12:Z) (x13:Z) := 0.
          
          Definition P_id_isNat (x11:Z) := 1* x11.
          
          Definition P_id_a__U23 (x11:Z) := 1* x11.
          
          Definition P_id_U94 (x11:Z) (x12:Z) := 1 + 2* x12.
          
          Definition P_id_U12 (x11:Z) (x12:Z) := 2* x11 + 2* x12.
          
          Definition P_id_a__U81 (x11:Z) (x12:Z) (x13:Z) := 
            2* x11 + 1* x12 + 2* x13.
          
          Definition P_id_U45 (x11:Z) (x12:Z) := 0.
          
          Definition P_id_zeros  := 0.
          
          Definition P_id_U71 (x11:Z) := 2* x11.
          
          Definition P_id_a__U91 (x11:Z) (x12:Z) (x13:Z) := 1 + 2* x12.
          
          Definition P_id_a__U42 (x11:Z) (x12:Z) (x13:Z) := 0.
          
          Definition P_id_isNatKind (x11:Z) := 0.
          
          Definition P_id_a__U22 (x11:Z) (x12:Z) := 2* x11 + 1* x12.
          
          Definition P_id_U92 (x11:Z) (x12:Z) (x13:Z) := 1 + 2* x12.
          
          Definition P_id_nil  := 0.
          
          Definition P_id_a__U61 (x11:Z) := 1* x11.
          
          Definition P_id_U43 (x11:Z) (x12:Z) (x13:Z) := 0.
          
          Definition P_id_a__isNatIListKind (x11:Z) := 0.
          
          Definition P_id_U84 (x11:Z) (x12:Z) (x13:Z) := 
            2* x11 + 1* x12 + 2* x13.
          
          Definition P_id_s (x11:Z) := 1* x11.
          
          Definition P_id_a__U46 (x11:Z) := 0.
          
          Definition P_id_U32 (x11:Z) (x12:Z) := 0.
          
          Definition P_id_a__U31 (x11:Z) (x12:Z) := 0.
          
          Definition P_id_U13 (x11:Z) := 1* x11.
          
          Definition P_id_a__U83 (x11:Z) (x12:Z) (x13:Z) := 
            2* x11 + 1* x12 + 2* x13.
          
          Definition P_id_isNatIList (x11:Z) := 3* x11.
          
          Lemma P_id_U51_monotonic :
           forall x12 x14 x13 x11, 
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->P_id_U51 x12 x14 <= P_id_U51 x11 x13.
          Proof.
            intros x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U84_monotonic :
           forall x16 x12 x14 x13 x11 x15, 
            (0 <= x16)/\ (x16 <= x15) ->
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_a__U84 x12 x14 x16 <= P_id_a__U84 x11 x13 x15.
          Proof.
            intros x16 x15 x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            intros [H_5 H_4].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U32_monotonic :
           forall x12 x14 x13 x11, 
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_a__U32 x12 x14 <= P_id_a__U32 x11 x13.
          Proof.
            intros x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_isNatList_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_isNatList x12 <= P_id_isNatList x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_0].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U13_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->P_id_a__U13 x12 <= P_id_a__U13 x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_0].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U85_monotonic :
           forall x12 x14 x13 x11, 
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->P_id_U85 x12 x14 <= P_id_U85 x11 x13.
          Proof.
            intros x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__length_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_a__length x12 <= P_id_a__length x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_0].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__isNatIList_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_a__isNatIList x12 <= P_id_a__isNatIList x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_0].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U33_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->P_id_U33 x12 <= P_id_U33 x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_0].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U11_monotonic :
           forall x12 x14 x13 x11, 
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_a__U11 x12 x14 <= P_id_a__U11 x11 x13.
          Proof.
            intros x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U81_monotonic :
           forall x16 x12 x14 x13 x11 x15, 
            (0 <= x16)/\ (x16 <= x15) ->
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_U81 x12 x14 x16 <= P_id_U81 x11 x13 x15.
          Proof.
            intros x16 x15 x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            intros [H_5 H_4].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U92_monotonic :
           forall x16 x12 x14 x13 x11 x15, 
            (0 <= x16)/\ (x16 <= x15) ->
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_a__U92 x12 x14 x16 <= P_id_a__U92 x11 x13 x15.
          Proof.
            intros x16 x15 x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            intros [H_5 H_4].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U43_monotonic :
           forall x16 x12 x14 x13 x11 x15, 
            (0 <= x16)/\ (x16 <= x15) ->
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_a__U43 x12 x14 x16 <= P_id_a__U43 x11 x13 x15.
          Proof.
            intros x16 x15 x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            intros [H_5 H_4].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U23_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->P_id_U23 x12 <= P_id_U23 x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_0].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__isNatKind_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_a__isNatKind x12 <= P_id_a__isNatKind x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_0].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U93_monotonic :
           forall x16 x12 x14 x13 x11 x15, 
            (0 <= x16)/\ (x16 <= x15) ->
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_U93 x12 x14 x16 <= P_id_U93 x11 x13 x15.
          Proof.
            intros x16 x15 x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            intros [H_5 H_4].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U11_monotonic :
           forall x12 x14 x13 x11, 
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->P_id_U11 x12 x14 <= P_id_U11 x11 x13.
          Proof.
            intros x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U71_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->P_id_a__U71 x12 <= P_id_a__U71 x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_0].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U44_monotonic :
           forall x16 x12 x14 x13 x11 x15, 
            (0 <= x16)/\ (x16 <= x15) ->
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_U44 x12 x14 x16 <= P_id_U44 x11 x13 x15.
          Proof.
            intros x16 x15 x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            intros [H_5 H_4].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U61_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->P_id_U61 x12 <= P_id_U61 x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_0].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U86_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->P_id_a__U86 x12 <= P_id_a__U86 x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_0].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U41_monotonic :
           forall x16 x12 x14 x13 x11 x15, 
            (0 <= x16)/\ (x16 <= x15) ->
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_a__U41 x12 x14 x16 <= P_id_a__U41 x11 x13 x15.
          Proof.
            intros x16 x15 x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            intros [H_5 H_4].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U22_monotonic :
           forall x12 x14 x13 x11, 
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->P_id_U22 x12 x14 <= P_id_U22 x11 x13.
          Proof.
            intros x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U21_monotonic :
           forall x12 x14 x13 x11, 
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_a__U21 x12 x14 <= P_id_a__U21 x11 x13.
          Proof.
            intros x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U91_monotonic :
           forall x16 x12 x14 x13 x11 x15, 
            (0 <= x16)/\ (x16 <= x15) ->
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_U91 x12 x14 x16 <= P_id_U91 x11 x13 x15.
          Proof.
            intros x16 x15 x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            intros [H_5 H_4].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_length_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->P_id_length x12 <= P_id_length x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_0].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U52_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->P_id_a__U52 x12 <= P_id_a__U52 x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_0].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U42_monotonic :
           forall x16 x12 x14 x13 x11 x15, 
            (0 <= x16)/\ (x16 <= x15) ->
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_U42 x12 x14 x16 <= P_id_U42 x11 x13 x15.
          Proof.
            intros x16 x15 x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            intros [H_5 H_4].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U12_monotonic :
           forall x12 x14 x13 x11, 
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_a__U12 x12 x14 <= P_id_a__U12 x11 x13.
          Proof.
            intros x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U83_monotonic :
           forall x16 x12 x14 x13 x11 x15, 
            (0 <= x16)/\ (x16 <= x15) ->
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_U83 x12 x14 x16 <= P_id_U83 x11 x13 x15.
          Proof.
            intros x16 x15 x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            intros [H_5 H_4].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U94_monotonic :
           forall x12 x14 x13 x11, 
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_a__U94 x12 x14 <= P_id_a__U94 x11 x13.
          Proof.
            intros x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U45_monotonic :
           forall x12 x14 x13 x11, 
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_a__U45 x12 x14 <= P_id_a__U45 x11 x13.
          Proof.
            intros x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U31_monotonic :
           forall x12 x14 x13 x11, 
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->P_id_U31 x12 x14 <= P_id_U31 x11 x13.
          Proof.
            intros x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__isNat_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_a__isNat x12 <= P_id_a__isNat x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_0].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_isNatIListKind_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_isNatIListKind x12 <= P_id_isNatIListKind x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_0].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U82_monotonic :
           forall x16 x12 x14 x13 x11 x15, 
            (0 <= x16)/\ (x16 <= x15) ->
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_a__U82 x12 x14 x16 <= P_id_a__U82 x11 x13 x15.
          Proof.
            intros x16 x15 x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            intros [H_5 H_4].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U46_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->P_id_U46 x12 <= P_id_U46 x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_0].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_cons_monotonic :
           forall x12 x14 x13 x11, 
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_cons x12 x14 <= P_id_cons x11 x13.
          Proof.
            intros x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U52_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->P_id_U52 x12 <= P_id_U52 x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_0].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U85_monotonic :
           forall x12 x14 x13 x11, 
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_a__U85 x12 x14 <= P_id_a__U85 x11 x13.
          Proof.
            intros x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U33_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->P_id_a__U33 x12 <= P_id_a__U33 x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_0].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U21_monotonic :
           forall x12 x14 x13 x11, 
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->P_id_U21 x12 x14 <= P_id_U21 x11 x13.
          Proof.
            intros x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__isNatList_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_a__isNatList x12 <= P_id_a__isNatList x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_0].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U86_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->P_id_U86 x12 <= P_id_U86 x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_0].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_mark_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->P_id_mark x12 <= P_id_mark x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_0].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U51_monotonic :
           forall x12 x14 x13 x11, 
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_a__U51 x12 x14 <= P_id_a__U51 x11 x13.
          Proof.
            intros x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U41_monotonic :
           forall x16 x12 x14 x13 x11 x15, 
            (0 <= x16)/\ (x16 <= x15) ->
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_U41 x12 x14 x16 <= P_id_U41 x11 x13 x15.
          Proof.
            intros x16 x15 x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            intros [H_5 H_4].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U82_monotonic :
           forall x16 x12 x14 x13 x11 x15, 
            (0 <= x16)/\ (x16 <= x15) ->
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_U82 x12 x14 x16 <= P_id_U82 x11 x13 x15.
          Proof.
            intros x16 x15 x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            intros [H_5 H_4].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U93_monotonic :
           forall x16 x12 x14 x13 x11 x15, 
            (0 <= x16)/\ (x16 <= x15) ->
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_a__U93 x12 x14 x16 <= P_id_a__U93 x11 x13 x15.
          Proof.
            intros x16 x15 x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            intros [H_5 H_4].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U44_monotonic :
           forall x16 x12 x14 x13 x11 x15, 
            (0 <= x16)/\ (x16 <= x15) ->
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_a__U44 x12 x14 x16 <= P_id_a__U44 x11 x13 x15.
          Proof.
            intros x16 x15 x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            intros [H_5 H_4].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_isNat_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->P_id_isNat x12 <= P_id_isNat x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_0].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U23_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->P_id_a__U23 x12 <= P_id_a__U23 x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_0].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U94_monotonic :
           forall x12 x14 x13 x11, 
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->P_id_U94 x12 x14 <= P_id_U94 x11 x13.
          Proof.
            intros x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U12_monotonic :
           forall x12 x14 x13 x11, 
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->P_id_U12 x12 x14 <= P_id_U12 x11 x13.
          Proof.
            intros x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U81_monotonic :
           forall x16 x12 x14 x13 x11 x15, 
            (0 <= x16)/\ (x16 <= x15) ->
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_a__U81 x12 x14 x16 <= P_id_a__U81 x11 x13 x15.
          Proof.
            intros x16 x15 x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            intros [H_5 H_4].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U45_monotonic :
           forall x12 x14 x13 x11, 
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->P_id_U45 x12 x14 <= P_id_U45 x11 x13.
          Proof.
            intros x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U71_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->P_id_U71 x12 <= P_id_U71 x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_0].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U91_monotonic :
           forall x16 x12 x14 x13 x11 x15, 
            (0 <= x16)/\ (x16 <= x15) ->
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_a__U91 x12 x14 x16 <= P_id_a__U91 x11 x13 x15.
          Proof.
            intros x16 x15 x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            intros [H_5 H_4].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U42_monotonic :
           forall x16 x12 x14 x13 x11 x15, 
            (0 <= x16)/\ (x16 <= x15) ->
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_a__U42 x12 x14 x16 <= P_id_a__U42 x11 x13 x15.
          Proof.
            intros x16 x15 x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            intros [H_5 H_4].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_isNatKind_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_isNatKind x12 <= P_id_isNatKind x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_0].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U22_monotonic :
           forall x12 x14 x13 x11, 
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_a__U22 x12 x14 <= P_id_a__U22 x11 x13.
          Proof.
            intros x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U92_monotonic :
           forall x16 x12 x14 x13 x11 x15, 
            (0 <= x16)/\ (x16 <= x15) ->
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_U92 x12 x14 x16 <= P_id_U92 x11 x13 x15.
          Proof.
            intros x16 x15 x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            intros [H_5 H_4].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U61_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->P_id_a__U61 x12 <= P_id_a__U61 x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_0].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U43_monotonic :
           forall x16 x12 x14 x13 x11 x15, 
            (0 <= x16)/\ (x16 <= x15) ->
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_U43 x12 x14 x16 <= P_id_U43 x11 x13 x15.
          Proof.
            intros x16 x15 x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            intros [H_5 H_4].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__isNatIListKind_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_a__isNatIListKind x12 <= P_id_a__isNatIListKind x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_0].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U84_monotonic :
           forall x16 x12 x14 x13 x11 x15, 
            (0 <= x16)/\ (x16 <= x15) ->
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_U84 x12 x14 x16 <= P_id_U84 x11 x13 x15.
          Proof.
            intros x16 x15 x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            intros [H_5 H_4].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_s_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->P_id_s x12 <= P_id_s x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_0].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U46_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->P_id_a__U46 x12 <= P_id_a__U46 x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_0].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U32_monotonic :
           forall x12 x14 x13 x11, 
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->P_id_U32 x12 x14 <= P_id_U32 x11 x13.
          Proof.
            intros x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U31_monotonic :
           forall x12 x14 x13 x11, 
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_a__U31 x12 x14 <= P_id_a__U31 x11 x13.
          Proof.
            intros x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U13_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->P_id_U13 x12 <= P_id_U13 x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_0].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U83_monotonic :
           forall x16 x12 x14 x13 x11 x15, 
            (0 <= x16)/\ (x16 <= x15) ->
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_a__U83 x12 x14 x16 <= P_id_a__U83 x11 x13 x15.
          Proof.
            intros x16 x15 x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            intros [H_5 H_4].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_isNatIList_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_isNatIList x12 <= P_id_isNatIList x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_0].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__zeros_bounded : 0 <= P_id_a__zeros .
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U51_bounded :
           forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U51 x12 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U84_bounded :
           forall x12 x13 x11, 
            (0 <= x11) ->
             (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U84 x13 x12 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U32_bounded :
           forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U32 x12 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_isNatList_bounded :
           forall x11, (0 <= x11) ->0 <= P_id_isNatList x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U13_bounded :
           forall x11, (0 <= x11) ->0 <= P_id_a__U13 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U85_bounded :
           forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U85 x12 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__length_bounded :
           forall x11, (0 <= x11) ->0 <= P_id_a__length x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__isNatIList_bounded :
           forall x11, (0 <= x11) ->0 <= P_id_a__isNatIList x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U33_bounded :
           forall x11, (0 <= x11) ->0 <= P_id_U33 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U11_bounded :
           forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U11 x12 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U81_bounded :
           forall x12 x13 x11, 
            (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U81 x13 x12 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U92_bounded :
           forall x12 x13 x11, 
            (0 <= x11) ->
             (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U92 x13 x12 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U43_bounded :
           forall x12 x13 x11, 
            (0 <= x11) ->
             (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U43 x13 x12 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U23_bounded :
           forall x11, (0 <= x11) ->0 <= P_id_U23 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__isNatKind_bounded :
           forall x11, (0 <= x11) ->0 <= P_id_a__isNatKind x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U93_bounded :
           forall x12 x13 x11, 
            (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U93 x13 x12 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U11_bounded :
           forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U11 x12 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U71_bounded :
           forall x11, (0 <= x11) ->0 <= P_id_a__U71 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U44_bounded :
           forall x12 x13 x11, 
            (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U44 x13 x12 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_0_bounded : 0 <= P_id_0 .
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U61_bounded :
           forall x11, (0 <= x11) ->0 <= P_id_U61 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U86_bounded :
           forall x11, (0 <= x11) ->0 <= P_id_a__U86 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U41_bounded :
           forall x12 x13 x11, 
            (0 <= x11) ->
             (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U41 x13 x12 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U22_bounded :
           forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U22 x12 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U21_bounded :
           forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U21 x12 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U91_bounded :
           forall x12 x13 x11, 
            (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U91 x13 x12 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_length_bounded :
           forall x11, (0 <= x11) ->0 <= P_id_length x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U52_bounded :
           forall x11, (0 <= x11) ->0 <= P_id_a__U52 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U42_bounded :
           forall x12 x13 x11, 
            (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U42 x13 x12 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U12_bounded :
           forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U12 x12 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U83_bounded :
           forall x12 x13 x11, 
            (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U83 x13 x12 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U94_bounded :
           forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U94 x12 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U45_bounded :
           forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U45 x12 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U31_bounded :
           forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U31 x12 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__isNat_bounded :
           forall x11, (0 <= x11) ->0 <= P_id_a__isNat x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_isNatIListKind_bounded :
           forall x11, (0 <= x11) ->0 <= P_id_isNatIListKind x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U82_bounded :
           forall x12 x13 x11, 
            (0 <= x11) ->
             (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U82 x13 x12 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U46_bounded :
           forall x11, (0 <= x11) ->0 <= P_id_U46 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_cons_bounded :
           forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_cons x12 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U52_bounded :
           forall x11, (0 <= x11) ->0 <= P_id_U52 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U85_bounded :
           forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U85 x12 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U33_bounded :
           forall x11, (0 <= x11) ->0 <= P_id_a__U33 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U21_bounded :
           forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U21 x12 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__isNatList_bounded :
           forall x11, (0 <= x11) ->0 <= P_id_a__isNatList x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U86_bounded :
           forall x11, (0 <= x11) ->0 <= P_id_U86 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_mark_bounded :
           forall x11, (0 <= x11) ->0 <= P_id_mark x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U51_bounded :
           forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U51 x12 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U41_bounded :
           forall x12 x13 x11, 
            (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U41 x13 x12 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_tt_bounded : 0 <= P_id_tt .
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U82_bounded :
           forall x12 x13 x11, 
            (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U82 x13 x12 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U93_bounded :
           forall x12 x13 x11, 
            (0 <= x11) ->
             (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U93 x13 x12 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U44_bounded :
           forall x12 x13 x11, 
            (0 <= x11) ->
             (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U44 x13 x12 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_isNat_bounded :
           forall x11, (0 <= x11) ->0 <= P_id_isNat x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U23_bounded :
           forall x11, (0 <= x11) ->0 <= P_id_a__U23 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U94_bounded :
           forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U94 x12 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U12_bounded :
           forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U12 x12 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U81_bounded :
           forall x12 x13 x11, 
            (0 <= x11) ->
             (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U81 x13 x12 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U45_bounded :
           forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U45 x12 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_zeros_bounded : 0 <= P_id_zeros .
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U71_bounded :
           forall x11, (0 <= x11) ->0 <= P_id_U71 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U91_bounded :
           forall x12 x13 x11, 
            (0 <= x11) ->
             (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U91 x13 x12 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U42_bounded :
           forall x12 x13 x11, 
            (0 <= x11) ->
             (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U42 x13 x12 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_isNatKind_bounded :
           forall x11, (0 <= x11) ->0 <= P_id_isNatKind x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U22_bounded :
           forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U22 x12 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U92_bounded :
           forall x12 x13 x11, 
            (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U92 x13 x12 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_nil_bounded : 0 <= P_id_nil .
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U61_bounded :
           forall x11, (0 <= x11) ->0 <= P_id_a__U61 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U43_bounded :
           forall x12 x13 x11, 
            (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U43 x13 x12 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__isNatIListKind_bounded :
           forall x11, (0 <= x11) ->0 <= P_id_a__isNatIListKind x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U84_bounded :
           forall x12 x13 x11, 
            (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U84 x13 x12 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_s_bounded : forall x11, (0 <= x11) ->0 <= P_id_s x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U46_bounded :
           forall x11, (0 <= x11) ->0 <= P_id_a__U46 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U32_bounded :
           forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U32 x12 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U31_bounded :
           forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U31 x12 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U13_bounded :
           forall x11, (0 <= x11) ->0 <= P_id_U13 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_a__U83_bounded :
           forall x12 x13 x11, 
            (0 <= x11) ->
             (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U83 x13 x12 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_isNatIList_bounded :
           forall x11, (0 <= x11) ->0 <= P_id_isNatIList x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Definition measure  := 
            InterpZ.measure 0 P_id_a__zeros P_id_U51 P_id_a__U84 P_id_a__U32 
             P_id_isNatList P_id_a__U13 P_id_U85 P_id_a__length 
             P_id_a__isNatIList P_id_U33 P_id_a__U11 P_id_U81 P_id_a__U92 
             P_id_a__U43 P_id_U23 P_id_a__isNatKind P_id_U93 P_id_U11 
             P_id_a__U71 P_id_U44 P_id_0 P_id_U61 P_id_a__U86 P_id_a__U41 
             P_id_U22 P_id_a__U21 P_id_U91 P_id_length P_id_a__U52 P_id_U42 
             P_id_a__U12 P_id_U83 P_id_a__U94 P_id_a__U45 P_id_U31 
             P_id_a__isNat P_id_isNatIListKind P_id_a__U82 P_id_U46 
             P_id_cons P_id_U52 P_id_a__U85 P_id_a__U33 P_id_U21 
             P_id_a__isNatList P_id_U86 P_id_mark P_id_a__U51 P_id_U41 
             P_id_tt P_id_U82 P_id_a__U93 P_id_a__U44 P_id_isNat P_id_a__U23 
             P_id_U94 P_id_U12 P_id_a__U81 P_id_U45 P_id_zeros P_id_U71 
             P_id_a__U91 P_id_a__U42 P_id_isNatKind P_id_a__U22 P_id_U92 
             P_id_nil P_id_a__U61 P_id_U43 P_id_a__isNatIListKind P_id_U84 
             P_id_s P_id_a__U46 P_id_U32 P_id_a__U31 P_id_U13 P_id_a__U83 
             P_id_isNatIList.
          
          Lemma measure_equation :
           forall t, 
            measure t = match t with
                          | (algebra.Alg.Term algebra.F.id_a__zeros nil) =>
                           P_id_a__zeros 
                          | (algebra.Alg.Term algebra.F.id_U51 (x12::
                             x11::nil)) =>
                           P_id_U51 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U84 (x13::x12::
                             x11::nil)) =>
                           P_id_a__U84 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U32 (x12::
                             x11::nil)) =>
                           P_id_a__U32 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_isNatList 
                             (x11::nil)) =>
                           P_id_isNatList (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U13 (x11::nil)) =>
                           P_id_a__U13 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_U85 (x12::
                             x11::nil)) =>
                           P_id_U85 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__length 
                             (x11::nil)) =>
                           P_id_a__length (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__isNatIList 
                             (x11::nil)) =>
                           P_id_a__isNatIList (measure x11)
                          | (algebra.Alg.Term algebra.F.id_U33 (x11::nil)) =>
                           P_id_U33 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U11 (x12::
                             x11::nil)) =>
                           P_id_a__U11 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_U81 (x13::x12::
                             x11::nil)) =>
                           P_id_U81 (measure x13) (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U92 (x13::x12::
                             x11::nil)) =>
                           P_id_a__U92 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U43 (x13::x12::
                             x11::nil)) =>
                           P_id_a__U43 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_U23 (x11::nil)) =>
                           P_id_U23 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__isNatKind 
                             (x11::nil)) =>
                           P_id_a__isNatKind (measure x11)
                          | (algebra.Alg.Term algebra.F.id_U93 (x13::x12::
                             x11::nil)) =>
                           P_id_U93 (measure x13) (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_U11 (x12::
                             x11::nil)) =>
                           P_id_U11 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U71 (x11::nil)) =>
                           P_id_a__U71 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_U44 (x13::x12::
                             x11::nil)) =>
                           P_id_U44 (measure x13) (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_0 nil) => 
                          P_id_0 
                          | (algebra.Alg.Term algebra.F.id_U61 (x11::nil)) =>
                           P_id_U61 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U86 (x11::nil)) =>
                           P_id_a__U86 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U41 (x13::x12::
                             x11::nil)) =>
                           P_id_a__U41 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_U22 (x12::
                             x11::nil)) =>
                           P_id_U22 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U21 (x12::
                             x11::nil)) =>
                           P_id_a__U21 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_U91 (x13::x12::
                             x11::nil)) =>
                           P_id_U91 (measure x13) (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_length (x11::nil)) =>
                           P_id_length (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U52 (x11::nil)) =>
                           P_id_a__U52 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_U42 (x13::x12::
                             x11::nil)) =>
                           P_id_U42 (measure x13) (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U12 (x12::
                             x11::nil)) =>
                           P_id_a__U12 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_U83 (x13::x12::
                             x11::nil)) =>
                           P_id_U83 (measure x13) (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U94 (x12::
                             x11::nil)) =>
                           P_id_a__U94 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U45 (x12::
                             x11::nil)) =>
                           P_id_a__U45 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_U31 (x12::
                             x11::nil)) =>
                           P_id_U31 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__isNat 
                             (x11::nil)) =>
                           P_id_a__isNat (measure x11)
                          | (algebra.Alg.Term algebra.F.id_isNatIListKind 
                             (x11::nil)) =>
                           P_id_isNatIListKind (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U82 (x13::x12::
                             x11::nil)) =>
                           P_id_a__U82 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_U46 (x11::nil)) =>
                           P_id_U46 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_cons (x12::
                             x11::nil)) =>
                           P_id_cons (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_U52 (x11::nil)) =>
                           P_id_U52 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U85 (x12::
                             x11::nil)) =>
                           P_id_a__U85 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U33 (x11::nil)) =>
                           P_id_a__U33 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_U21 (x12::
                             x11::nil)) =>
                           P_id_U21 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__isNatList 
                             (x11::nil)) =>
                           P_id_a__isNatList (measure x11)
                          | (algebra.Alg.Term algebra.F.id_U86 (x11::nil)) =>
                           P_id_U86 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_mark (x11::nil)) =>
                           P_id_mark (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U51 (x12::
                             x11::nil)) =>
                           P_id_a__U51 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_U41 (x13::x12::
                             x11::nil)) =>
                           P_id_U41 (measure x13) (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_tt nil) =>
                           P_id_tt 
                          | (algebra.Alg.Term algebra.F.id_U82 (x13::x12::
                             x11::nil)) =>
                           P_id_U82 (measure x13) (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U93 (x13::x12::
                             x11::nil)) =>
                           P_id_a__U93 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U44 (x13::x12::
                             x11::nil)) =>
                           P_id_a__U44 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_isNat (x11::nil)) =>
                           P_id_isNat (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U23 (x11::nil)) =>
                           P_id_a__U23 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_U94 (x12::
                             x11::nil)) =>
                           P_id_U94 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_U12 (x12::
                             x11::nil)) =>
                           P_id_U12 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U81 (x13::x12::
                             x11::nil)) =>
                           P_id_a__U81 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_U45 (x12::
                             x11::nil)) =>
                           P_id_U45 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_zeros nil) =>
                           P_id_zeros 
                          | (algebra.Alg.Term algebra.F.id_U71 (x11::nil)) =>
                           P_id_U71 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U91 (x13::x12::
                             x11::nil)) =>
                           P_id_a__U91 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U42 (x13::x12::
                             x11::nil)) =>
                           P_id_a__U42 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_isNatKind 
                             (x11::nil)) =>
                           P_id_isNatKind (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U22 (x12::
                             x11::nil)) =>
                           P_id_a__U22 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_U92 (x13::x12::
                             x11::nil)) =>
                           P_id_U92 (measure x13) (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_nil nil) =>
                           P_id_nil 
                          | (algebra.Alg.Term algebra.F.id_a__U61 (x11::nil)) =>
                           P_id_a__U61 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_U43 (x13::x12::
                             x11::nil)) =>
                           P_id_U43 (measure x13) (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
                             (x11::nil)) =>
                           P_id_a__isNatIListKind (measure x11)
                          | (algebra.Alg.Term algebra.F.id_U84 (x13::x12::
                             x11::nil)) =>
                           P_id_U84 (measure x13) (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_s (x11::nil)) =>
                           P_id_s (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U46 (x11::nil)) =>
                           P_id_a__U46 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_U32 (x12::
                             x11::nil)) =>
                           P_id_U32 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U31 (x12::
                             x11::nil)) =>
                           P_id_a__U31 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_U13 (x11::nil)) =>
                           P_id_U13 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U83 (x13::x12::
                             x11::nil)) =>
                           P_id_a__U83 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_isNatIList 
                             (x11::nil)) =>
                           P_id_isNatIList (measure x11)
                          | _ => 0
                          end.
          Proof.
            intros t;case t;intros ;apply refl_equal.
          Qed.
          
          Lemma measure_bounded : forall t, 0 <= measure t.
          Proof.
            unfold measure in |-*.
            
            apply InterpZ.measure_bounded;
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Ltac generate_pos_hyp  :=
           match goal with
             | H:context [measure ?x] |- _ =>
              let v := fresh "v" in 
               (let H := fresh "h" in 
                 (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
                   clearbody H;clearbody v))
             |  |- context [measure ?x] =>
              let v := fresh "v" in 
               (let H := fresh "h" in 
                 (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
                   clearbody H;clearbody v))
             end
           .
          
          Lemma rules_monotonic :
           forall l r, 
            (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) ->
             measure r <= measure l.
          Proof.
            intros l r H.
            fold measure in |-*.
            
            inversion H;clear H;subst;inversion H0;clear H0;subst;
             simpl algebra.EQT.T.apply_subst in |-*;
             repeat (
             match goal with
               |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
                rewrite (measure_equation (algebra.Alg.Term f t))
               end
             );repeat (generate_pos_hyp );
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma measure_star_monotonic :
           forall l r, 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                       r l) ->measure r <= measure l.
          Proof.
            unfold measure in *.
            apply InterpZ.measure_star_monotonic.
            intros ;apply P_id_U51_monotonic;assumption.
            intros ;apply P_id_a__U84_monotonic;assumption.
            intros ;apply P_id_a__U32_monotonic;assumption.
            intros ;apply P_id_isNatList_monotonic;assumption.
            intros ;apply P_id_a__U13_monotonic;assumption.
            intros ;apply P_id_U85_monotonic;assumption.
            intros ;apply P_id_a__length_monotonic;assumption.
            intros ;apply P_id_a__isNatIList_monotonic;assumption.
            intros ;apply P_id_U33_monotonic;assumption.
            intros ;apply P_id_a__U11_monotonic;assumption.
            intros ;apply P_id_U81_monotonic;assumption.
            intros ;apply P_id_a__U92_monotonic;assumption.
            intros ;apply P_id_a__U43_monotonic;assumption.
            intros ;apply P_id_U23_monotonic;assumption.
            intros ;apply P_id_a__isNatKind_monotonic;assumption.
            intros ;apply P_id_U93_monotonic;assumption.
            intros ;apply P_id_U11_monotonic;assumption.
            intros ;apply P_id_a__U71_monotonic;assumption.
            intros ;apply P_id_U44_monotonic;assumption.
            intros ;apply P_id_U61_monotonic;assumption.
            intros ;apply P_id_a__U86_monotonic;assumption.
            intros ;apply P_id_a__U41_monotonic;assumption.
            intros ;apply P_id_U22_monotonic;assumption.
            intros ;apply P_id_a__U21_monotonic;assumption.
            intros ;apply P_id_U91_monotonic;assumption.
            intros ;apply P_id_length_monotonic;assumption.
            intros ;apply P_id_a__U52_monotonic;assumption.
            intros ;apply P_id_U42_monotonic;assumption.
            intros ;apply P_id_a__U12_monotonic;assumption.
            intros ;apply P_id_U83_monotonic;assumption.
            intros ;apply P_id_a__U94_monotonic;assumption.
            intros ;apply P_id_a__U45_monotonic;assumption.
            intros ;apply P_id_U31_monotonic;assumption.
            intros ;apply P_id_a__isNat_monotonic;assumption.
            intros ;apply P_id_isNatIListKind_monotonic;assumption.
            intros ;apply P_id_a__U82_monotonic;assumption.
            intros ;apply P_id_U46_monotonic;assumption.
            intros ;apply P_id_cons_monotonic;assumption.
            intros ;apply P_id_U52_monotonic;assumption.
            intros ;apply P_id_a__U85_monotonic;assumption.
            intros ;apply P_id_a__U33_monotonic;assumption.
            intros ;apply P_id_U21_monotonic;assumption.
            intros ;apply P_id_a__isNatList_monotonic;assumption.
            intros ;apply P_id_U86_monotonic;assumption.
            intros ;apply P_id_mark_monotonic;assumption.
            intros ;apply P_id_a__U51_monotonic;assumption.
            intros ;apply P_id_U41_monotonic;assumption.
            intros ;apply P_id_U82_monotonic;assumption.
            intros ;apply P_id_a__U93_monotonic;assumption.
            intros ;apply P_id_a__U44_monotonic;assumption.
            intros ;apply P_id_isNat_monotonic;assumption.
            intros ;apply P_id_a__U23_monotonic;assumption.
            intros ;apply P_id_U94_monotonic;assumption.
            intros ;apply P_id_U12_monotonic;assumption.
            intros ;apply P_id_a__U81_monotonic;assumption.
            intros ;apply P_id_U45_monotonic;assumption.
            intros ;apply P_id_U71_monotonic;assumption.
            intros ;apply P_id_a__U91_monotonic;assumption.
            intros ;apply P_id_a__U42_monotonic;assumption.
            intros ;apply P_id_isNatKind_monotonic;assumption.
            intros ;apply P_id_a__U22_monotonic;assumption.
            intros ;apply P_id_U92_monotonic;assumption.
            intros ;apply P_id_a__U61_monotonic;assumption.
            intros ;apply P_id_U43_monotonic;assumption.
            intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
            intros ;apply P_id_U84_monotonic;assumption.
            intros ;apply P_id_s_monotonic;assumption.
            intros ;apply P_id_a__U46_monotonic;assumption.
            intros ;apply P_id_U32_monotonic;assumption.
            intros ;apply P_id_a__U31_monotonic;assumption.
            intros ;apply P_id_U13_monotonic;assumption.
            intros ;apply P_id_a__U83_monotonic;assumption.
            intros ;apply P_id_isNatIList_monotonic;assumption.
            intros ;apply P_id_a__zeros_bounded;assumption.
            intros ;apply P_id_U51_bounded;assumption.
            intros ;apply P_id_a__U84_bounded;assumption.
            intros ;apply P_id_a__U32_bounded;assumption.
            intros ;apply P_id_isNatList_bounded;assumption.
            intros ;apply P_id_a__U13_bounded;assumption.
            intros ;apply P_id_U85_bounded;assumption.
            intros ;apply P_id_a__length_bounded;assumption.
            intros ;apply P_id_a__isNatIList_bounded;assumption.
            intros ;apply P_id_U33_bounded;assumption.
            intros ;apply P_id_a__U11_bounded;assumption.
            intros ;apply P_id_U81_bounded;assumption.
            intros ;apply P_id_a__U92_bounded;assumption.
            intros ;apply P_id_a__U43_bounded;assumption.
            intros ;apply P_id_U23_bounded;assumption.
            intros ;apply P_id_a__isNatKind_bounded;assumption.
            intros ;apply P_id_U93_bounded;assumption.
            intros ;apply P_id_U11_bounded;assumption.
            intros ;apply P_id_a__U71_bounded;assumption.
            intros ;apply P_id_U44_bounded;assumption.
            intros ;apply P_id_0_bounded;assumption.
            intros ;apply P_id_U61_bounded;assumption.
            intros ;apply P_id_a__U86_bounded;assumption.
            intros ;apply P_id_a__U41_bounded;assumption.
            intros ;apply P_id_U22_bounded;assumption.
            intros ;apply P_id_a__U21_bounded;assumption.
            intros ;apply P_id_U91_bounded;assumption.
            intros ;apply P_id_length_bounded;assumption.
            intros ;apply P_id_a__U52_bounded;assumption.
            intros ;apply P_id_U42_bounded;assumption.
            intros ;apply P_id_a__U12_bounded;assumption.
            intros ;apply P_id_U83_bounded;assumption.
            intros ;apply P_id_a__U94_bounded;assumption.
            intros ;apply P_id_a__U45_bounded;assumption.
            intros ;apply P_id_U31_bounded;assumption.
            intros ;apply P_id_a__isNat_bounded;assumption.
            intros ;apply P_id_isNatIListKind_bounded;assumption.
            intros ;apply P_id_a__U82_bounded;assumption.
            intros ;apply P_id_U46_bounded;assumption.
            intros ;apply P_id_cons_bounded;assumption.
            intros ;apply P_id_U52_bounded;assumption.
            intros ;apply P_id_a__U85_bounded;assumption.
            intros ;apply P_id_a__U33_bounded;assumption.
            intros ;apply P_id_U21_bounded;assumption.
            intros ;apply P_id_a__isNatList_bounded;assumption.
            intros ;apply P_id_U86_bounded;assumption.
            intros ;apply P_id_mark_bounded;assumption.
            intros ;apply P_id_a__U51_bounded;assumption.
            intros ;apply P_id_U41_bounded;assumption.
            intros ;apply P_id_tt_bounded;assumption.
            intros ;apply P_id_U82_bounded;assumption.
            intros ;apply P_id_a__U93_bounded;assumption.
            intros ;apply P_id_a__U44_bounded;assumption.
            intros ;apply P_id_isNat_bounded;assumption.
            intros ;apply P_id_a__U23_bounded;assumption.
            intros ;apply P_id_U94_bounded;assumption.
            intros ;apply P_id_U12_bounded;assumption.
            intros ;apply P_id_a__U81_bounded;assumption.
            intros ;apply P_id_U45_bounded;assumption.
            intros ;apply P_id_zeros_bounded;assumption.
            intros ;apply P_id_U71_bounded;assumption.
            intros ;apply P_id_a__U91_bounded;assumption.
            intros ;apply P_id_a__U42_bounded;assumption.
            intros ;apply P_id_isNatKind_bounded;assumption.
            intros ;apply P_id_a__U22_bounded;assumption.
            intros ;apply P_id_U92_bounded;assumption.
            intros ;apply P_id_nil_bounded;assumption.
            intros ;apply P_id_a__U61_bounded;assumption.
            intros ;apply P_id_U43_bounded;assumption.
            intros ;apply P_id_a__isNatIListKind_bounded;assumption.
            intros ;apply P_id_U84_bounded;assumption.
            intros ;apply P_id_s_bounded;assumption.
            intros ;apply P_id_a__U46_bounded;assumption.
            intros ;apply P_id_U32_bounded;assumption.
            intros ;apply P_id_a__U31_bounded;assumption.
            intros ;apply P_id_U13_bounded;assumption.
            intros ;apply P_id_a__U83_bounded;assumption.
            intros ;apply P_id_isNatIList_bounded;assumption.
            apply rules_monotonic.
          Qed.
          
          Definition P_id_A__U46 (x11:Z) := 0.
          
          Definition P_id_A__ISNATILISTKIND (x11:Z) := 0.
          
          Definition P_id_A__U61 (x11:Z) := 0.
          
          Definition P_id_A__U85 (x11:Z) (x12:Z) := 0.
          
          Definition P_id_A__U31 (x11:Z) (x12:Z) := 0.
          
          Definition P_id_A__U81 (x11:Z) (x12:Z) (x13:Z) := 0.
          
          Definition P_id_A__U22 (x11:Z) (x12:Z) := 0.
          
          Definition P_id_A__U93 (x11:Z) (x12:Z) (x13:Z) := 0.
          
          Definition P_id_A__U42 (x11:Z) (x12:Z) (x13:Z) := 0.
          
          Definition P_id_A__U51 (x11:Z) (x12:Z) := 0.
          
          Definition P_id_A__ISNATLIST (x11:Z) := 0.
          
          Definition P_id_A__ZEROS  := 0.
          
          Definition P_id_A__U91 (x11:Z) (x12:Z) (x13:Z) := 0.
          
          Definition P_id_A__U33 (x11:Z) := 0.
          
          Definition P_id_A__U83 (x11:Z) (x12:Z) (x13:Z) := 0.
          
          Definition P_id_A__U23 (x11:Z) := 0.
          
          Definition P_id_A__U11 (x11:Z) (x12:Z) := 0.
          
          Definition P_id_A__LENGTH (x11:Z) := 0.
          
          Definition P_id_A__U44 (x11:Z) (x12:Z) (x13:Z) := 0.
          
          Definition P_id_A__ISNATILIST (x11:Z) := 0.
          
          Definition P_id_A__U13 (x11:Z) := 0.
          
          Definition P_id_A__U71 (x11:Z) := 0.
          
          Definition P_id_A__U86 (x11:Z) := 0.
          
          Definition P_id_A__U32 (x11:Z) (x12:Z) := 0.
          
          Definition P_id_A__U82 (x11:Z) (x12:Z) (x13:Z) := 0.
          
          Definition P_id_A__ISNATKIND (x11:Z) := 0.
          
          Definition P_id_A__U94 (x11:Z) (x12:Z) := 0.
          
          Definition P_id_A__U43 (x11:Z) (x12:Z) (x13:Z) := 0.
          
          Definition P_id_A__U52 (x11:Z) := 0.
          
          Definition P_id_A__U21 (x11:Z) (x12:Z) := 0.
          
          Definition P_id_A__U92 (x11:Z) (x12:Z) (x13:Z) := 0.
          
          Definition P_id_A__U41 (x11:Z) (x12:Z) (x13:Z) := 0.
          
          Definition P_id_A__U84 (x11:Z) (x12:Z) (x13:Z) := 0.
          
          Definition P_id_A__ISNAT (x11:Z) := 0.
          
          Definition P_id_A__U12 (x11:Z) (x12:Z) := 0.
          
          Definition P_id_MARK (x11:Z) := 2* x11.
          
          Definition P_id_A__U45 (x11:Z) (x12:Z) := 0.
          
          Lemma P_id_A__U46_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->P_id_A__U46 x12 <= P_id_A__U46 x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_0].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_A__ISNATILISTKIND_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_A__ISNATILISTKIND x12 <= P_id_A__ISNATILISTKIND x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_0].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_A__U61_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->P_id_A__U61 x12 <= P_id_A__U61 x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_0].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_A__U85_monotonic :
           forall x12 x14 x13 x11, 
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_A__U85 x12 x14 <= P_id_A__U85 x11 x13.
          Proof.
            intros x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_A__U31_monotonic :
           forall x12 x14 x13 x11, 
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_A__U31 x12 x14 <= P_id_A__U31 x11 x13.
          Proof.
            intros x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_A__U81_monotonic :
           forall x16 x12 x14 x13 x11 x15, 
            (0 <= x16)/\ (x16 <= x15) ->
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_A__U81 x12 x14 x16 <= P_id_A__U81 x11 x13 x15.
          Proof.
            intros x16 x15 x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            intros [H_5 H_4].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_A__U22_monotonic :
           forall x12 x14 x13 x11, 
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_A__U22 x12 x14 <= P_id_A__U22 x11 x13.
          Proof.
            intros x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_A__U93_monotonic :
           forall x16 x12 x14 x13 x11 x15, 
            (0 <= x16)/\ (x16 <= x15) ->
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_A__U93 x12 x14 x16 <= P_id_A__U93 x11 x13 x15.
          Proof.
            intros x16 x15 x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            intros [H_5 H_4].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_A__U42_monotonic :
           forall x16 x12 x14 x13 x11 x15, 
            (0 <= x16)/\ (x16 <= x15) ->
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_A__U42 x12 x14 x16 <= P_id_A__U42 x11 x13 x15.
          Proof.
            intros x16 x15 x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            intros [H_5 H_4].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_A__U51_monotonic :
           forall x12 x14 x13 x11, 
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_A__U51 x12 x14 <= P_id_A__U51 x11 x13.
          Proof.
            intros x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_A__ISNATLIST_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_A__ISNATLIST x12 <= P_id_A__ISNATLIST x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_0].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_A__U91_monotonic :
           forall x16 x12 x14 x13 x11 x15, 
            (0 <= x16)/\ (x16 <= x15) ->
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_A__U91 x12 x14 x16 <= P_id_A__U91 x11 x13 x15.
          Proof.
            intros x16 x15 x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            intros [H_5 H_4].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_A__U33_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->P_id_A__U33 x12 <= P_id_A__U33 x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_0].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_A__U83_monotonic :
           forall x16 x12 x14 x13 x11 x15, 
            (0 <= x16)/\ (x16 <= x15) ->
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_A__U83 x12 x14 x16 <= P_id_A__U83 x11 x13 x15.
          Proof.
            intros x16 x15 x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            intros [H_5 H_4].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_A__U23_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->P_id_A__U23 x12 <= P_id_A__U23 x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_0].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_A__U11_monotonic :
           forall x12 x14 x13 x11, 
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_A__U11 x12 x14 <= P_id_A__U11 x11 x13.
          Proof.
            intros x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_A__LENGTH_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_A__LENGTH x12 <= P_id_A__LENGTH x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_0].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_A__U44_monotonic :
           forall x16 x12 x14 x13 x11 x15, 
            (0 <= x16)/\ (x16 <= x15) ->
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_A__U44 x12 x14 x16 <= P_id_A__U44 x11 x13 x15.
          Proof.
            intros x16 x15 x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            intros [H_5 H_4].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_A__ISNATILIST_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_A__ISNATILIST x12 <= P_id_A__ISNATILIST x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_0].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_A__U13_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->P_id_A__U13 x12 <= P_id_A__U13 x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_0].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_A__U71_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->P_id_A__U71 x12 <= P_id_A__U71 x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_0].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_A__U86_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->P_id_A__U86 x12 <= P_id_A__U86 x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_0].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_A__U32_monotonic :
           forall x12 x14 x13 x11, 
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_A__U32 x12 x14 <= P_id_A__U32 x11 x13.
          Proof.
            intros x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_A__U82_monotonic :
           forall x16 x12 x14 x13 x11 x15, 
            (0 <= x16)/\ (x16 <= x15) ->
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_A__U82 x12 x14 x16 <= P_id_A__U82 x11 x13 x15.
          Proof.
            intros x16 x15 x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            intros [H_5 H_4].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_A__ISNATKIND_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_A__ISNATKIND x12 <= P_id_A__ISNATKIND x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_0].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_A__U94_monotonic :
           forall x12 x14 x13 x11, 
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_A__U94 x12 x14 <= P_id_A__U94 x11 x13.
          Proof.
            intros x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_A__U43_monotonic :
           forall x16 x12 x14 x13 x11 x15, 
            (0 <= x16)/\ (x16 <= x15) ->
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_A__U43 x12 x14 x16 <= P_id_A__U43 x11 x13 x15.
          Proof.
            intros x16 x15 x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            intros [H_5 H_4].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_A__U52_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->P_id_A__U52 x12 <= P_id_A__U52 x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_0].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_A__U21_monotonic :
           forall x12 x14 x13 x11, 
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_A__U21 x12 x14 <= P_id_A__U21 x11 x13.
          Proof.
            intros x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_A__U92_monotonic :
           forall x16 x12 x14 x13 x11 x15, 
            (0 <= x16)/\ (x16 <= x15) ->
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_A__U92 x12 x14 x16 <= P_id_A__U92 x11 x13 x15.
          Proof.
            intros x16 x15 x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            intros [H_5 H_4].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_A__U41_monotonic :
           forall x16 x12 x14 x13 x11 x15, 
            (0 <= x16)/\ (x16 <= x15) ->
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_A__U41 x12 x14 x16 <= P_id_A__U41 x11 x13 x15.
          Proof.
            intros x16 x15 x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            intros [H_5 H_4].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_A__U84_monotonic :
           forall x16 x12 x14 x13 x11 x15, 
            (0 <= x16)/\ (x16 <= x15) ->
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_A__U84 x12 x14 x16 <= P_id_A__U84 x11 x13 x15.
          Proof.
            intros x16 x15 x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            intros [H_5 H_4].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_A__ISNAT_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_A__ISNAT x12 <= P_id_A__ISNAT x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_0].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_A__U12_monotonic :
           forall x12 x14 x13 x11, 
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_A__U12 x12 x14 <= P_id_A__U12 x11 x13.
          Proof.
            intros x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_MARK_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->P_id_MARK x12 <= P_id_MARK x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_0].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_A__U45_monotonic :
           forall x12 x14 x13 x11, 
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_A__U45 x12 x14 <= P_id_A__U45 x11 x13.
          Proof.
            intros x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Definition marked_measure  := 
            InterpZ.marked_measure 0 P_id_a__zeros P_id_U51 P_id_a__U84 
             P_id_a__U32 P_id_isNatList P_id_a__U13 P_id_U85 P_id_a__length 
             P_id_a__isNatIList P_id_U33 P_id_a__U11 P_id_U81 P_id_a__U92 
             P_id_a__U43 P_id_U23 P_id_a__isNatKind P_id_U93 P_id_U11 
             P_id_a__U71 P_id_U44 P_id_0 P_id_U61 P_id_a__U86 P_id_a__U41 
             P_id_U22 P_id_a__U21 P_id_U91 P_id_length P_id_a__U52 P_id_U42 
             P_id_a__U12 P_id_U83 P_id_a__U94 P_id_a__U45 P_id_U31 
             P_id_a__isNat P_id_isNatIListKind P_id_a__U82 P_id_U46 
             P_id_cons P_id_U52 P_id_a__U85 P_id_a__U33 P_id_U21 
             P_id_a__isNatList P_id_U86 P_id_mark P_id_a__U51 P_id_U41 
             P_id_tt P_id_U82 P_id_a__U93 P_id_a__U44 P_id_isNat P_id_a__U23 
             P_id_U94 P_id_U12 P_id_a__U81 P_id_U45 P_id_zeros P_id_U71 
             P_id_a__U91 P_id_a__U42 P_id_isNatKind P_id_a__U22 P_id_U92 
             P_id_nil P_id_a__U61 P_id_U43 P_id_a__isNatIListKind P_id_U84 
             P_id_s P_id_a__U46 P_id_U32 P_id_a__U31 P_id_U13 P_id_a__U83 
             P_id_isNatIList P_id_A__U46 P_id_A__ISNATILISTKIND P_id_A__U61 
             P_id_A__U85 P_id_A__U31 P_id_A__U81 P_id_A__U22 P_id_A__U93 
             P_id_A__U42 P_id_A__U51 P_id_A__ISNATLIST P_id_A__ZEROS 
             P_id_A__U91 P_id_A__U33 P_id_A__U83 P_id_A__U23 P_id_A__U11 
             P_id_A__LENGTH P_id_A__U44 P_id_A__ISNATILIST P_id_A__U13 
             P_id_A__U71 P_id_A__U86 P_id_A__U32 P_id_A__U82 
             P_id_A__ISNATKIND P_id_A__U94 P_id_A__U43 P_id_A__U52 
             P_id_A__U21 P_id_A__U92 P_id_A__U41 P_id_A__U84 P_id_A__ISNAT 
             P_id_A__U12 P_id_MARK P_id_A__U45.
          
          Lemma marked_measure_equation :
           forall t, 
            marked_measure t = match t with
                                 | (algebra.Alg.Term algebra.F.id_a__U46 
                                    (x11::nil)) =>
                                  P_id_A__U46 (measure x11)
                                 | (algebra.Alg.Term 
                                    algebra.F.id_a__isNatIListKind 
                                    (x11::nil)) =>
                                  P_id_A__ISNATILISTKIND (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U61 
                                    (x11::nil)) =>
                                  P_id_A__U61 (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U85 
                                    (x12::x11::nil)) =>
                                  P_id_A__U85 (measure x12) (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U31 
                                    (x12::x11::nil)) =>
                                  P_id_A__U31 (measure x12) (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U81 
                                    (x13::x12::x11::nil)) =>
                                  P_id_A__U81 (measure x13) (measure x12) 
                                   (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U22 
                                    (x12::x11::nil)) =>
                                  P_id_A__U22 (measure x12) (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U93 
                                    (x13::x12::x11::nil)) =>
                                  P_id_A__U93 (measure x13) (measure x12) 
                                   (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U42 
                                    (x13::x12::x11::nil)) =>
                                  P_id_A__U42 (measure x13) (measure x12) 
                                   (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U51 
                                    (x12::x11::nil)) =>
                                  P_id_A__U51 (measure x12) (measure x11)
                                 | (algebra.Alg.Term 
                                    algebra.F.id_a__isNatList (x11::nil)) =>
                                  P_id_A__ISNATLIST (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__zeros 
                                    nil) =>
                                  P_id_A__ZEROS 
                                 | (algebra.Alg.Term algebra.F.id_a__U91 
                                    (x13::x12::x11::nil)) =>
                                  P_id_A__U91 (measure x13) (measure x12) 
                                   (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U33 
                                    (x11::nil)) =>
                                  P_id_A__U33 (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U83 
                                    (x13::x12::x11::nil)) =>
                                  P_id_A__U83 (measure x13) (measure x12) 
                                   (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U23 
                                    (x11::nil)) =>
                                  P_id_A__U23 (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U11 
                                    (x12::x11::nil)) =>
                                  P_id_A__U11 (measure x12) (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__length 
                                    (x11::nil)) =>
                                  P_id_A__LENGTH (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U44 
                                    (x13::x12::x11::nil)) =>
                                  P_id_A__U44 (measure x13) (measure x12) 
                                   (measure x11)
                                 | (algebra.Alg.Term 
                                    algebra.F.id_a__isNatIList (x11::nil)) =>
                                  P_id_A__ISNATILIST (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U13 
                                    (x11::nil)) =>
                                  P_id_A__U13 (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U71 
                                    (x11::nil)) =>
                                  P_id_A__U71 (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U86 
                                    (x11::nil)) =>
                                  P_id_A__U86 (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U32 
                                    (x12::x11::nil)) =>
                                  P_id_A__U32 (measure x12) (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U82 
                                    (x13::x12::x11::nil)) =>
                                  P_id_A__U82 (measure x13) (measure x12) 
                                   (measure x11)
                                 | (algebra.Alg.Term 
                                    algebra.F.id_a__isNatKind (x11::nil)) =>
                                  P_id_A__ISNATKIND (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U94 
                                    (x12::x11::nil)) =>
                                  P_id_A__U94 (measure x12) (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U43 
                                    (x13::x12::x11::nil)) =>
                                  P_id_A__U43 (measure x13) (measure x12) 
                                   (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U52 
                                    (x11::nil)) =>
                                  P_id_A__U52 (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U21 
                                    (x12::x11::nil)) =>
                                  P_id_A__U21 (measure x12) (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U92 
                                    (x13::x12::x11::nil)) =>
                                  P_id_A__U92 (measure x13) (measure x12) 
                                   (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U41 
                                    (x13::x12::x11::nil)) =>
                                  P_id_A__U41 (measure x13) (measure x12) 
                                   (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U84 
                                    (x13::x12::x11::nil)) =>
                                  P_id_A__U84 (measure x13) (measure x12) 
                                   (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__isNat 
                                    (x11::nil)) =>
                                  P_id_A__ISNAT (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U12 
                                    (x12::x11::nil)) =>
                                  P_id_A__U12 (measure x12) (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_mark 
                                    (x11::nil)) =>
                                  P_id_MARK (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U45 
                                    (x12::x11::nil)) =>
                                  P_id_A__U45 (measure x12) (measure x11)
                                 | _ => measure t
                                 end.
          Proof.
            reflexivity .
          Qed.
          
          Lemma marked_measure_star_monotonic :
           forall f l1 l2, 
            (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step 
                                                              R_xml_0_deep_rew.R_xml_0_rules)
                                                             ) l1 l2) ->
             marked_measure (algebra.Alg.Term f l1) <= marked_measure 
                                                        (algebra.Alg.Term 
                                                          f l2).
          Proof.
            unfold marked_measure in *.
            apply InterpZ.marked_measure_star_monotonic.
            intros ;apply P_id_U51_monotonic;assumption.
            intros ;apply P_id_a__U84_monotonic;assumption.
            intros ;apply P_id_a__U32_monotonic;assumption.
            intros ;apply P_id_isNatList_monotonic;assumption.
            intros ;apply P_id_a__U13_monotonic;assumption.
            intros ;apply P_id_U85_monotonic;assumption.
            intros ;apply P_id_a__length_monotonic;assumption.
            intros ;apply P_id_a__isNatIList_monotonic;assumption.
            intros ;apply P_id_U33_monotonic;assumption.
            intros ;apply P_id_a__U11_monotonic;assumption.
            intros ;apply P_id_U81_monotonic;assumption.
            intros ;apply P_id_a__U92_monotonic;assumption.
            intros ;apply P_id_a__U43_monotonic;assumption.
            intros ;apply P_id_U23_monotonic;assumption.
            intros ;apply P_id_a__isNatKind_monotonic;assumption.
            intros ;apply P_id_U93_monotonic;assumption.
            intros ;apply P_id_U11_monotonic;assumption.
            intros ;apply P_id_a__U71_monotonic;assumption.
            intros ;apply P_id_U44_monotonic;assumption.
            intros ;apply P_id_U61_monotonic;assumption.
            intros ;apply P_id_a__U86_monotonic;assumption.
            intros ;apply P_id_a__U41_monotonic;assumption.
            intros ;apply P_id_U22_monotonic;assumption.
            intros ;apply P_id_a__U21_monotonic;assumption.
            intros ;apply P_id_U91_monotonic;assumption.
            intros ;apply P_id_length_monotonic;assumption.
            intros ;apply P_id_a__U52_monotonic;assumption.
            intros ;apply P_id_U42_monotonic;assumption.
            intros ;apply P_id_a__U12_monotonic;assumption.
            intros ;apply P_id_U83_monotonic;assumption.
            intros ;apply P_id_a__U94_monotonic;assumption.
            intros ;apply P_id_a__U45_monotonic;assumption.
            intros ;apply P_id_U31_monotonic;assumption.
            intros ;apply P_id_a__isNat_monotonic;assumption.
            intros ;apply P_id_isNatIListKind_monotonic;assumption.
            intros ;apply P_id_a__U82_monotonic;assumption.
            intros ;apply P_id_U46_monotonic;assumption.
            intros ;apply P_id_cons_monotonic;assumption.
            intros ;apply P_id_U52_monotonic;assumption.
            intros ;apply P_id_a__U85_monotonic;assumption.
            intros ;apply P_id_a__U33_monotonic;assumption.
            intros ;apply P_id_U21_monotonic;assumption.
            intros ;apply P_id_a__isNatList_monotonic;assumption.
            intros ;apply P_id_U86_monotonic;assumption.
            intros ;apply P_id_mark_monotonic;assumption.
            intros ;apply P_id_a__U51_monotonic;assumption.
            intros ;apply P_id_U41_monotonic;assumption.
            intros ;apply P_id_U82_monotonic;assumption.
            intros ;apply P_id_a__U93_monotonic;assumption.
            intros ;apply P_id_a__U44_monotonic;assumption.
            intros ;apply P_id_isNat_monotonic;assumption.
            intros ;apply P_id_a__U23_monotonic;assumption.
            intros ;apply P_id_U94_monotonic;assumption.
            intros ;apply P_id_U12_monotonic;assumption.
            intros ;apply P_id_a__U81_monotonic;assumption.
            intros ;apply P_id_U45_monotonic;assumption.
            intros ;apply P_id_U71_monotonic;assumption.
            intros ;apply P_id_a__U91_monotonic;assumption.
            intros ;apply P_id_a__U42_monotonic;assumption.
            intros ;apply P_id_isNatKind_monotonic;assumption.
            intros ;apply P_id_a__U22_monotonic;assumption.
            intros ;apply P_id_U92_monotonic;assumption.
            intros ;apply P_id_a__U61_monotonic;assumption.
            intros ;apply P_id_U43_monotonic;assumption.
            intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
            intros ;apply P_id_U84_monotonic;assumption.
            intros ;apply P_id_s_monotonic;assumption.
            intros ;apply P_id_a__U46_monotonic;assumption.
            intros ;apply P_id_U32_monotonic;assumption.
            intros ;apply P_id_a__U31_monotonic;assumption.
            intros ;apply P_id_U13_monotonic;assumption.
            intros ;apply P_id_a__U83_monotonic;assumption.
            intros ;apply P_id_isNatIList_monotonic;assumption.
            intros ;apply P_id_a__zeros_bounded;assumption.
            intros ;apply P_id_U51_bounded;assumption.
            intros ;apply P_id_a__U84_bounded;assumption.
            intros ;apply P_id_a__U32_bounded;assumption.
            intros ;apply P_id_isNatList_bounded;assumption.
            intros ;apply P_id_a__U13_bounded;assumption.
            intros ;apply P_id_U85_bounded;assumption.
            intros ;apply P_id_a__length_bounded;assumption.
            intros ;apply P_id_a__isNatIList_bounded;assumption.
            intros ;apply P_id_U33_bounded;assumption.
            intros ;apply P_id_a__U11_bounded;assumption.
            intros ;apply P_id_U81_bounded;assumption.
            intros ;apply P_id_a__U92_bounded;assumption.
            intros ;apply P_id_a__U43_bounded;assumption.
            intros ;apply P_id_U23_bounded;assumption.
            intros ;apply P_id_a__isNatKind_bounded;assumption.
            intros ;apply P_id_U93_bounded;assumption.
            intros ;apply P_id_U11_bounded;assumption.
            intros ;apply P_id_a__U71_bounded;assumption.
            intros ;apply P_id_U44_bounded;assumption.
            intros ;apply P_id_0_bounded;assumption.
            intros ;apply P_id_U61_bounded;assumption.
            intros ;apply P_id_a__U86_bounded;assumption.
            intros ;apply P_id_a__U41_bounded;assumption.
            intros ;apply P_id_U22_bounded;assumption.
            intros ;apply P_id_a__U21_bounded;assumption.
            intros ;apply P_id_U91_bounded;assumption.
            intros ;apply P_id_length_bounded;assumption.
            intros ;apply P_id_a__U52_bounded;assumption.
            intros ;apply P_id_U42_bounded;assumption.
            intros ;apply P_id_a__U12_bounded;assumption.
            intros ;apply P_id_U83_bounded;assumption.
            intros ;apply P_id_a__U94_bounded;assumption.
            intros ;apply P_id_a__U45_bounded;assumption.
            intros ;apply P_id_U31_bounded;assumption.
            intros ;apply P_id_a__isNat_bounded;assumption.
            intros ;apply P_id_isNatIListKind_bounded;assumption.
            intros ;apply P_id_a__U82_bounded;assumption.
            intros ;apply P_id_U46_bounded;assumption.
            intros ;apply P_id_cons_bounded;assumption.
            intros ;apply P_id_U52_bounded;assumption.
            intros ;apply P_id_a__U85_bounded;assumption.
            intros ;apply P_id_a__U33_bounded;assumption.
            intros ;apply P_id_U21_bounded;assumption.
            intros ;apply P_id_a__isNatList_bounded;assumption.
            intros ;apply P_id_U86_bounded;assumption.
            intros ;apply P_id_mark_bounded;assumption.
            intros ;apply P_id_a__U51_bounded;assumption.
            intros ;apply P_id_U41_bounded;assumption.
            intros ;apply P_id_tt_bounded;assumption.
            intros ;apply P_id_U82_bounded;assumption.
            intros ;apply P_id_a__U93_bounded;assumption.
            intros ;apply P_id_a__U44_bounded;assumption.
            intros ;apply P_id_isNat_bounded;assumption.
            intros ;apply P_id_a__U23_bounded;assumption.
            intros ;apply P_id_U94_bounded;assumption.
            intros ;apply P_id_U12_bounded;assumption.
            intros ;apply P_id_a__U81_bounded;assumption.
            intros ;apply P_id_U45_bounded;assumption.
            intros ;apply P_id_zeros_bounded;assumption.
            intros ;apply P_id_U71_bounded;assumption.
            intros ;apply P_id_a__U91_bounded;assumption.
            intros ;apply P_id_a__U42_bounded;assumption.
            intros ;apply P_id_isNatKind_bounded;assumption.
            intros ;apply P_id_a__U22_bounded;assumption.
            intros ;apply P_id_U92_bounded;assumption.
            intros ;apply P_id_nil_bounded;assumption.
            intros ;apply P_id_a__U61_bounded;assumption.
            intros ;apply P_id_U43_bounded;assumption.
            intros ;apply P_id_a__isNatIListKind_bounded;assumption.
            intros ;apply P_id_U84_bounded;assumption.
            intros ;apply P_id_s_bounded;assumption.
            intros ;apply P_id_a__U46_bounded;assumption.
            intros ;apply P_id_U32_bounded;assumption.
            intros ;apply P_id_a__U31_bounded;assumption.
            intros ;apply P_id_U13_bounded;assumption.
            intros ;apply P_id_a__U83_bounded;assumption.
            intros ;apply P_id_isNatIList_bounded;assumption.
            apply rules_monotonic.
            intros ;apply P_id_A__U46_monotonic;assumption.
            intros ;apply P_id_A__ISNATILISTKIND_monotonic;assumption.
            intros ;apply P_id_A__U61_monotonic;assumption.
            intros ;apply P_id_A__U85_monotonic;assumption.
            intros ;apply P_id_A__U31_monotonic;assumption.
            intros ;apply P_id_A__U81_monotonic;assumption.
            intros ;apply P_id_A__U22_monotonic;assumption.
            intros ;apply P_id_A__U93_monotonic;assumption.
            intros ;apply P_id_A__U42_monotonic;assumption.
            intros ;apply P_id_A__U51_monotonic;assumption.
            intros ;apply P_id_A__ISNATLIST_monotonic;assumption.
            intros ;apply P_id_A__U91_monotonic;assumption.
            intros ;apply P_id_A__U33_monotonic;assumption.
            intros ;apply P_id_A__U83_monotonic;assumption.
            intros ;apply P_id_A__U23_monotonic;assumption.
            intros ;apply P_id_A__U11_monotonic;assumption.
            intros ;apply P_id_A__LENGTH_monotonic;assumption.
            intros ;apply P_id_A__U44_monotonic;assumption.
            intros ;apply P_id_A__ISNATILIST_monotonic;assumption.
            intros ;apply P_id_A__U13_monotonic;assumption.
            intros ;apply P_id_A__U71_monotonic;assumption.
            intros ;apply P_id_A__U86_monotonic;assumption.
            intros ;apply P_id_A__U32_monotonic;assumption.
            intros ;apply P_id_A__U82_monotonic;assumption.
            intros ;apply P_id_A__ISNATKIND_monotonic;assumption.
            intros ;apply P_id_A__U94_monotonic;assumption.
            intros ;apply P_id_A__U43_monotonic;assumption.
            intros ;apply P_id_A__U52_monotonic;assumption.
            intros ;apply P_id_A__U21_monotonic;assumption.
            intros ;apply P_id_A__U92_monotonic;assumption.
            intros ;apply P_id_A__U41_monotonic;assumption.
            intros ;apply P_id_A__U84_monotonic;assumption.
            intros ;apply P_id_A__ISNAT_monotonic;assumption.
            intros ;apply P_id_A__U12_monotonic;assumption.
            intros ;apply P_id_MARK_monotonic;assumption.
            intros ;apply P_id_A__U45_monotonic;assumption.
          Qed.
          
          Ltac rewrite_and_unfold  :=
           do 2 (rewrite marked_measure_equation);
            repeat (
            match goal with
              |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
               rewrite (measure_equation (algebra.Alg.Term f t))
              | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ =>
               rewrite (measure_equation (algebra.Alg.Term f t)) in H|-
              end
            ).
          
          Definition lt a b := 
            (Zwf.Zwf 0) (marked_measure a) (marked_measure b).
          
          Definition le a b := marked_measure a <= marked_measure b.
          
          Lemma lt_le_compat : forall a b c, (lt a b) ->(le b c) ->lt a c.
          Proof.
            unfold lt, le in *.
            intros a b c.
            apply (interp.le_lt_compat_right (interp.o_Z 0)).
          Qed.
          
          Lemma wf_lt : well_founded lt.
          Proof.
            unfold lt in *.
            apply Inverse_Image.wf_inverse_image with  (B:=Z).
            apply Zwf.Zwf_well_founded.
          Qed.
          
          Lemma DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_strict_in_lt :
           Relation_Definitions.inclusion _ 
            DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_strict 
            lt.
          Proof.
            unfold Relation_Definitions.inclusion, lt in *.
            
            intros a b H;destruct H;
             match goal with
               |  |- (Zwf.Zwf 0) _ (marked_measure (algebra.Alg.Term ?f ?l)) =>
                let l'' := algebra.Alg_ext.find_replacement l  in 
                 ((apply (interp.le_lt_compat_right (interp.o_Z 0)) with
                    (marked_measure (algebra.Alg.Term f l''));[idtac|
                   apply marked_measure_star_monotonic;
                    repeat (
                    apply algebra.EQT_ext.one_step_list_refl_trans_clos);
                    (assumption)||(constructor 1)]))
               end
             ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_in_le :
           Relation_Definitions.inclusion _ 
            DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large 
            le.
          Proof.
            unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *.
            
            intros a b H;destruct H;
             match goal with
               |  |- _ <= marked_measure (algebra.Alg.Term ?f ?l) =>
                let l'' := algebra.Alg_ext.find_replacement l  in 
                 ((apply (interp.le_trans (interp.o_Z 0)) with
                    (marked_measure (algebra.Alg.Term f l''));[idtac|
                   apply marked_measure_star_monotonic;
                    repeat (
                    apply algebra.EQT_ext.one_step_list_refl_trans_clos);
                    (assumption)||(constructor 1)]))
               end
             ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
             cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
              (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Definition wf_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large  := 
            WF_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large.wf
            .
          
          
          Lemma wf :
           well_founded WF_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large.DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large
            .
          Proof.
            intros x.
            apply (well_founded_ind wf_lt).
            clear x.
            intros x.
            pattern x.
            
            apply (@Acc_ind _ 
                    DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large)
                   .
            clear x.
            intros x _ IHx IHx'.
            constructor.
            intros y H.
            
            destruct H;
             (apply IHx';
               apply DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_strict_in_lt;
               econstructor eassumption)||
             ((apply IHx;[econstructor eassumption|
               intros y' Hlt;apply IHx';apply lt_le_compat with (1:=Hlt) ;
                apply DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large_in_le;
                econstructor eassumption])).
            
            apply wf_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_large.
          Qed.
         End WF_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large.
         
         Open Scope Z_scope.
         
         Import ring_extention.
         
         Notation Local "a <= b" := (Zle a b).
         
         Notation Local "a < b" := (Zlt a b).
         
         Definition P_id_a__zeros  := 1.
         
         Definition P_id_U51 (x11:Z) (x12:Z) := 1* x11.
         
         Definition P_id_a__U84 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
         
         Definition P_id_a__U32 (x11:Z) (x12:Z) := 1* x11.
         
         Definition P_id_isNatList (x11:Z) := 0.
         
         Definition P_id_a__U13 (x11:Z) := 1* x11.
         
         Definition P_id_U85 (x11:Z) (x12:Z) := 1* x11.
         
         Definition P_id_a__length (x11:Z) := 0.
         
         Definition P_id_a__isNatIList (x11:Z) := 1* x11.
         
         Definition P_id_U33 (x11:Z) := 0.
         
         Definition P_id_a__U11 (x11:Z) (x12:Z) := 1* x11.
         
         Definition P_id_U81 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
         
         Definition P_id_a__U92 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
         
         Definition P_id_a__U43 (x11:Z) (x12:Z) (x13:Z) := 1 + 3* x13.
         
         Definition P_id_U23 (x11:Z) := 1* x11.
         
         Definition P_id_a__isNatKind (x11:Z) := 0.
         
         Definition P_id_U93 (x11:Z) (x12:Z) (x13:Z) := 0.
         
         Definition P_id_U11 (x11:Z) (x12:Z) := 1* x11.
         
         Definition P_id_a__U71 (x11:Z) := 1* x11.
         
         Definition P_id_U44 (x11:Z) (x12:Z) (x13:Z) := 1 + 2* x13.
         
         Definition P_id_0  := 0.
         
         Definition P_id_U61 (x11:Z) := 1* x11.
         
         Definition P_id_a__U86 (x11:Z) := 1* x11.
         
         Definition P_id_a__U41 (x11:Z) (x12:Z) (x13:Z) := 1 + 3* x13.
         
         Definition P_id_U22 (x11:Z) (x12:Z) := 1* x11.
         
         Definition P_id_a__U21 (x11:Z) (x12:Z) := 1* x11.
         
         Definition P_id_U91 (x11:Z) (x12:Z) (x13:Z) := 0.
         
         Definition P_id_length (x11:Z) := 0.
         
         Definition P_id_a__U52 (x11:Z) := 1* x11.
         
         Definition P_id_U42 (x11:Z) (x12:Z) (x13:Z) := 2* x13.
         
         Definition P_id_a__U12 (x11:Z) (x12:Z) := 1* x11.
         
         Definition P_id_U83 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
         
         Definition P_id_a__U94 (x11:Z) (x12:Z) := 0.
         
         Definition P_id_a__U45 (x11:Z) (x12:Z) := 1 + 2* x12.
         
         Definition P_id_U31 (x11:Z) (x12:Z) := 0.
         
         Definition P_id_a__isNat (x11:Z) := 0.
         
         Definition P_id_isNatIListKind (x11:Z) := 0.
         
         Definition P_id_a__U82 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
         
         Definition P_id_U46 (x11:Z) := 1 + 2* x11.
         
         Definition P_id_cons (x11:Z) (x12:Z) := 1 + 3* x12.
         
         Definition P_id_U52 (x11:Z) := 1* x11.
         
         Definition P_id_a__U85 (x11:Z) (x12:Z) := 1* x11.
         
         Definition P_id_a__U33 (x11:Z) := 0.
         
         Definition P_id_U21 (x11:Z) (x12:Z) := 1* x11.
         
         Definition P_id_a__isNatList (x11:Z) := 0.
         
         Definition P_id_U86 (x11:Z) := 1* x11.
         
         Definition P_id_mark (x11:Z) := 1 + 2* x11.
         
         Definition P_id_a__U51 (x11:Z) (x12:Z) := 1* x11.
         
         Definition P_id_U41 (x11:Z) (x12:Z) (x13:Z) := 1 + 2* x13.
         
         Definition P_id_tt  := 0.
         
         Definition P_id_U82 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
         
         Definition P_id_a__U93 (x11:Z) (x12:Z) (x13:Z) := 0.
         
         Definition P_id_a__U44 (x11:Z) (x12:Z) (x13:Z) := 1 + 2* x13.
         
         Definition P_id_isNat (x11:Z) := 0.
         
         Definition P_id_a__U23 (x11:Z) := 1* x11.
         
         Definition P_id_U94 (x11:Z) (x12:Z) := 0.
         
         Definition P_id_U12 (x11:Z) (x12:Z) := 1* x11.
         
         Definition P_id_a__U81 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
         
         Definition P_id_U45 (x11:Z) (x12:Z) := 1 + 2* x12.
         
         Definition P_id_zeros  := 0.
         
         Definition P_id_U71 (x11:Z) := 1* x11.
         
         Definition P_id_a__U91 (x11:Z) (x12:Z) (x13:Z) := 0.
         
         Definition P_id_a__U42 (x11:Z) (x12:Z) (x13:Z) := 1 + 3* x13.
         
         Definition P_id_isNatKind (x11:Z) := 0.
         
         Definition P_id_a__U22 (x11:Z) (x12:Z) := 1* x11.
         
         Definition P_id_U92 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
         
         Definition P_id_nil  := 0.
         
         Definition P_id_a__U61 (x11:Z) := 1* x11.
         
         Definition P_id_U43 (x11:Z) (x12:Z) (x13:Z) := 2* x13.
         
         Definition P_id_a__isNatIListKind (x11:Z) := 0.
         
         Definition P_id_U84 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
         
         Definition P_id_s (x11:Z) := 1* x11.
         
         Definition P_id_a__U46 (x11:Z) := 1 + 2* x11.
         
         Definition P_id_U32 (x11:Z) (x12:Z) := 1* x11.
         
         Definition P_id_a__U31 (x11:Z) (x12:Z) := 0.
         
         Definition P_id_U13 (x11:Z) := 1* x11.
         
         Definition P_id_a__U83 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
         
         Definition P_id_isNatIList (x11:Z) := 1* x11.
         
         Lemma P_id_U51_monotonic :
          forall x12 x14 x13 x11, 
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->P_id_U51 x12 x14 <= P_id_U51 x11 x13.
         Proof.
           intros x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U84_monotonic :
          forall x16 x12 x14 x13 x11 x15, 
           (0 <= x16)/\ (x16 <= x15) ->
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_a__U84 x12 x14 x16 <= P_id_a__U84 x11 x13 x15.
         Proof.
           intros x16 x15 x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           intros [H_5 H_4].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U32_monotonic :
          forall x12 x14 x13 x11, 
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_a__U32 x12 x14 <= P_id_a__U32 x11 x13.
         Proof.
           intros x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_isNatList_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_isNatList x12 <= P_id_isNatList x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_0].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U13_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->P_id_a__U13 x12 <= P_id_a__U13 x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_0].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U85_monotonic :
          forall x12 x14 x13 x11, 
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->P_id_U85 x12 x14 <= P_id_U85 x11 x13.
         Proof.
           intros x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__length_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_a__length x12 <= P_id_a__length x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_0].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__isNatIList_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_a__isNatIList x12 <= P_id_a__isNatIList x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_0].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U33_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->P_id_U33 x12 <= P_id_U33 x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_0].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U11_monotonic :
          forall x12 x14 x13 x11, 
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_a__U11 x12 x14 <= P_id_a__U11 x11 x13.
         Proof.
           intros x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U81_monotonic :
          forall x16 x12 x14 x13 x11 x15, 
           (0 <= x16)/\ (x16 <= x15) ->
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_U81 x12 x14 x16 <= P_id_U81 x11 x13 x15.
         Proof.
           intros x16 x15 x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           intros [H_5 H_4].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U92_monotonic :
          forall x16 x12 x14 x13 x11 x15, 
           (0 <= x16)/\ (x16 <= x15) ->
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_a__U92 x12 x14 x16 <= P_id_a__U92 x11 x13 x15.
         Proof.
           intros x16 x15 x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           intros [H_5 H_4].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U43_monotonic :
          forall x16 x12 x14 x13 x11 x15, 
           (0 <= x16)/\ (x16 <= x15) ->
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_a__U43 x12 x14 x16 <= P_id_a__U43 x11 x13 x15.
         Proof.
           intros x16 x15 x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           intros [H_5 H_4].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U23_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->P_id_U23 x12 <= P_id_U23 x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_0].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__isNatKind_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_a__isNatKind x12 <= P_id_a__isNatKind x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_0].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U93_monotonic :
          forall x16 x12 x14 x13 x11 x15, 
           (0 <= x16)/\ (x16 <= x15) ->
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_U93 x12 x14 x16 <= P_id_U93 x11 x13 x15.
         Proof.
           intros x16 x15 x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           intros [H_5 H_4].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U11_monotonic :
          forall x12 x14 x13 x11, 
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->P_id_U11 x12 x14 <= P_id_U11 x11 x13.
         Proof.
           intros x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U71_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->P_id_a__U71 x12 <= P_id_a__U71 x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_0].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U44_monotonic :
          forall x16 x12 x14 x13 x11 x15, 
           (0 <= x16)/\ (x16 <= x15) ->
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_U44 x12 x14 x16 <= P_id_U44 x11 x13 x15.
         Proof.
           intros x16 x15 x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           intros [H_5 H_4].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U61_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->P_id_U61 x12 <= P_id_U61 x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_0].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U86_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->P_id_a__U86 x12 <= P_id_a__U86 x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_0].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U41_monotonic :
          forall x16 x12 x14 x13 x11 x15, 
           (0 <= x16)/\ (x16 <= x15) ->
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_a__U41 x12 x14 x16 <= P_id_a__U41 x11 x13 x15.
         Proof.
           intros x16 x15 x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           intros [H_5 H_4].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U22_monotonic :
          forall x12 x14 x13 x11, 
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->P_id_U22 x12 x14 <= P_id_U22 x11 x13.
         Proof.
           intros x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U21_monotonic :
          forall x12 x14 x13 x11, 
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_a__U21 x12 x14 <= P_id_a__U21 x11 x13.
         Proof.
           intros x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U91_monotonic :
          forall x16 x12 x14 x13 x11 x15, 
           (0 <= x16)/\ (x16 <= x15) ->
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_U91 x12 x14 x16 <= P_id_U91 x11 x13 x15.
         Proof.
           intros x16 x15 x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           intros [H_5 H_4].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_length_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->P_id_length x12 <= P_id_length x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_0].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U52_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->P_id_a__U52 x12 <= P_id_a__U52 x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_0].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U42_monotonic :
          forall x16 x12 x14 x13 x11 x15, 
           (0 <= x16)/\ (x16 <= x15) ->
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_U42 x12 x14 x16 <= P_id_U42 x11 x13 x15.
         Proof.
           intros x16 x15 x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           intros [H_5 H_4].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U12_monotonic :
          forall x12 x14 x13 x11, 
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_a__U12 x12 x14 <= P_id_a__U12 x11 x13.
         Proof.
           intros x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U83_monotonic :
          forall x16 x12 x14 x13 x11 x15, 
           (0 <= x16)/\ (x16 <= x15) ->
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_U83 x12 x14 x16 <= P_id_U83 x11 x13 x15.
         Proof.
           intros x16 x15 x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           intros [H_5 H_4].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U94_monotonic :
          forall x12 x14 x13 x11, 
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_a__U94 x12 x14 <= P_id_a__U94 x11 x13.
         Proof.
           intros x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U45_monotonic :
          forall x12 x14 x13 x11, 
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_a__U45 x12 x14 <= P_id_a__U45 x11 x13.
         Proof.
           intros x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U31_monotonic :
          forall x12 x14 x13 x11, 
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->P_id_U31 x12 x14 <= P_id_U31 x11 x13.
         Proof.
           intros x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__isNat_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->P_id_a__isNat x12 <= P_id_a__isNat x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_0].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_isNatIListKind_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_isNatIListKind x12 <= P_id_isNatIListKind x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_0].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U82_monotonic :
          forall x16 x12 x14 x13 x11 x15, 
           (0 <= x16)/\ (x16 <= x15) ->
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_a__U82 x12 x14 x16 <= P_id_a__U82 x11 x13 x15.
         Proof.
           intros x16 x15 x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           intros [H_5 H_4].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U46_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->P_id_U46 x12 <= P_id_U46 x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_0].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_cons_monotonic :
          forall x12 x14 x13 x11, 
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_cons x12 x14 <= P_id_cons x11 x13.
         Proof.
           intros x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U52_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->P_id_U52 x12 <= P_id_U52 x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_0].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U85_monotonic :
          forall x12 x14 x13 x11, 
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_a__U85 x12 x14 <= P_id_a__U85 x11 x13.
         Proof.
           intros x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U33_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->P_id_a__U33 x12 <= P_id_a__U33 x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_0].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U21_monotonic :
          forall x12 x14 x13 x11, 
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->P_id_U21 x12 x14 <= P_id_U21 x11 x13.
         Proof.
           intros x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__isNatList_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_a__isNatList x12 <= P_id_a__isNatList x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_0].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U86_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->P_id_U86 x12 <= P_id_U86 x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_0].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_mark_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->P_id_mark x12 <= P_id_mark x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_0].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U51_monotonic :
          forall x12 x14 x13 x11, 
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_a__U51 x12 x14 <= P_id_a__U51 x11 x13.
         Proof.
           intros x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U41_monotonic :
          forall x16 x12 x14 x13 x11 x15, 
           (0 <= x16)/\ (x16 <= x15) ->
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_U41 x12 x14 x16 <= P_id_U41 x11 x13 x15.
         Proof.
           intros x16 x15 x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           intros [H_5 H_4].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U82_monotonic :
          forall x16 x12 x14 x13 x11 x15, 
           (0 <= x16)/\ (x16 <= x15) ->
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_U82 x12 x14 x16 <= P_id_U82 x11 x13 x15.
         Proof.
           intros x16 x15 x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           intros [H_5 H_4].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U93_monotonic :
          forall x16 x12 x14 x13 x11 x15, 
           (0 <= x16)/\ (x16 <= x15) ->
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_a__U93 x12 x14 x16 <= P_id_a__U93 x11 x13 x15.
         Proof.
           intros x16 x15 x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           intros [H_5 H_4].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U44_monotonic :
          forall x16 x12 x14 x13 x11 x15, 
           (0 <= x16)/\ (x16 <= x15) ->
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_a__U44 x12 x14 x16 <= P_id_a__U44 x11 x13 x15.
         Proof.
           intros x16 x15 x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           intros [H_5 H_4].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_isNat_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->P_id_isNat x12 <= P_id_isNat x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_0].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U23_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->P_id_a__U23 x12 <= P_id_a__U23 x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_0].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U94_monotonic :
          forall x12 x14 x13 x11, 
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->P_id_U94 x12 x14 <= P_id_U94 x11 x13.
         Proof.
           intros x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U12_monotonic :
          forall x12 x14 x13 x11, 
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->P_id_U12 x12 x14 <= P_id_U12 x11 x13.
         Proof.
           intros x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U81_monotonic :
          forall x16 x12 x14 x13 x11 x15, 
           (0 <= x16)/\ (x16 <= x15) ->
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_a__U81 x12 x14 x16 <= P_id_a__U81 x11 x13 x15.
         Proof.
           intros x16 x15 x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           intros [H_5 H_4].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U45_monotonic :
          forall x12 x14 x13 x11, 
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->P_id_U45 x12 x14 <= P_id_U45 x11 x13.
         Proof.
           intros x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U71_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->P_id_U71 x12 <= P_id_U71 x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_0].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U91_monotonic :
          forall x16 x12 x14 x13 x11 x15, 
           (0 <= x16)/\ (x16 <= x15) ->
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_a__U91 x12 x14 x16 <= P_id_a__U91 x11 x13 x15.
         Proof.
           intros x16 x15 x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           intros [H_5 H_4].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U42_monotonic :
          forall x16 x12 x14 x13 x11 x15, 
           (0 <= x16)/\ (x16 <= x15) ->
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_a__U42 x12 x14 x16 <= P_id_a__U42 x11 x13 x15.
         Proof.
           intros x16 x15 x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           intros [H_5 H_4].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_isNatKind_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_isNatKind x12 <= P_id_isNatKind x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_0].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U22_monotonic :
          forall x12 x14 x13 x11, 
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_a__U22 x12 x14 <= P_id_a__U22 x11 x13.
         Proof.
           intros x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U92_monotonic :
          forall x16 x12 x14 x13 x11 x15, 
           (0 <= x16)/\ (x16 <= x15) ->
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_U92 x12 x14 x16 <= P_id_U92 x11 x13 x15.
         Proof.
           intros x16 x15 x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           intros [H_5 H_4].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U61_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->P_id_a__U61 x12 <= P_id_a__U61 x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_0].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U43_monotonic :
          forall x16 x12 x14 x13 x11 x15, 
           (0 <= x16)/\ (x16 <= x15) ->
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_U43 x12 x14 x16 <= P_id_U43 x11 x13 x15.
         Proof.
           intros x16 x15 x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           intros [H_5 H_4].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__isNatIListKind_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_a__isNatIListKind x12 <= P_id_a__isNatIListKind x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_0].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U84_monotonic :
          forall x16 x12 x14 x13 x11 x15, 
           (0 <= x16)/\ (x16 <= x15) ->
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_U84 x12 x14 x16 <= P_id_U84 x11 x13 x15.
         Proof.
           intros x16 x15 x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           intros [H_5 H_4].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_s_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->P_id_s x12 <= P_id_s x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_0].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U46_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->P_id_a__U46 x12 <= P_id_a__U46 x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_0].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U32_monotonic :
          forall x12 x14 x13 x11, 
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->P_id_U32 x12 x14 <= P_id_U32 x11 x13.
         Proof.
           intros x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U31_monotonic :
          forall x12 x14 x13 x11, 
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_a__U31 x12 x14 <= P_id_a__U31 x11 x13.
         Proof.
           intros x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U13_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->P_id_U13 x12 <= P_id_U13 x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_0].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U83_monotonic :
          forall x16 x12 x14 x13 x11 x15, 
           (0 <= x16)/\ (x16 <= x15) ->
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_a__U83 x12 x14 x16 <= P_id_a__U83 x11 x13 x15.
         Proof.
           intros x16 x15 x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           intros [H_5 H_4].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_isNatIList_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_isNatIList x12 <= P_id_isNatIList x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_0].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__zeros_bounded : 0 <= P_id_a__zeros .
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U51_bounded :
          forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U51 x12 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U84_bounded :
          forall x12 x13 x11, 
           (0 <= x11) ->
            (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U84 x13 x12 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U32_bounded :
          forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U32 x12 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_isNatList_bounded :
          forall x11, (0 <= x11) ->0 <= P_id_isNatList x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U13_bounded :
          forall x11, (0 <= x11) ->0 <= P_id_a__U13 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U85_bounded :
          forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U85 x12 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__length_bounded :
          forall x11, (0 <= x11) ->0 <= P_id_a__length x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__isNatIList_bounded :
          forall x11, (0 <= x11) ->0 <= P_id_a__isNatIList x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U33_bounded : forall x11, (0 <= x11) ->0 <= P_id_U33 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U11_bounded :
          forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U11 x12 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U81_bounded :
          forall x12 x13 x11, 
           (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U81 x13 x12 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U92_bounded :
          forall x12 x13 x11, 
           (0 <= x11) ->
            (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U92 x13 x12 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U43_bounded :
          forall x12 x13 x11, 
           (0 <= x11) ->
            (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U43 x13 x12 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U23_bounded : forall x11, (0 <= x11) ->0 <= P_id_U23 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__isNatKind_bounded :
          forall x11, (0 <= x11) ->0 <= P_id_a__isNatKind x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U93_bounded :
          forall x12 x13 x11, 
           (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U93 x13 x12 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U11_bounded :
          forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U11 x12 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U71_bounded :
          forall x11, (0 <= x11) ->0 <= P_id_a__U71 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U44_bounded :
          forall x12 x13 x11, 
           (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U44 x13 x12 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_0_bounded : 0 <= P_id_0 .
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U61_bounded : forall x11, (0 <= x11) ->0 <= P_id_U61 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U86_bounded :
          forall x11, (0 <= x11) ->0 <= P_id_a__U86 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U41_bounded :
          forall x12 x13 x11, 
           (0 <= x11) ->
            (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U41 x13 x12 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U22_bounded :
          forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U22 x12 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U21_bounded :
          forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U21 x12 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U91_bounded :
          forall x12 x13 x11, 
           (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U91 x13 x12 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_length_bounded :
          forall x11, (0 <= x11) ->0 <= P_id_length x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U52_bounded :
          forall x11, (0 <= x11) ->0 <= P_id_a__U52 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U42_bounded :
          forall x12 x13 x11, 
           (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U42 x13 x12 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U12_bounded :
          forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U12 x12 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U83_bounded :
          forall x12 x13 x11, 
           (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U83 x13 x12 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U94_bounded :
          forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U94 x12 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U45_bounded :
          forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U45 x12 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U31_bounded :
          forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U31 x12 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__isNat_bounded :
          forall x11, (0 <= x11) ->0 <= P_id_a__isNat x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_isNatIListKind_bounded :
          forall x11, (0 <= x11) ->0 <= P_id_isNatIListKind x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U82_bounded :
          forall x12 x13 x11, 
           (0 <= x11) ->
            (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U82 x13 x12 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U46_bounded : forall x11, (0 <= x11) ->0 <= P_id_U46 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_cons_bounded :
          forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_cons x12 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U52_bounded : forall x11, (0 <= x11) ->0 <= P_id_U52 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U85_bounded :
          forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U85 x12 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U33_bounded :
          forall x11, (0 <= x11) ->0 <= P_id_a__U33 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U21_bounded :
          forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U21 x12 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__isNatList_bounded :
          forall x11, (0 <= x11) ->0 <= P_id_a__isNatList x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U86_bounded : forall x11, (0 <= x11) ->0 <= P_id_U86 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_mark_bounded :
          forall x11, (0 <= x11) ->0 <= P_id_mark x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U51_bounded :
          forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U51 x12 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U41_bounded :
          forall x12 x13 x11, 
           (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U41 x13 x12 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_tt_bounded : 0 <= P_id_tt .
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U82_bounded :
          forall x12 x13 x11, 
           (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U82 x13 x12 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U93_bounded :
          forall x12 x13 x11, 
           (0 <= x11) ->
            (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U93 x13 x12 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U44_bounded :
          forall x12 x13 x11, 
           (0 <= x11) ->
            (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U44 x13 x12 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_isNat_bounded :
          forall x11, (0 <= x11) ->0 <= P_id_isNat x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U23_bounded :
          forall x11, (0 <= x11) ->0 <= P_id_a__U23 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U94_bounded :
          forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U94 x12 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U12_bounded :
          forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U12 x12 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U81_bounded :
          forall x12 x13 x11, 
           (0 <= x11) ->
            (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U81 x13 x12 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U45_bounded :
          forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U45 x12 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_zeros_bounded : 0 <= P_id_zeros .
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U71_bounded : forall x11, (0 <= x11) ->0 <= P_id_U71 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U91_bounded :
          forall x12 x13 x11, 
           (0 <= x11) ->
            (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U91 x13 x12 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U42_bounded :
          forall x12 x13 x11, 
           (0 <= x11) ->
            (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U42 x13 x12 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_isNatKind_bounded :
          forall x11, (0 <= x11) ->0 <= P_id_isNatKind x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U22_bounded :
          forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U22 x12 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U92_bounded :
          forall x12 x13 x11, 
           (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U92 x13 x12 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_nil_bounded : 0 <= P_id_nil .
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U61_bounded :
          forall x11, (0 <= x11) ->0 <= P_id_a__U61 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U43_bounded :
          forall x12 x13 x11, 
           (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U43 x13 x12 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__isNatIListKind_bounded :
          forall x11, (0 <= x11) ->0 <= P_id_a__isNatIListKind x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U84_bounded :
          forall x12 x13 x11, 
           (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U84 x13 x12 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_s_bounded : forall x11, (0 <= x11) ->0 <= P_id_s x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U46_bounded :
          forall x11, (0 <= x11) ->0 <= P_id_a__U46 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U32_bounded :
          forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U32 x12 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U31_bounded :
          forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U31 x12 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U13_bounded : forall x11, (0 <= x11) ->0 <= P_id_U13 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_a__U83_bounded :
          forall x12 x13 x11, 
           (0 <= x11) ->
            (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U83 x13 x12 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_isNatIList_bounded :
          forall x11, (0 <= x11) ->0 <= P_id_isNatIList x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Definition measure  := 
           InterpZ.measure 0 P_id_a__zeros P_id_U51 P_id_a__U84 P_id_a__U32 
            P_id_isNatList P_id_a__U13 P_id_U85 P_id_a__length 
            P_id_a__isNatIList P_id_U33 P_id_a__U11 P_id_U81 P_id_a__U92 
            P_id_a__U43 P_id_U23 P_id_a__isNatKind P_id_U93 P_id_U11 
            P_id_a__U71 P_id_U44 P_id_0 P_id_U61 P_id_a__U86 P_id_a__U41 
            P_id_U22 P_id_a__U21 P_id_U91 P_id_length P_id_a__U52 P_id_U42 
            P_id_a__U12 P_id_U83 P_id_a__U94 P_id_a__U45 P_id_U31 
            P_id_a__isNat P_id_isNatIListKind P_id_a__U82 P_id_U46 P_id_cons 
            P_id_U52 P_id_a__U85 P_id_a__U33 P_id_U21 P_id_a__isNatList 
            P_id_U86 P_id_mark P_id_a__U51 P_id_U41 P_id_tt P_id_U82 
            P_id_a__U93 P_id_a__U44 P_id_isNat P_id_a__U23 P_id_U94 P_id_U12 
            P_id_a__U81 P_id_U45 P_id_zeros P_id_U71 P_id_a__U91 P_id_a__U42 
            P_id_isNatKind P_id_a__U22 P_id_U92 P_id_nil P_id_a__U61 
            P_id_U43 P_id_a__isNatIListKind P_id_U84 P_id_s P_id_a__U46 
            P_id_U32 P_id_a__U31 P_id_U13 P_id_a__U83 P_id_isNatIList.
         
         Lemma measure_equation :
          forall t, 
           measure t = match t with
                         | (algebra.Alg.Term algebra.F.id_a__zeros nil) =>
                          P_id_a__zeros 
                         | (algebra.Alg.Term algebra.F.id_U51 (x12::
                            x11::nil)) =>
                          P_id_U51 (measure x12) (measure x11)
                         | (algebra.Alg.Term algebra.F.id_a__U84 (x13::x12::
                            x11::nil)) =>
                          P_id_a__U84 (measure x13) (measure x12) 
                           (measure x11)
                         | (algebra.Alg.Term algebra.F.id_a__U32 (x12::
                            x11::nil)) =>
                          P_id_a__U32 (measure x12) (measure x11)
                         | (algebra.Alg.Term algebra.F.id_isNatList 
                            (x11::nil)) =>
                          P_id_isNatList (measure x11)
                         | (algebra.Alg.Term algebra.F.id_a__U13 (x11::nil)) =>
                          P_id_a__U13 (measure x11)
                         | (algebra.Alg.Term algebra.F.id_U85 (x12::
                            x11::nil)) =>
                          P_id_U85 (measure x12) (measure x11)
                         | (algebra.Alg.Term algebra.F.id_a__length 
                            (x11::nil)) =>
                          P_id_a__length (measure x11)
                         | (algebra.Alg.Term algebra.F.id_a__isNatIList 
                            (x11::nil)) =>
                          P_id_a__isNatIList (measure x11)
                         | (algebra.Alg.Term algebra.F.id_U33 (x11::nil)) =>
                          P_id_U33 (measure x11)
                         | (algebra.Alg.Term algebra.F.id_a__U11 (x12::
                            x11::nil)) =>
                          P_id_a__U11 (measure x12) (measure x11)
                         | (algebra.Alg.Term algebra.F.id_U81 (x13::x12::
                            x11::nil)) =>
                          P_id_U81 (measure x13) (measure x12) (measure x11)
                         | (algebra.Alg.Term algebra.F.id_a__U92 (x13::x12::
                            x11::nil)) =>
                          P_id_a__U92 (measure x13) (measure x12) 
                           (measure x11)
                         | (algebra.Alg.Term algebra.F.id_a__U43 (x13::x12::
                            x11::nil)) =>
                          P_id_a__U43 (measure x13) (measure x12) 
                           (measure x11)
                         | (algebra.Alg.Term algebra.F.id_U23 (x11::nil)) =>
                          P_id_U23 (measure x11)
                         | (algebra.Alg.Term algebra.F.id_a__isNatKind 
                            (x11::nil)) =>
                          P_id_a__isNatKind (measure x11)
                         | (algebra.Alg.Term algebra.F.id_U93 (x13::x12::
                            x11::nil)) =>
                          P_id_U93 (measure x13) (measure x12) (measure x11)
                         | (algebra.Alg.Term algebra.F.id_U11 (x12::
                            x11::nil)) =>
                          P_id_U11 (measure x12) (measure x11)
                         | (algebra.Alg.Term algebra.F.id_a__U71 (x11::nil)) =>
                          P_id_a__U71 (measure x11)
                         | (algebra.Alg.Term algebra.F.id_U44 (x13::x12::
                            x11::nil)) =>
                          P_id_U44 (measure x13) (measure x12) (measure x11)
                         | (algebra.Alg.Term algebra.F.id_0 nil) => P_id_0 
                         | (algebra.Alg.Term algebra.F.id_U61 (x11::nil)) =>
                          P_id_U61 (measure x11)
                         | (algebra.Alg.Term algebra.F.id_a__U86 (x11::nil)) =>
                          P_id_a__U86 (measure x11)
                         | (algebra.Alg.Term algebra.F.id_a__U41 (x13::x12::
                            x11::nil)) =>
                          P_id_a__U41 (measure x13) (measure x12) 
                           (measure x11)
                         | (algebra.Alg.Term algebra.F.id_U22 (x12::
                            x11::nil)) =>
                          P_id_U22 (measure x12) (measure x11)
                         | (algebra.Alg.Term algebra.F.id_a__U21 (x12::
                            x11::nil)) =>
                          P_id_a__U21 (measure x12) (measure x11)
                         | (algebra.Alg.Term algebra.F.id_U91 (x13::x12::
                            x11::nil)) =>
                          P_id_U91 (measure x13) (measure x12) (measure x11)
                         | (algebra.Alg.Term algebra.F.id_length (x11::nil)) =>
                          P_id_length (measure x11)
                         | (algebra.Alg.Term algebra.F.id_a__U52 (x11::nil)) =>
                          P_id_a__U52 (measure x11)
                         | (algebra.Alg.Term algebra.F.id_U42 (x13::x12::
                            x11::nil)) =>
                          P_id_U42 (measure x13) (measure x12) (measure x11)
                         | (algebra.Alg.Term algebra.F.id_a__U12 (x12::
                            x11::nil)) =>
                          P_id_a__U12 (measure x12) (measure x11)
                         | (algebra.Alg.Term algebra.F.id_U83 (x13::x12::
                            x11::nil)) =>
                          P_id_U83 (measure x13) (measure x12) (measure x11)
                         | (algebra.Alg.Term algebra.F.id_a__U94 (x12::
                            x11::nil)) =>
                          P_id_a__U94 (measure x12) (measure x11)
                         | (algebra.Alg.Term algebra.F.id_a__U45 (x12::
                            x11::nil)) =>
                          P_id_a__U45 (measure x12) (measure x11)
                         | (algebra.Alg.Term algebra.F.id_U31 (x12::
                            x11::nil)) =>
                          P_id_U31 (measure x12) (measure x11)
                         | (algebra.Alg.Term algebra.F.id_a__isNat 
                            (x11::nil)) =>
                          P_id_a__isNat (measure x11)
                         | (algebra.Alg.Term algebra.F.id_isNatIListKind 
                            (x11::nil)) =>
                          P_id_isNatIListKind (measure x11)
                         | (algebra.Alg.Term algebra.F.id_a__U82 (x13::x12::
                            x11::nil)) =>
                          P_id_a__U82 (measure x13) (measure x12) 
                           (measure x11)
                         | (algebra.Alg.Term algebra.F.id_U46 (x11::nil)) =>
                          P_id_U46 (measure x11)
                         | (algebra.Alg.Term algebra.F.id_cons (x12::
                            x11::nil)) =>
                          P_id_cons (measure x12) (measure x11)
                         | (algebra.Alg.Term algebra.F.id_U52 (x11::nil)) =>
                          P_id_U52 (measure x11)
                         | (algebra.Alg.Term algebra.F.id_a__U85 (x12::
                            x11::nil)) =>
                          P_id_a__U85 (measure x12) (measure x11)
                         | (algebra.Alg.Term algebra.F.id_a__U33 (x11::nil)) =>
                          P_id_a__U33 (measure x11)
                         | (algebra.Alg.Term algebra.F.id_U21 (x12::
                            x11::nil)) =>
                          P_id_U21 (measure x12) (measure x11)
                         | (algebra.Alg.Term algebra.F.id_a__isNatList 
                            (x11::nil)) =>
                          P_id_a__isNatList (measure x11)
                         | (algebra.Alg.Term algebra.F.id_U86 (x11::nil)) =>
                          P_id_U86 (measure x11)
                         | (algebra.Alg.Term algebra.F.id_mark (x11::nil)) =>
                          P_id_mark (measure x11)
                         | (algebra.Alg.Term algebra.F.id_a__U51 (x12::
                            x11::nil)) =>
                          P_id_a__U51 (measure x12) (measure x11)
                         | (algebra.Alg.Term algebra.F.id_U41 (x13::x12::
                            x11::nil)) =>
                          P_id_U41 (measure x13) (measure x12) (measure x11)
                         | (algebra.Alg.Term algebra.F.id_tt nil) => 
                         P_id_tt 
                         | (algebra.Alg.Term algebra.F.id_U82 (x13::x12::
                            x11::nil)) =>
                          P_id_U82 (measure x13) (measure x12) (measure x11)
                         | (algebra.Alg.Term algebra.F.id_a__U93 (x13::x12::
                            x11::nil)) =>
                          P_id_a__U93 (measure x13) (measure x12) 
                           (measure x11)
                         | (algebra.Alg.Term algebra.F.id_a__U44 (x13::x12::
                            x11::nil)) =>
                          P_id_a__U44 (measure x13) (measure x12) 
                           (measure x11)
                         | (algebra.Alg.Term algebra.F.id_isNat (x11::nil)) =>
                          P_id_isNat (measure x11)
                         | (algebra.Alg.Term algebra.F.id_a__U23 (x11::nil)) =>
                          P_id_a__U23 (measure x11)
                         | (algebra.Alg.Term algebra.F.id_U94 (x12::
                            x11::nil)) =>
                          P_id_U94 (measure x12) (measure x11)
                         | (algebra.Alg.Term algebra.F.id_U12 (x12::
                            x11::nil)) =>
                          P_id_U12 (measure x12) (measure x11)
                         | (algebra.Alg.Term algebra.F.id_a__U81 (x13::x12::
                            x11::nil)) =>
                          P_id_a__U81 (measure x13) (measure x12) 
                           (measure x11)
                         | (algebra.Alg.Term algebra.F.id_U45 (x12::
                            x11::nil)) =>
                          P_id_U45 (measure x12) (measure x11)
                         | (algebra.Alg.Term algebra.F.id_zeros nil) =>
                          P_id_zeros 
                         | (algebra.Alg.Term algebra.F.id_U71 (x11::nil)) =>
                          P_id_U71 (measure x11)
                         | (algebra.Alg.Term algebra.F.id_a__U91 (x13::x12::
                            x11::nil)) =>
                          P_id_a__U91 (measure x13) (measure x12) 
                           (measure x11)
                         | (algebra.Alg.Term algebra.F.id_a__U42 (x13::x12::
                            x11::nil)) =>
                          P_id_a__U42 (measure x13) (measure x12) 
                           (measure x11)
                         | (algebra.Alg.Term algebra.F.id_isNatKind 
                            (x11::nil)) =>
                          P_id_isNatKind (measure x11)
                         | (algebra.Alg.Term algebra.F.id_a__U22 (x12::
                            x11::nil)) =>
                          P_id_a__U22 (measure x12) (measure x11)
                         | (algebra.Alg.Term algebra.F.id_U92 (x13::x12::
                            x11::nil)) =>
                          P_id_U92 (measure x13) (measure x12) (measure x11)
                         | (algebra.Alg.Term algebra.F.id_nil nil) =>
                          P_id_nil 
                         | (algebra.Alg.Term algebra.F.id_a__U61 (x11::nil)) =>
                          P_id_a__U61 (measure x11)
                         | (algebra.Alg.Term algebra.F.id_U43 (x13::x12::
                            x11::nil)) =>
                          P_id_U43 (measure x13) (measure x12) (measure x11)
                         | (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
                            (x11::nil)) =>
                          P_id_a__isNatIListKind (measure x11)
                         | (algebra.Alg.Term algebra.F.id_U84 (x13::x12::
                            x11::nil)) =>
                          P_id_U84 (measure x13) (measure x12) (measure x11)
                         | (algebra.Alg.Term algebra.F.id_s (x11::nil)) =>
                          P_id_s (measure x11)
                         | (algebra.Alg.Term algebra.F.id_a__U46 (x11::nil)) =>
                          P_id_a__U46 (measure x11)
                         | (algebra.Alg.Term algebra.F.id_U32 (x12::
                            x11::nil)) =>
                          P_id_U32 (measure x12) (measure x11)
                         | (algebra.Alg.Term algebra.F.id_a__U31 (x12::
                            x11::nil)) =>
                          P_id_a__U31 (measure x12) (measure x11)
                         | (algebra.Alg.Term algebra.F.id_U13 (x11::nil)) =>
                          P_id_U13 (measure x11)
                         | (algebra.Alg.Term algebra.F.id_a__U83 (x13::x12::
                            x11::nil)) =>
                          P_id_a__U83 (measure x13) (measure x12) 
                           (measure x11)
                         | (algebra.Alg.Term algebra.F.id_isNatIList 
                            (x11::nil)) =>
                          P_id_isNatIList (measure x11)
                         | _ => 0
                         end.
         Proof.
           intros t;case t;intros ;apply refl_equal.
         Qed.
         
         Lemma measure_bounded : forall t, 0 <= measure t.
         Proof.
           unfold measure in |-*.
           
           apply InterpZ.measure_bounded;
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Ltac generate_pos_hyp  :=
          match goal with
            | H:context [measure ?x] |- _ =>
             let v := fresh "v" in 
              (let H := fresh "h" in 
                (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
                  clearbody H;clearbody v))
            |  |- context [measure ?x] =>
             let v := fresh "v" in 
              (let H := fresh "h" in 
                (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
                  clearbody H;clearbody v))
            end
          .
         
         Lemma rules_monotonic :
          forall l r, 
           (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) ->
            measure r <= measure l.
         Proof.
           intros l r H.
           fold measure in |-*.
           
           inversion H;clear H;subst;inversion H0;clear H0;subst;
            simpl algebra.EQT.T.apply_subst in |-*;
            repeat (
            match goal with
              |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
               rewrite (measure_equation (algebra.Alg.Term f t))
              end
            );repeat (generate_pos_hyp );
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma measure_star_monotonic :
          forall l r, 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                      r l) ->measure r <= measure l.
         Proof.
           unfold measure in *.
           apply InterpZ.measure_star_monotonic.
           intros ;apply P_id_U51_monotonic;assumption.
           intros ;apply P_id_a__U84_monotonic;assumption.
           intros ;apply P_id_a__U32_monotonic;assumption.
           intros ;apply P_id_isNatList_monotonic;assumption.
           intros ;apply P_id_a__U13_monotonic;assumption.
           intros ;apply P_id_U85_monotonic;assumption.
           intros ;apply P_id_a__length_monotonic;assumption.
           intros ;apply P_id_a__isNatIList_monotonic;assumption.
           intros ;apply P_id_U33_monotonic;assumption.
           intros ;apply P_id_a__U11_monotonic;assumption.
           intros ;apply P_id_U81_monotonic;assumption.
           intros ;apply P_id_a__U92_monotonic;assumption.
           intros ;apply P_id_a__U43_monotonic;assumption.
           intros ;apply P_id_U23_monotonic;assumption.
           intros ;apply P_id_a__isNatKind_monotonic;assumption.
           intros ;apply P_id_U93_monotonic;assumption.
           intros ;apply P_id_U11_monotonic;assumption.
           intros ;apply P_id_a__U71_monotonic;assumption.
           intros ;apply P_id_U44_monotonic;assumption.
           intros ;apply P_id_U61_monotonic;assumption.
           intros ;apply P_id_a__U86_monotonic;assumption.
           intros ;apply P_id_a__U41_monotonic;assumption.
           intros ;apply P_id_U22_monotonic;assumption.
           intros ;apply P_id_a__U21_monotonic;assumption.
           intros ;apply P_id_U91_monotonic;assumption.
           intros ;apply P_id_length_monotonic;assumption.
           intros ;apply P_id_a__U52_monotonic;assumption.
           intros ;apply P_id_U42_monotonic;assumption.
           intros ;apply P_id_a__U12_monotonic;assumption.
           intros ;apply P_id_U83_monotonic;assumption.
           intros ;apply P_id_a__U94_monotonic;assumption.
           intros ;apply P_id_a__U45_monotonic;assumption.
           intros ;apply P_id_U31_monotonic;assumption.
           intros ;apply P_id_a__isNat_monotonic;assumption.
           intros ;apply P_id_isNatIListKind_monotonic;assumption.
           intros ;apply P_id_a__U82_monotonic;assumption.
           intros ;apply P_id_U46_monotonic;assumption.
           intros ;apply P_id_cons_monotonic;assumption.
           intros ;apply P_id_U52_monotonic;assumption.
           intros ;apply P_id_a__U85_monotonic;assumption.
           intros ;apply P_id_a__U33_monotonic;assumption.
           intros ;apply P_id_U21_monotonic;assumption.
           intros ;apply P_id_a__isNatList_monotonic;assumption.
           intros ;apply P_id_U86_monotonic;assumption.
           intros ;apply P_id_mark_monotonic;assumption.
           intros ;apply P_id_a__U51_monotonic;assumption.
           intros ;apply P_id_U41_monotonic;assumption.
           intros ;apply P_id_U82_monotonic;assumption.
           intros ;apply P_id_a__U93_monotonic;assumption.
           intros ;apply P_id_a__U44_monotonic;assumption.
           intros ;apply P_id_isNat_monotonic;assumption.
           intros ;apply P_id_a__U23_monotonic;assumption.
           intros ;apply P_id_U94_monotonic;assumption.
           intros ;apply P_id_U12_monotonic;assumption.
           intros ;apply P_id_a__U81_monotonic;assumption.
           intros ;apply P_id_U45_monotonic;assumption.
           intros ;apply P_id_U71_monotonic;assumption.
           intros ;apply P_id_a__U91_monotonic;assumption.
           intros ;apply P_id_a__U42_monotonic;assumption.
           intros ;apply P_id_isNatKind_monotonic;assumption.
           intros ;apply P_id_a__U22_monotonic;assumption.
           intros ;apply P_id_U92_monotonic;assumption.
           intros ;apply P_id_a__U61_monotonic;assumption.
           intros ;apply P_id_U43_monotonic;assumption.
           intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
           intros ;apply P_id_U84_monotonic;assumption.
           intros ;apply P_id_s_monotonic;assumption.
           intros ;apply P_id_a__U46_monotonic;assumption.
           intros ;apply P_id_U32_monotonic;assumption.
           intros ;apply P_id_a__U31_monotonic;assumption.
           intros ;apply P_id_U13_monotonic;assumption.
           intros ;apply P_id_a__U83_monotonic;assumption.
           intros ;apply P_id_isNatIList_monotonic;assumption.
           intros ;apply P_id_a__zeros_bounded;assumption.
           intros ;apply P_id_U51_bounded;assumption.
           intros ;apply P_id_a__U84_bounded;assumption.
           intros ;apply P_id_a__U32_bounded;assumption.
           intros ;apply P_id_isNatList_bounded;assumption.
           intros ;apply P_id_a__U13_bounded;assumption.
           intros ;apply P_id_U85_bounded;assumption.
           intros ;apply P_id_a__length_bounded;assumption.
           intros ;apply P_id_a__isNatIList_bounded;assumption.
           intros ;apply P_id_U33_bounded;assumption.
           intros ;apply P_id_a__U11_bounded;assumption.
           intros ;apply P_id_U81_bounded;assumption.
           intros ;apply P_id_a__U92_bounded;assumption.
           intros ;apply P_id_a__U43_bounded;assumption.
           intros ;apply P_id_U23_bounded;assumption.
           intros ;apply P_id_a__isNatKind_bounded;assumption.
           intros ;apply P_id_U93_bounded;assumption.
           intros ;apply P_id_U11_bounded;assumption.
           intros ;apply P_id_a__U71_bounded;assumption.
           intros ;apply P_id_U44_bounded;assumption.
           intros ;apply P_id_0_bounded;assumption.
           intros ;apply P_id_U61_bounded;assumption.
           intros ;apply P_id_a__U86_bounded;assumption.
           intros ;apply P_id_a__U41_bounded;assumption.
           intros ;apply P_id_U22_bounded;assumption.
           intros ;apply P_id_a__U21_bounded;assumption.
           intros ;apply P_id_U91_bounded;assumption.
           intros ;apply P_id_length_bounded;assumption.
           intros ;apply P_id_a__U52_bounded;assumption.
           intros ;apply P_id_U42_bounded;assumption.
           intros ;apply P_id_a__U12_bounded;assumption.
           intros ;apply P_id_U83_bounded;assumption.
           intros ;apply P_id_a__U94_bounded;assumption.
           intros ;apply P_id_a__U45_bounded;assumption.
           intros ;apply P_id_U31_bounded;assumption.
           intros ;apply P_id_a__isNat_bounded;assumption.
           intros ;apply P_id_isNatIListKind_bounded;assumption.
           intros ;apply P_id_a__U82_bounded;assumption.
           intros ;apply P_id_U46_bounded;assumption.
           intros ;apply P_id_cons_bounded;assumption.
           intros ;apply P_id_U52_bounded;assumption.
           intros ;apply P_id_a__U85_bounded;assumption.
           intros ;apply P_id_a__U33_bounded;assumption.
           intros ;apply P_id_U21_bounded;assumption.
           intros ;apply P_id_a__isNatList_bounded;assumption.
           intros ;apply P_id_U86_bounded;assumption.
           intros ;apply P_id_mark_bounded;assumption.
           intros ;apply P_id_a__U51_bounded;assumption.
           intros ;apply P_id_U41_bounded;assumption.
           intros ;apply P_id_tt_bounded;assumption.
           intros ;apply P_id_U82_bounded;assumption.
           intros ;apply P_id_a__U93_bounded;assumption.
           intros ;apply P_id_a__U44_bounded;assumption.
           intros ;apply P_id_isNat_bounded;assumption.
           intros ;apply P_id_a__U23_bounded;assumption.
           intros ;apply P_id_U94_bounded;assumption.
           intros ;apply P_id_U12_bounded;assumption.
           intros ;apply P_id_a__U81_bounded;assumption.
           intros ;apply P_id_U45_bounded;assumption.
           intros ;apply P_id_zeros_bounded;assumption.
           intros ;apply P_id_U71_bounded;assumption.
           intros ;apply P_id_a__U91_bounded;assumption.
           intros ;apply P_id_a__U42_bounded;assumption.
           intros ;apply P_id_isNatKind_bounded;assumption.
           intros ;apply P_id_a__U22_bounded;assumption.
           intros ;apply P_id_U92_bounded;assumption.
           intros ;apply P_id_nil_bounded;assumption.
           intros ;apply P_id_a__U61_bounded;assumption.
           intros ;apply P_id_U43_bounded;assumption.
           intros ;apply P_id_a__isNatIListKind_bounded;assumption.
           intros ;apply P_id_U84_bounded;assumption.
           intros ;apply P_id_s_bounded;assumption.
           intros ;apply P_id_a__U46_bounded;assumption.
           intros ;apply P_id_U32_bounded;assumption.
           intros ;apply P_id_a__U31_bounded;assumption.
           intros ;apply P_id_U13_bounded;assumption.
           intros ;apply P_id_a__U83_bounded;assumption.
           intros ;apply P_id_isNatIList_bounded;assumption.
           apply rules_monotonic.
         Qed.
         
         Definition P_id_A__U46 (x11:Z) := 0.
         
         Definition P_id_A__ISNATILISTKIND (x11:Z) := 0.
         
         Definition P_id_A__U61 (x11:Z) := 0.
         
         Definition P_id_A__U85 (x11:Z) (x12:Z) := 0.
         
         Definition P_id_A__U31 (x11:Z) (x12:Z) := 0.
         
         Definition P_id_A__U81 (x11:Z) (x12:Z) (x13:Z) := 0.
         
         Definition P_id_A__U22 (x11:Z) (x12:Z) := 0.
         
         Definition P_id_A__U93 (x11:Z) (x12:Z) (x13:Z) := 0.
         
         Definition P_id_A__U42 (x11:Z) (x12:Z) (x13:Z) := 0.
         
         Definition P_id_A__U51 (x11:Z) (x12:Z) := 0.
         
         Definition P_id_A__ISNATLIST (x11:Z) := 0.
         
         Definition P_id_A__ZEROS  := 0.
         
         Definition P_id_A__U91 (x11:Z) (x12:Z) (x13:Z) := 0.
         
         Definition P_id_A__U33 (x11:Z) := 0.
         
         Definition P_id_A__U83 (x11:Z) (x12:Z) (x13:Z) := 0.
         
         Definition P_id_A__U23 (x11:Z) := 0.
         
         Definition P_id_A__U11 (x11:Z) (x12:Z) := 0.
         
         Definition P_id_A__LENGTH (x11:Z) := 0.
         
         Definition P_id_A__U44 (x11:Z) (x12:Z) (x13:Z) := 0.
         
         Definition P_id_A__ISNATILIST (x11:Z) := 0.
         
         Definition P_id_A__U13 (x11:Z) := 0.
         
         Definition P_id_A__U71 (x11:Z) := 0.
         
         Definition P_id_A__U86 (x11:Z) := 0.
         
         Definition P_id_A__U32 (x11:Z) (x12:Z) := 0.
         
         Definition P_id_A__U82 (x11:Z) (x12:Z) (x13:Z) := 0.
         
         Definition P_id_A__ISNATKIND (x11:Z) := 0.
         
         Definition P_id_A__U94 (x11:Z) (x12:Z) := 0.
         
         Definition P_id_A__U43 (x11:Z) (x12:Z) (x13:Z) := 0.
         
         Definition P_id_A__U52 (x11:Z) := 0.
         
         Definition P_id_A__U21 (x11:Z) (x12:Z) := 0.
         
         Definition P_id_A__U92 (x11:Z) (x12:Z) (x13:Z) := 0.
         
         Definition P_id_A__U41 (x11:Z) (x12:Z) (x13:Z) := 0.
         
         Definition P_id_A__U84 (x11:Z) (x12:Z) (x13:Z) := 0.
         
         Definition P_id_A__ISNAT (x11:Z) := 0.
         
         Definition P_id_A__U12 (x11:Z) (x12:Z) := 0.
         
         Definition P_id_MARK (x11:Z) := 2* x11.
         
         Definition P_id_A__U45 (x11:Z) (x12:Z) := 0.
         
         Lemma P_id_A__U46_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->P_id_A__U46 x12 <= P_id_A__U46 x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_0].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_A__ISNATILISTKIND_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_A__ISNATILISTKIND x12 <= P_id_A__ISNATILISTKIND x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_0].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_A__U61_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->P_id_A__U61 x12 <= P_id_A__U61 x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_0].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_A__U85_monotonic :
          forall x12 x14 x13 x11, 
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_A__U85 x12 x14 <= P_id_A__U85 x11 x13.
         Proof.
           intros x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_A__U31_monotonic :
          forall x12 x14 x13 x11, 
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_A__U31 x12 x14 <= P_id_A__U31 x11 x13.
         Proof.
           intros x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_A__U81_monotonic :
          forall x16 x12 x14 x13 x11 x15, 
           (0 <= x16)/\ (x16 <= x15) ->
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_A__U81 x12 x14 x16 <= P_id_A__U81 x11 x13 x15.
         Proof.
           intros x16 x15 x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           intros [H_5 H_4].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_A__U22_monotonic :
          forall x12 x14 x13 x11, 
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_A__U22 x12 x14 <= P_id_A__U22 x11 x13.
         Proof.
           intros x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_A__U93_monotonic :
          forall x16 x12 x14 x13 x11 x15, 
           (0 <= x16)/\ (x16 <= x15) ->
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_A__U93 x12 x14 x16 <= P_id_A__U93 x11 x13 x15.
         Proof.
           intros x16 x15 x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           intros [H_5 H_4].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_A__U42_monotonic :
          forall x16 x12 x14 x13 x11 x15, 
           (0 <= x16)/\ (x16 <= x15) ->
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_A__U42 x12 x14 x16 <= P_id_A__U42 x11 x13 x15.
         Proof.
           intros x16 x15 x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           intros [H_5 H_4].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_A__U51_monotonic :
          forall x12 x14 x13 x11, 
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_A__U51 x12 x14 <= P_id_A__U51 x11 x13.
         Proof.
           intros x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_A__ISNATLIST_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_A__ISNATLIST x12 <= P_id_A__ISNATLIST x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_0].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_A__U91_monotonic :
          forall x16 x12 x14 x13 x11 x15, 
           (0 <= x16)/\ (x16 <= x15) ->
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_A__U91 x12 x14 x16 <= P_id_A__U91 x11 x13 x15.
         Proof.
           intros x16 x15 x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           intros [H_5 H_4].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_A__U33_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->P_id_A__U33 x12 <= P_id_A__U33 x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_0].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_A__U83_monotonic :
          forall x16 x12 x14 x13 x11 x15, 
           (0 <= x16)/\ (x16 <= x15) ->
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_A__U83 x12 x14 x16 <= P_id_A__U83 x11 x13 x15.
         Proof.
           intros x16 x15 x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           intros [H_5 H_4].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_A__U23_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->P_id_A__U23 x12 <= P_id_A__U23 x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_0].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_A__U11_monotonic :
          forall x12 x14 x13 x11, 
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_A__U11 x12 x14 <= P_id_A__U11 x11 x13.
         Proof.
           intros x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_A__LENGTH_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_A__LENGTH x12 <= P_id_A__LENGTH x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_0].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_A__U44_monotonic :
          forall x16 x12 x14 x13 x11 x15, 
           (0 <= x16)/\ (x16 <= x15) ->
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_A__U44 x12 x14 x16 <= P_id_A__U44 x11 x13 x15.
         Proof.
           intros x16 x15 x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           intros [H_5 H_4].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_A__ISNATILIST_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_A__ISNATILIST x12 <= P_id_A__ISNATILIST x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_0].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_A__U13_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->P_id_A__U13 x12 <= P_id_A__U13 x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_0].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_A__U71_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->P_id_A__U71 x12 <= P_id_A__U71 x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_0].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_A__U86_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->P_id_A__U86 x12 <= P_id_A__U86 x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_0].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_A__U32_monotonic :
          forall x12 x14 x13 x11, 
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_A__U32 x12 x14 <= P_id_A__U32 x11 x13.
         Proof.
           intros x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_A__U82_monotonic :
          forall x16 x12 x14 x13 x11 x15, 
           (0 <= x16)/\ (x16 <= x15) ->
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_A__U82 x12 x14 x16 <= P_id_A__U82 x11 x13 x15.
         Proof.
           intros x16 x15 x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           intros [H_5 H_4].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_A__ISNATKIND_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_A__ISNATKIND x12 <= P_id_A__ISNATKIND x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_0].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_A__U94_monotonic :
          forall x12 x14 x13 x11, 
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_A__U94 x12 x14 <= P_id_A__U94 x11 x13.
         Proof.
           intros x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_A__U43_monotonic :
          forall x16 x12 x14 x13 x11 x15, 
           (0 <= x16)/\ (x16 <= x15) ->
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_A__U43 x12 x14 x16 <= P_id_A__U43 x11 x13 x15.
         Proof.
           intros x16 x15 x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           intros [H_5 H_4].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_A__U52_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->P_id_A__U52 x12 <= P_id_A__U52 x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_0].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_A__U21_monotonic :
          forall x12 x14 x13 x11, 
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_A__U21 x12 x14 <= P_id_A__U21 x11 x13.
         Proof.
           intros x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_A__U92_monotonic :
          forall x16 x12 x14 x13 x11 x15, 
           (0 <= x16)/\ (x16 <= x15) ->
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_A__U92 x12 x14 x16 <= P_id_A__U92 x11 x13 x15.
         Proof.
           intros x16 x15 x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           intros [H_5 H_4].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_A__U41_monotonic :
          forall x16 x12 x14 x13 x11 x15, 
           (0 <= x16)/\ (x16 <= x15) ->
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_A__U41 x12 x14 x16 <= P_id_A__U41 x11 x13 x15.
         Proof.
           intros x16 x15 x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           intros [H_5 H_4].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_A__U84_monotonic :
          forall x16 x12 x14 x13 x11 x15, 
           (0 <= x16)/\ (x16 <= x15) ->
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_A__U84 x12 x14 x16 <= P_id_A__U84 x11 x13 x15.
         Proof.
           intros x16 x15 x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           intros [H_5 H_4].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_A__ISNAT_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->P_id_A__ISNAT x12 <= P_id_A__ISNAT x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_0].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_A__U12_monotonic :
          forall x12 x14 x13 x11, 
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_A__U12 x12 x14 <= P_id_A__U12 x11 x13.
         Proof.
           intros x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_MARK_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->P_id_MARK x12 <= P_id_MARK x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_0].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_A__U45_monotonic :
          forall x12 x14 x13 x11, 
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_A__U45 x12 x14 <= P_id_A__U45 x11 x13.
         Proof.
           intros x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Definition marked_measure  := 
           InterpZ.marked_measure 0 P_id_a__zeros P_id_U51 P_id_a__U84 
            P_id_a__U32 P_id_isNatList P_id_a__U13 P_id_U85 P_id_a__length 
            P_id_a__isNatIList P_id_U33 P_id_a__U11 P_id_U81 P_id_a__U92 
            P_id_a__U43 P_id_U23 P_id_a__isNatKind P_id_U93 P_id_U11 
            P_id_a__U71 P_id_U44 P_id_0 P_id_U61 P_id_a__U86 P_id_a__U41 
            P_id_U22 P_id_a__U21 P_id_U91 P_id_length P_id_a__U52 P_id_U42 
            P_id_a__U12 P_id_U83 P_id_a__U94 P_id_a__U45 P_id_U31 
            P_id_a__isNat P_id_isNatIListKind P_id_a__U82 P_id_U46 P_id_cons 
            P_id_U52 P_id_a__U85 P_id_a__U33 P_id_U21 P_id_a__isNatList 
            P_id_U86 P_id_mark P_id_a__U51 P_id_U41 P_id_tt P_id_U82 
            P_id_a__U93 P_id_a__U44 P_id_isNat P_id_a__U23 P_id_U94 P_id_U12 
            P_id_a__U81 P_id_U45 P_id_zeros P_id_U71 P_id_a__U91 P_id_a__U42 
            P_id_isNatKind P_id_a__U22 P_id_U92 P_id_nil P_id_a__U61 
            P_id_U43 P_id_a__isNatIListKind P_id_U84 P_id_s P_id_a__U46 
            P_id_U32 P_id_a__U31 P_id_U13 P_id_a__U83 P_id_isNatIList 
            P_id_A__U46 P_id_A__ISNATILISTKIND P_id_A__U61 P_id_A__U85 
            P_id_A__U31 P_id_A__U81 P_id_A__U22 P_id_A__U93 P_id_A__U42 
            P_id_A__U51 P_id_A__ISNATLIST P_id_A__ZEROS P_id_A__U91 
            P_id_A__U33 P_id_A__U83 P_id_A__U23 P_id_A__U11 P_id_A__LENGTH 
            P_id_A__U44 P_id_A__ISNATILIST P_id_A__U13 P_id_A__U71 
            P_id_A__U86 P_id_A__U32 P_id_A__U82 P_id_A__ISNATKIND 
            P_id_A__U94 P_id_A__U43 P_id_A__U52 P_id_A__U21 P_id_A__U92 
            P_id_A__U41 P_id_A__U84 P_id_A__ISNAT P_id_A__U12 P_id_MARK 
            P_id_A__U45.
         
         Lemma marked_measure_equation :
          forall t, 
           marked_measure t = match t with
                                | (algebra.Alg.Term algebra.F.id_a__U46 
                                   (x11::nil)) =>
                                 P_id_A__U46 (measure x11)
                                | (algebra.Alg.Term 
                                   algebra.F.id_a__isNatIListKind (x11::nil)) =>
                                 P_id_A__ISNATILISTKIND (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U61 
                                   (x11::nil)) =>
                                 P_id_A__U61 (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U85 
                                   (x12::x11::nil)) =>
                                 P_id_A__U85 (measure x12) (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U31 
                                   (x12::x11::nil)) =>
                                 P_id_A__U31 (measure x12) (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U81 
                                   (x13::x12::x11::nil)) =>
                                 P_id_A__U81 (measure x13) (measure x12) 
                                  (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U22 
                                   (x12::x11::nil)) =>
                                 P_id_A__U22 (measure x12) (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U93 
                                   (x13::x12::x11::nil)) =>
                                 P_id_A__U93 (measure x13) (measure x12) 
                                  (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U42 
                                   (x13::x12::x11::nil)) =>
                                 P_id_A__U42 (measure x13) (measure x12) 
                                  (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U51 
                                   (x12::x11::nil)) =>
                                 P_id_A__U51 (measure x12) (measure x11)
                                | (algebra.Alg.Term 
                                   algebra.F.id_a__isNatList (x11::nil)) =>
                                 P_id_A__ISNATLIST (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__zeros 
                                   nil) =>
                                 P_id_A__ZEROS 
                                | (algebra.Alg.Term algebra.F.id_a__U91 
                                   (x13::x12::x11::nil)) =>
                                 P_id_A__U91 (measure x13) (measure x12) 
                                  (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U33 
                                   (x11::nil)) =>
                                 P_id_A__U33 (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U83 
                                   (x13::x12::x11::nil)) =>
                                 P_id_A__U83 (measure x13) (measure x12) 
                                  (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U23 
                                   (x11::nil)) =>
                                 P_id_A__U23 (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U11 
                                   (x12::x11::nil)) =>
                                 P_id_A__U11 (measure x12) (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__length 
                                   (x11::nil)) =>
                                 P_id_A__LENGTH (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U44 
                                   (x13::x12::x11::nil)) =>
                                 P_id_A__U44 (measure x13) (measure x12) 
                                  (measure x11)
                                | (algebra.Alg.Term 
                                   algebra.F.id_a__isNatIList (x11::nil)) =>
                                 P_id_A__ISNATILIST (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U13 
                                   (x11::nil)) =>
                                 P_id_A__U13 (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U71 
                                   (x11::nil)) =>
                                 P_id_A__U71 (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U86 
                                   (x11::nil)) =>
                                 P_id_A__U86 (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U32 
                                   (x12::x11::nil)) =>
                                 P_id_A__U32 (measure x12) (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U82 
                                   (x13::x12::x11::nil)) =>
                                 P_id_A__U82 (measure x13) (measure x12) 
                                  (measure x11)
                                | (algebra.Alg.Term 
                                   algebra.F.id_a__isNatKind (x11::nil)) =>
                                 P_id_A__ISNATKIND (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U94 
                                   (x12::x11::nil)) =>
                                 P_id_A__U94 (measure x12) (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U43 
                                   (x13::x12::x11::nil)) =>
                                 P_id_A__U43 (measure x13) (measure x12) 
                                  (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U52 
                                   (x11::nil)) =>
                                 P_id_A__U52 (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U21 
                                   (x12::x11::nil)) =>
                                 P_id_A__U21 (measure x12) (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U92 
                                   (x13::x12::x11::nil)) =>
                                 P_id_A__U92 (measure x13) (measure x12) 
                                  (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U41 
                                   (x13::x12::x11::nil)) =>
                                 P_id_A__U41 (measure x13) (measure x12) 
                                  (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U84 
                                   (x13::x12::x11::nil)) =>
                                 P_id_A__U84 (measure x13) (measure x12) 
                                  (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__isNat 
                                   (x11::nil)) =>
                                 P_id_A__ISNAT (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U12 
                                   (x12::x11::nil)) =>
                                 P_id_A__U12 (measure x12) (measure x11)
                                | (algebra.Alg.Term algebra.F.id_mark 
                                   (x11::nil)) =>
                                 P_id_MARK (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U45 
                                   (x12::x11::nil)) =>
                                 P_id_A__U45 (measure x12) (measure x11)
                                | _ => measure t
                                end.
         Proof.
           reflexivity .
         Qed.
         
         Lemma marked_measure_star_monotonic :
          forall f l1 l2, 
           (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step 
                                                             R_xml_0_deep_rew.R_xml_0_rules)
                                                            ) l1 l2) ->
            marked_measure (algebra.Alg.Term f l1) <= marked_measure 
                                                       (algebra.Alg.Term f l2).
         Proof.
           unfold marked_measure in *.
           apply InterpZ.marked_measure_star_monotonic.
           intros ;apply P_id_U51_monotonic;assumption.
           intros ;apply P_id_a__U84_monotonic;assumption.
           intros ;apply P_id_a__U32_monotonic;assumption.
           intros ;apply P_id_isNatList_monotonic;assumption.
           intros ;apply P_id_a__U13_monotonic;assumption.
           intros ;apply P_id_U85_monotonic;assumption.
           intros ;apply P_id_a__length_monotonic;assumption.
           intros ;apply P_id_a__isNatIList_monotonic;assumption.
           intros ;apply P_id_U33_monotonic;assumption.
           intros ;apply P_id_a__U11_monotonic;assumption.
           intros ;apply P_id_U81_monotonic;assumption.
           intros ;apply P_id_a__U92_monotonic;assumption.
           intros ;apply P_id_a__U43_monotonic;assumption.
           intros ;apply P_id_U23_monotonic;assumption.
           intros ;apply P_id_a__isNatKind_monotonic;assumption.
           intros ;apply P_id_U93_monotonic;assumption.
           intros ;apply P_id_U11_monotonic;assumption.
           intros ;apply P_id_a__U71_monotonic;assumption.
           intros ;apply P_id_U44_monotonic;assumption.
           intros ;apply P_id_U61_monotonic;assumption.
           intros ;apply P_id_a__U86_monotonic;assumption.
           intros ;apply P_id_a__U41_monotonic;assumption.
           intros ;apply P_id_U22_monotonic;assumption.
           intros ;apply P_id_a__U21_monotonic;assumption.
           intros ;apply P_id_U91_monotonic;assumption.
           intros ;apply P_id_length_monotonic;assumption.
           intros ;apply P_id_a__U52_monotonic;assumption.
           intros ;apply P_id_U42_monotonic;assumption.
           intros ;apply P_id_a__U12_monotonic;assumption.
           intros ;apply P_id_U83_monotonic;assumption.
           intros ;apply P_id_a__U94_monotonic;assumption.
           intros ;apply P_id_a__U45_monotonic;assumption.
           intros ;apply P_id_U31_monotonic;assumption.
           intros ;apply P_id_a__isNat_monotonic;assumption.
           intros ;apply P_id_isNatIListKind_monotonic;assumption.
           intros ;apply P_id_a__U82_monotonic;assumption.
           intros ;apply P_id_U46_monotonic;assumption.
           intros ;apply P_id_cons_monotonic;assumption.
           intros ;apply P_id_U52_monotonic;assumption.
           intros ;apply P_id_a__U85_monotonic;assumption.
           intros ;apply P_id_a__U33_monotonic;assumption.
           intros ;apply P_id_U21_monotonic;assumption.
           intros ;apply P_id_a__isNatList_monotonic;assumption.
           intros ;apply P_id_U86_monotonic;assumption.
           intros ;apply P_id_mark_monotonic;assumption.
           intros ;apply P_id_a__U51_monotonic;assumption.
           intros ;apply P_id_U41_monotonic;assumption.
           intros ;apply P_id_U82_monotonic;assumption.
           intros ;apply P_id_a__U93_monotonic;assumption.
           intros ;apply P_id_a__U44_monotonic;assumption.
           intros ;apply P_id_isNat_monotonic;assumption.
           intros ;apply P_id_a__U23_monotonic;assumption.
           intros ;apply P_id_U94_monotonic;assumption.
           intros ;apply P_id_U12_monotonic;assumption.
           intros ;apply P_id_a__U81_monotonic;assumption.
           intros ;apply P_id_U45_monotonic;assumption.
           intros ;apply P_id_U71_monotonic;assumption.
           intros ;apply P_id_a__U91_monotonic;assumption.
           intros ;apply P_id_a__U42_monotonic;assumption.
           intros ;apply P_id_isNatKind_monotonic;assumption.
           intros ;apply P_id_a__U22_monotonic;assumption.
           intros ;apply P_id_U92_monotonic;assumption.
           intros ;apply P_id_a__U61_monotonic;assumption.
           intros ;apply P_id_U43_monotonic;assumption.
           intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
           intros ;apply P_id_U84_monotonic;assumption.
           intros ;apply P_id_s_monotonic;assumption.
           intros ;apply P_id_a__U46_monotonic;assumption.
           intros ;apply P_id_U32_monotonic;assumption.
           intros ;apply P_id_a__U31_monotonic;assumption.
           intros ;apply P_id_U13_monotonic;assumption.
           intros ;apply P_id_a__U83_monotonic;assumption.
           intros ;apply P_id_isNatIList_monotonic;assumption.
           intros ;apply P_id_a__zeros_bounded;assumption.
           intros ;apply P_id_U51_bounded;assumption.
           intros ;apply P_id_a__U84_bounded;assumption.
           intros ;apply P_id_a__U32_bounded;assumption.
           intros ;apply P_id_isNatList_bounded;assumption.
           intros ;apply P_id_a__U13_bounded;assumption.
           intros ;apply P_id_U85_bounded;assumption.
           intros ;apply P_id_a__length_bounded;assumption.
           intros ;apply P_id_a__isNatIList_bounded;assumption.
           intros ;apply P_id_U33_bounded;assumption.
           intros ;apply P_id_a__U11_bounded;assumption.
           intros ;apply P_id_U81_bounded;assumption.
           intros ;apply P_id_a__U92_bounded;assumption.
           intros ;apply P_id_a__U43_bounded;assumption.
           intros ;apply P_id_U23_bounded;assumption.
           intros ;apply P_id_a__isNatKind_bounded;assumption.
           intros ;apply P_id_U93_bounded;assumption.
           intros ;apply P_id_U11_bounded;assumption.
           intros ;apply P_id_a__U71_bounded;assumption.
           intros ;apply P_id_U44_bounded;assumption.
           intros ;apply P_id_0_bounded;assumption.
           intros ;apply P_id_U61_bounded;assumption.
           intros ;apply P_id_a__U86_bounded;assumption.
           intros ;apply P_id_a__U41_bounded;assumption.
           intros ;apply P_id_U22_bounded;assumption.
           intros ;apply P_id_a__U21_bounded;assumption.
           intros ;apply P_id_U91_bounded;assumption.
           intros ;apply P_id_length_bounded;assumption.
           intros ;apply P_id_a__U52_bounded;assumption.
           intros ;apply P_id_U42_bounded;assumption.
           intros ;apply P_id_a__U12_bounded;assumption.
           intros ;apply P_id_U83_bounded;assumption.
           intros ;apply P_id_a__U94_bounded;assumption.
           intros ;apply P_id_a__U45_bounded;assumption.
           intros ;apply P_id_U31_bounded;assumption.
           intros ;apply P_id_a__isNat_bounded;assumption.
           intros ;apply P_id_isNatIListKind_bounded;assumption.
           intros ;apply P_id_a__U82_bounded;assumption.
           intros ;apply P_id_U46_bounded;assumption.
           intros ;apply P_id_cons_bounded;assumption.
           intros ;apply P_id_U52_bounded;assumption.
           intros ;apply P_id_a__U85_bounded;assumption.
           intros ;apply P_id_a__U33_bounded;assumption.
           intros ;apply P_id_U21_bounded;assumption.
           intros ;apply P_id_a__isNatList_bounded;assumption.
           intros ;apply P_id_U86_bounded;assumption.
           intros ;apply P_id_mark_bounded;assumption.
           intros ;apply P_id_a__U51_bounded;assumption.
           intros ;apply P_id_U41_bounded;assumption.
           intros ;apply P_id_tt_bounded;assumption.
           intros ;apply P_id_U82_bounded;assumption.
           intros ;apply P_id_a__U93_bounded;assumption.
           intros ;apply P_id_a__U44_bounded;assumption.
           intros ;apply P_id_isNat_bounded;assumption.
           intros ;apply P_id_a__U23_bounded;assumption.
           intros ;apply P_id_U94_bounded;assumption.
           intros ;apply P_id_U12_bounded;assumption.
           intros ;apply P_id_a__U81_bounded;assumption.
           intros ;apply P_id_U45_bounded;assumption.
           intros ;apply P_id_zeros_bounded;assumption.
           intros ;apply P_id_U71_bounded;assumption.
           intros ;apply P_id_a__U91_bounded;assumption.
           intros ;apply P_id_a__U42_bounded;assumption.
           intros ;apply P_id_isNatKind_bounded;assumption.
           intros ;apply P_id_a__U22_bounded;assumption.
           intros ;apply P_id_U92_bounded;assumption.
           intros ;apply P_id_nil_bounded;assumption.
           intros ;apply P_id_a__U61_bounded;assumption.
           intros ;apply P_id_U43_bounded;assumption.
           intros ;apply P_id_a__isNatIListKind_bounded;assumption.
           intros ;apply P_id_U84_bounded;assumption.
           intros ;apply P_id_s_bounded;assumption.
           intros ;apply P_id_a__U46_bounded;assumption.
           intros ;apply P_id_U32_bounded;assumption.
           intros ;apply P_id_a__U31_bounded;assumption.
           intros ;apply P_id_U13_bounded;assumption.
           intros ;apply P_id_a__U83_bounded;assumption.
           intros ;apply P_id_isNatIList_bounded;assumption.
           apply rules_monotonic.
           intros ;apply P_id_A__U46_monotonic;assumption.
           intros ;apply P_id_A__ISNATILISTKIND_monotonic;assumption.
           intros ;apply P_id_A__U61_monotonic;assumption.
           intros ;apply P_id_A__U85_monotonic;assumption.
           intros ;apply P_id_A__U31_monotonic;assumption.
           intros ;apply P_id_A__U81_monotonic;assumption.
           intros ;apply P_id_A__U22_monotonic;assumption.
           intros ;apply P_id_A__U93_monotonic;assumption.
           intros ;apply P_id_A__U42_monotonic;assumption.
           intros ;apply P_id_A__U51_monotonic;assumption.
           intros ;apply P_id_A__ISNATLIST_monotonic;assumption.
           intros ;apply P_id_A__U91_monotonic;assumption.
           intros ;apply P_id_A__U33_monotonic;assumption.
           intros ;apply P_id_A__U83_monotonic;assumption.
           intros ;apply P_id_A__U23_monotonic;assumption.
           intros ;apply P_id_A__U11_monotonic;assumption.
           intros ;apply P_id_A__LENGTH_monotonic;assumption.
           intros ;apply P_id_A__U44_monotonic;assumption.
           intros ;apply P_id_A__ISNATILIST_monotonic;assumption.
           intros ;apply P_id_A__U13_monotonic;assumption.
           intros ;apply P_id_A__U71_monotonic;assumption.
           intros ;apply P_id_A__U86_monotonic;assumption.
           intros ;apply P_id_A__U32_monotonic;assumption.
           intros ;apply P_id_A__U82_monotonic;assumption.
           intros ;apply P_id_A__ISNATKIND_monotonic;assumption.
           intros ;apply P_id_A__U94_monotonic;assumption.
           intros ;apply P_id_A__U43_monotonic;assumption.
           intros ;apply P_id_A__U52_monotonic;assumption.
           intros ;apply P_id_A__U21_monotonic;assumption.
           intros ;apply P_id_A__U92_monotonic;assumption.
           intros ;apply P_id_A__U41_monotonic;assumption.
           intros ;apply P_id_A__U84_monotonic;assumption.
           intros ;apply P_id_A__ISNAT_monotonic;assumption.
           intros ;apply P_id_A__U12_monotonic;assumption.
           intros ;apply P_id_MARK_monotonic;assumption.
           intros ;apply P_id_A__U45_monotonic;assumption.
         Qed.
         
         Ltac rewrite_and_unfold  :=
          do 2 (rewrite marked_measure_equation);
           repeat (
           match goal with
             |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
              rewrite (measure_equation (algebra.Alg.Term f t))
             | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ =>
              rewrite (measure_equation (algebra.Alg.Term f t)) in H|-
             end
           ).
         
         Definition lt a b := 
           (Zwf.Zwf 0) (marked_measure a) (marked_measure b).
         
         Definition le a b := marked_measure a <= marked_measure b.
         
         Lemma lt_le_compat : forall a b c, (lt a b) ->(le b c) ->lt a c.
         Proof.
           unfold lt, le in *.
           intros a b c.
           apply (interp.le_lt_compat_right (interp.o_Z 0)).
         Qed.
         
         Lemma wf_lt : well_founded lt.
         Proof.
           unfold lt in *.
           apply Inverse_Image.wf_inverse_image with  (B:=Z).
           apply Zwf.Zwf_well_founded.
         Qed.
         
         Lemma DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_strict_in_lt :
          Relation_Definitions.inclusion _ 
           DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_strict 
           lt.
         Proof.
           unfold Relation_Definitions.inclusion, lt in *.
           
           intros a b H;destruct H;
            match goal with
              |  |- (Zwf.Zwf 0) _ (marked_measure (algebra.Alg.Term ?f ?l)) =>
               let l'' := algebra.Alg_ext.find_replacement l  in 
                ((apply (interp.le_lt_compat_right (interp.o_Z 0)) with
                   (marked_measure (algebra.Alg.Term f l''));[idtac|
                  apply marked_measure_star_monotonic;
                   repeat (
                   apply algebra.EQT_ext.one_step_list_refl_trans_clos);
                   (assumption)||(constructor 1)]))
              end
            ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_in_le :
          Relation_Definitions.inclusion _ 
           DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large 
           le.
         Proof.
           unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *.
           
           intros a b H;destruct H;
            match goal with
              |  |- _ <= marked_measure (algebra.Alg.Term ?f ?l) =>
               let l'' := algebra.Alg_ext.find_replacement l  in 
                ((apply (interp.le_trans (interp.o_Z 0)) with
                   (marked_measure (algebra.Alg.Term f l''));[idtac|
                  apply marked_measure_star_monotonic;
                   repeat (
                   apply algebra.EQT_ext.one_step_list_refl_trans_clos);
                   (assumption)||(constructor 1)]))
              end
            ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Definition wf_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large  := 
           WF_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large.wf.
         
         
         Lemma wf :
          well_founded WF_DP_R_xml_0_scc_65_large_scc_1_large_large_large.DP_R_xml_0_scc_65_large_scc_1_large_large_large_large
           .
         Proof.
           intros x.
           apply (well_founded_ind wf_lt).
           clear x.
           intros x.
           pattern x.
           
           apply (@Acc_ind _ 
                   DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large)
                  .
           clear x.
           intros x _ IHx IHx'.
           constructor.
           intros y H.
           
           destruct H;
            (apply IHx';
              apply DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_strict_in_lt;
              econstructor eassumption)||
            ((apply IHx;[econstructor eassumption|
              intros y' Hlt;apply IHx';apply lt_le_compat with (1:=Hlt) ;
               apply DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large_in_le;
               econstructor eassumption])).
           
           apply wf_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_large.
         Qed.
        End WF_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large.
        
        Open Scope Z_scope.
        
        Import ring_extention.
        
        Notation Local "a <= b" := (Zle a b).
        
        Notation Local "a < b" := (Zlt a b).
        
        Definition P_id_a__zeros  := 0.
        
        Definition P_id_U51 (x11:Z) (x12:Z) := 1* x11.
        
        Definition P_id_a__U84 (x11:Z) (x12:Z) (x13:Z) := 2* x11.
        
        Definition P_id_a__U32 (x11:Z) (x12:Z) := 1.
        
        Definition P_id_isNatList (x11:Z) := 0.
        
        Definition P_id_a__U13 (x11:Z) := 1* x11.
        
        Definition P_id_U85 (x11:Z) (x12:Z) := 2* x11.
        
        Definition P_id_a__length (x11:Z) := 0.
        
        Definition P_id_a__isNatIList (x11:Z) := 1.
        
        Definition P_id_U33 (x11:Z) := 1 + 1* x11.
        
        Definition P_id_a__U11 (x11:Z) (x12:Z) := 2* x11.
        
        Definition P_id_U81 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
        
        Definition P_id_a__U92 (x11:Z) (x12:Z) (x13:Z) := 0.
        
        Definition P_id_a__U43 (x11:Z) (x12:Z) (x13:Z) := 1.
        
        Definition P_id_U23 (x11:Z) := 2* x11.
        
        Definition P_id_a__isNatKind (x11:Z) := 0.
        
        Definition P_id_U93 (x11:Z) (x12:Z) (x13:Z) := 0.
        
        Definition P_id_U11 (x11:Z) (x12:Z) := 2* x11.
        
        Definition P_id_a__U71 (x11:Z) := 1* x11.
        
        Definition P_id_U44 (x11:Z) (x12:Z) (x13:Z) := 1.
        
        Definition P_id_0  := 0.
        
        Definition P_id_U61 (x11:Z) := 2* x11.
        
        Definition P_id_a__U86 (x11:Z) := 2* x11.
        
        Definition P_id_a__U41 (x11:Z) (x12:Z) (x13:Z) := 1.
        
        Definition P_id_U22 (x11:Z) (x12:Z) := 1* x11.
        
        Definition P_id_a__U21 (x11:Z) (x12:Z) := 1* x11.
        
        Definition P_id_U91 (x11:Z) (x12:Z) (x13:Z) := 0.
        
        Definition P_id_length (x11:Z) := 0.
        
        Definition P_id_a__U52 (x11:Z) := 1* x11.
        
        Definition P_id_U42 (x11:Z) (x12:Z) (x13:Z) := 1.
        
        Definition P_id_a__U12 (x11:Z) (x12:Z) := 2* x11.
        
        Definition P_id_U83 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
        
        Definition P_id_a__U94 (x11:Z) (x12:Z) := 0.
        
        Definition P_id_a__U45 (x11:Z) (x12:Z) := 1.
        
        Definition P_id_U31 (x11:Z) (x12:Z) := 1.
        
        Definition P_id_a__isNat (x11:Z) := 0.
        
        Definition P_id_isNatIListKind (x11:Z) := 0.
        
        Definition P_id_a__U82 (x11:Z) (x12:Z) (x13:Z) := 2* x11.
        
        Definition P_id_U46 (x11:Z) := 1* x11.
        
        Definition P_id_cons (x11:Z) (x12:Z) := 0.
        
        Definition P_id_U52 (x11:Z) := 1* x11.
        
        Definition P_id_a__U85 (x11:Z) (x12:Z) := 2* x11.
        
        Definition P_id_a__U33 (x11:Z) := 1 + 1* x11.
        
        Definition P_id_U21 (x11:Z) (x12:Z) := 1* x11.
        
        Definition P_id_a__isNatList (x11:Z) := 0.
        
        Definition P_id_U86 (x11:Z) := 2* x11.
        
        Definition P_id_mark (x11:Z) := 1* x11.
        
        Definition P_id_a__U51 (x11:Z) (x12:Z) := 1* x11.
        
        Definition P_id_U41 (x11:Z) (x12:Z) (x13:Z) := 1.
        
        Definition P_id_tt  := 0.
        
        Definition P_id_U82 (x11:Z) (x12:Z) (x13:Z) := 2* x11.
        
        Definition P_id_a__U93 (x11:Z) (x12:Z) (x13:Z) := 0.
        
        Definition P_id_a__U44 (x11:Z) (x12:Z) (x13:Z) := 1.
        
        Definition P_id_isNat (x11:Z) := 0.
        
        Definition P_id_a__U23 (x11:Z) := 2* x11.
        
        Definition P_id_U94 (x11:Z) (x12:Z) := 0.
        
        Definition P_id_U12 (x11:Z) (x12:Z) := 2* x11.
        
        Definition P_id_a__U81 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
        
        Definition P_id_U45 (x11:Z) (x12:Z) := 1.
        
        Definition P_id_zeros  := 0.
        
        Definition P_id_U71 (x11:Z) := 1* x11.
        
        Definition P_id_a__U91 (x11:Z) (x12:Z) (x13:Z) := 0.
        
        Definition P_id_a__U42 (x11:Z) (x12:Z) (x13:Z) := 1.
        
        Definition P_id_isNatKind (x11:Z) := 0.
        
        Definition P_id_a__U22 (x11:Z) (x12:Z) := 1* x11.
        
        Definition P_id_U92 (x11:Z) (x12:Z) (x13:Z) := 0.
        
        Definition P_id_nil  := 0.
        
        Definition P_id_a__U61 (x11:Z) := 2* x11.
        
        Definition P_id_U43 (x11:Z) (x12:Z) (x13:Z) := 1.
        
        Definition P_id_a__isNatIListKind (x11:Z) := 0.
        
        Definition P_id_U84 (x11:Z) (x12:Z) (x13:Z) := 2* x11.
        
        Definition P_id_s (x11:Z) := 1* x11.
        
        Definition P_id_a__U46 (x11:Z) := 1* x11.
        
        Definition P_id_U32 (x11:Z) (x12:Z) := 1.
        
        Definition P_id_a__U31 (x11:Z) (x12:Z) := 1.
        
        Definition P_id_U13 (x11:Z) := 1* x11.
        
        Definition P_id_a__U83 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
        
        Definition P_id_isNatIList (x11:Z) := 1.
        
        Lemma P_id_U51_monotonic :
         forall x12 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->P_id_U51 x12 x14 <= P_id_U51 x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U84_monotonic :
         forall x16 x12 x14 x13 x11 x15, 
          (0 <= x16)/\ (x16 <= x15) ->
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_a__U84 x12 x14 x16 <= P_id_a__U84 x11 x13 x15.
        Proof.
          intros x16 x15 x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          intros [H_5 H_4].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U32_monotonic :
         forall x12 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_a__U32 x12 x14 <= P_id_a__U32 x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_isNatList_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_isNatList x12 <= P_id_isNatList x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U13_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_a__U13 x12 <= P_id_a__U13 x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U85_monotonic :
         forall x12 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->P_id_U85 x12 x14 <= P_id_U85 x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__length_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_a__length x12 <= P_id_a__length x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__isNatIList_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_a__isNatIList x12 <= P_id_a__isNatIList x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U33_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_U33 x12 <= P_id_U33 x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U11_monotonic :
         forall x12 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_a__U11 x12 x14 <= P_id_a__U11 x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U81_monotonic :
         forall x16 x12 x14 x13 x11 x15, 
          (0 <= x16)/\ (x16 <= x15) ->
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_U81 x12 x14 x16 <= P_id_U81 x11 x13 x15.
        Proof.
          intros x16 x15 x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          intros [H_5 H_4].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U92_monotonic :
         forall x16 x12 x14 x13 x11 x15, 
          (0 <= x16)/\ (x16 <= x15) ->
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_a__U92 x12 x14 x16 <= P_id_a__U92 x11 x13 x15.
        Proof.
          intros x16 x15 x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          intros [H_5 H_4].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U43_monotonic :
         forall x16 x12 x14 x13 x11 x15, 
          (0 <= x16)/\ (x16 <= x15) ->
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_a__U43 x12 x14 x16 <= P_id_a__U43 x11 x13 x15.
        Proof.
          intros x16 x15 x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          intros [H_5 H_4].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U23_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_U23 x12 <= P_id_U23 x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__isNatKind_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_a__isNatKind x12 <= P_id_a__isNatKind x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U93_monotonic :
         forall x16 x12 x14 x13 x11 x15, 
          (0 <= x16)/\ (x16 <= x15) ->
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_U93 x12 x14 x16 <= P_id_U93 x11 x13 x15.
        Proof.
          intros x16 x15 x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          intros [H_5 H_4].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U11_monotonic :
         forall x12 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->P_id_U11 x12 x14 <= P_id_U11 x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U71_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_a__U71 x12 <= P_id_a__U71 x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U44_monotonic :
         forall x16 x12 x14 x13 x11 x15, 
          (0 <= x16)/\ (x16 <= x15) ->
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_U44 x12 x14 x16 <= P_id_U44 x11 x13 x15.
        Proof.
          intros x16 x15 x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          intros [H_5 H_4].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U61_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_U61 x12 <= P_id_U61 x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U86_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_a__U86 x12 <= P_id_a__U86 x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U41_monotonic :
         forall x16 x12 x14 x13 x11 x15, 
          (0 <= x16)/\ (x16 <= x15) ->
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_a__U41 x12 x14 x16 <= P_id_a__U41 x11 x13 x15.
        Proof.
          intros x16 x15 x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          intros [H_5 H_4].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U22_monotonic :
         forall x12 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->P_id_U22 x12 x14 <= P_id_U22 x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U21_monotonic :
         forall x12 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_a__U21 x12 x14 <= P_id_a__U21 x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U91_monotonic :
         forall x16 x12 x14 x13 x11 x15, 
          (0 <= x16)/\ (x16 <= x15) ->
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_U91 x12 x14 x16 <= P_id_U91 x11 x13 x15.
        Proof.
          intros x16 x15 x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          intros [H_5 H_4].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_length_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_length x12 <= P_id_length x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U52_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_a__U52 x12 <= P_id_a__U52 x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U42_monotonic :
         forall x16 x12 x14 x13 x11 x15, 
          (0 <= x16)/\ (x16 <= x15) ->
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_U42 x12 x14 x16 <= P_id_U42 x11 x13 x15.
        Proof.
          intros x16 x15 x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          intros [H_5 H_4].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U12_monotonic :
         forall x12 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_a__U12 x12 x14 <= P_id_a__U12 x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U83_monotonic :
         forall x16 x12 x14 x13 x11 x15, 
          (0 <= x16)/\ (x16 <= x15) ->
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_U83 x12 x14 x16 <= P_id_U83 x11 x13 x15.
        Proof.
          intros x16 x15 x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          intros [H_5 H_4].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U94_monotonic :
         forall x12 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_a__U94 x12 x14 <= P_id_a__U94 x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U45_monotonic :
         forall x12 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_a__U45 x12 x14 <= P_id_a__U45 x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U31_monotonic :
         forall x12 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->P_id_U31 x12 x14 <= P_id_U31 x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__isNat_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_a__isNat x12 <= P_id_a__isNat x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_isNatIListKind_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_isNatIListKind x12 <= P_id_isNatIListKind x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U82_monotonic :
         forall x16 x12 x14 x13 x11 x15, 
          (0 <= x16)/\ (x16 <= x15) ->
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_a__U82 x12 x14 x16 <= P_id_a__U82 x11 x13 x15.
        Proof.
          intros x16 x15 x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          intros [H_5 H_4].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U46_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_U46 x12 <= P_id_U46 x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_cons_monotonic :
         forall x12 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->P_id_cons x12 x14 <= P_id_cons x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U52_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_U52 x12 <= P_id_U52 x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U85_monotonic :
         forall x12 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_a__U85 x12 x14 <= P_id_a__U85 x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U33_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_a__U33 x12 <= P_id_a__U33 x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U21_monotonic :
         forall x12 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->P_id_U21 x12 x14 <= P_id_U21 x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__isNatList_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_a__isNatList x12 <= P_id_a__isNatList x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U86_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_U86 x12 <= P_id_U86 x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_mark_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_mark x12 <= P_id_mark x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U51_monotonic :
         forall x12 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_a__U51 x12 x14 <= P_id_a__U51 x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U41_monotonic :
         forall x16 x12 x14 x13 x11 x15, 
          (0 <= x16)/\ (x16 <= x15) ->
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_U41 x12 x14 x16 <= P_id_U41 x11 x13 x15.
        Proof.
          intros x16 x15 x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          intros [H_5 H_4].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U82_monotonic :
         forall x16 x12 x14 x13 x11 x15, 
          (0 <= x16)/\ (x16 <= x15) ->
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_U82 x12 x14 x16 <= P_id_U82 x11 x13 x15.
        Proof.
          intros x16 x15 x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          intros [H_5 H_4].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U93_monotonic :
         forall x16 x12 x14 x13 x11 x15, 
          (0 <= x16)/\ (x16 <= x15) ->
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_a__U93 x12 x14 x16 <= P_id_a__U93 x11 x13 x15.
        Proof.
          intros x16 x15 x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          intros [H_5 H_4].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U44_monotonic :
         forall x16 x12 x14 x13 x11 x15, 
          (0 <= x16)/\ (x16 <= x15) ->
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_a__U44 x12 x14 x16 <= P_id_a__U44 x11 x13 x15.
        Proof.
          intros x16 x15 x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          intros [H_5 H_4].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_isNat_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_isNat x12 <= P_id_isNat x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U23_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_a__U23 x12 <= P_id_a__U23 x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U94_monotonic :
         forall x12 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->P_id_U94 x12 x14 <= P_id_U94 x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U12_monotonic :
         forall x12 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->P_id_U12 x12 x14 <= P_id_U12 x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U81_monotonic :
         forall x16 x12 x14 x13 x11 x15, 
          (0 <= x16)/\ (x16 <= x15) ->
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_a__U81 x12 x14 x16 <= P_id_a__U81 x11 x13 x15.
        Proof.
          intros x16 x15 x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          intros [H_5 H_4].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U45_monotonic :
         forall x12 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->P_id_U45 x12 x14 <= P_id_U45 x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U71_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_U71 x12 <= P_id_U71 x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U91_monotonic :
         forall x16 x12 x14 x13 x11 x15, 
          (0 <= x16)/\ (x16 <= x15) ->
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_a__U91 x12 x14 x16 <= P_id_a__U91 x11 x13 x15.
        Proof.
          intros x16 x15 x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          intros [H_5 H_4].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U42_monotonic :
         forall x16 x12 x14 x13 x11 x15, 
          (0 <= x16)/\ (x16 <= x15) ->
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_a__U42 x12 x14 x16 <= P_id_a__U42 x11 x13 x15.
        Proof.
          intros x16 x15 x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          intros [H_5 H_4].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_isNatKind_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_isNatKind x12 <= P_id_isNatKind x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U22_monotonic :
         forall x12 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_a__U22 x12 x14 <= P_id_a__U22 x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U92_monotonic :
         forall x16 x12 x14 x13 x11 x15, 
          (0 <= x16)/\ (x16 <= x15) ->
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_U92 x12 x14 x16 <= P_id_U92 x11 x13 x15.
        Proof.
          intros x16 x15 x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          intros [H_5 H_4].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U61_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_a__U61 x12 <= P_id_a__U61 x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U43_monotonic :
         forall x16 x12 x14 x13 x11 x15, 
          (0 <= x16)/\ (x16 <= x15) ->
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_U43 x12 x14 x16 <= P_id_U43 x11 x13 x15.
        Proof.
          intros x16 x15 x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          intros [H_5 H_4].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__isNatIListKind_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_a__isNatIListKind x12 <= P_id_a__isNatIListKind x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U84_monotonic :
         forall x16 x12 x14 x13 x11 x15, 
          (0 <= x16)/\ (x16 <= x15) ->
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_U84 x12 x14 x16 <= P_id_U84 x11 x13 x15.
        Proof.
          intros x16 x15 x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          intros [H_5 H_4].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_s_monotonic :
         forall x12 x11, (0 <= x12)/\ (x12 <= x11) ->P_id_s x12 <= P_id_s x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U46_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_a__U46 x12 <= P_id_a__U46 x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U32_monotonic :
         forall x12 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->P_id_U32 x12 x14 <= P_id_U32 x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U31_monotonic :
         forall x12 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_a__U31 x12 x14 <= P_id_a__U31 x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U13_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_U13 x12 <= P_id_U13 x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U83_monotonic :
         forall x16 x12 x14 x13 x11 x15, 
          (0 <= x16)/\ (x16 <= x15) ->
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_a__U83 x12 x14 x16 <= P_id_a__U83 x11 x13 x15.
        Proof.
          intros x16 x15 x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          intros [H_5 H_4].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_isNatIList_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_isNatIList x12 <= P_id_isNatIList x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__zeros_bounded : 0 <= P_id_a__zeros .
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U51_bounded :
         forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U51 x12 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U84_bounded :
         forall x12 x13 x11, 
          (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U84 x13 x12 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U32_bounded :
         forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U32 x12 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_isNatList_bounded :
         forall x11, (0 <= x11) ->0 <= P_id_isNatList x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U13_bounded :
         forall x11, (0 <= x11) ->0 <= P_id_a__U13 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U85_bounded :
         forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U85 x12 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__length_bounded :
         forall x11, (0 <= x11) ->0 <= P_id_a__length x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__isNatIList_bounded :
         forall x11, (0 <= x11) ->0 <= P_id_a__isNatIList x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U33_bounded : forall x11, (0 <= x11) ->0 <= P_id_U33 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U11_bounded :
         forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U11 x12 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U81_bounded :
         forall x12 x13 x11, 
          (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U81 x13 x12 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U92_bounded :
         forall x12 x13 x11, 
          (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U92 x13 x12 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U43_bounded :
         forall x12 x13 x11, 
          (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U43 x13 x12 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U23_bounded : forall x11, (0 <= x11) ->0 <= P_id_U23 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__isNatKind_bounded :
         forall x11, (0 <= x11) ->0 <= P_id_a__isNatKind x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U93_bounded :
         forall x12 x13 x11, 
          (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U93 x13 x12 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U11_bounded :
         forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U11 x12 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U71_bounded :
         forall x11, (0 <= x11) ->0 <= P_id_a__U71 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U44_bounded :
         forall x12 x13 x11, 
          (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U44 x13 x12 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_0_bounded : 0 <= P_id_0 .
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U61_bounded : forall x11, (0 <= x11) ->0 <= P_id_U61 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U86_bounded :
         forall x11, (0 <= x11) ->0 <= P_id_a__U86 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U41_bounded :
         forall x12 x13 x11, 
          (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U41 x13 x12 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U22_bounded :
         forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U22 x12 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U21_bounded :
         forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U21 x12 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U91_bounded :
         forall x12 x13 x11, 
          (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U91 x13 x12 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_length_bounded :
         forall x11, (0 <= x11) ->0 <= P_id_length x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U52_bounded :
         forall x11, (0 <= x11) ->0 <= P_id_a__U52 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U42_bounded :
         forall x12 x13 x11, 
          (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U42 x13 x12 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U12_bounded :
         forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U12 x12 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U83_bounded :
         forall x12 x13 x11, 
          (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U83 x13 x12 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U94_bounded :
         forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U94 x12 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U45_bounded :
         forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U45 x12 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U31_bounded :
         forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U31 x12 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__isNat_bounded :
         forall x11, (0 <= x11) ->0 <= P_id_a__isNat x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_isNatIListKind_bounded :
         forall x11, (0 <= x11) ->0 <= P_id_isNatIListKind x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U82_bounded :
         forall x12 x13 x11, 
          (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U82 x13 x12 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U46_bounded : forall x11, (0 <= x11) ->0 <= P_id_U46 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_cons_bounded :
         forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_cons x12 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U52_bounded : forall x11, (0 <= x11) ->0 <= P_id_U52 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U85_bounded :
         forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U85 x12 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U33_bounded :
         forall x11, (0 <= x11) ->0 <= P_id_a__U33 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U21_bounded :
         forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U21 x12 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__isNatList_bounded :
         forall x11, (0 <= x11) ->0 <= P_id_a__isNatList x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U86_bounded : forall x11, (0 <= x11) ->0 <= P_id_U86 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_mark_bounded :
         forall x11, (0 <= x11) ->0 <= P_id_mark x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U51_bounded :
         forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U51 x12 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U41_bounded :
         forall x12 x13 x11, 
          (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U41 x13 x12 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_tt_bounded : 0 <= P_id_tt .
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U82_bounded :
         forall x12 x13 x11, 
          (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U82 x13 x12 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U93_bounded :
         forall x12 x13 x11, 
          (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U93 x13 x12 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U44_bounded :
         forall x12 x13 x11, 
          (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U44 x13 x12 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_isNat_bounded :
         forall x11, (0 <= x11) ->0 <= P_id_isNat x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U23_bounded :
         forall x11, (0 <= x11) ->0 <= P_id_a__U23 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U94_bounded :
         forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U94 x12 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U12_bounded :
         forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U12 x12 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U81_bounded :
         forall x12 x13 x11, 
          (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U81 x13 x12 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U45_bounded :
         forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U45 x12 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_zeros_bounded : 0 <= P_id_zeros .
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U71_bounded : forall x11, (0 <= x11) ->0 <= P_id_U71 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U91_bounded :
         forall x12 x13 x11, 
          (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U91 x13 x12 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U42_bounded :
         forall x12 x13 x11, 
          (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U42 x13 x12 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_isNatKind_bounded :
         forall x11, (0 <= x11) ->0 <= P_id_isNatKind x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U22_bounded :
         forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U22 x12 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U92_bounded :
         forall x12 x13 x11, 
          (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U92 x13 x12 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_nil_bounded : 0 <= P_id_nil .
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U61_bounded :
         forall x11, (0 <= x11) ->0 <= P_id_a__U61 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U43_bounded :
         forall x12 x13 x11, 
          (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U43 x13 x12 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__isNatIListKind_bounded :
         forall x11, (0 <= x11) ->0 <= P_id_a__isNatIListKind x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U84_bounded :
         forall x12 x13 x11, 
          (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U84 x13 x12 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_s_bounded : forall x11, (0 <= x11) ->0 <= P_id_s x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U46_bounded :
         forall x11, (0 <= x11) ->0 <= P_id_a__U46 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U32_bounded :
         forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U32 x12 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U31_bounded :
         forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U31 x12 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U13_bounded : forall x11, (0 <= x11) ->0 <= P_id_U13 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_a__U83_bounded :
         forall x12 x13 x11, 
          (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U83 x13 x12 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_isNatIList_bounded :
         forall x11, (0 <= x11) ->0 <= P_id_isNatIList x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Definition measure  := 
          InterpZ.measure 0 P_id_a__zeros P_id_U51 P_id_a__U84 P_id_a__U32 
           P_id_isNatList P_id_a__U13 P_id_U85 P_id_a__length 
           P_id_a__isNatIList P_id_U33 P_id_a__U11 P_id_U81 P_id_a__U92 
           P_id_a__U43 P_id_U23 P_id_a__isNatKind P_id_U93 P_id_U11 
           P_id_a__U71 P_id_U44 P_id_0 P_id_U61 P_id_a__U86 P_id_a__U41 
           P_id_U22 P_id_a__U21 P_id_U91 P_id_length P_id_a__U52 P_id_U42 
           P_id_a__U12 P_id_U83 P_id_a__U94 P_id_a__U45 P_id_U31 
           P_id_a__isNat P_id_isNatIListKind P_id_a__U82 P_id_U46 P_id_cons 
           P_id_U52 P_id_a__U85 P_id_a__U33 P_id_U21 P_id_a__isNatList 
           P_id_U86 P_id_mark P_id_a__U51 P_id_U41 P_id_tt P_id_U82 
           P_id_a__U93 P_id_a__U44 P_id_isNat P_id_a__U23 P_id_U94 P_id_U12 
           P_id_a__U81 P_id_U45 P_id_zeros P_id_U71 P_id_a__U91 P_id_a__U42 
           P_id_isNatKind P_id_a__U22 P_id_U92 P_id_nil P_id_a__U61 P_id_U43 
           P_id_a__isNatIListKind P_id_U84 P_id_s P_id_a__U46 P_id_U32 
           P_id_a__U31 P_id_U13 P_id_a__U83 P_id_isNatIList.
        
        Lemma measure_equation :
         forall t, 
          measure t = match t with
                        | (algebra.Alg.Term algebra.F.id_a__zeros nil) =>
                         P_id_a__zeros 
                        | (algebra.Alg.Term algebra.F.id_U51 (x12::x11::nil)) =>
                         P_id_U51 (measure x12) (measure x11)
                        | (algebra.Alg.Term algebra.F.id_a__U84 (x13::x12::
                           x11::nil)) =>
                         P_id_a__U84 (measure x13) (measure x12) 
                          (measure x11)
                        | (algebra.Alg.Term algebra.F.id_a__U32 (x12::
                           x11::nil)) =>
                         P_id_a__U32 (measure x12) (measure x11)
                        | (algebra.Alg.Term algebra.F.id_isNatList 
                           (x11::nil)) =>
                         P_id_isNatList (measure x11)
                        | (algebra.Alg.Term algebra.F.id_a__U13 (x11::nil)) =>
                         P_id_a__U13 (measure x11)
                        | (algebra.Alg.Term algebra.F.id_U85 (x12::x11::nil)) =>
                         P_id_U85 (measure x12) (measure x11)
                        | (algebra.Alg.Term algebra.F.id_a__length 
                           (x11::nil)) =>
                         P_id_a__length (measure x11)
                        | (algebra.Alg.Term algebra.F.id_a__isNatIList 
                           (x11::nil)) =>
                         P_id_a__isNatIList (measure x11)
                        | (algebra.Alg.Term algebra.F.id_U33 (x11::nil)) =>
                         P_id_U33 (measure x11)
                        | (algebra.Alg.Term algebra.F.id_a__U11 (x12::
                           x11::nil)) =>
                         P_id_a__U11 (measure x12) (measure x11)
                        | (algebra.Alg.Term algebra.F.id_U81 (x13::x12::
                           x11::nil)) =>
                         P_id_U81 (measure x13) (measure x12) (measure x11)
                        | (algebra.Alg.Term algebra.F.id_a__U92 (x13::x12::
                           x11::nil)) =>
                         P_id_a__U92 (measure x13) (measure x12) 
                          (measure x11)
                        | (algebra.Alg.Term algebra.F.id_a__U43 (x13::x12::
                           x11::nil)) =>
                         P_id_a__U43 (measure x13) (measure x12) 
                          (measure x11)
                        | (algebra.Alg.Term algebra.F.id_U23 (x11::nil)) =>
                         P_id_U23 (measure x11)
                        | (algebra.Alg.Term algebra.F.id_a__isNatKind 
                           (x11::nil)) =>
                         P_id_a__isNatKind (measure x11)
                        | (algebra.Alg.Term algebra.F.id_U93 (x13::x12::
                           x11::nil)) =>
                         P_id_U93 (measure x13) (measure x12) (measure x11)
                        | (algebra.Alg.Term algebra.F.id_U11 (x12::x11::nil)) =>
                         P_id_U11 (measure x12) (measure x11)
                        | (algebra.Alg.Term algebra.F.id_a__U71 (x11::nil)) =>
                         P_id_a__U71 (measure x11)
                        | (algebra.Alg.Term algebra.F.id_U44 (x13::x12::
                           x11::nil)) =>
                         P_id_U44 (measure x13) (measure x12) (measure x11)
                        | (algebra.Alg.Term algebra.F.id_0 nil) => P_id_0 
                        | (algebra.Alg.Term algebra.F.id_U61 (x11::nil)) =>
                         P_id_U61 (measure x11)
                        | (algebra.Alg.Term algebra.F.id_a__U86 (x11::nil)) =>
                         P_id_a__U86 (measure x11)
                        | (algebra.Alg.Term algebra.F.id_a__U41 (x13::x12::
                           x11::nil)) =>
                         P_id_a__U41 (measure x13) (measure x12) 
                          (measure x11)
                        | (algebra.Alg.Term algebra.F.id_U22 (x12::x11::nil)) =>
                         P_id_U22 (measure x12) (measure x11)
                        | (algebra.Alg.Term algebra.F.id_a__U21 (x12::
                           x11::nil)) =>
                         P_id_a__U21 (measure x12) (measure x11)
                        | (algebra.Alg.Term algebra.F.id_U91 (x13::x12::
                           x11::nil)) =>
                         P_id_U91 (measure x13) (measure x12) (measure x11)
                        | (algebra.Alg.Term algebra.F.id_length (x11::nil)) =>
                         P_id_length (measure x11)
                        | (algebra.Alg.Term algebra.F.id_a__U52 (x11::nil)) =>
                         P_id_a__U52 (measure x11)
                        | (algebra.Alg.Term algebra.F.id_U42 (x13::x12::
                           x11::nil)) =>
                         P_id_U42 (measure x13) (measure x12) (measure x11)
                        | (algebra.Alg.Term algebra.F.id_a__U12 (x12::
                           x11::nil)) =>
                         P_id_a__U12 (measure x12) (measure x11)
                        | (algebra.Alg.Term algebra.F.id_U83 (x13::x12::
                           x11::nil)) =>
                         P_id_U83 (measure x13) (measure x12) (measure x11)
                        | (algebra.Alg.Term algebra.F.id_a__U94 (x12::
                           x11::nil)) =>
                         P_id_a__U94 (measure x12) (measure x11)
                        | (algebra.Alg.Term algebra.F.id_a__U45 (x12::
                           x11::nil)) =>
                         P_id_a__U45 (measure x12) (measure x11)
                        | (algebra.Alg.Term algebra.F.id_U31 (x12::x11::nil)) =>
                         P_id_U31 (measure x12) (measure x11)
                        | (algebra.Alg.Term algebra.F.id_a__isNat (x11::nil)) =>
                         P_id_a__isNat (measure x11)
                        | (algebra.Alg.Term algebra.F.id_isNatIListKind 
                           (x11::nil)) =>
                         P_id_isNatIListKind (measure x11)
                        | (algebra.Alg.Term algebra.F.id_a__U82 (x13::x12::
                           x11::nil)) =>
                         P_id_a__U82 (measure x13) (measure x12) 
                          (measure x11)
                        | (algebra.Alg.Term algebra.F.id_U46 (x11::nil)) =>
                         P_id_U46 (measure x11)
                        | (algebra.Alg.Term algebra.F.id_cons (x12::
                           x11::nil)) =>
                         P_id_cons (measure x12) (measure x11)
                        | (algebra.Alg.Term algebra.F.id_U52 (x11::nil)) =>
                         P_id_U52 (measure x11)
                        | (algebra.Alg.Term algebra.F.id_a__U85 (x12::
                           x11::nil)) =>
                         P_id_a__U85 (measure x12) (measure x11)
                        | (algebra.Alg.Term algebra.F.id_a__U33 (x11::nil)) =>
                         P_id_a__U33 (measure x11)
                        | (algebra.Alg.Term algebra.F.id_U21 (x12::x11::nil)) =>
                         P_id_U21 (measure x12) (measure x11)
                        | (algebra.Alg.Term algebra.F.id_a__isNatList 
                           (x11::nil)) =>
                         P_id_a__isNatList (measure x11)
                        | (algebra.Alg.Term algebra.F.id_U86 (x11::nil)) =>
                         P_id_U86 (measure x11)
                        | (algebra.Alg.Term algebra.F.id_mark (x11::nil)) =>
                         P_id_mark (measure x11)
                        | (algebra.Alg.Term algebra.F.id_a__U51 (x12::
                           x11::nil)) =>
                         P_id_a__U51 (measure x12) (measure x11)
                        | (algebra.Alg.Term algebra.F.id_U41 (x13::x12::
                           x11::nil)) =>
                         P_id_U41 (measure x13) (measure x12) (measure x11)
                        | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt 
                        | (algebra.Alg.Term algebra.F.id_U82 (x13::x12::
                           x11::nil)) =>
                         P_id_U82 (measure x13) (measure x12) (measure x11)
                        | (algebra.Alg.Term algebra.F.id_a__U93 (x13::x12::
                           x11::nil)) =>
                         P_id_a__U93 (measure x13) (measure x12) 
                          (measure x11)
                        | (algebra.Alg.Term algebra.F.id_a__U44 (x13::x12::
                           x11::nil)) =>
                         P_id_a__U44 (measure x13) (measure x12) 
                          (measure x11)
                        | (algebra.Alg.Term algebra.F.id_isNat (x11::nil)) =>
                         P_id_isNat (measure x11)
                        | (algebra.Alg.Term algebra.F.id_a__U23 (x11::nil)) =>
                         P_id_a__U23 (measure x11)
                        | (algebra.Alg.Term algebra.F.id_U94 (x12::x11::nil)) =>
                         P_id_U94 (measure x12) (measure x11)
                        | (algebra.Alg.Term algebra.F.id_U12 (x12::x11::nil)) =>
                         P_id_U12 (measure x12) (measure x11)
                        | (algebra.Alg.Term algebra.F.id_a__U81 (x13::x12::
                           x11::nil)) =>
                         P_id_a__U81 (measure x13) (measure x12) 
                          (measure x11)
                        | (algebra.Alg.Term algebra.F.id_U45 (x12::x11::nil)) =>
                         P_id_U45 (measure x12) (measure x11)
                        | (algebra.Alg.Term algebra.F.id_zeros nil) =>
                         P_id_zeros 
                        | (algebra.Alg.Term algebra.F.id_U71 (x11::nil)) =>
                         P_id_U71 (measure x11)
                        | (algebra.Alg.Term algebra.F.id_a__U91 (x13::x12::
                           x11::nil)) =>
                         P_id_a__U91 (measure x13) (measure x12) 
                          (measure x11)
                        | (algebra.Alg.Term algebra.F.id_a__U42 (x13::x12::
                           x11::nil)) =>
                         P_id_a__U42 (measure x13) (measure x12) 
                          (measure x11)
                        | (algebra.Alg.Term algebra.F.id_isNatKind 
                           (x11::nil)) =>
                         P_id_isNatKind (measure x11)
                        | (algebra.Alg.Term algebra.F.id_a__U22 (x12::
                           x11::nil)) =>
                         P_id_a__U22 (measure x12) (measure x11)
                        | (algebra.Alg.Term algebra.F.id_U92 (x13::x12::
                           x11::nil)) =>
                         P_id_U92 (measure x13) (measure x12) (measure x11)
                        | (algebra.Alg.Term algebra.F.id_nil nil) =>
                         P_id_nil 
                        | (algebra.Alg.Term algebra.F.id_a__U61 (x11::nil)) =>
                         P_id_a__U61 (measure x11)
                        | (algebra.Alg.Term algebra.F.id_U43 (x13::x12::
                           x11::nil)) =>
                         P_id_U43 (measure x13) (measure x12) (measure x11)
                        | (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
                           (x11::nil)) =>
                         P_id_a__isNatIListKind (measure x11)
                        | (algebra.Alg.Term algebra.F.id_U84 (x13::x12::
                           x11::nil)) =>
                         P_id_U84 (measure x13) (measure x12) (measure x11)
                        | (algebra.Alg.Term algebra.F.id_s (x11::nil)) =>
                         P_id_s (measure x11)
                        | (algebra.Alg.Term algebra.F.id_a__U46 (x11::nil)) =>
                         P_id_a__U46 (measure x11)
                        | (algebra.Alg.Term algebra.F.id_U32 (x12::x11::nil)) =>
                         P_id_U32 (measure x12) (measure x11)
                        | (algebra.Alg.Term algebra.F.id_a__U31 (x12::
                           x11::nil)) =>
                         P_id_a__U31 (measure x12) (measure x11)
                        | (algebra.Alg.Term algebra.F.id_U13 (x11::nil)) =>
                         P_id_U13 (measure x11)
                        | (algebra.Alg.Term algebra.F.id_a__U83 (x13::x12::
                           x11::nil)) =>
                         P_id_a__U83 (measure x13) (measure x12) 
                          (measure x11)
                        | (algebra.Alg.Term algebra.F.id_isNatIList 
                           (x11::nil)) =>
                         P_id_isNatIList (measure x11)
                        | _ => 0
                        end.
        Proof.
          intros t;case t;intros ;apply refl_equal.
        Qed.
        
        Lemma measure_bounded : forall t, 0 <= measure t.
        Proof.
          unfold measure in |-*.
          
          apply InterpZ.measure_bounded;
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Ltac generate_pos_hyp  :=
         match goal with
           | H:context [measure ?x] |- _ =>
            let v := fresh "v" in 
             (let H := fresh "h" in 
               (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
                 clearbody H;clearbody v))
           |  |- context [measure ?x] =>
            let v := fresh "v" in 
             (let H := fresh "h" in 
               (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
                 clearbody H;clearbody v))
           end
         .
        
        Lemma rules_monotonic :
         forall l r, 
          (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) ->
           measure r <= measure l.
        Proof.
          intros l r H.
          fold measure in |-*.
          
          inversion H;clear H;subst;inversion H0;clear H0;subst;
           simpl algebra.EQT.T.apply_subst in |-*;
           repeat (
           match goal with
             |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
              rewrite (measure_equation (algebra.Alg.Term f t))
             end
           );repeat (generate_pos_hyp );
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma measure_star_monotonic :
         forall l r, 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     r l) ->measure r <= measure l.
        Proof.
          unfold measure in *.
          apply InterpZ.measure_star_monotonic.
          intros ;apply P_id_U51_monotonic;assumption.
          intros ;apply P_id_a__U84_monotonic;assumption.
          intros ;apply P_id_a__U32_monotonic;assumption.
          intros ;apply P_id_isNatList_monotonic;assumption.
          intros ;apply P_id_a__U13_monotonic;assumption.
          intros ;apply P_id_U85_monotonic;assumption.
          intros ;apply P_id_a__length_monotonic;assumption.
          intros ;apply P_id_a__isNatIList_monotonic;assumption.
          intros ;apply P_id_U33_monotonic;assumption.
          intros ;apply P_id_a__U11_monotonic;assumption.
          intros ;apply P_id_U81_monotonic;assumption.
          intros ;apply P_id_a__U92_monotonic;assumption.
          intros ;apply P_id_a__U43_monotonic;assumption.
          intros ;apply P_id_U23_monotonic;assumption.
          intros ;apply P_id_a__isNatKind_monotonic;assumption.
          intros ;apply P_id_U93_monotonic;assumption.
          intros ;apply P_id_U11_monotonic;assumption.
          intros ;apply P_id_a__U71_monotonic;assumption.
          intros ;apply P_id_U44_monotonic;assumption.
          intros ;apply P_id_U61_monotonic;assumption.
          intros ;apply P_id_a__U86_monotonic;assumption.
          intros ;apply P_id_a__U41_monotonic;assumption.
          intros ;apply P_id_U22_monotonic;assumption.
          intros ;apply P_id_a__U21_monotonic;assumption.
          intros ;apply P_id_U91_monotonic;assumption.
          intros ;apply P_id_length_monotonic;assumption.
          intros ;apply P_id_a__U52_monotonic;assumption.
          intros ;apply P_id_U42_monotonic;assumption.
          intros ;apply P_id_a__U12_monotonic;assumption.
          intros ;apply P_id_U83_monotonic;assumption.
          intros ;apply P_id_a__U94_monotonic;assumption.
          intros ;apply P_id_a__U45_monotonic;assumption.
          intros ;apply P_id_U31_monotonic;assumption.
          intros ;apply P_id_a__isNat_monotonic;assumption.
          intros ;apply P_id_isNatIListKind_monotonic;assumption.
          intros ;apply P_id_a__U82_monotonic;assumption.
          intros ;apply P_id_U46_monotonic;assumption.
          intros ;apply P_id_cons_monotonic;assumption.
          intros ;apply P_id_U52_monotonic;assumption.
          intros ;apply P_id_a__U85_monotonic;assumption.
          intros ;apply P_id_a__U33_monotonic;assumption.
          intros ;apply P_id_U21_monotonic;assumption.
          intros ;apply P_id_a__isNatList_monotonic;assumption.
          intros ;apply P_id_U86_monotonic;assumption.
          intros ;apply P_id_mark_monotonic;assumption.
          intros ;apply P_id_a__U51_monotonic;assumption.
          intros ;apply P_id_U41_monotonic;assumption.
          intros ;apply P_id_U82_monotonic;assumption.
          intros ;apply P_id_a__U93_monotonic;assumption.
          intros ;apply P_id_a__U44_monotonic;assumption.
          intros ;apply P_id_isNat_monotonic;assumption.
          intros ;apply P_id_a__U23_monotonic;assumption.
          intros ;apply P_id_U94_monotonic;assumption.
          intros ;apply P_id_U12_monotonic;assumption.
          intros ;apply P_id_a__U81_monotonic;assumption.
          intros ;apply P_id_U45_monotonic;assumption.
          intros ;apply P_id_U71_monotonic;assumption.
          intros ;apply P_id_a__U91_monotonic;assumption.
          intros ;apply P_id_a__U42_monotonic;assumption.
          intros ;apply P_id_isNatKind_monotonic;assumption.
          intros ;apply P_id_a__U22_monotonic;assumption.
          intros ;apply P_id_U92_monotonic;assumption.
          intros ;apply P_id_a__U61_monotonic;assumption.
          intros ;apply P_id_U43_monotonic;assumption.
          intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
          intros ;apply P_id_U84_monotonic;assumption.
          intros ;apply P_id_s_monotonic;assumption.
          intros ;apply P_id_a__U46_monotonic;assumption.
          intros ;apply P_id_U32_monotonic;assumption.
          intros ;apply P_id_a__U31_monotonic;assumption.
          intros ;apply P_id_U13_monotonic;assumption.
          intros ;apply P_id_a__U83_monotonic;assumption.
          intros ;apply P_id_isNatIList_monotonic;assumption.
          intros ;apply P_id_a__zeros_bounded;assumption.
          intros ;apply P_id_U51_bounded;assumption.
          intros ;apply P_id_a__U84_bounded;assumption.
          intros ;apply P_id_a__U32_bounded;assumption.
          intros ;apply P_id_isNatList_bounded;assumption.
          intros ;apply P_id_a__U13_bounded;assumption.
          intros ;apply P_id_U85_bounded;assumption.
          intros ;apply P_id_a__length_bounded;assumption.
          intros ;apply P_id_a__isNatIList_bounded;assumption.
          intros ;apply P_id_U33_bounded;assumption.
          intros ;apply P_id_a__U11_bounded;assumption.
          intros ;apply P_id_U81_bounded;assumption.
          intros ;apply P_id_a__U92_bounded;assumption.
          intros ;apply P_id_a__U43_bounded;assumption.
          intros ;apply P_id_U23_bounded;assumption.
          intros ;apply P_id_a__isNatKind_bounded;assumption.
          intros ;apply P_id_U93_bounded;assumption.
          intros ;apply P_id_U11_bounded;assumption.
          intros ;apply P_id_a__U71_bounded;assumption.
          intros ;apply P_id_U44_bounded;assumption.
          intros ;apply P_id_0_bounded;assumption.
          intros ;apply P_id_U61_bounded;assumption.
          intros ;apply P_id_a__U86_bounded;assumption.
          intros ;apply P_id_a__U41_bounded;assumption.
          intros ;apply P_id_U22_bounded;assumption.
          intros ;apply P_id_a__U21_bounded;assumption.
          intros ;apply P_id_U91_bounded;assumption.
          intros ;apply P_id_length_bounded;assumption.
          intros ;apply P_id_a__U52_bounded;assumption.
          intros ;apply P_id_U42_bounded;assumption.
          intros ;apply P_id_a__U12_bounded;assumption.
          intros ;apply P_id_U83_bounded;assumption.
          intros ;apply P_id_a__U94_bounded;assumption.
          intros ;apply P_id_a__U45_bounded;assumption.
          intros ;apply P_id_U31_bounded;assumption.
          intros ;apply P_id_a__isNat_bounded;assumption.
          intros ;apply P_id_isNatIListKind_bounded;assumption.
          intros ;apply P_id_a__U82_bounded;assumption.
          intros ;apply P_id_U46_bounded;assumption.
          intros ;apply P_id_cons_bounded;assumption.
          intros ;apply P_id_U52_bounded;assumption.
          intros ;apply P_id_a__U85_bounded;assumption.
          intros ;apply P_id_a__U33_bounded;assumption.
          intros ;apply P_id_U21_bounded;assumption.
          intros ;apply P_id_a__isNatList_bounded;assumption.
          intros ;apply P_id_U86_bounded;assumption.
          intros ;apply P_id_mark_bounded;assumption.
          intros ;apply P_id_a__U51_bounded;assumption.
          intros ;apply P_id_U41_bounded;assumption.
          intros ;apply P_id_tt_bounded;assumption.
          intros ;apply P_id_U82_bounded;assumption.
          intros ;apply P_id_a__U93_bounded;assumption.
          intros ;apply P_id_a__U44_bounded;assumption.
          intros ;apply P_id_isNat_bounded;assumption.
          intros ;apply P_id_a__U23_bounded;assumption.
          intros ;apply P_id_U94_bounded;assumption.
          intros ;apply P_id_U12_bounded;assumption.
          intros ;apply P_id_a__U81_bounded;assumption.
          intros ;apply P_id_U45_bounded;assumption.
          intros ;apply P_id_zeros_bounded;assumption.
          intros ;apply P_id_U71_bounded;assumption.
          intros ;apply P_id_a__U91_bounded;assumption.
          intros ;apply P_id_a__U42_bounded;assumption.
          intros ;apply P_id_isNatKind_bounded;assumption.
          intros ;apply P_id_a__U22_bounded;assumption.
          intros ;apply P_id_U92_bounded;assumption.
          intros ;apply P_id_nil_bounded;assumption.
          intros ;apply P_id_a__U61_bounded;assumption.
          intros ;apply P_id_U43_bounded;assumption.
          intros ;apply P_id_a__isNatIListKind_bounded;assumption.
          intros ;apply P_id_U84_bounded;assumption.
          intros ;apply P_id_s_bounded;assumption.
          intros ;apply P_id_a__U46_bounded;assumption.
          intros ;apply P_id_U32_bounded;assumption.
          intros ;apply P_id_a__U31_bounded;assumption.
          intros ;apply P_id_U13_bounded;assumption.
          intros ;apply P_id_a__U83_bounded;assumption.
          intros ;apply P_id_isNatIList_bounded;assumption.
          apply rules_monotonic.
        Qed.
        
        Definition P_id_A__U46 (x11:Z) := 0.
        
        Definition P_id_A__ISNATILISTKIND (x11:Z) := 0.
        
        Definition P_id_A__U61 (x11:Z) := 0.
        
        Definition P_id_A__U85 (x11:Z) (x12:Z) := 0.
        
        Definition P_id_A__U31 (x11:Z) (x12:Z) := 0.
        
        Definition P_id_A__U81 (x11:Z) (x12:Z) (x13:Z) := 0.
        
        Definition P_id_A__U22 (x11:Z) (x12:Z) := 0.
        
        Definition P_id_A__U93 (x11:Z) (x12:Z) (x13:Z) := 0.
        
        Definition P_id_A__U42 (x11:Z) (x12:Z) (x13:Z) := 0.
        
        Definition P_id_A__U51 (x11:Z) (x12:Z) := 0.
        
        Definition P_id_A__ISNATLIST (x11:Z) := 0.
        
        Definition P_id_A__ZEROS  := 0.
        
        Definition P_id_A__U91 (x11:Z) (x12:Z) (x13:Z) := 0.
        
        Definition P_id_A__U33 (x11:Z) := 0.
        
        Definition P_id_A__U83 (x11:Z) (x12:Z) (x13:Z) := 0.
        
        Definition P_id_A__U23 (x11:Z) := 0.
        
        Definition P_id_A__U11 (x11:Z) (x12:Z) := 0.
        
        Definition P_id_A__LENGTH (x11:Z) := 0.
        
        Definition P_id_A__U44 (x11:Z) (x12:Z) (x13:Z) := 0.
        
        Definition P_id_A__ISNATILIST (x11:Z) := 0.
        
        Definition P_id_A__U13 (x11:Z) := 0.
        
        Definition P_id_A__U71 (x11:Z) := 0.
        
        Definition P_id_A__U86 (x11:Z) := 0.
        
        Definition P_id_A__U32 (x11:Z) (x12:Z) := 0.
        
        Definition P_id_A__U82 (x11:Z) (x12:Z) (x13:Z) := 0.
        
        Definition P_id_A__ISNATKIND (x11:Z) := 0.
        
        Definition P_id_A__U94 (x11:Z) (x12:Z) := 0.
        
        Definition P_id_A__U43 (x11:Z) (x12:Z) (x13:Z) := 0.
        
        Definition P_id_A__U52 (x11:Z) := 0.
        
        Definition P_id_A__U21 (x11:Z) (x12:Z) := 0.
        
        Definition P_id_A__U92 (x11:Z) (x12:Z) (x13:Z) := 0.
        
        Definition P_id_A__U41 (x11:Z) (x12:Z) (x13:Z) := 0.
        
        Definition P_id_A__U84 (x11:Z) (x12:Z) (x13:Z) := 0.
        
        Definition P_id_A__ISNAT (x11:Z) := 0.
        
        Definition P_id_A__U12 (x11:Z) (x12:Z) := 0.
        
        Definition P_id_MARK (x11:Z) := 1* x11.
        
        Definition P_id_A__U45 (x11:Z) (x12:Z) := 0.
        
        Lemma P_id_A__U46_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_A__U46 x12 <= P_id_A__U46 x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_A__ISNATILISTKIND_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_A__ISNATILISTKIND x12 <= P_id_A__ISNATILISTKIND x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_A__U61_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_A__U61 x12 <= P_id_A__U61 x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_A__U85_monotonic :
         forall x12 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_A__U85 x12 x14 <= P_id_A__U85 x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_A__U31_monotonic :
         forall x12 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_A__U31 x12 x14 <= P_id_A__U31 x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_A__U81_monotonic :
         forall x16 x12 x14 x13 x11 x15, 
          (0 <= x16)/\ (x16 <= x15) ->
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_A__U81 x12 x14 x16 <= P_id_A__U81 x11 x13 x15.
        Proof.
          intros x16 x15 x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          intros [H_5 H_4].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_A__U22_monotonic :
         forall x12 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_A__U22 x12 x14 <= P_id_A__U22 x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_A__U93_monotonic :
         forall x16 x12 x14 x13 x11 x15, 
          (0 <= x16)/\ (x16 <= x15) ->
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_A__U93 x12 x14 x16 <= P_id_A__U93 x11 x13 x15.
        Proof.
          intros x16 x15 x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          intros [H_5 H_4].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_A__U42_monotonic :
         forall x16 x12 x14 x13 x11 x15, 
          (0 <= x16)/\ (x16 <= x15) ->
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_A__U42 x12 x14 x16 <= P_id_A__U42 x11 x13 x15.
        Proof.
          intros x16 x15 x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          intros [H_5 H_4].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_A__U51_monotonic :
         forall x12 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_A__U51 x12 x14 <= P_id_A__U51 x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_A__ISNATLIST_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_A__ISNATLIST x12 <= P_id_A__ISNATLIST x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_A__U91_monotonic :
         forall x16 x12 x14 x13 x11 x15, 
          (0 <= x16)/\ (x16 <= x15) ->
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_A__U91 x12 x14 x16 <= P_id_A__U91 x11 x13 x15.
        Proof.
          intros x16 x15 x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          intros [H_5 H_4].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_A__U33_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_A__U33 x12 <= P_id_A__U33 x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_A__U83_monotonic :
         forall x16 x12 x14 x13 x11 x15, 
          (0 <= x16)/\ (x16 <= x15) ->
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_A__U83 x12 x14 x16 <= P_id_A__U83 x11 x13 x15.
        Proof.
          intros x16 x15 x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          intros [H_5 H_4].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_A__U23_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_A__U23 x12 <= P_id_A__U23 x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_A__U11_monotonic :
         forall x12 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_A__U11 x12 x14 <= P_id_A__U11 x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_A__LENGTH_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_A__LENGTH x12 <= P_id_A__LENGTH x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_A__U44_monotonic :
         forall x16 x12 x14 x13 x11 x15, 
          (0 <= x16)/\ (x16 <= x15) ->
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_A__U44 x12 x14 x16 <= P_id_A__U44 x11 x13 x15.
        Proof.
          intros x16 x15 x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          intros [H_5 H_4].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_A__ISNATILIST_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_A__ISNATILIST x12 <= P_id_A__ISNATILIST x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_A__U13_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_A__U13 x12 <= P_id_A__U13 x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_A__U71_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_A__U71 x12 <= P_id_A__U71 x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_A__U86_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_A__U86 x12 <= P_id_A__U86 x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_A__U32_monotonic :
         forall x12 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_A__U32 x12 x14 <= P_id_A__U32 x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_A__U82_monotonic :
         forall x16 x12 x14 x13 x11 x15, 
          (0 <= x16)/\ (x16 <= x15) ->
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_A__U82 x12 x14 x16 <= P_id_A__U82 x11 x13 x15.
        Proof.
          intros x16 x15 x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          intros [H_5 H_4].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_A__ISNATKIND_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_A__ISNATKIND x12 <= P_id_A__ISNATKIND x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_A__U94_monotonic :
         forall x12 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_A__U94 x12 x14 <= P_id_A__U94 x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_A__U43_monotonic :
         forall x16 x12 x14 x13 x11 x15, 
          (0 <= x16)/\ (x16 <= x15) ->
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_A__U43 x12 x14 x16 <= P_id_A__U43 x11 x13 x15.
        Proof.
          intros x16 x15 x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          intros [H_5 H_4].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_A__U52_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_A__U52 x12 <= P_id_A__U52 x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_A__U21_monotonic :
         forall x12 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_A__U21 x12 x14 <= P_id_A__U21 x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_A__U92_monotonic :
         forall x16 x12 x14 x13 x11 x15, 
          (0 <= x16)/\ (x16 <= x15) ->
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_A__U92 x12 x14 x16 <= P_id_A__U92 x11 x13 x15.
        Proof.
          intros x16 x15 x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          intros [H_5 H_4].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_A__U41_monotonic :
         forall x16 x12 x14 x13 x11 x15, 
          (0 <= x16)/\ (x16 <= x15) ->
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_A__U41 x12 x14 x16 <= P_id_A__U41 x11 x13 x15.
        Proof.
          intros x16 x15 x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          intros [H_5 H_4].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_A__U84_monotonic :
         forall x16 x12 x14 x13 x11 x15, 
          (0 <= x16)/\ (x16 <= x15) ->
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_A__U84 x12 x14 x16 <= P_id_A__U84 x11 x13 x15.
        Proof.
          intros x16 x15 x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          intros [H_5 H_4].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_A__ISNAT_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_A__ISNAT x12 <= P_id_A__ISNAT x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_A__U12_monotonic :
         forall x12 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_A__U12 x12 x14 <= P_id_A__U12 x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_MARK_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_MARK x12 <= P_id_MARK x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_A__U45_monotonic :
         forall x12 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_A__U45 x12 x14 <= P_id_A__U45 x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Definition marked_measure  := 
          InterpZ.marked_measure 0 P_id_a__zeros P_id_U51 P_id_a__U84 
           P_id_a__U32 P_id_isNatList P_id_a__U13 P_id_U85 P_id_a__length 
           P_id_a__isNatIList P_id_U33 P_id_a__U11 P_id_U81 P_id_a__U92 
           P_id_a__U43 P_id_U23 P_id_a__isNatKind P_id_U93 P_id_U11 
           P_id_a__U71 P_id_U44 P_id_0 P_id_U61 P_id_a__U86 P_id_a__U41 
           P_id_U22 P_id_a__U21 P_id_U91 P_id_length P_id_a__U52 P_id_U42 
           P_id_a__U12 P_id_U83 P_id_a__U94 P_id_a__U45 P_id_U31 
           P_id_a__isNat P_id_isNatIListKind P_id_a__U82 P_id_U46 P_id_cons 
           P_id_U52 P_id_a__U85 P_id_a__U33 P_id_U21 P_id_a__isNatList 
           P_id_U86 P_id_mark P_id_a__U51 P_id_U41 P_id_tt P_id_U82 
           P_id_a__U93 P_id_a__U44 P_id_isNat P_id_a__U23 P_id_U94 P_id_U12 
           P_id_a__U81 P_id_U45 P_id_zeros P_id_U71 P_id_a__U91 P_id_a__U42 
           P_id_isNatKind P_id_a__U22 P_id_U92 P_id_nil P_id_a__U61 P_id_U43 
           P_id_a__isNatIListKind P_id_U84 P_id_s P_id_a__U46 P_id_U32 
           P_id_a__U31 P_id_U13 P_id_a__U83 P_id_isNatIList P_id_A__U46 
           P_id_A__ISNATILISTKIND P_id_A__U61 P_id_A__U85 P_id_A__U31 
           P_id_A__U81 P_id_A__U22 P_id_A__U93 P_id_A__U42 P_id_A__U51 
           P_id_A__ISNATLIST P_id_A__ZEROS P_id_A__U91 P_id_A__U33 
           P_id_A__U83 P_id_A__U23 P_id_A__U11 P_id_A__LENGTH P_id_A__U44 
           P_id_A__ISNATILIST P_id_A__U13 P_id_A__U71 P_id_A__U86 
           P_id_A__U32 P_id_A__U82 P_id_A__ISNATKIND P_id_A__U94 P_id_A__U43 
           P_id_A__U52 P_id_A__U21 P_id_A__U92 P_id_A__U41 P_id_A__U84 
           P_id_A__ISNAT P_id_A__U12 P_id_MARK P_id_A__U45.
        
        Lemma marked_measure_equation :
         forall t, 
          marked_measure t = match t with
                               | (algebra.Alg.Term algebra.F.id_a__U46 
                                  (x11::nil)) =>
                                P_id_A__U46 (measure x11)
                               | (algebra.Alg.Term 
                                  algebra.F.id_a__isNatIListKind (x11::nil)) =>
                                P_id_A__ISNATILISTKIND (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U61 
                                  (x11::nil)) =>
                                P_id_A__U61 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U85 (x12::
                                  x11::nil)) =>
                                P_id_A__U85 (measure x12) (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U31 (x12::
                                  x11::nil)) =>
                                P_id_A__U31 (measure x12) (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U81 (x13::
                                  x12::x11::nil)) =>
                                P_id_A__U81 (measure x13) (measure x12) 
                                 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U22 (x12::
                                  x11::nil)) =>
                                P_id_A__U22 (measure x12) (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U93 (x13::
                                  x12::x11::nil)) =>
                                P_id_A__U93 (measure x13) (measure x12) 
                                 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U42 (x13::
                                  x12::x11::nil)) =>
                                P_id_A__U42 (measure x13) (measure x12) 
                                 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U51 (x12::
                                  x11::nil)) =>
                                P_id_A__U51 (measure x12) (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__isNatList 
                                  (x11::nil)) =>
                                P_id_A__ISNATLIST (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__zeros nil) =>
                                P_id_A__ZEROS 
                               | (algebra.Alg.Term algebra.F.id_a__U91 (x13::
                                  x12::x11::nil)) =>
                                P_id_A__U91 (measure x13) (measure x12) 
                                 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U33 
                                  (x11::nil)) =>
                                P_id_A__U33 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U83 (x13::
                                  x12::x11::nil)) =>
                                P_id_A__U83 (measure x13) (measure x12) 
                                 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U23 
                                  (x11::nil)) =>
                                P_id_A__U23 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U11 (x12::
                                  x11::nil)) =>
                                P_id_A__U11 (measure x12) (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__length 
                                  (x11::nil)) =>
                                P_id_A__LENGTH (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U44 (x13::
                                  x12::x11::nil)) =>
                                P_id_A__U44 (measure x13) (measure x12) 
                                 (measure x11)
                               | (algebra.Alg.Term 
                                  algebra.F.id_a__isNatIList (x11::nil)) =>
                                P_id_A__ISNATILIST (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U13 
                                  (x11::nil)) =>
                                P_id_A__U13 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U71 
                                  (x11::nil)) =>
                                P_id_A__U71 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U86 
                                  (x11::nil)) =>
                                P_id_A__U86 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U32 (x12::
                                  x11::nil)) =>
                                P_id_A__U32 (measure x12) (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U82 (x13::
                                  x12::x11::nil)) =>
                                P_id_A__U82 (measure x13) (measure x12) 
                                 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__isNatKind 
                                  (x11::nil)) =>
                                P_id_A__ISNATKIND (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U94 (x12::
                                  x11::nil)) =>
                                P_id_A__U94 (measure x12) (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U43 (x13::
                                  x12::x11::nil)) =>
                                P_id_A__U43 (measure x13) (measure x12) 
                                 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U52 
                                  (x11::nil)) =>
                                P_id_A__U52 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U21 (x12::
                                  x11::nil)) =>
                                P_id_A__U21 (measure x12) (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U92 (x13::
                                  x12::x11::nil)) =>
                                P_id_A__U92 (measure x13) (measure x12) 
                                 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U41 (x13::
                                  x12::x11::nil)) =>
                                P_id_A__U41 (measure x13) (measure x12) 
                                 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U84 (x13::
                                  x12::x11::nil)) =>
                                P_id_A__U84 (measure x13) (measure x12) 
                                 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__isNat 
                                  (x11::nil)) =>
                                P_id_A__ISNAT (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U12 (x12::
                                  x11::nil)) =>
                                P_id_A__U12 (measure x12) (measure x11)
                               | (algebra.Alg.Term algebra.F.id_mark 
                                  (x11::nil)) =>
                                P_id_MARK (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U45 (x12::
                                  x11::nil)) =>
                                P_id_A__U45 (measure x12) (measure x11)
                               | _ => measure t
                               end.
        Proof.
          reflexivity .
        Qed.
        
        Lemma marked_measure_star_monotonic :
         forall f l1 l2, 
          (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step 
                                                            R_xml_0_deep_rew.R_xml_0_rules)
                                                           ) l1 l2) ->
           marked_measure (algebra.Alg.Term f l1) <= marked_measure (algebra.Alg.Term 
                                                                    f l2).
        Proof.
          unfold marked_measure in *.
          apply InterpZ.marked_measure_star_monotonic.
          intros ;apply P_id_U51_monotonic;assumption.
          intros ;apply P_id_a__U84_monotonic;assumption.
          intros ;apply P_id_a__U32_monotonic;assumption.
          intros ;apply P_id_isNatList_monotonic;assumption.
          intros ;apply P_id_a__U13_monotonic;assumption.
          intros ;apply P_id_U85_monotonic;assumption.
          intros ;apply P_id_a__length_monotonic;assumption.
          intros ;apply P_id_a__isNatIList_monotonic;assumption.
          intros ;apply P_id_U33_monotonic;assumption.
          intros ;apply P_id_a__U11_monotonic;assumption.
          intros ;apply P_id_U81_monotonic;assumption.
          intros ;apply P_id_a__U92_monotonic;assumption.
          intros ;apply P_id_a__U43_monotonic;assumption.
          intros ;apply P_id_U23_monotonic;assumption.
          intros ;apply P_id_a__isNatKind_monotonic;assumption.
          intros ;apply P_id_U93_monotonic;assumption.
          intros ;apply P_id_U11_monotonic;assumption.
          intros ;apply P_id_a__U71_monotonic;assumption.
          intros ;apply P_id_U44_monotonic;assumption.
          intros ;apply P_id_U61_monotonic;assumption.
          intros ;apply P_id_a__U86_monotonic;assumption.
          intros ;apply P_id_a__U41_monotonic;assumption.
          intros ;apply P_id_U22_monotonic;assumption.
          intros ;apply P_id_a__U21_monotonic;assumption.
          intros ;apply P_id_U91_monotonic;assumption.
          intros ;apply P_id_length_monotonic;assumption.
          intros ;apply P_id_a__U52_monotonic;assumption.
          intros ;apply P_id_U42_monotonic;assumption.
          intros ;apply P_id_a__U12_monotonic;assumption.
          intros ;apply P_id_U83_monotonic;assumption.
          intros ;apply P_id_a__U94_monotonic;assumption.
          intros ;apply P_id_a__U45_monotonic;assumption.
          intros ;apply P_id_U31_monotonic;assumption.
          intros ;apply P_id_a__isNat_monotonic;assumption.
          intros ;apply P_id_isNatIListKind_monotonic;assumption.
          intros ;apply P_id_a__U82_monotonic;assumption.
          intros ;apply P_id_U46_monotonic;assumption.
          intros ;apply P_id_cons_monotonic;assumption.
          intros ;apply P_id_U52_monotonic;assumption.
          intros ;apply P_id_a__U85_monotonic;assumption.
          intros ;apply P_id_a__U33_monotonic;assumption.
          intros ;apply P_id_U21_monotonic;assumption.
          intros ;apply P_id_a__isNatList_monotonic;assumption.
          intros ;apply P_id_U86_monotonic;assumption.
          intros ;apply P_id_mark_monotonic;assumption.
          intros ;apply P_id_a__U51_monotonic;assumption.
          intros ;apply P_id_U41_monotonic;assumption.
          intros ;apply P_id_U82_monotonic;assumption.
          intros ;apply P_id_a__U93_monotonic;assumption.
          intros ;apply P_id_a__U44_monotonic;assumption.
          intros ;apply P_id_isNat_monotonic;assumption.
          intros ;apply P_id_a__U23_monotonic;assumption.
          intros ;apply P_id_U94_monotonic;assumption.
          intros ;apply P_id_U12_monotonic;assumption.
          intros ;apply P_id_a__U81_monotonic;assumption.
          intros ;apply P_id_U45_monotonic;assumption.
          intros ;apply P_id_U71_monotonic;assumption.
          intros ;apply P_id_a__U91_monotonic;assumption.
          intros ;apply P_id_a__U42_monotonic;assumption.
          intros ;apply P_id_isNatKind_monotonic;assumption.
          intros ;apply P_id_a__U22_monotonic;assumption.
          intros ;apply P_id_U92_monotonic;assumption.
          intros ;apply P_id_a__U61_monotonic;assumption.
          intros ;apply P_id_U43_monotonic;assumption.
          intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
          intros ;apply P_id_U84_monotonic;assumption.
          intros ;apply P_id_s_monotonic;assumption.
          intros ;apply P_id_a__U46_monotonic;assumption.
          intros ;apply P_id_U32_monotonic;assumption.
          intros ;apply P_id_a__U31_monotonic;assumption.
          intros ;apply P_id_U13_monotonic;assumption.
          intros ;apply P_id_a__U83_monotonic;assumption.
          intros ;apply P_id_isNatIList_monotonic;assumption.
          intros ;apply P_id_a__zeros_bounded;assumption.
          intros ;apply P_id_U51_bounded;assumption.
          intros ;apply P_id_a__U84_bounded;assumption.
          intros ;apply P_id_a__U32_bounded;assumption.
          intros ;apply P_id_isNatList_bounded;assumption.
          intros ;apply P_id_a__U13_bounded;assumption.
          intros ;apply P_id_U85_bounded;assumption.
          intros ;apply P_id_a__length_bounded;assumption.
          intros ;apply P_id_a__isNatIList_bounded;assumption.
          intros ;apply P_id_U33_bounded;assumption.
          intros ;apply P_id_a__U11_bounded;assumption.
          intros ;apply P_id_U81_bounded;assumption.
          intros ;apply P_id_a__U92_bounded;assumption.
          intros ;apply P_id_a__U43_bounded;assumption.
          intros ;apply P_id_U23_bounded;assumption.
          intros ;apply P_id_a__isNatKind_bounded;assumption.
          intros ;apply P_id_U93_bounded;assumption.
          intros ;apply P_id_U11_bounded;assumption.
          intros ;apply P_id_a__U71_bounded;assumption.
          intros ;apply P_id_U44_bounded;assumption.
          intros ;apply P_id_0_bounded;assumption.
          intros ;apply P_id_U61_bounded;assumption.
          intros ;apply P_id_a__U86_bounded;assumption.
          intros ;apply P_id_a__U41_bounded;assumption.
          intros ;apply P_id_U22_bounded;assumption.
          intros ;apply P_id_a__U21_bounded;assumption.
          intros ;apply P_id_U91_bounded;assumption.
          intros ;apply P_id_length_bounded;assumption.
          intros ;apply P_id_a__U52_bounded;assumption.
          intros ;apply P_id_U42_bounded;assumption.
          intros ;apply P_id_a__U12_bounded;assumption.
          intros ;apply P_id_U83_bounded;assumption.
          intros ;apply P_id_a__U94_bounded;assumption.
          intros ;apply P_id_a__U45_bounded;assumption.
          intros ;apply P_id_U31_bounded;assumption.
          intros ;apply P_id_a__isNat_bounded;assumption.
          intros ;apply P_id_isNatIListKind_bounded;assumption.
          intros ;apply P_id_a__U82_bounded;assumption.
          intros ;apply P_id_U46_bounded;assumption.
          intros ;apply P_id_cons_bounded;assumption.
          intros ;apply P_id_U52_bounded;assumption.
          intros ;apply P_id_a__U85_bounded;assumption.
          intros ;apply P_id_a__U33_bounded;assumption.
          intros ;apply P_id_U21_bounded;assumption.
          intros ;apply P_id_a__isNatList_bounded;assumption.
          intros ;apply P_id_U86_bounded;assumption.
          intros ;apply P_id_mark_bounded;assumption.
          intros ;apply P_id_a__U51_bounded;assumption.
          intros ;apply P_id_U41_bounded;assumption.
          intros ;apply P_id_tt_bounded;assumption.
          intros ;apply P_id_U82_bounded;assumption.
          intros ;apply P_id_a__U93_bounded;assumption.
          intros ;apply P_id_a__U44_bounded;assumption.
          intros ;apply P_id_isNat_bounded;assumption.
          intros ;apply P_id_a__U23_bounded;assumption.
          intros ;apply P_id_U94_bounded;assumption.
          intros ;apply P_id_U12_bounded;assumption.
          intros ;apply P_id_a__U81_bounded;assumption.
          intros ;apply P_id_U45_bounded;assumption.
          intros ;apply P_id_zeros_bounded;assumption.
          intros ;apply P_id_U71_bounded;assumption.
          intros ;apply P_id_a__U91_bounded;assumption.
          intros ;apply P_id_a__U42_bounded;assumption.
          intros ;apply P_id_isNatKind_bounded;assumption.
          intros ;apply P_id_a__U22_bounded;assumption.
          intros ;apply P_id_U92_bounded;assumption.
          intros ;apply P_id_nil_bounded;assumption.
          intros ;apply P_id_a__U61_bounded;assumption.
          intros ;apply P_id_U43_bounded;assumption.
          intros ;apply P_id_a__isNatIListKind_bounded;assumption.
          intros ;apply P_id_U84_bounded;assumption.
          intros ;apply P_id_s_bounded;assumption.
          intros ;apply P_id_a__U46_bounded;assumption.
          intros ;apply P_id_U32_bounded;assumption.
          intros ;apply P_id_a__U31_bounded;assumption.
          intros ;apply P_id_U13_bounded;assumption.
          intros ;apply P_id_a__U83_bounded;assumption.
          intros ;apply P_id_isNatIList_bounded;assumption.
          apply rules_monotonic.
          intros ;apply P_id_A__U46_monotonic;assumption.
          intros ;apply P_id_A__ISNATILISTKIND_monotonic;assumption.
          intros ;apply P_id_A__U61_monotonic;assumption.
          intros ;apply P_id_A__U85_monotonic;assumption.
          intros ;apply P_id_A__U31_monotonic;assumption.
          intros ;apply P_id_A__U81_monotonic;assumption.
          intros ;apply P_id_A__U22_monotonic;assumption.
          intros ;apply P_id_A__U93_monotonic;assumption.
          intros ;apply P_id_A__U42_monotonic;assumption.
          intros ;apply P_id_A__U51_monotonic;assumption.
          intros ;apply P_id_A__ISNATLIST_monotonic;assumption.
          intros ;apply P_id_A__U91_monotonic;assumption.
          intros ;apply P_id_A__U33_monotonic;assumption.
          intros ;apply P_id_A__U83_monotonic;assumption.
          intros ;apply P_id_A__U23_monotonic;assumption.
          intros ;apply P_id_A__U11_monotonic;assumption.
          intros ;apply P_id_A__LENGTH_monotonic;assumption.
          intros ;apply P_id_A__U44_monotonic;assumption.
          intros ;apply P_id_A__ISNATILIST_monotonic;assumption.
          intros ;apply P_id_A__U13_monotonic;assumption.
          intros ;apply P_id_A__U71_monotonic;assumption.
          intros ;apply P_id_A__U86_monotonic;assumption.
          intros ;apply P_id_A__U32_monotonic;assumption.
          intros ;apply P_id_A__U82_monotonic;assumption.
          intros ;apply P_id_A__ISNATKIND_monotonic;assumption.
          intros ;apply P_id_A__U94_monotonic;assumption.
          intros ;apply P_id_A__U43_monotonic;assumption.
          intros ;apply P_id_A__U52_monotonic;assumption.
          intros ;apply P_id_A__U21_monotonic;assumption.
          intros ;apply P_id_A__U92_monotonic;assumption.
          intros ;apply P_id_A__U41_monotonic;assumption.
          intros ;apply P_id_A__U84_monotonic;assumption.
          intros ;apply P_id_A__ISNAT_monotonic;assumption.
          intros ;apply P_id_A__U12_monotonic;assumption.
          intros ;apply P_id_MARK_monotonic;assumption.
          intros ;apply P_id_A__U45_monotonic;assumption.
        Qed.
        
        Ltac rewrite_and_unfold  :=
         do 2 (rewrite marked_measure_equation);
          repeat (
          match goal with
            |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
             rewrite (measure_equation (algebra.Alg.Term f t))
            | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ =>
             rewrite (measure_equation (algebra.Alg.Term f t)) in H|-
            end
          ).
        
        Definition lt a b := 
          (Zwf.Zwf 0) (marked_measure a) (marked_measure b).
        
        Definition le a b := marked_measure a <= marked_measure b.
        
        Lemma lt_le_compat : forall a b c, (lt a b) ->(le b c) ->lt a c.
        Proof.
          unfold lt, le in *.
          intros a b c.
          apply (interp.le_lt_compat_right (interp.o_Z 0)).
        Qed.
        
        Lemma wf_lt : well_founded lt.
        Proof.
          unfold lt in *.
          apply Inverse_Image.wf_inverse_image with  (B:=Z).
          apply Zwf.Zwf_well_founded.
        Qed.
        
        Lemma DP_R_xml_0_scc_65_large_scc_1_large_large_large_strict_in_lt :
         Relation_Definitions.inclusion _ 
          DP_R_xml_0_scc_65_large_scc_1_large_large_large_strict lt.
        Proof.
          unfold Relation_Definitions.inclusion, lt in *.
          
          intros a b H;destruct H;
           match goal with
             |  |- (Zwf.Zwf 0) _ (marked_measure (algebra.Alg.Term ?f ?l)) =>
              let l'' := algebra.Alg_ext.find_replacement l  in 
               ((apply (interp.le_lt_compat_right (interp.o_Z 0)) with
                  (marked_measure (algebra.Alg.Term f l''));[idtac|
                 apply marked_measure_star_monotonic;
                  repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos
                  );(assumption)||(constructor 1)]))
             end
           ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_in_le :
         Relation_Definitions.inclusion _ 
          DP_R_xml_0_scc_65_large_scc_1_large_large_large_large le.
        Proof.
          unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *.
          
          intros a b H;destruct H;
           match goal with
             |  |- _ <= marked_measure (algebra.Alg.Term ?f ?l) =>
              let l'' := algebra.Alg_ext.find_replacement l  in 
               ((apply (interp.le_trans (interp.o_Z 0)) with
                  (marked_measure (algebra.Alg.Term f l''));[idtac|
                 apply marked_measure_star_monotonic;
                  repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos
                  );(assumption)||(constructor 1)]))
             end
           ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Definition wf_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large  := 
          WF_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large.wf.
        
        
        Lemma wf :
         well_founded WF_DP_R_xml_0_scc_65_large_scc_1_large_large.DP_R_xml_0_scc_65_large_scc_1_large_large_large
          .
        Proof.
          intros x.
          apply (well_founded_ind wf_lt).
          clear x.
          intros x.
          pattern x.
          
          apply (@Acc_ind _ 
                  DP_R_xml_0_scc_65_large_scc_1_large_large_large_large).
          clear x.
          intros x _ IHx IHx'.
          constructor.
          intros y H.
          
          destruct H;
           (apply IHx';
             apply DP_R_xml_0_scc_65_large_scc_1_large_large_large_strict_in_lt;
             econstructor eassumption)||
           ((apply IHx;[econstructor eassumption|
             intros y' Hlt;apply IHx';apply lt_le_compat with (1:=Hlt) ;
              apply DP_R_xml_0_scc_65_large_scc_1_large_large_large_large_in_le;
              econstructor eassumption])).
          apply wf_DP_R_xml_0_scc_65_large_scc_1_large_large_large_large.
        Qed.
       End WF_DP_R_xml_0_scc_65_large_scc_1_large_large_large.
       
       Open Scope Z_scope.
       
       Import ring_extention.
       
       Notation Local "a <= b" := (Zle a b).
       
       Notation Local "a < b" := (Zlt a b).
       
       Definition P_id_a__zeros  := 1.
       
       Definition P_id_U51 (x11:Z) (x12:Z) := 1* x11.
       
       Definition P_id_a__U84 (x11:Z) (x12:Z) (x13:Z) := 2* x11.
       
       Definition P_id_a__U32 (x11:Z) (x12:Z) := 1 + 1* x11.
       
       Definition P_id_isNatList (x11:Z) := 0.
       
       Definition P_id_a__U13 (x11:Z) := 2* x11.
       
       Definition P_id_U85 (x11:Z) (x12:Z) := 1* x11.
       
       Definition P_id_a__length (x11:Z) := 0.
       
       Definition P_id_a__isNatIList (x11:Z) := 1.
       
       Definition P_id_U33 (x11:Z) := 2* x11.
       
       Definition P_id_a__U11 (x11:Z) (x12:Z) := 2* x11.
       
       Definition P_id_U81 (x11:Z) (x12:Z) (x13:Z) := 2* x11.
       
       Definition P_id_a__U92 (x11:Z) (x12:Z) (x13:Z) := 0.
       
       Definition P_id_a__U43 (x11:Z) (x12:Z) (x13:Z) := 1.
       
       Definition P_id_U23 (x11:Z) := 1* x11.
       
       Definition P_id_a__isNatKind (x11:Z) := 0.
       
       Definition P_id_U93 (x11:Z) (x12:Z) (x13:Z) := 0.
       
       Definition P_id_U11 (x11:Z) (x12:Z) := 2* x11.
       
       Definition P_id_a__U71 (x11:Z) := 1* x11.
       
       Definition P_id_U44 (x11:Z) (x12:Z) (x13:Z) := 1.
       
       Definition P_id_0  := 0.
       
       Definition P_id_U61 (x11:Z) := 2* x11.
       
       Definition P_id_a__U86 (x11:Z) := 2* x11.
       
       Definition P_id_a__U41 (x11:Z) (x12:Z) (x13:Z) := 1.
       
       Definition P_id_U22 (x11:Z) (x12:Z) := 1* x11.
       
       Definition P_id_a__U21 (x11:Z) (x12:Z) := 2* x11.
       
       Definition P_id_U91 (x11:Z) (x12:Z) (x13:Z) := 0.
       
       Definition P_id_length (x11:Z) := 0.
       
       Definition P_id_a__U52 (x11:Z) := 1* x11.
       
       Definition P_id_U42 (x11:Z) (x12:Z) (x13:Z) := 1.
       
       Definition P_id_a__U12 (x11:Z) (x12:Z) := 1* x11.
       
       Definition P_id_U83 (x11:Z) (x12:Z) (x13:Z) := 2* x11.
       
       Definition P_id_a__U94 (x11:Z) (x12:Z) := 0.
       
       Definition P_id_a__U45 (x11:Z) (x12:Z) := 1.
       
       Definition P_id_U31 (x11:Z) (x12:Z) := 1.
       
       Definition P_id_a__isNat (x11:Z) := 0.
       
       Definition P_id_isNatIListKind (x11:Z) := 0.
       
       Definition P_id_a__U82 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
       
       Definition P_id_U46 (x11:Z) := 1* x11.
       
       Definition P_id_cons (x11:Z) (x12:Z) := 0.
       
       Definition P_id_U52 (x11:Z) := 1* x11.
       
       Definition P_id_a__U85 (x11:Z) (x12:Z) := 1* x11.
       
       Definition P_id_a__U33 (x11:Z) := 2* x11.
       
       Definition P_id_U21 (x11:Z) (x12:Z) := 2* x11.
       
       Definition P_id_a__isNatList (x11:Z) := 0.
       
       Definition P_id_U86 (x11:Z) := 2* x11.
       
       Definition P_id_mark (x11:Z) := 1* x11.
       
       Definition P_id_a__U51 (x11:Z) (x12:Z) := 1* x11.
       
       Definition P_id_U41 (x11:Z) (x12:Z) (x13:Z) := 1.
       
       Definition P_id_tt  := 0.
       
       Definition P_id_U82 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
       
       Definition P_id_a__U93 (x11:Z) (x12:Z) (x13:Z) := 0.
       
       Definition P_id_a__U44 (x11:Z) (x12:Z) (x13:Z) := 1.
       
       Definition P_id_isNat (x11:Z) := 0.
       
       Definition P_id_a__U23 (x11:Z) := 1* x11.
       
       Definition P_id_U94 (x11:Z) (x12:Z) := 0.
       
       Definition P_id_U12 (x11:Z) (x12:Z) := 1* x11.
       
       Definition P_id_a__U81 (x11:Z) (x12:Z) (x13:Z) := 2* x11.
       
       Definition P_id_U45 (x11:Z) (x12:Z) := 1.
       
       Definition P_id_zeros  := 1.
       
       Definition P_id_U71 (x11:Z) := 1* x11.
       
       Definition P_id_a__U91 (x11:Z) (x12:Z) (x13:Z) := 0.
       
       Definition P_id_a__U42 (x11:Z) (x12:Z) (x13:Z) := 1.
       
       Definition P_id_isNatKind (x11:Z) := 0.
       
       Definition P_id_a__U22 (x11:Z) (x12:Z) := 1* x11.
       
       Definition P_id_U92 (x11:Z) (x12:Z) (x13:Z) := 0.
       
       Definition P_id_nil  := 0.
       
       Definition P_id_a__U61 (x11:Z) := 2* x11.
       
       Definition P_id_U43 (x11:Z) (x12:Z) (x13:Z) := 1.
       
       Definition P_id_a__isNatIListKind (x11:Z) := 0.
       
       Definition P_id_U84 (x11:Z) (x12:Z) (x13:Z) := 2* x11.
       
       Definition P_id_s (x11:Z) := 1* x11.
       
       Definition P_id_a__U46 (x11:Z) := 1* x11.
       
       Definition P_id_U32 (x11:Z) (x12:Z) := 1 + 1* x11.
       
       Definition P_id_a__U31 (x11:Z) (x12:Z) := 1.
       
       Definition P_id_U13 (x11:Z) := 2* x11.
       
       Definition P_id_a__U83 (x11:Z) (x12:Z) (x13:Z) := 2* x11.
       
       Definition P_id_isNatIList (x11:Z) := 1.
       
       Lemma P_id_U51_monotonic :
        forall x12 x14 x13 x11, 
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->P_id_U51 x12 x14 <= P_id_U51 x11 x13.
       Proof.
         intros x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U84_monotonic :
        forall x16 x12 x14 x13 x11 x15, 
         (0 <= x16)/\ (x16 <= x15) ->
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_a__U84 x12 x14 x16 <= P_id_a__U84 x11 x13 x15.
       Proof.
         intros x16 x15 x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         intros [H_5 H_4].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U32_monotonic :
        forall x12 x14 x13 x11, 
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_a__U32 x12 x14 <= P_id_a__U32 x11 x13.
       Proof.
         intros x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_isNatList_monotonic :
        forall x12 x11, 
         (0 <= x12)/\ (x12 <= x11) ->P_id_isNatList x12 <= P_id_isNatList x11.
       Proof.
         intros x12 x11.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U13_monotonic :
        forall x12 x11, 
         (0 <= x12)/\ (x12 <= x11) ->P_id_a__U13 x12 <= P_id_a__U13 x11.
       Proof.
         intros x12 x11.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U85_monotonic :
        forall x12 x14 x13 x11, 
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->P_id_U85 x12 x14 <= P_id_U85 x11 x13.
       Proof.
         intros x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__length_monotonic :
        forall x12 x11, 
         (0 <= x12)/\ (x12 <= x11) ->P_id_a__length x12 <= P_id_a__length x11.
       Proof.
         intros x12 x11.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__isNatIList_monotonic :
        forall x12 x11, 
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__isNatIList x12 <= P_id_a__isNatIList x11.
       Proof.
         intros x12 x11.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U33_monotonic :
        forall x12 x11, 
         (0 <= x12)/\ (x12 <= x11) ->P_id_U33 x12 <= P_id_U33 x11.
       Proof.
         intros x12 x11.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U11_monotonic :
        forall x12 x14 x13 x11, 
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_a__U11 x12 x14 <= P_id_a__U11 x11 x13.
       Proof.
         intros x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U81_monotonic :
        forall x16 x12 x14 x13 x11 x15, 
         (0 <= x16)/\ (x16 <= x15) ->
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_U81 x12 x14 x16 <= P_id_U81 x11 x13 x15.
       Proof.
         intros x16 x15 x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         intros [H_5 H_4].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U92_monotonic :
        forall x16 x12 x14 x13 x11 x15, 
         (0 <= x16)/\ (x16 <= x15) ->
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_a__U92 x12 x14 x16 <= P_id_a__U92 x11 x13 x15.
       Proof.
         intros x16 x15 x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         intros [H_5 H_4].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U43_monotonic :
        forall x16 x12 x14 x13 x11 x15, 
         (0 <= x16)/\ (x16 <= x15) ->
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_a__U43 x12 x14 x16 <= P_id_a__U43 x11 x13 x15.
       Proof.
         intros x16 x15 x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         intros [H_5 H_4].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U23_monotonic :
        forall x12 x11, 
         (0 <= x12)/\ (x12 <= x11) ->P_id_U23 x12 <= P_id_U23 x11.
       Proof.
         intros x12 x11.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__isNatKind_monotonic :
        forall x12 x11, 
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__isNatKind x12 <= P_id_a__isNatKind x11.
       Proof.
         intros x12 x11.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U93_monotonic :
        forall x16 x12 x14 x13 x11 x15, 
         (0 <= x16)/\ (x16 <= x15) ->
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_U93 x12 x14 x16 <= P_id_U93 x11 x13 x15.
       Proof.
         intros x16 x15 x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         intros [H_5 H_4].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U11_monotonic :
        forall x12 x14 x13 x11, 
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->P_id_U11 x12 x14 <= P_id_U11 x11 x13.
       Proof.
         intros x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U71_monotonic :
        forall x12 x11, 
         (0 <= x12)/\ (x12 <= x11) ->P_id_a__U71 x12 <= P_id_a__U71 x11.
       Proof.
         intros x12 x11.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U44_monotonic :
        forall x16 x12 x14 x13 x11 x15, 
         (0 <= x16)/\ (x16 <= x15) ->
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_U44 x12 x14 x16 <= P_id_U44 x11 x13 x15.
       Proof.
         intros x16 x15 x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         intros [H_5 H_4].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U61_monotonic :
        forall x12 x11, 
         (0 <= x12)/\ (x12 <= x11) ->P_id_U61 x12 <= P_id_U61 x11.
       Proof.
         intros x12 x11.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U86_monotonic :
        forall x12 x11, 
         (0 <= x12)/\ (x12 <= x11) ->P_id_a__U86 x12 <= P_id_a__U86 x11.
       Proof.
         intros x12 x11.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U41_monotonic :
        forall x16 x12 x14 x13 x11 x15, 
         (0 <= x16)/\ (x16 <= x15) ->
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_a__U41 x12 x14 x16 <= P_id_a__U41 x11 x13 x15.
       Proof.
         intros x16 x15 x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         intros [H_5 H_4].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U22_monotonic :
        forall x12 x14 x13 x11, 
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->P_id_U22 x12 x14 <= P_id_U22 x11 x13.
       Proof.
         intros x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U21_monotonic :
        forall x12 x14 x13 x11, 
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_a__U21 x12 x14 <= P_id_a__U21 x11 x13.
       Proof.
         intros x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U91_monotonic :
        forall x16 x12 x14 x13 x11 x15, 
         (0 <= x16)/\ (x16 <= x15) ->
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_U91 x12 x14 x16 <= P_id_U91 x11 x13 x15.
       Proof.
         intros x16 x15 x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         intros [H_5 H_4].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_length_monotonic :
        forall x12 x11, 
         (0 <= x12)/\ (x12 <= x11) ->P_id_length x12 <= P_id_length x11.
       Proof.
         intros x12 x11.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U52_monotonic :
        forall x12 x11, 
         (0 <= x12)/\ (x12 <= x11) ->P_id_a__U52 x12 <= P_id_a__U52 x11.
       Proof.
         intros x12 x11.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U42_monotonic :
        forall x16 x12 x14 x13 x11 x15, 
         (0 <= x16)/\ (x16 <= x15) ->
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_U42 x12 x14 x16 <= P_id_U42 x11 x13 x15.
       Proof.
         intros x16 x15 x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         intros [H_5 H_4].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U12_monotonic :
        forall x12 x14 x13 x11, 
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_a__U12 x12 x14 <= P_id_a__U12 x11 x13.
       Proof.
         intros x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U83_monotonic :
        forall x16 x12 x14 x13 x11 x15, 
         (0 <= x16)/\ (x16 <= x15) ->
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_U83 x12 x14 x16 <= P_id_U83 x11 x13 x15.
       Proof.
         intros x16 x15 x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         intros [H_5 H_4].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U94_monotonic :
        forall x12 x14 x13 x11, 
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_a__U94 x12 x14 <= P_id_a__U94 x11 x13.
       Proof.
         intros x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U45_monotonic :
        forall x12 x14 x13 x11, 
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_a__U45 x12 x14 <= P_id_a__U45 x11 x13.
       Proof.
         intros x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U31_monotonic :
        forall x12 x14 x13 x11, 
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->P_id_U31 x12 x14 <= P_id_U31 x11 x13.
       Proof.
         intros x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__isNat_monotonic :
        forall x12 x11, 
         (0 <= x12)/\ (x12 <= x11) ->P_id_a__isNat x12 <= P_id_a__isNat x11.
       Proof.
         intros x12 x11.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_isNatIListKind_monotonic :
        forall x12 x11, 
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_isNatIListKind x12 <= P_id_isNatIListKind x11.
       Proof.
         intros x12 x11.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U82_monotonic :
        forall x16 x12 x14 x13 x11 x15, 
         (0 <= x16)/\ (x16 <= x15) ->
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_a__U82 x12 x14 x16 <= P_id_a__U82 x11 x13 x15.
       Proof.
         intros x16 x15 x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         intros [H_5 H_4].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U46_monotonic :
        forall x12 x11, 
         (0 <= x12)/\ (x12 <= x11) ->P_id_U46 x12 <= P_id_U46 x11.
       Proof.
         intros x12 x11.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_cons_monotonic :
        forall x12 x14 x13 x11, 
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->P_id_cons x12 x14 <= P_id_cons x11 x13.
       Proof.
         intros x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U52_monotonic :
        forall x12 x11, 
         (0 <= x12)/\ (x12 <= x11) ->P_id_U52 x12 <= P_id_U52 x11.
       Proof.
         intros x12 x11.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U85_monotonic :
        forall x12 x14 x13 x11, 
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_a__U85 x12 x14 <= P_id_a__U85 x11 x13.
       Proof.
         intros x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U33_monotonic :
        forall x12 x11, 
         (0 <= x12)/\ (x12 <= x11) ->P_id_a__U33 x12 <= P_id_a__U33 x11.
       Proof.
         intros x12 x11.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U21_monotonic :
        forall x12 x14 x13 x11, 
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->P_id_U21 x12 x14 <= P_id_U21 x11 x13.
       Proof.
         intros x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__isNatList_monotonic :
        forall x12 x11, 
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__isNatList x12 <= P_id_a__isNatList x11.
       Proof.
         intros x12 x11.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U86_monotonic :
        forall x12 x11, 
         (0 <= x12)/\ (x12 <= x11) ->P_id_U86 x12 <= P_id_U86 x11.
       Proof.
         intros x12 x11.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_mark_monotonic :
        forall x12 x11, 
         (0 <= x12)/\ (x12 <= x11) ->P_id_mark x12 <= P_id_mark x11.
       Proof.
         intros x12 x11.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U51_monotonic :
        forall x12 x14 x13 x11, 
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_a__U51 x12 x14 <= P_id_a__U51 x11 x13.
       Proof.
         intros x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U41_monotonic :
        forall x16 x12 x14 x13 x11 x15, 
         (0 <= x16)/\ (x16 <= x15) ->
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_U41 x12 x14 x16 <= P_id_U41 x11 x13 x15.
       Proof.
         intros x16 x15 x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         intros [H_5 H_4].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U82_monotonic :
        forall x16 x12 x14 x13 x11 x15, 
         (0 <= x16)/\ (x16 <= x15) ->
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_U82 x12 x14 x16 <= P_id_U82 x11 x13 x15.
       Proof.
         intros x16 x15 x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         intros [H_5 H_4].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U93_monotonic :
        forall x16 x12 x14 x13 x11 x15, 
         (0 <= x16)/\ (x16 <= x15) ->
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_a__U93 x12 x14 x16 <= P_id_a__U93 x11 x13 x15.
       Proof.
         intros x16 x15 x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         intros [H_5 H_4].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U44_monotonic :
        forall x16 x12 x14 x13 x11 x15, 
         (0 <= x16)/\ (x16 <= x15) ->
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_a__U44 x12 x14 x16 <= P_id_a__U44 x11 x13 x15.
       Proof.
         intros x16 x15 x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         intros [H_5 H_4].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_isNat_monotonic :
        forall x12 x11, 
         (0 <= x12)/\ (x12 <= x11) ->P_id_isNat x12 <= P_id_isNat x11.
       Proof.
         intros x12 x11.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U23_monotonic :
        forall x12 x11, 
         (0 <= x12)/\ (x12 <= x11) ->P_id_a__U23 x12 <= P_id_a__U23 x11.
       Proof.
         intros x12 x11.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U94_monotonic :
        forall x12 x14 x13 x11, 
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->P_id_U94 x12 x14 <= P_id_U94 x11 x13.
       Proof.
         intros x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U12_monotonic :
        forall x12 x14 x13 x11, 
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->P_id_U12 x12 x14 <= P_id_U12 x11 x13.
       Proof.
         intros x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U81_monotonic :
        forall x16 x12 x14 x13 x11 x15, 
         (0 <= x16)/\ (x16 <= x15) ->
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_a__U81 x12 x14 x16 <= P_id_a__U81 x11 x13 x15.
       Proof.
         intros x16 x15 x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         intros [H_5 H_4].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U45_monotonic :
        forall x12 x14 x13 x11, 
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->P_id_U45 x12 x14 <= P_id_U45 x11 x13.
       Proof.
         intros x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U71_monotonic :
        forall x12 x11, 
         (0 <= x12)/\ (x12 <= x11) ->P_id_U71 x12 <= P_id_U71 x11.
       Proof.
         intros x12 x11.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U91_monotonic :
        forall x16 x12 x14 x13 x11 x15, 
         (0 <= x16)/\ (x16 <= x15) ->
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_a__U91 x12 x14 x16 <= P_id_a__U91 x11 x13 x15.
       Proof.
         intros x16 x15 x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         intros [H_5 H_4].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U42_monotonic :
        forall x16 x12 x14 x13 x11 x15, 
         (0 <= x16)/\ (x16 <= x15) ->
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_a__U42 x12 x14 x16 <= P_id_a__U42 x11 x13 x15.
       Proof.
         intros x16 x15 x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         intros [H_5 H_4].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_isNatKind_monotonic :
        forall x12 x11, 
         (0 <= x12)/\ (x12 <= x11) ->P_id_isNatKind x12 <= P_id_isNatKind x11.
       Proof.
         intros x12 x11.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U22_monotonic :
        forall x12 x14 x13 x11, 
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_a__U22 x12 x14 <= P_id_a__U22 x11 x13.
       Proof.
         intros x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U92_monotonic :
        forall x16 x12 x14 x13 x11 x15, 
         (0 <= x16)/\ (x16 <= x15) ->
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_U92 x12 x14 x16 <= P_id_U92 x11 x13 x15.
       Proof.
         intros x16 x15 x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         intros [H_5 H_4].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U61_monotonic :
        forall x12 x11, 
         (0 <= x12)/\ (x12 <= x11) ->P_id_a__U61 x12 <= P_id_a__U61 x11.
       Proof.
         intros x12 x11.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U43_monotonic :
        forall x16 x12 x14 x13 x11 x15, 
         (0 <= x16)/\ (x16 <= x15) ->
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_U43 x12 x14 x16 <= P_id_U43 x11 x13 x15.
       Proof.
         intros x16 x15 x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         intros [H_5 H_4].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__isNatIListKind_monotonic :
        forall x12 x11, 
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__isNatIListKind x12 <= P_id_a__isNatIListKind x11.
       Proof.
         intros x12 x11.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U84_monotonic :
        forall x16 x12 x14 x13 x11 x15, 
         (0 <= x16)/\ (x16 <= x15) ->
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_U84 x12 x14 x16 <= P_id_U84 x11 x13 x15.
       Proof.
         intros x16 x15 x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         intros [H_5 H_4].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_s_monotonic :
        forall x12 x11, (0 <= x12)/\ (x12 <= x11) ->P_id_s x12 <= P_id_s x11.
       Proof.
         intros x12 x11.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U46_monotonic :
        forall x12 x11, 
         (0 <= x12)/\ (x12 <= x11) ->P_id_a__U46 x12 <= P_id_a__U46 x11.
       Proof.
         intros x12 x11.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U32_monotonic :
        forall x12 x14 x13 x11, 
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->P_id_U32 x12 x14 <= P_id_U32 x11 x13.
       Proof.
         intros x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U31_monotonic :
        forall x12 x14 x13 x11, 
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_a__U31 x12 x14 <= P_id_a__U31 x11 x13.
       Proof.
         intros x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U13_monotonic :
        forall x12 x11, 
         (0 <= x12)/\ (x12 <= x11) ->P_id_U13 x12 <= P_id_U13 x11.
       Proof.
         intros x12 x11.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U83_monotonic :
        forall x16 x12 x14 x13 x11 x15, 
         (0 <= x16)/\ (x16 <= x15) ->
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_a__U83 x12 x14 x16 <= P_id_a__U83 x11 x13 x15.
       Proof.
         intros x16 x15 x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         intros [H_5 H_4].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_isNatIList_monotonic :
        forall x12 x11, 
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_isNatIList x12 <= P_id_isNatIList x11.
       Proof.
         intros x12 x11.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__zeros_bounded : 0 <= P_id_a__zeros .
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U51_bounded :
        forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U51 x12 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U84_bounded :
        forall x12 x13 x11, 
         (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U84 x13 x12 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U32_bounded :
        forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U32 x12 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_isNatList_bounded :
        forall x11, (0 <= x11) ->0 <= P_id_isNatList x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U13_bounded :
        forall x11, (0 <= x11) ->0 <= P_id_a__U13 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U85_bounded :
        forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U85 x12 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__length_bounded :
        forall x11, (0 <= x11) ->0 <= P_id_a__length x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__isNatIList_bounded :
        forall x11, (0 <= x11) ->0 <= P_id_a__isNatIList x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U33_bounded : forall x11, (0 <= x11) ->0 <= P_id_U33 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U11_bounded :
        forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U11 x12 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U81_bounded :
        forall x12 x13 x11, 
         (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U81 x13 x12 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U92_bounded :
        forall x12 x13 x11, 
         (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U92 x13 x12 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U43_bounded :
        forall x12 x13 x11, 
         (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U43 x13 x12 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U23_bounded : forall x11, (0 <= x11) ->0 <= P_id_U23 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__isNatKind_bounded :
        forall x11, (0 <= x11) ->0 <= P_id_a__isNatKind x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U93_bounded :
        forall x12 x13 x11, 
         (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U93 x13 x12 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U11_bounded :
        forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U11 x12 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U71_bounded :
        forall x11, (0 <= x11) ->0 <= P_id_a__U71 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U44_bounded :
        forall x12 x13 x11, 
         (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U44 x13 x12 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_0_bounded : 0 <= P_id_0 .
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U61_bounded : forall x11, (0 <= x11) ->0 <= P_id_U61 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U86_bounded :
        forall x11, (0 <= x11) ->0 <= P_id_a__U86 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U41_bounded :
        forall x12 x13 x11, 
         (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U41 x13 x12 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U22_bounded :
        forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U22 x12 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U21_bounded :
        forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U21 x12 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U91_bounded :
        forall x12 x13 x11, 
         (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U91 x13 x12 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_length_bounded :
        forall x11, (0 <= x11) ->0 <= P_id_length x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U52_bounded :
        forall x11, (0 <= x11) ->0 <= P_id_a__U52 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U42_bounded :
        forall x12 x13 x11, 
         (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U42 x13 x12 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U12_bounded :
        forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U12 x12 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U83_bounded :
        forall x12 x13 x11, 
         (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U83 x13 x12 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U94_bounded :
        forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U94 x12 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U45_bounded :
        forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U45 x12 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U31_bounded :
        forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U31 x12 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__isNat_bounded :
        forall x11, (0 <= x11) ->0 <= P_id_a__isNat x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_isNatIListKind_bounded :
        forall x11, (0 <= x11) ->0 <= P_id_isNatIListKind x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U82_bounded :
        forall x12 x13 x11, 
         (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U82 x13 x12 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U46_bounded : forall x11, (0 <= x11) ->0 <= P_id_U46 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_cons_bounded :
        forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_cons x12 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U52_bounded : forall x11, (0 <= x11) ->0 <= P_id_U52 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U85_bounded :
        forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U85 x12 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U33_bounded :
        forall x11, (0 <= x11) ->0 <= P_id_a__U33 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U21_bounded :
        forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U21 x12 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__isNatList_bounded :
        forall x11, (0 <= x11) ->0 <= P_id_a__isNatList x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U86_bounded : forall x11, (0 <= x11) ->0 <= P_id_U86 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_mark_bounded : forall x11, (0 <= x11) ->0 <= P_id_mark x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U51_bounded :
        forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U51 x12 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U41_bounded :
        forall x12 x13 x11, 
         (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U41 x13 x12 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_tt_bounded : 0 <= P_id_tt .
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U82_bounded :
        forall x12 x13 x11, 
         (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U82 x13 x12 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U93_bounded :
        forall x12 x13 x11, 
         (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U93 x13 x12 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U44_bounded :
        forall x12 x13 x11, 
         (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U44 x13 x12 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_isNat_bounded :
        forall x11, (0 <= x11) ->0 <= P_id_isNat x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U23_bounded :
        forall x11, (0 <= x11) ->0 <= P_id_a__U23 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U94_bounded :
        forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U94 x12 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U12_bounded :
        forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U12 x12 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U81_bounded :
        forall x12 x13 x11, 
         (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U81 x13 x12 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U45_bounded :
        forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U45 x12 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_zeros_bounded : 0 <= P_id_zeros .
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U71_bounded : forall x11, (0 <= x11) ->0 <= P_id_U71 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U91_bounded :
        forall x12 x13 x11, 
         (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U91 x13 x12 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U42_bounded :
        forall x12 x13 x11, 
         (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U42 x13 x12 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_isNatKind_bounded :
        forall x11, (0 <= x11) ->0 <= P_id_isNatKind x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U22_bounded :
        forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U22 x12 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U92_bounded :
        forall x12 x13 x11, 
         (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U92 x13 x12 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_nil_bounded : 0 <= P_id_nil .
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U61_bounded :
        forall x11, (0 <= x11) ->0 <= P_id_a__U61 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U43_bounded :
        forall x12 x13 x11, 
         (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U43 x13 x12 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__isNatIListKind_bounded :
        forall x11, (0 <= x11) ->0 <= P_id_a__isNatIListKind x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U84_bounded :
        forall x12 x13 x11, 
         (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U84 x13 x12 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_s_bounded : forall x11, (0 <= x11) ->0 <= P_id_s x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U46_bounded :
        forall x11, (0 <= x11) ->0 <= P_id_a__U46 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U32_bounded :
        forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U32 x12 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U31_bounded :
        forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U31 x12 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U13_bounded : forall x11, (0 <= x11) ->0 <= P_id_U13 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__U83_bounded :
        forall x12 x13 x11, 
         (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U83 x13 x12 x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_isNatIList_bounded :
        forall x11, (0 <= x11) ->0 <= P_id_isNatIList x11.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Definition measure  := 
         InterpZ.measure 0 P_id_a__zeros P_id_U51 P_id_a__U84 P_id_a__U32 
          P_id_isNatList P_id_a__U13 P_id_U85 P_id_a__length 
          P_id_a__isNatIList P_id_U33 P_id_a__U11 P_id_U81 P_id_a__U92 
          P_id_a__U43 P_id_U23 P_id_a__isNatKind P_id_U93 P_id_U11 
          P_id_a__U71 P_id_U44 P_id_0 P_id_U61 P_id_a__U86 P_id_a__U41 
          P_id_U22 P_id_a__U21 P_id_U91 P_id_length P_id_a__U52 P_id_U42 
          P_id_a__U12 P_id_U83 P_id_a__U94 P_id_a__U45 P_id_U31 
          P_id_a__isNat P_id_isNatIListKind P_id_a__U82 P_id_U46 P_id_cons 
          P_id_U52 P_id_a__U85 P_id_a__U33 P_id_U21 P_id_a__isNatList 
          P_id_U86 P_id_mark P_id_a__U51 P_id_U41 P_id_tt P_id_U82 
          P_id_a__U93 P_id_a__U44 P_id_isNat P_id_a__U23 P_id_U94 P_id_U12 
          P_id_a__U81 P_id_U45 P_id_zeros P_id_U71 P_id_a__U91 P_id_a__U42 
          P_id_isNatKind P_id_a__U22 P_id_U92 P_id_nil P_id_a__U61 P_id_U43 
          P_id_a__isNatIListKind P_id_U84 P_id_s P_id_a__U46 P_id_U32 
          P_id_a__U31 P_id_U13 P_id_a__U83 P_id_isNatIList.
       
       Lemma measure_equation :
        forall t, 
         measure t = match t with
                       | (algebra.Alg.Term algebra.F.id_a__zeros nil) =>
                        P_id_a__zeros 
                       | (algebra.Alg.Term algebra.F.id_U51 (x12::x11::nil)) =>
                        P_id_U51 (measure x12) (measure x11)
                       | (algebra.Alg.Term algebra.F.id_a__U84 (x13::x12::
                          x11::nil)) =>
                        P_id_a__U84 (measure x13) (measure x12) (measure x11)
                       | (algebra.Alg.Term algebra.F.id_a__U32 (x12::
                          x11::nil)) =>
                        P_id_a__U32 (measure x12) (measure x11)
                       | (algebra.Alg.Term algebra.F.id_isNatList (x11::nil)) =>
                        P_id_isNatList (measure x11)
                       | (algebra.Alg.Term algebra.F.id_a__U13 (x11::nil)) =>
                        P_id_a__U13 (measure x11)
                       | (algebra.Alg.Term algebra.F.id_U85 (x12::x11::nil)) =>
                        P_id_U85 (measure x12) (measure x11)
                       | (algebra.Alg.Term algebra.F.id_a__length (x11::nil)) =>
                        P_id_a__length (measure x11)
                       | (algebra.Alg.Term algebra.F.id_a__isNatIList 
                          (x11::nil)) =>
                        P_id_a__isNatIList (measure x11)
                       | (algebra.Alg.Term algebra.F.id_U33 (x11::nil)) =>
                        P_id_U33 (measure x11)
                       | (algebra.Alg.Term algebra.F.id_a__U11 (x12::
                          x11::nil)) =>
                        P_id_a__U11 (measure x12) (measure x11)
                       | (algebra.Alg.Term algebra.F.id_U81 (x13::x12::
                          x11::nil)) =>
                        P_id_U81 (measure x13) (measure x12) (measure x11)
                       | (algebra.Alg.Term algebra.F.id_a__U92 (x13::x12::
                          x11::nil)) =>
                        P_id_a__U92 (measure x13) (measure x12) (measure x11)
                       | (algebra.Alg.Term algebra.F.id_a__U43 (x13::x12::
                          x11::nil)) =>
                        P_id_a__U43 (measure x13) (measure x12) (measure x11)
                       | (algebra.Alg.Term algebra.F.id_U23 (x11::nil)) =>
                        P_id_U23 (measure x11)
                       | (algebra.Alg.Term algebra.F.id_a__isNatKind 
                          (x11::nil)) =>
                        P_id_a__isNatKind (measure x11)
                       | (algebra.Alg.Term algebra.F.id_U93 (x13::x12::
                          x11::nil)) =>
                        P_id_U93 (measure x13) (measure x12) (measure x11)
                       | (algebra.Alg.Term algebra.F.id_U11 (x12::x11::nil)) =>
                        P_id_U11 (measure x12) (measure x11)
                       | (algebra.Alg.Term algebra.F.id_a__U71 (x11::nil)) =>
                        P_id_a__U71 (measure x11)
                       | (algebra.Alg.Term algebra.F.id_U44 (x13::x12::
                          x11::nil)) =>
                        P_id_U44 (measure x13) (measure x12) (measure x11)
                       | (algebra.Alg.Term algebra.F.id_0 nil) => P_id_0 
                       | (algebra.Alg.Term algebra.F.id_U61 (x11::nil)) =>
                        P_id_U61 (measure x11)
                       | (algebra.Alg.Term algebra.F.id_a__U86 (x11::nil)) =>
                        P_id_a__U86 (measure x11)
                       | (algebra.Alg.Term algebra.F.id_a__U41 (x13::x12::
                          x11::nil)) =>
                        P_id_a__U41 (measure x13) (measure x12) (measure x11)
                       | (algebra.Alg.Term algebra.F.id_U22 (x12::x11::nil)) =>
                        P_id_U22 (measure x12) (measure x11)
                       | (algebra.Alg.Term algebra.F.id_a__U21 (x12::
                          x11::nil)) =>
                        P_id_a__U21 (measure x12) (measure x11)
                       | (algebra.Alg.Term algebra.F.id_U91 (x13::x12::
                          x11::nil)) =>
                        P_id_U91 (measure x13) (measure x12) (measure x11)
                       | (algebra.Alg.Term algebra.F.id_length (x11::nil)) =>
                        P_id_length (measure x11)
                       | (algebra.Alg.Term algebra.F.id_a__U52 (x11::nil)) =>
                        P_id_a__U52 (measure x11)
                       | (algebra.Alg.Term algebra.F.id_U42 (x13::x12::
                          x11::nil)) =>
                        P_id_U42 (measure x13) (measure x12) (measure x11)
                       | (algebra.Alg.Term algebra.F.id_a__U12 (x12::
                          x11::nil)) =>
                        P_id_a__U12 (measure x12) (measure x11)
                       | (algebra.Alg.Term algebra.F.id_U83 (x13::x12::
                          x11::nil)) =>
                        P_id_U83 (measure x13) (measure x12) (measure x11)
                       | (algebra.Alg.Term algebra.F.id_a__U94 (x12::
                          x11::nil)) =>
                        P_id_a__U94 (measure x12) (measure x11)
                       | (algebra.Alg.Term algebra.F.id_a__U45 (x12::
                          x11::nil)) =>
                        P_id_a__U45 (measure x12) (measure x11)
                       | (algebra.Alg.Term algebra.F.id_U31 (x12::x11::nil)) =>
                        P_id_U31 (measure x12) (measure x11)
                       | (algebra.Alg.Term algebra.F.id_a__isNat (x11::nil)) =>
                        P_id_a__isNat (measure x11)
                       | (algebra.Alg.Term algebra.F.id_isNatIListKind 
                          (x11::nil)) =>
                        P_id_isNatIListKind (measure x11)
                       | (algebra.Alg.Term algebra.F.id_a__U82 (x13::x12::
                          x11::nil)) =>
                        P_id_a__U82 (measure x13) (measure x12) (measure x11)
                       | (algebra.Alg.Term algebra.F.id_U46 (x11::nil)) =>
                        P_id_U46 (measure x11)
                       | (algebra.Alg.Term algebra.F.id_cons (x12::x11::nil)) =>
                        P_id_cons (measure x12) (measure x11)
                       | (algebra.Alg.Term algebra.F.id_U52 (x11::nil)) =>
                        P_id_U52 (measure x11)
                       | (algebra.Alg.Term algebra.F.id_a__U85 (x12::
                          x11::nil)) =>
                        P_id_a__U85 (measure x12) (measure x11)
                       | (algebra.Alg.Term algebra.F.id_a__U33 (x11::nil)) =>
                        P_id_a__U33 (measure x11)
                       | (algebra.Alg.Term algebra.F.id_U21 (x12::x11::nil)) =>
                        P_id_U21 (measure x12) (measure x11)
                       | (algebra.Alg.Term algebra.F.id_a__isNatList 
                          (x11::nil)) =>
                        P_id_a__isNatList (measure x11)
                       | (algebra.Alg.Term algebra.F.id_U86 (x11::nil)) =>
                        P_id_U86 (measure x11)
                       | (algebra.Alg.Term algebra.F.id_mark (x11::nil)) =>
                        P_id_mark (measure x11)
                       | (algebra.Alg.Term algebra.F.id_a__U51 (x12::
                          x11::nil)) =>
                        P_id_a__U51 (measure x12) (measure x11)
                       | (algebra.Alg.Term algebra.F.id_U41 (x13::x12::
                          x11::nil)) =>
                        P_id_U41 (measure x13) (measure x12) (measure x11)
                       | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt 
                       | (algebra.Alg.Term algebra.F.id_U82 (x13::x12::
                          x11::nil)) =>
                        P_id_U82 (measure x13) (measure x12) (measure x11)
                       | (algebra.Alg.Term algebra.F.id_a__U93 (x13::x12::
                          x11::nil)) =>
                        P_id_a__U93 (measure x13) (measure x12) (measure x11)
                       | (algebra.Alg.Term algebra.F.id_a__U44 (x13::x12::
                          x11::nil)) =>
                        P_id_a__U44 (measure x13) (measure x12) (measure x11)
                       | (algebra.Alg.Term algebra.F.id_isNat (x11::nil)) =>
                        P_id_isNat (measure x11)
                       | (algebra.Alg.Term algebra.F.id_a__U23 (x11::nil)) =>
                        P_id_a__U23 (measure x11)
                       | (algebra.Alg.Term algebra.F.id_U94 (x12::x11::nil)) =>
                        P_id_U94 (measure x12) (measure x11)
                       | (algebra.Alg.Term algebra.F.id_U12 (x12::x11::nil)) =>
                        P_id_U12 (measure x12) (measure x11)
                       | (algebra.Alg.Term algebra.F.id_a__U81 (x13::x12::
                          x11::nil)) =>
                        P_id_a__U81 (measure x13) (measure x12) (measure x11)
                       | (algebra.Alg.Term algebra.F.id_U45 (x12::x11::nil)) =>
                        P_id_U45 (measure x12) (measure x11)
                       | (algebra.Alg.Term algebra.F.id_zeros nil) =>
                        P_id_zeros 
                       | (algebra.Alg.Term algebra.F.id_U71 (x11::nil)) =>
                        P_id_U71 (measure x11)
                       | (algebra.Alg.Term algebra.F.id_a__U91 (x13::x12::
                          x11::nil)) =>
                        P_id_a__U91 (measure x13) (measure x12) (measure x11)
                       | (algebra.Alg.Term algebra.F.id_a__U42 (x13::x12::
                          x11::nil)) =>
                        P_id_a__U42 (measure x13) (measure x12) (measure x11)
                       | (algebra.Alg.Term algebra.F.id_isNatKind (x11::nil)) =>
                        P_id_isNatKind (measure x11)
                       | (algebra.Alg.Term algebra.F.id_a__U22 (x12::
                          x11::nil)) =>
                        P_id_a__U22 (measure x12) (measure x11)
                       | (algebra.Alg.Term algebra.F.id_U92 (x13::x12::
                          x11::nil)) =>
                        P_id_U92 (measure x13) (measure x12) (measure x11)
                       | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil 
                       | (algebra.Alg.Term algebra.F.id_a__U61 (x11::nil)) =>
                        P_id_a__U61 (measure x11)
                       | (algebra.Alg.Term algebra.F.id_U43 (x13::x12::
                          x11::nil)) =>
                        P_id_U43 (measure x13) (measure x12) (measure x11)
                       | (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
                          (x11::nil)) =>
                        P_id_a__isNatIListKind (measure x11)
                       | (algebra.Alg.Term algebra.F.id_U84 (x13::x12::
                          x11::nil)) =>
                        P_id_U84 (measure x13) (measure x12) (measure x11)
                       | (algebra.Alg.Term algebra.F.id_s (x11::nil)) =>
                        P_id_s (measure x11)
                       | (algebra.Alg.Term algebra.F.id_a__U46 (x11::nil)) =>
                        P_id_a__U46 (measure x11)
                       | (algebra.Alg.Term algebra.F.id_U32 (x12::x11::nil)) =>
                        P_id_U32 (measure x12) (measure x11)
                       | (algebra.Alg.Term algebra.F.id_a__U31 (x12::
                          x11::nil)) =>
                        P_id_a__U31 (measure x12) (measure x11)
                       | (algebra.Alg.Term algebra.F.id_U13 (x11::nil)) =>
                        P_id_U13 (measure x11)
                       | (algebra.Alg.Term algebra.F.id_a__U83 (x13::x12::
                          x11::nil)) =>
                        P_id_a__U83 (measure x13) (measure x12) (measure x11)
                       | (algebra.Alg.Term algebra.F.id_isNatIList 
                          (x11::nil)) =>
                        P_id_isNatIList (measure x11)
                       | _ => 0
                       end.
       Proof.
         intros t;case t;intros ;apply refl_equal.
       Qed.
       
       Lemma measure_bounded : forall t, 0 <= measure t.
       Proof.
         unfold measure in |-*.
         
         apply InterpZ.measure_bounded;
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Ltac generate_pos_hyp  :=
        match goal with
          | H:context [measure ?x] |- _ =>
           let v := fresh "v" in 
            (let H := fresh "h" in 
              (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
                clearbody H;clearbody v))
          |  |- context [measure ?x] =>
           let v := fresh "v" in 
            (let H := fresh "h" in 
              (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
                clearbody H;clearbody v))
          end
        .
       
       Lemma rules_monotonic :
        forall l r, 
         (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) ->
          measure r <= measure l.
       Proof.
         intros l r H.
         fold measure in |-*.
         
         inversion H;clear H;subst;inversion H0;clear H0;subst;
          simpl algebra.EQT.T.apply_subst in |-*;
          repeat (
          match goal with
            |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
             rewrite (measure_equation (algebra.Alg.Term f t))
            end
          );repeat (generate_pos_hyp );
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma measure_star_monotonic :
        forall l r, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    r l) ->measure r <= measure l.
       Proof.
         unfold measure in *.
         apply InterpZ.measure_star_monotonic.
         intros ;apply P_id_U51_monotonic;assumption.
         intros ;apply P_id_a__U84_monotonic;assumption.
         intros ;apply P_id_a__U32_monotonic;assumption.
         intros ;apply P_id_isNatList_monotonic;assumption.
         intros ;apply P_id_a__U13_monotonic;assumption.
         intros ;apply P_id_U85_monotonic;assumption.
         intros ;apply P_id_a__length_monotonic;assumption.
         intros ;apply P_id_a__isNatIList_monotonic;assumption.
         intros ;apply P_id_U33_monotonic;assumption.
         intros ;apply P_id_a__U11_monotonic;assumption.
         intros ;apply P_id_U81_monotonic;assumption.
         intros ;apply P_id_a__U92_monotonic;assumption.
         intros ;apply P_id_a__U43_monotonic;assumption.
         intros ;apply P_id_U23_monotonic;assumption.
         intros ;apply P_id_a__isNatKind_monotonic;assumption.
         intros ;apply P_id_U93_monotonic;assumption.
         intros ;apply P_id_U11_monotonic;assumption.
         intros ;apply P_id_a__U71_monotonic;assumption.
         intros ;apply P_id_U44_monotonic;assumption.
         intros ;apply P_id_U61_monotonic;assumption.
         intros ;apply P_id_a__U86_monotonic;assumption.
         intros ;apply P_id_a__U41_monotonic;assumption.
         intros ;apply P_id_U22_monotonic;assumption.
         intros ;apply P_id_a__U21_monotonic;assumption.
         intros ;apply P_id_U91_monotonic;assumption.
         intros ;apply P_id_length_monotonic;assumption.
         intros ;apply P_id_a__U52_monotonic;assumption.
         intros ;apply P_id_U42_monotonic;assumption.
         intros ;apply P_id_a__U12_monotonic;assumption.
         intros ;apply P_id_U83_monotonic;assumption.
         intros ;apply P_id_a__U94_monotonic;assumption.
         intros ;apply P_id_a__U45_monotonic;assumption.
         intros ;apply P_id_U31_monotonic;assumption.
         intros ;apply P_id_a__isNat_monotonic;assumption.
         intros ;apply P_id_isNatIListKind_monotonic;assumption.
         intros ;apply P_id_a__U82_monotonic;assumption.
         intros ;apply P_id_U46_monotonic;assumption.
         intros ;apply P_id_cons_monotonic;assumption.
         intros ;apply P_id_U52_monotonic;assumption.
         intros ;apply P_id_a__U85_monotonic;assumption.
         intros ;apply P_id_a__U33_monotonic;assumption.
         intros ;apply P_id_U21_monotonic;assumption.
         intros ;apply P_id_a__isNatList_monotonic;assumption.
         intros ;apply P_id_U86_monotonic;assumption.
         intros ;apply P_id_mark_monotonic;assumption.
         intros ;apply P_id_a__U51_monotonic;assumption.
         intros ;apply P_id_U41_monotonic;assumption.
         intros ;apply P_id_U82_monotonic;assumption.
         intros ;apply P_id_a__U93_monotonic;assumption.
         intros ;apply P_id_a__U44_monotonic;assumption.
         intros ;apply P_id_isNat_monotonic;assumption.
         intros ;apply P_id_a__U23_monotonic;assumption.
         intros ;apply P_id_U94_monotonic;assumption.
         intros ;apply P_id_U12_monotonic;assumption.
         intros ;apply P_id_a__U81_monotonic;assumption.
         intros ;apply P_id_U45_monotonic;assumption.
         intros ;apply P_id_U71_monotonic;assumption.
         intros ;apply P_id_a__U91_monotonic;assumption.
         intros ;apply P_id_a__U42_monotonic;assumption.
         intros ;apply P_id_isNatKind_monotonic;assumption.
         intros ;apply P_id_a__U22_monotonic;assumption.
         intros ;apply P_id_U92_monotonic;assumption.
         intros ;apply P_id_a__U61_monotonic;assumption.
         intros ;apply P_id_U43_monotonic;assumption.
         intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
         intros ;apply P_id_U84_monotonic;assumption.
         intros ;apply P_id_s_monotonic;assumption.
         intros ;apply P_id_a__U46_monotonic;assumption.
         intros ;apply P_id_U32_monotonic;assumption.
         intros ;apply P_id_a__U31_monotonic;assumption.
         intros ;apply P_id_U13_monotonic;assumption.
         intros ;apply P_id_a__U83_monotonic;assumption.
         intros ;apply P_id_isNatIList_monotonic;assumption.
         intros ;apply P_id_a__zeros_bounded;assumption.
         intros ;apply P_id_U51_bounded;assumption.
         intros ;apply P_id_a__U84_bounded;assumption.
         intros ;apply P_id_a__U32_bounded;assumption.
         intros ;apply P_id_isNatList_bounded;assumption.
         intros ;apply P_id_a__U13_bounded;assumption.
         intros ;apply P_id_U85_bounded;assumption.
         intros ;apply P_id_a__length_bounded;assumption.
         intros ;apply P_id_a__isNatIList_bounded;assumption.
         intros ;apply P_id_U33_bounded;assumption.
         intros ;apply P_id_a__U11_bounded;assumption.
         intros ;apply P_id_U81_bounded;assumption.
         intros ;apply P_id_a__U92_bounded;assumption.
         intros ;apply P_id_a__U43_bounded;assumption.
         intros ;apply P_id_U23_bounded;assumption.
         intros ;apply P_id_a__isNatKind_bounded;assumption.
         intros ;apply P_id_U93_bounded;assumption.
         intros ;apply P_id_U11_bounded;assumption.
         intros ;apply P_id_a__U71_bounded;assumption.
         intros ;apply P_id_U44_bounded;assumption.
         intros ;apply P_id_0_bounded;assumption.
         intros ;apply P_id_U61_bounded;assumption.
         intros ;apply P_id_a__U86_bounded;assumption.
         intros ;apply P_id_a__U41_bounded;assumption.
         intros ;apply P_id_U22_bounded;assumption.
         intros ;apply P_id_a__U21_bounded;assumption.
         intros ;apply P_id_U91_bounded;assumption.
         intros ;apply P_id_length_bounded;assumption.
         intros ;apply P_id_a__U52_bounded;assumption.
         intros ;apply P_id_U42_bounded;assumption.
         intros ;apply P_id_a__U12_bounded;assumption.
         intros ;apply P_id_U83_bounded;assumption.
         intros ;apply P_id_a__U94_bounded;assumption.
         intros ;apply P_id_a__U45_bounded;assumption.
         intros ;apply P_id_U31_bounded;assumption.
         intros ;apply P_id_a__isNat_bounded;assumption.
         intros ;apply P_id_isNatIListKind_bounded;assumption.
         intros ;apply P_id_a__U82_bounded;assumption.
         intros ;apply P_id_U46_bounded;assumption.
         intros ;apply P_id_cons_bounded;assumption.
         intros ;apply P_id_U52_bounded;assumption.
         intros ;apply P_id_a__U85_bounded;assumption.
         intros ;apply P_id_a__U33_bounded;assumption.
         intros ;apply P_id_U21_bounded;assumption.
         intros ;apply P_id_a__isNatList_bounded;assumption.
         intros ;apply P_id_U86_bounded;assumption.
         intros ;apply P_id_mark_bounded;assumption.
         intros ;apply P_id_a__U51_bounded;assumption.
         intros ;apply P_id_U41_bounded;assumption.
         intros ;apply P_id_tt_bounded;assumption.
         intros ;apply P_id_U82_bounded;assumption.
         intros ;apply P_id_a__U93_bounded;assumption.
         intros ;apply P_id_a__U44_bounded;assumption.
         intros ;apply P_id_isNat_bounded;assumption.
         intros ;apply P_id_a__U23_bounded;assumption.
         intros ;apply P_id_U94_bounded;assumption.
         intros ;apply P_id_U12_bounded;assumption.
         intros ;apply P_id_a__U81_bounded;assumption.
         intros ;apply P_id_U45_bounded;assumption.
         intros ;apply P_id_zeros_bounded;assumption.
         intros ;apply P_id_U71_bounded;assumption.
         intros ;apply P_id_a__U91_bounded;assumption.
         intros ;apply P_id_a__U42_bounded;assumption.
         intros ;apply P_id_isNatKind_bounded;assumption.
         intros ;apply P_id_a__U22_bounded;assumption.
         intros ;apply P_id_U92_bounded;assumption.
         intros ;apply P_id_nil_bounded;assumption.
         intros ;apply P_id_a__U61_bounded;assumption.
         intros ;apply P_id_U43_bounded;assumption.
         intros ;apply P_id_a__isNatIListKind_bounded;assumption.
         intros ;apply P_id_U84_bounded;assumption.
         intros ;apply P_id_s_bounded;assumption.
         intros ;apply P_id_a__U46_bounded;assumption.
         intros ;apply P_id_U32_bounded;assumption.
         intros ;apply P_id_a__U31_bounded;assumption.
         intros ;apply P_id_U13_bounded;assumption.
         intros ;apply P_id_a__U83_bounded;assumption.
         intros ;apply P_id_isNatIList_bounded;assumption.
         apply rules_monotonic.
       Qed.
       
       Definition P_id_A__U46 (x11:Z) := 0.
       
       Definition P_id_A__ISNATILISTKIND (x11:Z) := 0.
       
       Definition P_id_A__U61 (x11:Z) := 0.
       
       Definition P_id_A__U85 (x11:Z) (x12:Z) := 0.
       
       Definition P_id_A__U31 (x11:Z) (x12:Z) := 0.
       
       Definition P_id_A__U81 (x11:Z) (x12:Z) (x13:Z) := 0.
       
       Definition P_id_A__U22 (x11:Z) (x12:Z) := 0.
       
       Definition P_id_A__U93 (x11:Z) (x12:Z) (x13:Z) := 0.
       
       Definition P_id_A__U42 (x11:Z) (x12:Z) (x13:Z) := 0.
       
       Definition P_id_A__U51 (x11:Z) (x12:Z) := 0.
       
       Definition P_id_A__ISNATLIST (x11:Z) := 0.
       
       Definition P_id_A__ZEROS  := 0.
       
       Definition P_id_A__U91 (x11:Z) (x12:Z) (x13:Z) := 0.
       
       Definition P_id_A__U33 (x11:Z) := 0.
       
       Definition P_id_A__U83 (x11:Z) (x12:Z) (x13:Z) := 0.
       
       Definition P_id_A__U23 (x11:Z) := 0.
       
       Definition P_id_A__U11 (x11:Z) (x12:Z) := 0.
       
       Definition P_id_A__LENGTH (x11:Z) := 0.
       
       Definition P_id_A__U44 (x11:Z) (x12:Z) (x13:Z) := 0.
       
       Definition P_id_A__ISNATILIST (x11:Z) := 0.
       
       Definition P_id_A__U13 (x11:Z) := 0.
       
       Definition P_id_A__U71 (x11:Z) := 0.
       
       Definition P_id_A__U86 (x11:Z) := 0.
       
       Definition P_id_A__U32 (x11:Z) (x12:Z) := 0.
       
       Definition P_id_A__U82 (x11:Z) (x12:Z) (x13:Z) := 0.
       
       Definition P_id_A__ISNATKIND (x11:Z) := 0.
       
       Definition P_id_A__U94 (x11:Z) (x12:Z) := 0.
       
       Definition P_id_A__U43 (x11:Z) (x12:Z) (x13:Z) := 0.
       
       Definition P_id_A__U52 (x11:Z) := 0.
       
       Definition P_id_A__U21 (x11:Z) (x12:Z) := 0.
       
       Definition P_id_A__U92 (x11:Z) (x12:Z) (x13:Z) := 0.
       
       Definition P_id_A__U41 (x11:Z) (x12:Z) (x13:Z) := 0.
       
       Definition P_id_A__U84 (x11:Z) (x12:Z) (x13:Z) := 0.
       
       Definition P_id_A__ISNAT (x11:Z) := 0.
       
       Definition P_id_A__U12 (x11:Z) (x12:Z) := 0.
       
       Definition P_id_MARK (x11:Z) := 2* x11.
       
       Definition P_id_A__U45 (x11:Z) (x12:Z) := 0.
       
       Lemma P_id_A__U46_monotonic :
        forall x12 x11, 
         (0 <= x12)/\ (x12 <= x11) ->P_id_A__U46 x12 <= P_id_A__U46 x11.
       Proof.
         intros x12 x11.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_A__ISNATILISTKIND_monotonic :
        forall x12 x11, 
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__ISNATILISTKIND x12 <= P_id_A__ISNATILISTKIND x11.
       Proof.
         intros x12 x11.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_A__U61_monotonic :
        forall x12 x11, 
         (0 <= x12)/\ (x12 <= x11) ->P_id_A__U61 x12 <= P_id_A__U61 x11.
       Proof.
         intros x12 x11.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_A__U85_monotonic :
        forall x12 x14 x13 x11, 
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_A__U85 x12 x14 <= P_id_A__U85 x11 x13.
       Proof.
         intros x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_A__U31_monotonic :
        forall x12 x14 x13 x11, 
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_A__U31 x12 x14 <= P_id_A__U31 x11 x13.
       Proof.
         intros x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_A__U81_monotonic :
        forall x16 x12 x14 x13 x11 x15, 
         (0 <= x16)/\ (x16 <= x15) ->
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_A__U81 x12 x14 x16 <= P_id_A__U81 x11 x13 x15.
       Proof.
         intros x16 x15 x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         intros [H_5 H_4].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_A__U22_monotonic :
        forall x12 x14 x13 x11, 
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_A__U22 x12 x14 <= P_id_A__U22 x11 x13.
       Proof.
         intros x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_A__U93_monotonic :
        forall x16 x12 x14 x13 x11 x15, 
         (0 <= x16)/\ (x16 <= x15) ->
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_A__U93 x12 x14 x16 <= P_id_A__U93 x11 x13 x15.
       Proof.
         intros x16 x15 x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         intros [H_5 H_4].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_A__U42_monotonic :
        forall x16 x12 x14 x13 x11 x15, 
         (0 <= x16)/\ (x16 <= x15) ->
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_A__U42 x12 x14 x16 <= P_id_A__U42 x11 x13 x15.
       Proof.
         intros x16 x15 x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         intros [H_5 H_4].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_A__U51_monotonic :
        forall x12 x14 x13 x11, 
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_A__U51 x12 x14 <= P_id_A__U51 x11 x13.
       Proof.
         intros x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_A__ISNATLIST_monotonic :
        forall x12 x11, 
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__ISNATLIST x12 <= P_id_A__ISNATLIST x11.
       Proof.
         intros x12 x11.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_A__U91_monotonic :
        forall x16 x12 x14 x13 x11 x15, 
         (0 <= x16)/\ (x16 <= x15) ->
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_A__U91 x12 x14 x16 <= P_id_A__U91 x11 x13 x15.
       Proof.
         intros x16 x15 x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         intros [H_5 H_4].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_A__U33_monotonic :
        forall x12 x11, 
         (0 <= x12)/\ (x12 <= x11) ->P_id_A__U33 x12 <= P_id_A__U33 x11.
       Proof.
         intros x12 x11.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_A__U83_monotonic :
        forall x16 x12 x14 x13 x11 x15, 
         (0 <= x16)/\ (x16 <= x15) ->
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_A__U83 x12 x14 x16 <= P_id_A__U83 x11 x13 x15.
       Proof.
         intros x16 x15 x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         intros [H_5 H_4].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_A__U23_monotonic :
        forall x12 x11, 
         (0 <= x12)/\ (x12 <= x11) ->P_id_A__U23 x12 <= P_id_A__U23 x11.
       Proof.
         intros x12 x11.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_A__U11_monotonic :
        forall x12 x14 x13 x11, 
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_A__U11 x12 x14 <= P_id_A__U11 x11 x13.
       Proof.
         intros x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_A__LENGTH_monotonic :
        forall x12 x11, 
         (0 <= x12)/\ (x12 <= x11) ->P_id_A__LENGTH x12 <= P_id_A__LENGTH x11.
       Proof.
         intros x12 x11.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_A__U44_monotonic :
        forall x16 x12 x14 x13 x11 x15, 
         (0 <= x16)/\ (x16 <= x15) ->
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_A__U44 x12 x14 x16 <= P_id_A__U44 x11 x13 x15.
       Proof.
         intros x16 x15 x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         intros [H_5 H_4].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_A__ISNATILIST_monotonic :
        forall x12 x11, 
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__ISNATILIST x12 <= P_id_A__ISNATILIST x11.
       Proof.
         intros x12 x11.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_A__U13_monotonic :
        forall x12 x11, 
         (0 <= x12)/\ (x12 <= x11) ->P_id_A__U13 x12 <= P_id_A__U13 x11.
       Proof.
         intros x12 x11.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_A__U71_monotonic :
        forall x12 x11, 
         (0 <= x12)/\ (x12 <= x11) ->P_id_A__U71 x12 <= P_id_A__U71 x11.
       Proof.
         intros x12 x11.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_A__U86_monotonic :
        forall x12 x11, 
         (0 <= x12)/\ (x12 <= x11) ->P_id_A__U86 x12 <= P_id_A__U86 x11.
       Proof.
         intros x12 x11.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_A__U32_monotonic :
        forall x12 x14 x13 x11, 
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_A__U32 x12 x14 <= P_id_A__U32 x11 x13.
       Proof.
         intros x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_A__U82_monotonic :
        forall x16 x12 x14 x13 x11 x15, 
         (0 <= x16)/\ (x16 <= x15) ->
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_A__U82 x12 x14 x16 <= P_id_A__U82 x11 x13 x15.
       Proof.
         intros x16 x15 x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         intros [H_5 H_4].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_A__ISNATKIND_monotonic :
        forall x12 x11, 
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__ISNATKIND x12 <= P_id_A__ISNATKIND x11.
       Proof.
         intros x12 x11.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_A__U94_monotonic :
        forall x12 x14 x13 x11, 
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_A__U94 x12 x14 <= P_id_A__U94 x11 x13.
       Proof.
         intros x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_A__U43_monotonic :
        forall x16 x12 x14 x13 x11 x15, 
         (0 <= x16)/\ (x16 <= x15) ->
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_A__U43 x12 x14 x16 <= P_id_A__U43 x11 x13 x15.
       Proof.
         intros x16 x15 x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         intros [H_5 H_4].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_A__U52_monotonic :
        forall x12 x11, 
         (0 <= x12)/\ (x12 <= x11) ->P_id_A__U52 x12 <= P_id_A__U52 x11.
       Proof.
         intros x12 x11.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_A__U21_monotonic :
        forall x12 x14 x13 x11, 
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_A__U21 x12 x14 <= P_id_A__U21 x11 x13.
       Proof.
         intros x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_A__U92_monotonic :
        forall x16 x12 x14 x13 x11 x15, 
         (0 <= x16)/\ (x16 <= x15) ->
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_A__U92 x12 x14 x16 <= P_id_A__U92 x11 x13 x15.
       Proof.
         intros x16 x15 x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         intros [H_5 H_4].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_A__U41_monotonic :
        forall x16 x12 x14 x13 x11 x15, 
         (0 <= x16)/\ (x16 <= x15) ->
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_A__U41 x12 x14 x16 <= P_id_A__U41 x11 x13 x15.
       Proof.
         intros x16 x15 x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         intros [H_5 H_4].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_A__U84_monotonic :
        forall x16 x12 x14 x13 x11 x15, 
         (0 <= x16)/\ (x16 <= x15) ->
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_A__U84 x12 x14 x16 <= P_id_A__U84 x11 x13 x15.
       Proof.
         intros x16 x15 x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         intros [H_5 H_4].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_A__ISNAT_monotonic :
        forall x12 x11, 
         (0 <= x12)/\ (x12 <= x11) ->P_id_A__ISNAT x12 <= P_id_A__ISNAT x11.
       Proof.
         intros x12 x11.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_A__U12_monotonic :
        forall x12 x14 x13 x11, 
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_A__U12 x12 x14 <= P_id_A__U12 x11 x13.
       Proof.
         intros x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_MARK_monotonic :
        forall x12 x11, 
         (0 <= x12)/\ (x12 <= x11) ->P_id_MARK x12 <= P_id_MARK x11.
       Proof.
         intros x12 x11.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_A__U45_monotonic :
        forall x12 x14 x13 x11, 
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_A__U45 x12 x14 <= P_id_A__U45 x11 x13.
       Proof.
         intros x14 x13 x12 x11.
         intros [H_1 H_0].
         intros [H_3 H_2].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Definition marked_measure  := 
         InterpZ.marked_measure 0 P_id_a__zeros P_id_U51 P_id_a__U84 
          P_id_a__U32 P_id_isNatList P_id_a__U13 P_id_U85 P_id_a__length 
          P_id_a__isNatIList P_id_U33 P_id_a__U11 P_id_U81 P_id_a__U92 
          P_id_a__U43 P_id_U23 P_id_a__isNatKind P_id_U93 P_id_U11 
          P_id_a__U71 P_id_U44 P_id_0 P_id_U61 P_id_a__U86 P_id_a__U41 
          P_id_U22 P_id_a__U21 P_id_U91 P_id_length P_id_a__U52 P_id_U42 
          P_id_a__U12 P_id_U83 P_id_a__U94 P_id_a__U45 P_id_U31 
          P_id_a__isNat P_id_isNatIListKind P_id_a__U82 P_id_U46 P_id_cons 
          P_id_U52 P_id_a__U85 P_id_a__U33 P_id_U21 P_id_a__isNatList 
          P_id_U86 P_id_mark P_id_a__U51 P_id_U41 P_id_tt P_id_U82 
          P_id_a__U93 P_id_a__U44 P_id_isNat P_id_a__U23 P_id_U94 P_id_U12 
          P_id_a__U81 P_id_U45 P_id_zeros P_id_U71 P_id_a__U91 P_id_a__U42 
          P_id_isNatKind P_id_a__U22 P_id_U92 P_id_nil P_id_a__U61 P_id_U43 
          P_id_a__isNatIListKind P_id_U84 P_id_s P_id_a__U46 P_id_U32 
          P_id_a__U31 P_id_U13 P_id_a__U83 P_id_isNatIList P_id_A__U46 
          P_id_A__ISNATILISTKIND P_id_A__U61 P_id_A__U85 P_id_A__U31 
          P_id_A__U81 P_id_A__U22 P_id_A__U93 P_id_A__U42 P_id_A__U51 
          P_id_A__ISNATLIST P_id_A__ZEROS P_id_A__U91 P_id_A__U33 
          P_id_A__U83 P_id_A__U23 P_id_A__U11 P_id_A__LENGTH P_id_A__U44 
          P_id_A__ISNATILIST P_id_A__U13 P_id_A__U71 P_id_A__U86 P_id_A__U32 
          P_id_A__U82 P_id_A__ISNATKIND P_id_A__U94 P_id_A__U43 P_id_A__U52 
          P_id_A__U21 P_id_A__U92 P_id_A__U41 P_id_A__U84 P_id_A__ISNAT 
          P_id_A__U12 P_id_MARK P_id_A__U45.
       
       Lemma marked_measure_equation :
        forall t, 
         marked_measure t = match t with
                              | (algebra.Alg.Term algebra.F.id_a__U46 
                                 (x11::nil)) =>
                               P_id_A__U46 (measure x11)
                              | (algebra.Alg.Term 
                                 algebra.F.id_a__isNatIListKind (x11::nil)) =>
                               P_id_A__ISNATILISTKIND (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U61 
                                 (x11::nil)) =>
                               P_id_A__U61 (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U85 (x12::
                                 x11::nil)) =>
                               P_id_A__U85 (measure x12) (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U31 (x12::
                                 x11::nil)) =>
                               P_id_A__U31 (measure x12) (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U81 (x13::
                                 x12::x11::nil)) =>
                               P_id_A__U81 (measure x13) (measure x12) 
                                (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U22 (x12::
                                 x11::nil)) =>
                               P_id_A__U22 (measure x12) (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U93 (x13::
                                 x12::x11::nil)) =>
                               P_id_A__U93 (measure x13) (measure x12) 
                                (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U42 (x13::
                                 x12::x11::nil)) =>
                               P_id_A__U42 (measure x13) (measure x12) 
                                (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U51 (x12::
                                 x11::nil)) =>
                               P_id_A__U51 (measure x12) (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__isNatList 
                                 (x11::nil)) =>
                               P_id_A__ISNATLIST (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__zeros nil) =>
                               P_id_A__ZEROS 
                              | (algebra.Alg.Term algebra.F.id_a__U91 (x13::
                                 x12::x11::nil)) =>
                               P_id_A__U91 (measure x13) (measure x12) 
                                (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U33 
                                 (x11::nil)) =>
                               P_id_A__U33 (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U83 (x13::
                                 x12::x11::nil)) =>
                               P_id_A__U83 (measure x13) (measure x12) 
                                (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U23 
                                 (x11::nil)) =>
                               P_id_A__U23 (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U11 (x12::
                                 x11::nil)) =>
                               P_id_A__U11 (measure x12) (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__length 
                                 (x11::nil)) =>
                               P_id_A__LENGTH (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U44 (x13::
                                 x12::x11::nil)) =>
                               P_id_A__U44 (measure x13) (measure x12) 
                                (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__isNatIList 
                                 (x11::nil)) =>
                               P_id_A__ISNATILIST (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U13 
                                 (x11::nil)) =>
                               P_id_A__U13 (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U71 
                                 (x11::nil)) =>
                               P_id_A__U71 (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U86 
                                 (x11::nil)) =>
                               P_id_A__U86 (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U32 (x12::
                                 x11::nil)) =>
                               P_id_A__U32 (measure x12) (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U82 (x13::
                                 x12::x11::nil)) =>
                               P_id_A__U82 (measure x13) (measure x12) 
                                (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__isNatKind 
                                 (x11::nil)) =>
                               P_id_A__ISNATKIND (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U94 (x12::
                                 x11::nil)) =>
                               P_id_A__U94 (measure x12) (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U43 (x13::
                                 x12::x11::nil)) =>
                               P_id_A__U43 (measure x13) (measure x12) 
                                (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U52 
                                 (x11::nil)) =>
                               P_id_A__U52 (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U21 (x12::
                                 x11::nil)) =>
                               P_id_A__U21 (measure x12) (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U92 (x13::
                                 x12::x11::nil)) =>
                               P_id_A__U92 (measure x13) (measure x12) 
                                (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U41 (x13::
                                 x12::x11::nil)) =>
                               P_id_A__U41 (measure x13) (measure x12) 
                                (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U84 (x13::
                                 x12::x11::nil)) =>
                               P_id_A__U84 (measure x13) (measure x12) 
                                (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__isNat 
                                 (x11::nil)) =>
                               P_id_A__ISNAT (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U12 (x12::
                                 x11::nil)) =>
                               P_id_A__U12 (measure x12) (measure x11)
                              | (algebra.Alg.Term algebra.F.id_mark 
                                 (x11::nil)) =>
                               P_id_MARK (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U45 (x12::
                                 x11::nil)) =>
                               P_id_A__U45 (measure x12) (measure x11)
                              | _ => measure t
                              end.
       Proof.
         reflexivity .
       Qed.
       
       Lemma marked_measure_star_monotonic :
        forall f l1 l2, 
         (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step 
                                                           R_xml_0_deep_rew.R_xml_0_rules)
                                                          ) l1 l2) ->
          marked_measure (algebra.Alg.Term f l1) <= marked_measure (algebra.Alg.Term 
                                                                    f 
                                                                    l2).
       Proof.
         unfold marked_measure in *.
         apply InterpZ.marked_measure_star_monotonic.
         intros ;apply P_id_U51_monotonic;assumption.
         intros ;apply P_id_a__U84_monotonic;assumption.
         intros ;apply P_id_a__U32_monotonic;assumption.
         intros ;apply P_id_isNatList_monotonic;assumption.
         intros ;apply P_id_a__U13_monotonic;assumption.
         intros ;apply P_id_U85_monotonic;assumption.
         intros ;apply P_id_a__length_monotonic;assumption.
         intros ;apply P_id_a__isNatIList_monotonic;assumption.
         intros ;apply P_id_U33_monotonic;assumption.
         intros ;apply P_id_a__U11_monotonic;assumption.
         intros ;apply P_id_U81_monotonic;assumption.
         intros ;apply P_id_a__U92_monotonic;assumption.
         intros ;apply P_id_a__U43_monotonic;assumption.
         intros ;apply P_id_U23_monotonic;assumption.
         intros ;apply P_id_a__isNatKind_monotonic;assumption.
         intros ;apply P_id_U93_monotonic;assumption.
         intros ;apply P_id_U11_monotonic;assumption.
         intros ;apply P_id_a__U71_monotonic;assumption.
         intros ;apply P_id_U44_monotonic;assumption.
         intros ;apply P_id_U61_monotonic;assumption.
         intros ;apply P_id_a__U86_monotonic;assumption.
         intros ;apply P_id_a__U41_monotonic;assumption.
         intros ;apply P_id_U22_monotonic;assumption.
         intros ;apply P_id_a__U21_monotonic;assumption.
         intros ;apply P_id_U91_monotonic;assumption.
         intros ;apply P_id_length_monotonic;assumption.
         intros ;apply P_id_a__U52_monotonic;assumption.
         intros ;apply P_id_U42_monotonic;assumption.
         intros ;apply P_id_a__U12_monotonic;assumption.
         intros ;apply P_id_U83_monotonic;assumption.
         intros ;apply P_id_a__U94_monotonic;assumption.
         intros ;apply P_id_a__U45_monotonic;assumption.
         intros ;apply P_id_U31_monotonic;assumption.
         intros ;apply P_id_a__isNat_monotonic;assumption.
         intros ;apply P_id_isNatIListKind_monotonic;assumption.
         intros ;apply P_id_a__U82_monotonic;assumption.
         intros ;apply P_id_U46_monotonic;assumption.
         intros ;apply P_id_cons_monotonic;assumption.
         intros ;apply P_id_U52_monotonic;assumption.
         intros ;apply P_id_a__U85_monotonic;assumption.
         intros ;apply P_id_a__U33_monotonic;assumption.
         intros ;apply P_id_U21_monotonic;assumption.
         intros ;apply P_id_a__isNatList_monotonic;assumption.
         intros ;apply P_id_U86_monotonic;assumption.
         intros ;apply P_id_mark_monotonic;assumption.
         intros ;apply P_id_a__U51_monotonic;assumption.
         intros ;apply P_id_U41_monotonic;assumption.
         intros ;apply P_id_U82_monotonic;assumption.
         intros ;apply P_id_a__U93_monotonic;assumption.
         intros ;apply P_id_a__U44_monotonic;assumption.
         intros ;apply P_id_isNat_monotonic;assumption.
         intros ;apply P_id_a__U23_monotonic;assumption.
         intros ;apply P_id_U94_monotonic;assumption.
         intros ;apply P_id_U12_monotonic;assumption.
         intros ;apply P_id_a__U81_monotonic;assumption.
         intros ;apply P_id_U45_monotonic;assumption.
         intros ;apply P_id_U71_monotonic;assumption.
         intros ;apply P_id_a__U91_monotonic;assumption.
         intros ;apply P_id_a__U42_monotonic;assumption.
         intros ;apply P_id_isNatKind_monotonic;assumption.
         intros ;apply P_id_a__U22_monotonic;assumption.
         intros ;apply P_id_U92_monotonic;assumption.
         intros ;apply P_id_a__U61_monotonic;assumption.
         intros ;apply P_id_U43_monotonic;assumption.
         intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
         intros ;apply P_id_U84_monotonic;assumption.
         intros ;apply P_id_s_monotonic;assumption.
         intros ;apply P_id_a__U46_monotonic;assumption.
         intros ;apply P_id_U32_monotonic;assumption.
         intros ;apply P_id_a__U31_monotonic;assumption.
         intros ;apply P_id_U13_monotonic;assumption.
         intros ;apply P_id_a__U83_monotonic;assumption.
         intros ;apply P_id_isNatIList_monotonic;assumption.
         intros ;apply P_id_a__zeros_bounded;assumption.
         intros ;apply P_id_U51_bounded;assumption.
         intros ;apply P_id_a__U84_bounded;assumption.
         intros ;apply P_id_a__U32_bounded;assumption.
         intros ;apply P_id_isNatList_bounded;assumption.
         intros ;apply P_id_a__U13_bounded;assumption.
         intros ;apply P_id_U85_bounded;assumption.
         intros ;apply P_id_a__length_bounded;assumption.
         intros ;apply P_id_a__isNatIList_bounded;assumption.
         intros ;apply P_id_U33_bounded;assumption.
         intros ;apply P_id_a__U11_bounded;assumption.
         intros ;apply P_id_U81_bounded;assumption.
         intros ;apply P_id_a__U92_bounded;assumption.
         intros ;apply P_id_a__U43_bounded;assumption.
         intros ;apply P_id_U23_bounded;assumption.
         intros ;apply P_id_a__isNatKind_bounded;assumption.
         intros ;apply P_id_U93_bounded;assumption.
         intros ;apply P_id_U11_bounded;assumption.
         intros ;apply P_id_a__U71_bounded;assumption.
         intros ;apply P_id_U44_bounded;assumption.
         intros ;apply P_id_0_bounded;assumption.
         intros ;apply P_id_U61_bounded;assumption.
         intros ;apply P_id_a__U86_bounded;assumption.
         intros ;apply P_id_a__U41_bounded;assumption.
         intros ;apply P_id_U22_bounded;assumption.
         intros ;apply P_id_a__U21_bounded;assumption.
         intros ;apply P_id_U91_bounded;assumption.
         intros ;apply P_id_length_bounded;assumption.
         intros ;apply P_id_a__U52_bounded;assumption.
         intros ;apply P_id_U42_bounded;assumption.
         intros ;apply P_id_a__U12_bounded;assumption.
         intros ;apply P_id_U83_bounded;assumption.
         intros ;apply P_id_a__U94_bounded;assumption.
         intros ;apply P_id_a__U45_bounded;assumption.
         intros ;apply P_id_U31_bounded;assumption.
         intros ;apply P_id_a__isNat_bounded;assumption.
         intros ;apply P_id_isNatIListKind_bounded;assumption.
         intros ;apply P_id_a__U82_bounded;assumption.
         intros ;apply P_id_U46_bounded;assumption.
         intros ;apply P_id_cons_bounded;assumption.
         intros ;apply P_id_U52_bounded;assumption.
         intros ;apply P_id_a__U85_bounded;assumption.
         intros ;apply P_id_a__U33_bounded;assumption.
         intros ;apply P_id_U21_bounded;assumption.
         intros ;apply P_id_a__isNatList_bounded;assumption.
         intros ;apply P_id_U86_bounded;assumption.
         intros ;apply P_id_mark_bounded;assumption.
         intros ;apply P_id_a__U51_bounded;assumption.
         intros ;apply P_id_U41_bounded;assumption.
         intros ;apply P_id_tt_bounded;assumption.
         intros ;apply P_id_U82_bounded;assumption.
         intros ;apply P_id_a__U93_bounded;assumption.
         intros ;apply P_id_a__U44_bounded;assumption.
         intros ;apply P_id_isNat_bounded;assumption.
         intros ;apply P_id_a__U23_bounded;assumption.
         intros ;apply P_id_U94_bounded;assumption.
         intros ;apply P_id_U12_bounded;assumption.
         intros ;apply P_id_a__U81_bounded;assumption.
         intros ;apply P_id_U45_bounded;assumption.
         intros ;apply P_id_zeros_bounded;assumption.
         intros ;apply P_id_U71_bounded;assumption.
         intros ;apply P_id_a__U91_bounded;assumption.
         intros ;apply P_id_a__U42_bounded;assumption.
         intros ;apply P_id_isNatKind_bounded;assumption.
         intros ;apply P_id_a__U22_bounded;assumption.
         intros ;apply P_id_U92_bounded;assumption.
         intros ;apply P_id_nil_bounded;assumption.
         intros ;apply P_id_a__U61_bounded;assumption.
         intros ;apply P_id_U43_bounded;assumption.
         intros ;apply P_id_a__isNatIListKind_bounded;assumption.
         intros ;apply P_id_U84_bounded;assumption.
         intros ;apply P_id_s_bounded;assumption.
         intros ;apply P_id_a__U46_bounded;assumption.
         intros ;apply P_id_U32_bounded;assumption.
         intros ;apply P_id_a__U31_bounded;assumption.
         intros ;apply P_id_U13_bounded;assumption.
         intros ;apply P_id_a__U83_bounded;assumption.
         intros ;apply P_id_isNatIList_bounded;assumption.
         apply rules_monotonic.
         intros ;apply P_id_A__U46_monotonic;assumption.
         intros ;apply P_id_A__ISNATILISTKIND_monotonic;assumption.
         intros ;apply P_id_A__U61_monotonic;assumption.
         intros ;apply P_id_A__U85_monotonic;assumption.
         intros ;apply P_id_A__U31_monotonic;assumption.
         intros ;apply P_id_A__U81_monotonic;assumption.
         intros ;apply P_id_A__U22_monotonic;assumption.
         intros ;apply P_id_A__U93_monotonic;assumption.
         intros ;apply P_id_A__U42_monotonic;assumption.
         intros ;apply P_id_A__U51_monotonic;assumption.
         intros ;apply P_id_A__ISNATLIST_monotonic;assumption.
         intros ;apply P_id_A__U91_monotonic;assumption.
         intros ;apply P_id_A__U33_monotonic;assumption.
         intros ;apply P_id_A__U83_monotonic;assumption.
         intros ;apply P_id_A__U23_monotonic;assumption.
         intros ;apply P_id_A__U11_monotonic;assumption.
         intros ;apply P_id_A__LENGTH_monotonic;assumption.
         intros ;apply P_id_A__U44_monotonic;assumption.
         intros ;apply P_id_A__ISNATILIST_monotonic;assumption.
         intros ;apply P_id_A__U13_monotonic;assumption.
         intros ;apply P_id_A__U71_monotonic;assumption.
         intros ;apply P_id_A__U86_monotonic;assumption.
         intros ;apply P_id_A__U32_monotonic;assumption.
         intros ;apply P_id_A__U82_monotonic;assumption.
         intros ;apply P_id_A__ISNATKIND_monotonic;assumption.
         intros ;apply P_id_A__U94_monotonic;assumption.
         intros ;apply P_id_A__U43_monotonic;assumption.
         intros ;apply P_id_A__U52_monotonic;assumption.
         intros ;apply P_id_A__U21_monotonic;assumption.
         intros ;apply P_id_A__U92_monotonic;assumption.
         intros ;apply P_id_A__U41_monotonic;assumption.
         intros ;apply P_id_A__U84_monotonic;assumption.
         intros ;apply P_id_A__ISNAT_monotonic;assumption.
         intros ;apply P_id_A__U12_monotonic;assumption.
         intros ;apply P_id_MARK_monotonic;assumption.
         intros ;apply P_id_A__U45_monotonic;assumption.
       Qed.
       
       Ltac rewrite_and_unfold  :=
        do 2 (rewrite marked_measure_equation);
         repeat (
         match goal with
           |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
            rewrite (measure_equation (algebra.Alg.Term f t))
           | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ =>
            rewrite (measure_equation (algebra.Alg.Term f t)) in H|-
           end
         ).
       
       Definition lt a b := (Zwf.Zwf 0) (marked_measure a) (marked_measure b)
         .
       
       Definition le a b := marked_measure a <= marked_measure b.
       
       Lemma lt_le_compat : forall a b c, (lt a b) ->(le b c) ->lt a c.
       Proof.
         unfold lt, le in *.
         intros a b c.
         apply (interp.le_lt_compat_right (interp.o_Z 0)).
       Qed.
       
       Lemma wf_lt : well_founded lt.
       Proof.
         unfold lt in *.
         apply Inverse_Image.wf_inverse_image with  (B:=Z).
         apply Zwf.Zwf_well_founded.
       Qed.
       
       Lemma DP_R_xml_0_scc_65_large_scc_1_large_large_strict_in_lt :
        Relation_Definitions.inclusion _ 
         DP_R_xml_0_scc_65_large_scc_1_large_large_strict lt.
       Proof.
         unfold Relation_Definitions.inclusion, lt in *.
         
         intros a b H;destruct H;
          match goal with
            |  |- (Zwf.Zwf 0) _ (marked_measure (algebra.Alg.Term ?f ?l)) =>
             let l'' := algebra.Alg_ext.find_replacement l  in 
              ((apply (interp.le_lt_compat_right (interp.o_Z 0)) with
                 (marked_measure (algebra.Alg.Term f l''));[idtac|
                apply marked_measure_star_monotonic;
                 repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos);
                 (assumption)||(constructor 1)]))
            end
          ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma DP_R_xml_0_scc_65_large_scc_1_large_large_large_in_le :
        Relation_Definitions.inclusion _ 
         DP_R_xml_0_scc_65_large_scc_1_large_large_large le.
       Proof.
         unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *.
         
         intros a b H;destruct H;
          match goal with
            |  |- _ <= marked_measure (algebra.Alg.Term ?f ?l) =>
             let l'' := algebra.Alg_ext.find_replacement l  in 
              ((apply (interp.le_trans (interp.o_Z 0)) with
                 (marked_measure (algebra.Alg.Term f l''));[idtac|
                apply marked_measure_star_monotonic;
                 repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos);
                 (assumption)||(constructor 1)]))
            end
          ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Definition wf_DP_R_xml_0_scc_65_large_scc_1_large_large_large  := 
         WF_DP_R_xml_0_scc_65_large_scc_1_large_large_large.wf.
       
       
       Lemma wf :
        well_founded WF_DP_R_xml_0_scc_65_large_scc_1_large.DP_R_xml_0_scc_65_large_scc_1_large_large
         .
       Proof.
         intros x.
         apply (well_founded_ind wf_lt).
         clear x.
         intros x.
         pattern x.
         apply (@Acc_ind _ DP_R_xml_0_scc_65_large_scc_1_large_large_large).
         clear x.
         intros x _ IHx IHx'.
         constructor.
         intros y H.
         
         destruct H;
          (apply IHx';
            apply DP_R_xml_0_scc_65_large_scc_1_large_large_strict_in_lt;
            econstructor eassumption)||
          ((apply IHx;[econstructor eassumption|
            intros y' Hlt;apply IHx';apply lt_le_compat with (1:=Hlt) ;
             apply DP_R_xml_0_scc_65_large_scc_1_large_large_large_in_le;
             econstructor eassumption])).
         apply wf_DP_R_xml_0_scc_65_large_scc_1_large_large_large.
       Qed.
      End WF_DP_R_xml_0_scc_65_large_scc_1_large_large.
      
      Open Scope Z_scope.
      
      Import ring_extention.
      
      Notation Local "a <= b" := (Zle a b).
      
      Notation Local "a < b" := (Zlt a b).
      
      Definition P_id_a__zeros  := 0.
      
      Definition P_id_U51 (x11:Z) (x12:Z) := 2* x11.
      
      Definition P_id_a__U84 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
      
      Definition P_id_a__U32 (x11:Z) (x12:Z) := 2* x11.
      
      Definition P_id_isNatList (x11:Z) := 0.
      
      Definition P_id_a__U13 (x11:Z) := 2* x11.
      
      Definition P_id_U85 (x11:Z) (x12:Z) := 2* x11.
      
      Definition P_id_a__length (x11:Z) := 0.
      
      Definition P_id_a__isNatIList (x11:Z) := 1.
      
      Definition P_id_U33 (x11:Z) := 1* x11.
      
      Definition P_id_a__U11 (x11:Z) (x12:Z) := 2* x11.
      
      Definition P_id_U81 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
      
      Definition P_id_a__U92 (x11:Z) (x12:Z) (x13:Z) := 0.
      
      Definition P_id_a__U43 (x11:Z) (x12:Z) (x13:Z) := 1 + 2* x11.
      
      Definition P_id_U23 (x11:Z) := 1* x11.
      
      Definition P_id_a__isNatKind (x11:Z) := 0.
      
      Definition P_id_U93 (x11:Z) (x12:Z) (x13:Z) := 0.
      
      Definition P_id_U11 (x11:Z) (x12:Z) := 2* x11.
      
      Definition P_id_a__U71 (x11:Z) := 1* x11.
      
      Definition P_id_U44 (x11:Z) (x12:Z) (x13:Z) := 1 + 2* x11.
      
      Definition P_id_0  := 0.
      
      Definition P_id_U61 (x11:Z) := 1* x11.
      
      Definition P_id_a__U86 (x11:Z) := 1* x11.
      
      Definition P_id_a__U41 (x11:Z) (x12:Z) (x13:Z) := 1 + 1* x11.
      
      Definition P_id_U22 (x11:Z) (x12:Z) := 1* x11.
      
      Definition P_id_a__U21 (x11:Z) (x12:Z) := 2* x11.
      
      Definition P_id_U91 (x11:Z) (x12:Z) (x13:Z) := 0.
      
      Definition P_id_length (x11:Z) := 0.
      
      Definition P_id_a__U52 (x11:Z) := 2* x11.
      
      Definition P_id_U42 (x11:Z) (x12:Z) (x13:Z) := 1 + 1* x11.
      
      Definition P_id_a__U12 (x11:Z) (x12:Z) := 2* x11.
      
      Definition P_id_U83 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
      
      Definition P_id_a__U94 (x11:Z) (x12:Z) := 0.
      
      Definition P_id_a__U45 (x11:Z) (x12:Z) := 1 + 1* x11.
      
      Definition P_id_U31 (x11:Z) (x12:Z) := 1 + 2* x11.
      
      Definition P_id_a__isNat (x11:Z) := 0.
      
      Definition P_id_isNatIListKind (x11:Z) := 0.
      
      Definition P_id_a__U82 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
      
      Definition P_id_U46 (x11:Z) := 1* x11.
      
      Definition P_id_cons (x11:Z) (x12:Z) := 0.
      
      Definition P_id_U52 (x11:Z) := 2* x11.
      
      Definition P_id_a__U85 (x11:Z) (x12:Z) := 2* x11.
      
      Definition P_id_a__U33 (x11:Z) := 1* x11.
      
      Definition P_id_U21 (x11:Z) (x12:Z) := 2* x11.
      
      Definition P_id_a__isNatList (x11:Z) := 0.
      
      Definition P_id_U86 (x11:Z) := 1* x11.
      
      Definition P_id_mark (x11:Z) := 1* x11.
      
      Definition P_id_a__U51 (x11:Z) (x12:Z) := 2* x11.
      
      Definition P_id_U41 (x11:Z) (x12:Z) (x13:Z) := 1 + 1* x11.
      
      Definition P_id_tt  := 0.
      
      Definition P_id_U82 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
      
      Definition P_id_a__U93 (x11:Z) (x12:Z) (x13:Z) := 0.
      
      Definition P_id_a__U44 (x11:Z) (x12:Z) (x13:Z) := 1 + 2* x11.
      
      Definition P_id_isNat (x11:Z) := 0.
      
      Definition P_id_a__U23 (x11:Z) := 1* x11.
      
      Definition P_id_U94 (x11:Z) (x12:Z) := 0.
      
      Definition P_id_U12 (x11:Z) (x12:Z) := 2* x11.
      
      Definition P_id_a__U81 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
      
      Definition P_id_U45 (x11:Z) (x12:Z) := 1 + 1* x11.
      
      Definition P_id_zeros  := 0.
      
      Definition P_id_U71 (x11:Z) := 1* x11.
      
      Definition P_id_a__U91 (x11:Z) (x12:Z) (x13:Z) := 0.
      
      Definition P_id_a__U42 (x11:Z) (x12:Z) (x13:Z) := 1 + 1* x11.
      
      Definition P_id_isNatKind (x11:Z) := 0.
      
      Definition P_id_a__U22 (x11:Z) (x12:Z) := 1* x11.
      
      Definition P_id_U92 (x11:Z) (x12:Z) (x13:Z) := 0.
      
      Definition P_id_nil  := 2.
      
      Definition P_id_a__U61 (x11:Z) := 1* x11.
      
      Definition P_id_U43 (x11:Z) (x12:Z) (x13:Z) := 1 + 2* x11.
      
      Definition P_id_a__isNatIListKind (x11:Z) := 0.
      
      Definition P_id_U84 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
      
      Definition P_id_s (x11:Z) := 2* x11.
      
      Definition P_id_a__U46 (x11:Z) := 1* x11.
      
      Definition P_id_U32 (x11:Z) (x12:Z) := 2* x11.
      
      Definition P_id_a__U31 (x11:Z) (x12:Z) := 1 + 2* x11.
      
      Definition P_id_U13 (x11:Z) := 2* x11.
      
      Definition P_id_a__U83 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
      
      Definition P_id_isNatIList (x11:Z) := 1.
      
      Lemma P_id_U51_monotonic :
       forall x12 x14 x13 x11, 
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->P_id_U51 x12 x14 <= P_id_U51 x11 x13.
      Proof.
        intros x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U84_monotonic :
       forall x16 x12 x14 x13 x11 x15, 
        (0 <= x16)/\ (x16 <= x15) ->
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_a__U84 x12 x14 x16 <= P_id_a__U84 x11 x13 x15.
      Proof.
        intros x16 x15 x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        intros [H_5 H_4].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U32_monotonic :
       forall x12 x14 x13 x11, 
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U32 x12 x14 <= P_id_a__U32 x11 x13.
      Proof.
        intros x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_isNatList_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->P_id_isNatList x12 <= P_id_isNatList x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U13_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->P_id_a__U13 x12 <= P_id_a__U13 x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U85_monotonic :
       forall x12 x14 x13 x11, 
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->P_id_U85 x12 x14 <= P_id_U85 x11 x13.
      Proof.
        intros x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__length_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->P_id_a__length x12 <= P_id_a__length x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__isNatIList_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__isNatIList x12 <= P_id_a__isNatIList x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U33_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->P_id_U33 x12 <= P_id_U33 x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U11_monotonic :
       forall x12 x14 x13 x11, 
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U11 x12 x14 <= P_id_a__U11 x11 x13.
      Proof.
        intros x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U81_monotonic :
       forall x16 x12 x14 x13 x11 x15, 
        (0 <= x16)/\ (x16 <= x15) ->
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_U81 x12 x14 x16 <= P_id_U81 x11 x13 x15.
      Proof.
        intros x16 x15 x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        intros [H_5 H_4].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U92_monotonic :
       forall x16 x12 x14 x13 x11 x15, 
        (0 <= x16)/\ (x16 <= x15) ->
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_a__U92 x12 x14 x16 <= P_id_a__U92 x11 x13 x15.
      Proof.
        intros x16 x15 x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        intros [H_5 H_4].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U43_monotonic :
       forall x16 x12 x14 x13 x11 x15, 
        (0 <= x16)/\ (x16 <= x15) ->
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_a__U43 x12 x14 x16 <= P_id_a__U43 x11 x13 x15.
      Proof.
        intros x16 x15 x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        intros [H_5 H_4].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U23_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->P_id_U23 x12 <= P_id_U23 x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__isNatKind_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__isNatKind x12 <= P_id_a__isNatKind x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U93_monotonic :
       forall x16 x12 x14 x13 x11 x15, 
        (0 <= x16)/\ (x16 <= x15) ->
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_U93 x12 x14 x16 <= P_id_U93 x11 x13 x15.
      Proof.
        intros x16 x15 x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        intros [H_5 H_4].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U11_monotonic :
       forall x12 x14 x13 x11, 
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->P_id_U11 x12 x14 <= P_id_U11 x11 x13.
      Proof.
        intros x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U71_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->P_id_a__U71 x12 <= P_id_a__U71 x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U44_monotonic :
       forall x16 x12 x14 x13 x11 x15, 
        (0 <= x16)/\ (x16 <= x15) ->
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_U44 x12 x14 x16 <= P_id_U44 x11 x13 x15.
      Proof.
        intros x16 x15 x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        intros [H_5 H_4].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U61_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->P_id_U61 x12 <= P_id_U61 x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U86_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->P_id_a__U86 x12 <= P_id_a__U86 x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U41_monotonic :
       forall x16 x12 x14 x13 x11 x15, 
        (0 <= x16)/\ (x16 <= x15) ->
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_a__U41 x12 x14 x16 <= P_id_a__U41 x11 x13 x15.
      Proof.
        intros x16 x15 x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        intros [H_5 H_4].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U22_monotonic :
       forall x12 x14 x13 x11, 
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->P_id_U22 x12 x14 <= P_id_U22 x11 x13.
      Proof.
        intros x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U21_monotonic :
       forall x12 x14 x13 x11, 
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U21 x12 x14 <= P_id_a__U21 x11 x13.
      Proof.
        intros x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U91_monotonic :
       forall x16 x12 x14 x13 x11 x15, 
        (0 <= x16)/\ (x16 <= x15) ->
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_U91 x12 x14 x16 <= P_id_U91 x11 x13 x15.
      Proof.
        intros x16 x15 x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        intros [H_5 H_4].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_length_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->P_id_length x12 <= P_id_length x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U52_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->P_id_a__U52 x12 <= P_id_a__U52 x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U42_monotonic :
       forall x16 x12 x14 x13 x11 x15, 
        (0 <= x16)/\ (x16 <= x15) ->
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_U42 x12 x14 x16 <= P_id_U42 x11 x13 x15.
      Proof.
        intros x16 x15 x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        intros [H_5 H_4].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U12_monotonic :
       forall x12 x14 x13 x11, 
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U12 x12 x14 <= P_id_a__U12 x11 x13.
      Proof.
        intros x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U83_monotonic :
       forall x16 x12 x14 x13 x11 x15, 
        (0 <= x16)/\ (x16 <= x15) ->
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_U83 x12 x14 x16 <= P_id_U83 x11 x13 x15.
      Proof.
        intros x16 x15 x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        intros [H_5 H_4].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U94_monotonic :
       forall x12 x14 x13 x11, 
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U94 x12 x14 <= P_id_a__U94 x11 x13.
      Proof.
        intros x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U45_monotonic :
       forall x12 x14 x13 x11, 
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U45 x12 x14 <= P_id_a__U45 x11 x13.
      Proof.
        intros x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U31_monotonic :
       forall x12 x14 x13 x11, 
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->P_id_U31 x12 x14 <= P_id_U31 x11 x13.
      Proof.
        intros x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__isNat_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->P_id_a__isNat x12 <= P_id_a__isNat x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_isNatIListKind_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_isNatIListKind x12 <= P_id_isNatIListKind x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U82_monotonic :
       forall x16 x12 x14 x13 x11 x15, 
        (0 <= x16)/\ (x16 <= x15) ->
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_a__U82 x12 x14 x16 <= P_id_a__U82 x11 x13 x15.
      Proof.
        intros x16 x15 x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        intros [H_5 H_4].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U46_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->P_id_U46 x12 <= P_id_U46 x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_cons_monotonic :
       forall x12 x14 x13 x11, 
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->P_id_cons x12 x14 <= P_id_cons x11 x13.
      Proof.
        intros x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U52_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->P_id_U52 x12 <= P_id_U52 x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U85_monotonic :
       forall x12 x14 x13 x11, 
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U85 x12 x14 <= P_id_a__U85 x11 x13.
      Proof.
        intros x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U33_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->P_id_a__U33 x12 <= P_id_a__U33 x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U21_monotonic :
       forall x12 x14 x13 x11, 
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->P_id_U21 x12 x14 <= P_id_U21 x11 x13.
      Proof.
        intros x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__isNatList_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__isNatList x12 <= P_id_a__isNatList x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U86_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->P_id_U86 x12 <= P_id_U86 x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_mark_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->P_id_mark x12 <= P_id_mark x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U51_monotonic :
       forall x12 x14 x13 x11, 
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U51 x12 x14 <= P_id_a__U51 x11 x13.
      Proof.
        intros x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U41_monotonic :
       forall x16 x12 x14 x13 x11 x15, 
        (0 <= x16)/\ (x16 <= x15) ->
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_U41 x12 x14 x16 <= P_id_U41 x11 x13 x15.
      Proof.
        intros x16 x15 x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        intros [H_5 H_4].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U82_monotonic :
       forall x16 x12 x14 x13 x11 x15, 
        (0 <= x16)/\ (x16 <= x15) ->
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_U82 x12 x14 x16 <= P_id_U82 x11 x13 x15.
      Proof.
        intros x16 x15 x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        intros [H_5 H_4].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U93_monotonic :
       forall x16 x12 x14 x13 x11 x15, 
        (0 <= x16)/\ (x16 <= x15) ->
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_a__U93 x12 x14 x16 <= P_id_a__U93 x11 x13 x15.
      Proof.
        intros x16 x15 x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        intros [H_5 H_4].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U44_monotonic :
       forall x16 x12 x14 x13 x11 x15, 
        (0 <= x16)/\ (x16 <= x15) ->
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_a__U44 x12 x14 x16 <= P_id_a__U44 x11 x13 x15.
      Proof.
        intros x16 x15 x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        intros [H_5 H_4].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_isNat_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->P_id_isNat x12 <= P_id_isNat x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U23_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->P_id_a__U23 x12 <= P_id_a__U23 x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U94_monotonic :
       forall x12 x14 x13 x11, 
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->P_id_U94 x12 x14 <= P_id_U94 x11 x13.
      Proof.
        intros x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U12_monotonic :
       forall x12 x14 x13 x11, 
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->P_id_U12 x12 x14 <= P_id_U12 x11 x13.
      Proof.
        intros x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U81_monotonic :
       forall x16 x12 x14 x13 x11 x15, 
        (0 <= x16)/\ (x16 <= x15) ->
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_a__U81 x12 x14 x16 <= P_id_a__U81 x11 x13 x15.
      Proof.
        intros x16 x15 x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        intros [H_5 H_4].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U45_monotonic :
       forall x12 x14 x13 x11, 
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->P_id_U45 x12 x14 <= P_id_U45 x11 x13.
      Proof.
        intros x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U71_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->P_id_U71 x12 <= P_id_U71 x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U91_monotonic :
       forall x16 x12 x14 x13 x11 x15, 
        (0 <= x16)/\ (x16 <= x15) ->
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_a__U91 x12 x14 x16 <= P_id_a__U91 x11 x13 x15.
      Proof.
        intros x16 x15 x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        intros [H_5 H_4].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U42_monotonic :
       forall x16 x12 x14 x13 x11 x15, 
        (0 <= x16)/\ (x16 <= x15) ->
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_a__U42 x12 x14 x16 <= P_id_a__U42 x11 x13 x15.
      Proof.
        intros x16 x15 x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        intros [H_5 H_4].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_isNatKind_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->P_id_isNatKind x12 <= P_id_isNatKind x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U22_monotonic :
       forall x12 x14 x13 x11, 
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U22 x12 x14 <= P_id_a__U22 x11 x13.
      Proof.
        intros x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U92_monotonic :
       forall x16 x12 x14 x13 x11 x15, 
        (0 <= x16)/\ (x16 <= x15) ->
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_U92 x12 x14 x16 <= P_id_U92 x11 x13 x15.
      Proof.
        intros x16 x15 x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        intros [H_5 H_4].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U61_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->P_id_a__U61 x12 <= P_id_a__U61 x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U43_monotonic :
       forall x16 x12 x14 x13 x11 x15, 
        (0 <= x16)/\ (x16 <= x15) ->
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_U43 x12 x14 x16 <= P_id_U43 x11 x13 x15.
      Proof.
        intros x16 x15 x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        intros [H_5 H_4].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__isNatIListKind_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__isNatIListKind x12 <= P_id_a__isNatIListKind x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U84_monotonic :
       forall x16 x12 x14 x13 x11 x15, 
        (0 <= x16)/\ (x16 <= x15) ->
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_U84 x12 x14 x16 <= P_id_U84 x11 x13 x15.
      Proof.
        intros x16 x15 x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        intros [H_5 H_4].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_s_monotonic :
       forall x12 x11, (0 <= x12)/\ (x12 <= x11) ->P_id_s x12 <= P_id_s x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U46_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->P_id_a__U46 x12 <= P_id_a__U46 x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U32_monotonic :
       forall x12 x14 x13 x11, 
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->P_id_U32 x12 x14 <= P_id_U32 x11 x13.
      Proof.
        intros x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U31_monotonic :
       forall x12 x14 x13 x11, 
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U31 x12 x14 <= P_id_a__U31 x11 x13.
      Proof.
        intros x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U13_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->P_id_U13 x12 <= P_id_U13 x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U83_monotonic :
       forall x16 x12 x14 x13 x11 x15, 
        (0 <= x16)/\ (x16 <= x15) ->
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_a__U83 x12 x14 x16 <= P_id_a__U83 x11 x13 x15.
      Proof.
        intros x16 x15 x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        intros [H_5 H_4].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_isNatIList_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_isNatIList x12 <= P_id_isNatIList x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__zeros_bounded : 0 <= P_id_a__zeros .
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U51_bounded :
       forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U51 x12 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U84_bounded :
       forall x12 x13 x11, 
        (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U84 x13 x12 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U32_bounded :
       forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U32 x12 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_isNatList_bounded :
       forall x11, (0 <= x11) ->0 <= P_id_isNatList x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U13_bounded :
       forall x11, (0 <= x11) ->0 <= P_id_a__U13 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U85_bounded :
       forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U85 x12 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__length_bounded :
       forall x11, (0 <= x11) ->0 <= P_id_a__length x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__isNatIList_bounded :
       forall x11, (0 <= x11) ->0 <= P_id_a__isNatIList x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U33_bounded : forall x11, (0 <= x11) ->0 <= P_id_U33 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U11_bounded :
       forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U11 x12 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U81_bounded :
       forall x12 x13 x11, 
        (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U81 x13 x12 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U92_bounded :
       forall x12 x13 x11, 
        (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U92 x13 x12 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U43_bounded :
       forall x12 x13 x11, 
        (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U43 x13 x12 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U23_bounded : forall x11, (0 <= x11) ->0 <= P_id_U23 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__isNatKind_bounded :
       forall x11, (0 <= x11) ->0 <= P_id_a__isNatKind x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U93_bounded :
       forall x12 x13 x11, 
        (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U93 x13 x12 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U11_bounded :
       forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U11 x12 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U71_bounded :
       forall x11, (0 <= x11) ->0 <= P_id_a__U71 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U44_bounded :
       forall x12 x13 x11, 
        (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U44 x13 x12 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_0_bounded : 0 <= P_id_0 .
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U61_bounded : forall x11, (0 <= x11) ->0 <= P_id_U61 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U86_bounded :
       forall x11, (0 <= x11) ->0 <= P_id_a__U86 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U41_bounded :
       forall x12 x13 x11, 
        (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U41 x13 x12 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U22_bounded :
       forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U22 x12 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U21_bounded :
       forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U21 x12 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U91_bounded :
       forall x12 x13 x11, 
        (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U91 x13 x12 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_length_bounded :
       forall x11, (0 <= x11) ->0 <= P_id_length x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U52_bounded :
       forall x11, (0 <= x11) ->0 <= P_id_a__U52 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U42_bounded :
       forall x12 x13 x11, 
        (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U42 x13 x12 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U12_bounded :
       forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U12 x12 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U83_bounded :
       forall x12 x13 x11, 
        (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U83 x13 x12 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U94_bounded :
       forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U94 x12 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U45_bounded :
       forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U45 x12 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U31_bounded :
       forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U31 x12 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__isNat_bounded :
       forall x11, (0 <= x11) ->0 <= P_id_a__isNat x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_isNatIListKind_bounded :
       forall x11, (0 <= x11) ->0 <= P_id_isNatIListKind x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U82_bounded :
       forall x12 x13 x11, 
        (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U82 x13 x12 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U46_bounded : forall x11, (0 <= x11) ->0 <= P_id_U46 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_cons_bounded :
       forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_cons x12 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U52_bounded : forall x11, (0 <= x11) ->0 <= P_id_U52 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U85_bounded :
       forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U85 x12 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U33_bounded :
       forall x11, (0 <= x11) ->0 <= P_id_a__U33 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U21_bounded :
       forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U21 x12 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__isNatList_bounded :
       forall x11, (0 <= x11) ->0 <= P_id_a__isNatList x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U86_bounded : forall x11, (0 <= x11) ->0 <= P_id_U86 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_mark_bounded : forall x11, (0 <= x11) ->0 <= P_id_mark x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U51_bounded :
       forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U51 x12 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U41_bounded :
       forall x12 x13 x11, 
        (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U41 x13 x12 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_tt_bounded : 0 <= P_id_tt .
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U82_bounded :
       forall x12 x13 x11, 
        (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U82 x13 x12 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U93_bounded :
       forall x12 x13 x11, 
        (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U93 x13 x12 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U44_bounded :
       forall x12 x13 x11, 
        (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U44 x13 x12 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_isNat_bounded :
       forall x11, (0 <= x11) ->0 <= P_id_isNat x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U23_bounded :
       forall x11, (0 <= x11) ->0 <= P_id_a__U23 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U94_bounded :
       forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U94 x12 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U12_bounded :
       forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U12 x12 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U81_bounded :
       forall x12 x13 x11, 
        (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U81 x13 x12 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U45_bounded :
       forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U45 x12 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_zeros_bounded : 0 <= P_id_zeros .
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U71_bounded : forall x11, (0 <= x11) ->0 <= P_id_U71 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U91_bounded :
       forall x12 x13 x11, 
        (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U91 x13 x12 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U42_bounded :
       forall x12 x13 x11, 
        (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U42 x13 x12 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_isNatKind_bounded :
       forall x11, (0 <= x11) ->0 <= P_id_isNatKind x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U22_bounded :
       forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U22 x12 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U92_bounded :
       forall x12 x13 x11, 
        (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U92 x13 x12 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_nil_bounded : 0 <= P_id_nil .
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U61_bounded :
       forall x11, (0 <= x11) ->0 <= P_id_a__U61 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U43_bounded :
       forall x12 x13 x11, 
        (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U43 x13 x12 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__isNatIListKind_bounded :
       forall x11, (0 <= x11) ->0 <= P_id_a__isNatIListKind x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U84_bounded :
       forall x12 x13 x11, 
        (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U84 x13 x12 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_s_bounded : forall x11, (0 <= x11) ->0 <= P_id_s x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U46_bounded :
       forall x11, (0 <= x11) ->0 <= P_id_a__U46 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U32_bounded :
       forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U32 x12 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U31_bounded :
       forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U31 x12 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U13_bounded : forall x11, (0 <= x11) ->0 <= P_id_U13 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_a__U83_bounded :
       forall x12 x13 x11, 
        (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U83 x13 x12 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_isNatIList_bounded :
       forall x11, (0 <= x11) ->0 <= P_id_isNatIList x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Definition measure  := 
        InterpZ.measure 0 P_id_a__zeros P_id_U51 P_id_a__U84 P_id_a__U32 
         P_id_isNatList P_id_a__U13 P_id_U85 P_id_a__length 
         P_id_a__isNatIList P_id_U33 P_id_a__U11 P_id_U81 P_id_a__U92 
         P_id_a__U43 P_id_U23 P_id_a__isNatKind P_id_U93 P_id_U11 
         P_id_a__U71 P_id_U44 P_id_0 P_id_U61 P_id_a__U86 P_id_a__U41 
         P_id_U22 P_id_a__U21 P_id_U91 P_id_length P_id_a__U52 P_id_U42 
         P_id_a__U12 P_id_U83 P_id_a__U94 P_id_a__U45 P_id_U31 P_id_a__isNat 
         P_id_isNatIListKind P_id_a__U82 P_id_U46 P_id_cons P_id_U52 
         P_id_a__U85 P_id_a__U33 P_id_U21 P_id_a__isNatList P_id_U86 
         P_id_mark P_id_a__U51 P_id_U41 P_id_tt P_id_U82 P_id_a__U93 
         P_id_a__U44 P_id_isNat P_id_a__U23 P_id_U94 P_id_U12 P_id_a__U81 
         P_id_U45 P_id_zeros P_id_U71 P_id_a__U91 P_id_a__U42 P_id_isNatKind 
         P_id_a__U22 P_id_U92 P_id_nil P_id_a__U61 P_id_U43 
         P_id_a__isNatIListKind P_id_U84 P_id_s P_id_a__U46 P_id_U32 
         P_id_a__U31 P_id_U13 P_id_a__U83 P_id_isNatIList.
      
      Lemma measure_equation :
       forall t, 
        measure t = match t with
                      | (algebra.Alg.Term algebra.F.id_a__zeros nil) =>
                       P_id_a__zeros 
                      | (algebra.Alg.Term algebra.F.id_U51 (x12::x11::nil)) =>
                       P_id_U51 (measure x12) (measure x11)
                      | (algebra.Alg.Term algebra.F.id_a__U84 (x13::x12::
                         x11::nil)) =>
                       P_id_a__U84 (measure x13) (measure x12) (measure x11)
                      | (algebra.Alg.Term algebra.F.id_a__U32 (x12::
                         x11::nil)) =>
                       P_id_a__U32 (measure x12) (measure x11)
                      | (algebra.Alg.Term algebra.F.id_isNatList (x11::nil)) =>
                       P_id_isNatList (measure x11)
                      | (algebra.Alg.Term algebra.F.id_a__U13 (x11::nil)) =>
                       P_id_a__U13 (measure x11)
                      | (algebra.Alg.Term algebra.F.id_U85 (x12::x11::nil)) =>
                       P_id_U85 (measure x12) (measure x11)
                      | (algebra.Alg.Term algebra.F.id_a__length (x11::nil)) =>
                       P_id_a__length (measure x11)
                      | (algebra.Alg.Term algebra.F.id_a__isNatIList 
                         (x11::nil)) =>
                       P_id_a__isNatIList (measure x11)
                      | (algebra.Alg.Term algebra.F.id_U33 (x11::nil)) =>
                       P_id_U33 (measure x11)
                      | (algebra.Alg.Term algebra.F.id_a__U11 (x12::
                         x11::nil)) =>
                       P_id_a__U11 (measure x12) (measure x11)
                      | (algebra.Alg.Term algebra.F.id_U81 (x13::x12::
                         x11::nil)) =>
                       P_id_U81 (measure x13) (measure x12) (measure x11)
                      | (algebra.Alg.Term algebra.F.id_a__U92 (x13::x12::
                         x11::nil)) =>
                       P_id_a__U92 (measure x13) (measure x12) (measure x11)
                      | (algebra.Alg.Term algebra.F.id_a__U43 (x13::x12::
                         x11::nil)) =>
                       P_id_a__U43 (measure x13) (measure x12) (measure x11)
                      | (algebra.Alg.Term algebra.F.id_U23 (x11::nil)) =>
                       P_id_U23 (measure x11)
                      | (algebra.Alg.Term algebra.F.id_a__isNatKind 
                         (x11::nil)) =>
                       P_id_a__isNatKind (measure x11)
                      | (algebra.Alg.Term algebra.F.id_U93 (x13::x12::
                         x11::nil)) =>
                       P_id_U93 (measure x13) (measure x12) (measure x11)
                      | (algebra.Alg.Term algebra.F.id_U11 (x12::x11::nil)) =>
                       P_id_U11 (measure x12) (measure x11)
                      | (algebra.Alg.Term algebra.F.id_a__U71 (x11::nil)) =>
                       P_id_a__U71 (measure x11)
                      | (algebra.Alg.Term algebra.F.id_U44 (x13::x12::
                         x11::nil)) =>
                       P_id_U44 (measure x13) (measure x12) (measure x11)
                      | (algebra.Alg.Term algebra.F.id_0 nil) => P_id_0 
                      | (algebra.Alg.Term algebra.F.id_U61 (x11::nil)) =>
                       P_id_U61 (measure x11)
                      | (algebra.Alg.Term algebra.F.id_a__U86 (x11::nil)) =>
                       P_id_a__U86 (measure x11)
                      | (algebra.Alg.Term algebra.F.id_a__U41 (x13::x12::
                         x11::nil)) =>
                       P_id_a__U41 (measure x13) (measure x12) (measure x11)
                      | (algebra.Alg.Term algebra.F.id_U22 (x12::x11::nil)) =>
                       P_id_U22 (measure x12) (measure x11)
                      | (algebra.Alg.Term algebra.F.id_a__U21 (x12::
                         x11::nil)) =>
                       P_id_a__U21 (measure x12) (measure x11)
                      | (algebra.Alg.Term algebra.F.id_U91 (x13::x12::
                         x11::nil)) =>
                       P_id_U91 (measure x13) (measure x12) (measure x11)
                      | (algebra.Alg.Term algebra.F.id_length (x11::nil)) =>
                       P_id_length (measure x11)
                      | (algebra.Alg.Term algebra.F.id_a__U52 (x11::nil)) =>
                       P_id_a__U52 (measure x11)
                      | (algebra.Alg.Term algebra.F.id_U42 (x13::x12::
                         x11::nil)) =>
                       P_id_U42 (measure x13) (measure x12) (measure x11)
                      | (algebra.Alg.Term algebra.F.id_a__U12 (x12::
                         x11::nil)) =>
                       P_id_a__U12 (measure x12) (measure x11)
                      | (algebra.Alg.Term algebra.F.id_U83 (x13::x12::
                         x11::nil)) =>
                       P_id_U83 (measure x13) (measure x12) (measure x11)
                      | (algebra.Alg.Term algebra.F.id_a__U94 (x12::
                         x11::nil)) =>
                       P_id_a__U94 (measure x12) (measure x11)
                      | (algebra.Alg.Term algebra.F.id_a__U45 (x12::
                         x11::nil)) =>
                       P_id_a__U45 (measure x12) (measure x11)
                      | (algebra.Alg.Term algebra.F.id_U31 (x12::x11::nil)) =>
                       P_id_U31 (measure x12) (measure x11)
                      | (algebra.Alg.Term algebra.F.id_a__isNat (x11::nil)) =>
                       P_id_a__isNat (measure x11)
                      | (algebra.Alg.Term algebra.F.id_isNatIListKind 
                         (x11::nil)) =>
                       P_id_isNatIListKind (measure x11)
                      | (algebra.Alg.Term algebra.F.id_a__U82 (x13::x12::
                         x11::nil)) =>
                       P_id_a__U82 (measure x13) (measure x12) (measure x11)
                      | (algebra.Alg.Term algebra.F.id_U46 (x11::nil)) =>
                       P_id_U46 (measure x11)
                      | (algebra.Alg.Term algebra.F.id_cons (x12::x11::nil)) =>
                       P_id_cons (measure x12) (measure x11)
                      | (algebra.Alg.Term algebra.F.id_U52 (x11::nil)) =>
                       P_id_U52 (measure x11)
                      | (algebra.Alg.Term algebra.F.id_a__U85 (x12::
                         x11::nil)) =>
                       P_id_a__U85 (measure x12) (measure x11)
                      | (algebra.Alg.Term algebra.F.id_a__U33 (x11::nil)) =>
                       P_id_a__U33 (measure x11)
                      | (algebra.Alg.Term algebra.F.id_U21 (x12::x11::nil)) =>
                       P_id_U21 (measure x12) (measure x11)
                      | (algebra.Alg.Term algebra.F.id_a__isNatList 
                         (x11::nil)) =>
                       P_id_a__isNatList (measure x11)
                      | (algebra.Alg.Term algebra.F.id_U86 (x11::nil)) =>
                       P_id_U86 (measure x11)
                      | (algebra.Alg.Term algebra.F.id_mark (x11::nil)) =>
                       P_id_mark (measure x11)
                      | (algebra.Alg.Term algebra.F.id_a__U51 (x12::
                         x11::nil)) =>
                       P_id_a__U51 (measure x12) (measure x11)
                      | (algebra.Alg.Term algebra.F.id_U41 (x13::x12::
                         x11::nil)) =>
                       P_id_U41 (measure x13) (measure x12) (measure x11)
                      | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt 
                      | (algebra.Alg.Term algebra.F.id_U82 (x13::x12::
                         x11::nil)) =>
                       P_id_U82 (measure x13) (measure x12) (measure x11)
                      | (algebra.Alg.Term algebra.F.id_a__U93 (x13::x12::
                         x11::nil)) =>
                       P_id_a__U93 (measure x13) (measure x12) (measure x11)
                      | (algebra.Alg.Term algebra.F.id_a__U44 (x13::x12::
                         x11::nil)) =>
                       P_id_a__U44 (measure x13) (measure x12) (measure x11)
                      | (algebra.Alg.Term algebra.F.id_isNat (x11::nil)) =>
                       P_id_isNat (measure x11)
                      | (algebra.Alg.Term algebra.F.id_a__U23 (x11::nil)) =>
                       P_id_a__U23 (measure x11)
                      | (algebra.Alg.Term algebra.F.id_U94 (x12::x11::nil)) =>
                       P_id_U94 (measure x12) (measure x11)
                      | (algebra.Alg.Term algebra.F.id_U12 (x12::x11::nil)) =>
                       P_id_U12 (measure x12) (measure x11)
                      | (algebra.Alg.Term algebra.F.id_a__U81 (x13::x12::
                         x11::nil)) =>
                       P_id_a__U81 (measure x13) (measure x12) (measure x11)
                      | (algebra.Alg.Term algebra.F.id_U45 (x12::x11::nil)) =>
                       P_id_U45 (measure x12) (measure x11)
                      | (algebra.Alg.Term algebra.F.id_zeros nil) =>
                       P_id_zeros 
                      | (algebra.Alg.Term algebra.F.id_U71 (x11::nil)) =>
                       P_id_U71 (measure x11)
                      | (algebra.Alg.Term algebra.F.id_a__U91 (x13::x12::
                         x11::nil)) =>
                       P_id_a__U91 (measure x13) (measure x12) (measure x11)
                      | (algebra.Alg.Term algebra.F.id_a__U42 (x13::x12::
                         x11::nil)) =>
                       P_id_a__U42 (measure x13) (measure x12) (measure x11)
                      | (algebra.Alg.Term algebra.F.id_isNatKind (x11::nil)) =>
                       P_id_isNatKind (measure x11)
                      | (algebra.Alg.Term algebra.F.id_a__U22 (x12::
                         x11::nil)) =>
                       P_id_a__U22 (measure x12) (measure x11)
                      | (algebra.Alg.Term algebra.F.id_U92 (x13::x12::
                         x11::nil)) =>
                       P_id_U92 (measure x13) (measure x12) (measure x11)
                      | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil 
                      | (algebra.Alg.Term algebra.F.id_a__U61 (x11::nil)) =>
                       P_id_a__U61 (measure x11)
                      | (algebra.Alg.Term algebra.F.id_U43 (x13::x12::
                         x11::nil)) =>
                       P_id_U43 (measure x13) (measure x12) (measure x11)
                      | (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
                         (x11::nil)) =>
                       P_id_a__isNatIListKind (measure x11)
                      | (algebra.Alg.Term algebra.F.id_U84 (x13::x12::
                         x11::nil)) =>
                       P_id_U84 (measure x13) (measure x12) (measure x11)
                      | (algebra.Alg.Term algebra.F.id_s (x11::nil)) =>
                       P_id_s (measure x11)
                      | (algebra.Alg.Term algebra.F.id_a__U46 (x11::nil)) =>
                       P_id_a__U46 (measure x11)
                      | (algebra.Alg.Term algebra.F.id_U32 (x12::x11::nil)) =>
                       P_id_U32 (measure x12) (measure x11)
                      | (algebra.Alg.Term algebra.F.id_a__U31 (x12::
                         x11::nil)) =>
                       P_id_a__U31 (measure x12) (measure x11)
                      | (algebra.Alg.Term algebra.F.id_U13 (x11::nil)) =>
                       P_id_U13 (measure x11)
                      | (algebra.Alg.Term algebra.F.id_a__U83 (x13::x12::
                         x11::nil)) =>
                       P_id_a__U83 (measure x13) (measure x12) (measure x11)
                      | (algebra.Alg.Term algebra.F.id_isNatIList (x11::nil)) =>
                       P_id_isNatIList (measure x11)
                      | _ => 0
                      end.
      Proof.
        intros t;case t;intros ;apply refl_equal.
      Qed.
      
      Lemma measure_bounded : forall t, 0 <= measure t.
      Proof.
        unfold measure in |-*.
        
        apply InterpZ.measure_bounded;
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Ltac generate_pos_hyp  :=
       match goal with
         | H:context [measure ?x] |- _ =>
          let v := fresh "v" in 
           (let H := fresh "h" in 
             (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
               clearbody H;clearbody v))
         |  |- context [measure ?x] =>
          let v := fresh "v" in 
           (let H := fresh "h" in 
             (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
               clearbody H;clearbody v))
         end
       .
      
      Lemma rules_monotonic :
       forall l r, 
        (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) ->
         measure r <= measure l.
      Proof.
        intros l r H.
        fold measure in |-*.
        
        inversion H;clear H;subst;inversion H0;clear H0;subst;
         simpl algebra.EQT.T.apply_subst in |-*;
         repeat (
         match goal with
           |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
            rewrite (measure_equation (algebra.Alg.Term f t))
           end
         );repeat (generate_pos_hyp );
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma measure_star_monotonic :
       forall l r, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   r l) ->measure r <= measure l.
      Proof.
        unfold measure in *.
        apply InterpZ.measure_star_monotonic.
        intros ;apply P_id_U51_monotonic;assumption.
        intros ;apply P_id_a__U84_monotonic;assumption.
        intros ;apply P_id_a__U32_monotonic;assumption.
        intros ;apply P_id_isNatList_monotonic;assumption.
        intros ;apply P_id_a__U13_monotonic;assumption.
        intros ;apply P_id_U85_monotonic;assumption.
        intros ;apply P_id_a__length_monotonic;assumption.
        intros ;apply P_id_a__isNatIList_monotonic;assumption.
        intros ;apply P_id_U33_monotonic;assumption.
        intros ;apply P_id_a__U11_monotonic;assumption.
        intros ;apply P_id_U81_monotonic;assumption.
        intros ;apply P_id_a__U92_monotonic;assumption.
        intros ;apply P_id_a__U43_monotonic;assumption.
        intros ;apply P_id_U23_monotonic;assumption.
        intros ;apply P_id_a__isNatKind_monotonic;assumption.
        intros ;apply P_id_U93_monotonic;assumption.
        intros ;apply P_id_U11_monotonic;assumption.
        intros ;apply P_id_a__U71_monotonic;assumption.
        intros ;apply P_id_U44_monotonic;assumption.
        intros ;apply P_id_U61_monotonic;assumption.
        intros ;apply P_id_a__U86_monotonic;assumption.
        intros ;apply P_id_a__U41_monotonic;assumption.
        intros ;apply P_id_U22_monotonic;assumption.
        intros ;apply P_id_a__U21_monotonic;assumption.
        intros ;apply P_id_U91_monotonic;assumption.
        intros ;apply P_id_length_monotonic;assumption.
        intros ;apply P_id_a__U52_monotonic;assumption.
        intros ;apply P_id_U42_monotonic;assumption.
        intros ;apply P_id_a__U12_monotonic;assumption.
        intros ;apply P_id_U83_monotonic;assumption.
        intros ;apply P_id_a__U94_monotonic;assumption.
        intros ;apply P_id_a__U45_monotonic;assumption.
        intros ;apply P_id_U31_monotonic;assumption.
        intros ;apply P_id_a__isNat_monotonic;assumption.
        intros ;apply P_id_isNatIListKind_monotonic;assumption.
        intros ;apply P_id_a__U82_monotonic;assumption.
        intros ;apply P_id_U46_monotonic;assumption.
        intros ;apply P_id_cons_monotonic;assumption.
        intros ;apply P_id_U52_monotonic;assumption.
        intros ;apply P_id_a__U85_monotonic;assumption.
        intros ;apply P_id_a__U33_monotonic;assumption.
        intros ;apply P_id_U21_monotonic;assumption.
        intros ;apply P_id_a__isNatList_monotonic;assumption.
        intros ;apply P_id_U86_monotonic;assumption.
        intros ;apply P_id_mark_monotonic;assumption.
        intros ;apply P_id_a__U51_monotonic;assumption.
        intros ;apply P_id_U41_monotonic;assumption.
        intros ;apply P_id_U82_monotonic;assumption.
        intros ;apply P_id_a__U93_monotonic;assumption.
        intros ;apply P_id_a__U44_monotonic;assumption.
        intros ;apply P_id_isNat_monotonic;assumption.
        intros ;apply P_id_a__U23_monotonic;assumption.
        intros ;apply P_id_U94_monotonic;assumption.
        intros ;apply P_id_U12_monotonic;assumption.
        intros ;apply P_id_a__U81_monotonic;assumption.
        intros ;apply P_id_U45_monotonic;assumption.
        intros ;apply P_id_U71_monotonic;assumption.
        intros ;apply P_id_a__U91_monotonic;assumption.
        intros ;apply P_id_a__U42_monotonic;assumption.
        intros ;apply P_id_isNatKind_monotonic;assumption.
        intros ;apply P_id_a__U22_monotonic;assumption.
        intros ;apply P_id_U92_monotonic;assumption.
        intros ;apply P_id_a__U61_monotonic;assumption.
        intros ;apply P_id_U43_monotonic;assumption.
        intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
        intros ;apply P_id_U84_monotonic;assumption.
        intros ;apply P_id_s_monotonic;assumption.
        intros ;apply P_id_a__U46_monotonic;assumption.
        intros ;apply P_id_U32_monotonic;assumption.
        intros ;apply P_id_a__U31_monotonic;assumption.
        intros ;apply P_id_U13_monotonic;assumption.
        intros ;apply P_id_a__U83_monotonic;assumption.
        intros ;apply P_id_isNatIList_monotonic;assumption.
        intros ;apply P_id_a__zeros_bounded;assumption.
        intros ;apply P_id_U51_bounded;assumption.
        intros ;apply P_id_a__U84_bounded;assumption.
        intros ;apply P_id_a__U32_bounded;assumption.
        intros ;apply P_id_isNatList_bounded;assumption.
        intros ;apply P_id_a__U13_bounded;assumption.
        intros ;apply P_id_U85_bounded;assumption.
        intros ;apply P_id_a__length_bounded;assumption.
        intros ;apply P_id_a__isNatIList_bounded;assumption.
        intros ;apply P_id_U33_bounded;assumption.
        intros ;apply P_id_a__U11_bounded;assumption.
        intros ;apply P_id_U81_bounded;assumption.
        intros ;apply P_id_a__U92_bounded;assumption.
        intros ;apply P_id_a__U43_bounded;assumption.
        intros ;apply P_id_U23_bounded;assumption.
        intros ;apply P_id_a__isNatKind_bounded;assumption.
        intros ;apply P_id_U93_bounded;assumption.
        intros ;apply P_id_U11_bounded;assumption.
        intros ;apply P_id_a__U71_bounded;assumption.
        intros ;apply P_id_U44_bounded;assumption.
        intros ;apply P_id_0_bounded;assumption.
        intros ;apply P_id_U61_bounded;assumption.
        intros ;apply P_id_a__U86_bounded;assumption.
        intros ;apply P_id_a__U41_bounded;assumption.
        intros ;apply P_id_U22_bounded;assumption.
        intros ;apply P_id_a__U21_bounded;assumption.
        intros ;apply P_id_U91_bounded;assumption.
        intros ;apply P_id_length_bounded;assumption.
        intros ;apply P_id_a__U52_bounded;assumption.
        intros ;apply P_id_U42_bounded;assumption.
        intros ;apply P_id_a__U12_bounded;assumption.
        intros ;apply P_id_U83_bounded;assumption.
        intros ;apply P_id_a__U94_bounded;assumption.
        intros ;apply P_id_a__U45_bounded;assumption.
        intros ;apply P_id_U31_bounded;assumption.
        intros ;apply P_id_a__isNat_bounded;assumption.
        intros ;apply P_id_isNatIListKind_bounded;assumption.
        intros ;apply P_id_a__U82_bounded;assumption.
        intros ;apply P_id_U46_bounded;assumption.
        intros ;apply P_id_cons_bounded;assumption.
        intros ;apply P_id_U52_bounded;assumption.
        intros ;apply P_id_a__U85_bounded;assumption.
        intros ;apply P_id_a__U33_bounded;assumption.
        intros ;apply P_id_U21_bounded;assumption.
        intros ;apply P_id_a__isNatList_bounded;assumption.
        intros ;apply P_id_U86_bounded;assumption.
        intros ;apply P_id_mark_bounded;assumption.
        intros ;apply P_id_a__U51_bounded;assumption.
        intros ;apply P_id_U41_bounded;assumption.
        intros ;apply P_id_tt_bounded;assumption.
        intros ;apply P_id_U82_bounded;assumption.
        intros ;apply P_id_a__U93_bounded;assumption.
        intros ;apply P_id_a__U44_bounded;assumption.
        intros ;apply P_id_isNat_bounded;assumption.
        intros ;apply P_id_a__U23_bounded;assumption.
        intros ;apply P_id_U94_bounded;assumption.
        intros ;apply P_id_U12_bounded;assumption.
        intros ;apply P_id_a__U81_bounded;assumption.
        intros ;apply P_id_U45_bounded;assumption.
        intros ;apply P_id_zeros_bounded;assumption.
        intros ;apply P_id_U71_bounded;assumption.
        intros ;apply P_id_a__U91_bounded;assumption.
        intros ;apply P_id_a__U42_bounded;assumption.
        intros ;apply P_id_isNatKind_bounded;assumption.
        intros ;apply P_id_a__U22_bounded;assumption.
        intros ;apply P_id_U92_bounded;assumption.
        intros ;apply P_id_nil_bounded;assumption.
        intros ;apply P_id_a__U61_bounded;assumption.
        intros ;apply P_id_U43_bounded;assumption.
        intros ;apply P_id_a__isNatIListKind_bounded;assumption.
        intros ;apply P_id_U84_bounded;assumption.
        intros ;apply P_id_s_bounded;assumption.
        intros ;apply P_id_a__U46_bounded;assumption.
        intros ;apply P_id_U32_bounded;assumption.
        intros ;apply P_id_a__U31_bounded;assumption.
        intros ;apply P_id_U13_bounded;assumption.
        intros ;apply P_id_a__U83_bounded;assumption.
        intros ;apply P_id_isNatIList_bounded;assumption.
        apply rules_monotonic.
      Qed.
      
      Definition P_id_A__U46 (x11:Z) := 0.
      
      Definition P_id_A__ISNATILISTKIND (x11:Z) := 0.
      
      Definition P_id_A__U61 (x11:Z) := 0.
      
      Definition P_id_A__U85 (x11:Z) (x12:Z) := 0.
      
      Definition P_id_A__U31 (x11:Z) (x12:Z) := 0.
      
      Definition P_id_A__U81 (x11:Z) (x12:Z) (x13:Z) := 0.
      
      Definition P_id_A__U22 (x11:Z) (x12:Z) := 0.
      
      Definition P_id_A__U93 (x11:Z) (x12:Z) (x13:Z) := 0.
      
      Definition P_id_A__U42 (x11:Z) (x12:Z) (x13:Z) := 0.
      
      Definition P_id_A__U51 (x11:Z) (x12:Z) := 0.
      
      Definition P_id_A__ISNATLIST (x11:Z) := 0.
      
      Definition P_id_A__ZEROS  := 0.
      
      Definition P_id_A__U91 (x11:Z) (x12:Z) (x13:Z) := 0.
      
      Definition P_id_A__U33 (x11:Z) := 0.
      
      Definition P_id_A__U83 (x11:Z) (x12:Z) (x13:Z) := 0.
      
      Definition P_id_A__U23 (x11:Z) := 0.
      
      Definition P_id_A__U11 (x11:Z) (x12:Z) := 0.
      
      Definition P_id_A__LENGTH (x11:Z) := 0.
      
      Definition P_id_A__U44 (x11:Z) (x12:Z) (x13:Z) := 0.
      
      Definition P_id_A__ISNATILIST (x11:Z) := 0.
      
      Definition P_id_A__U13 (x11:Z) := 0.
      
      Definition P_id_A__U71 (x11:Z) := 0.
      
      Definition P_id_A__U86 (x11:Z) := 0.
      
      Definition P_id_A__U32 (x11:Z) (x12:Z) := 0.
      
      Definition P_id_A__U82 (x11:Z) (x12:Z) (x13:Z) := 0.
      
      Definition P_id_A__ISNATKIND (x11:Z) := 0.
      
      Definition P_id_A__U94 (x11:Z) (x12:Z) := 0.
      
      Definition P_id_A__U43 (x11:Z) (x12:Z) (x13:Z) := 0.
      
      Definition P_id_A__U52 (x11:Z) := 0.
      
      Definition P_id_A__U21 (x11:Z) (x12:Z) := 0.
      
      Definition P_id_A__U92 (x11:Z) (x12:Z) (x13:Z) := 0.
      
      Definition P_id_A__U41 (x11:Z) (x12:Z) (x13:Z) := 0.
      
      Definition P_id_A__U84 (x11:Z) (x12:Z) (x13:Z) := 0.
      
      Definition P_id_A__ISNAT (x11:Z) := 0.
      
      Definition P_id_A__U12 (x11:Z) (x12:Z) := 0.
      
      Definition P_id_MARK (x11:Z) := 2* x11.
      
      Definition P_id_A__U45 (x11:Z) (x12:Z) := 0.
      
      Lemma P_id_A__U46_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->P_id_A__U46 x12 <= P_id_A__U46 x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_A__ISNATILISTKIND_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__ISNATILISTKIND x12 <= P_id_A__ISNATILISTKIND x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_A__U61_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->P_id_A__U61 x12 <= P_id_A__U61 x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_A__U85_monotonic :
       forall x12 x14 x13 x11, 
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U85 x12 x14 <= P_id_A__U85 x11 x13.
      Proof.
        intros x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_A__U31_monotonic :
       forall x12 x14 x13 x11, 
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U31 x12 x14 <= P_id_A__U31 x11 x13.
      Proof.
        intros x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_A__U81_monotonic :
       forall x16 x12 x14 x13 x11 x15, 
        (0 <= x16)/\ (x16 <= x15) ->
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_A__U81 x12 x14 x16 <= P_id_A__U81 x11 x13 x15.
      Proof.
        intros x16 x15 x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        intros [H_5 H_4].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_A__U22_monotonic :
       forall x12 x14 x13 x11, 
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U22 x12 x14 <= P_id_A__U22 x11 x13.
      Proof.
        intros x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_A__U93_monotonic :
       forall x16 x12 x14 x13 x11 x15, 
        (0 <= x16)/\ (x16 <= x15) ->
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_A__U93 x12 x14 x16 <= P_id_A__U93 x11 x13 x15.
      Proof.
        intros x16 x15 x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        intros [H_5 H_4].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_A__U42_monotonic :
       forall x16 x12 x14 x13 x11 x15, 
        (0 <= x16)/\ (x16 <= x15) ->
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_A__U42 x12 x14 x16 <= P_id_A__U42 x11 x13 x15.
      Proof.
        intros x16 x15 x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        intros [H_5 H_4].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_A__U51_monotonic :
       forall x12 x14 x13 x11, 
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U51 x12 x14 <= P_id_A__U51 x11 x13.
      Proof.
        intros x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_A__ISNATLIST_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__ISNATLIST x12 <= P_id_A__ISNATLIST x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_A__U91_monotonic :
       forall x16 x12 x14 x13 x11 x15, 
        (0 <= x16)/\ (x16 <= x15) ->
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_A__U91 x12 x14 x16 <= P_id_A__U91 x11 x13 x15.
      Proof.
        intros x16 x15 x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        intros [H_5 H_4].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_A__U33_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->P_id_A__U33 x12 <= P_id_A__U33 x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_A__U83_monotonic :
       forall x16 x12 x14 x13 x11 x15, 
        (0 <= x16)/\ (x16 <= x15) ->
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_A__U83 x12 x14 x16 <= P_id_A__U83 x11 x13 x15.
      Proof.
        intros x16 x15 x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        intros [H_5 H_4].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_A__U23_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->P_id_A__U23 x12 <= P_id_A__U23 x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_A__U11_monotonic :
       forall x12 x14 x13 x11, 
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U11 x12 x14 <= P_id_A__U11 x11 x13.
      Proof.
        intros x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_A__LENGTH_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->P_id_A__LENGTH x12 <= P_id_A__LENGTH x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_A__U44_monotonic :
       forall x16 x12 x14 x13 x11 x15, 
        (0 <= x16)/\ (x16 <= x15) ->
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_A__U44 x12 x14 x16 <= P_id_A__U44 x11 x13 x15.
      Proof.
        intros x16 x15 x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        intros [H_5 H_4].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_A__ISNATILIST_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__ISNATILIST x12 <= P_id_A__ISNATILIST x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_A__U13_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->P_id_A__U13 x12 <= P_id_A__U13 x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_A__U71_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->P_id_A__U71 x12 <= P_id_A__U71 x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_A__U86_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->P_id_A__U86 x12 <= P_id_A__U86 x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_A__U32_monotonic :
       forall x12 x14 x13 x11, 
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U32 x12 x14 <= P_id_A__U32 x11 x13.
      Proof.
        intros x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_A__U82_monotonic :
       forall x16 x12 x14 x13 x11 x15, 
        (0 <= x16)/\ (x16 <= x15) ->
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_A__U82 x12 x14 x16 <= P_id_A__U82 x11 x13 x15.
      Proof.
        intros x16 x15 x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        intros [H_5 H_4].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_A__ISNATKIND_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__ISNATKIND x12 <= P_id_A__ISNATKIND x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_A__U94_monotonic :
       forall x12 x14 x13 x11, 
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U94 x12 x14 <= P_id_A__U94 x11 x13.
      Proof.
        intros x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_A__U43_monotonic :
       forall x16 x12 x14 x13 x11 x15, 
        (0 <= x16)/\ (x16 <= x15) ->
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_A__U43 x12 x14 x16 <= P_id_A__U43 x11 x13 x15.
      Proof.
        intros x16 x15 x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        intros [H_5 H_4].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_A__U52_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->P_id_A__U52 x12 <= P_id_A__U52 x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_A__U21_monotonic :
       forall x12 x14 x13 x11, 
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U21 x12 x14 <= P_id_A__U21 x11 x13.
      Proof.
        intros x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_A__U92_monotonic :
       forall x16 x12 x14 x13 x11 x15, 
        (0 <= x16)/\ (x16 <= x15) ->
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_A__U92 x12 x14 x16 <= P_id_A__U92 x11 x13 x15.
      Proof.
        intros x16 x15 x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        intros [H_5 H_4].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_A__U41_monotonic :
       forall x16 x12 x14 x13 x11 x15, 
        (0 <= x16)/\ (x16 <= x15) ->
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_A__U41 x12 x14 x16 <= P_id_A__U41 x11 x13 x15.
      Proof.
        intros x16 x15 x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        intros [H_5 H_4].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_A__U84_monotonic :
       forall x16 x12 x14 x13 x11 x15, 
        (0 <= x16)/\ (x16 <= x15) ->
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_A__U84 x12 x14 x16 <= P_id_A__U84 x11 x13 x15.
      Proof.
        intros x16 x15 x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        intros [H_5 H_4].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_A__ISNAT_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->P_id_A__ISNAT x12 <= P_id_A__ISNAT x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_A__U12_monotonic :
       forall x12 x14 x13 x11, 
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U12 x12 x14 <= P_id_A__U12 x11 x13.
      Proof.
        intros x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_MARK_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->P_id_MARK x12 <= P_id_MARK x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_A__U45_monotonic :
       forall x12 x14 x13 x11, 
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U45 x12 x14 <= P_id_A__U45 x11 x13.
      Proof.
        intros x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Definition marked_measure  := 
        InterpZ.marked_measure 0 P_id_a__zeros P_id_U51 P_id_a__U84 
         P_id_a__U32 P_id_isNatList P_id_a__U13 P_id_U85 P_id_a__length 
         P_id_a__isNatIList P_id_U33 P_id_a__U11 P_id_U81 P_id_a__U92 
         P_id_a__U43 P_id_U23 P_id_a__isNatKind P_id_U93 P_id_U11 
         P_id_a__U71 P_id_U44 P_id_0 P_id_U61 P_id_a__U86 P_id_a__U41 
         P_id_U22 P_id_a__U21 P_id_U91 P_id_length P_id_a__U52 P_id_U42 
         P_id_a__U12 P_id_U83 P_id_a__U94 P_id_a__U45 P_id_U31 P_id_a__isNat 
         P_id_isNatIListKind P_id_a__U82 P_id_U46 P_id_cons P_id_U52 
         P_id_a__U85 P_id_a__U33 P_id_U21 P_id_a__isNatList P_id_U86 
         P_id_mark P_id_a__U51 P_id_U41 P_id_tt P_id_U82 P_id_a__U93 
         P_id_a__U44 P_id_isNat P_id_a__U23 P_id_U94 P_id_U12 P_id_a__U81 
         P_id_U45 P_id_zeros P_id_U71 P_id_a__U91 P_id_a__U42 P_id_isNatKind 
         P_id_a__U22 P_id_U92 P_id_nil P_id_a__U61 P_id_U43 
         P_id_a__isNatIListKind P_id_U84 P_id_s P_id_a__U46 P_id_U32 
         P_id_a__U31 P_id_U13 P_id_a__U83 P_id_isNatIList P_id_A__U46 
         P_id_A__ISNATILISTKIND P_id_A__U61 P_id_A__U85 P_id_A__U31 
         P_id_A__U81 P_id_A__U22 P_id_A__U93 P_id_A__U42 P_id_A__U51 
         P_id_A__ISNATLIST P_id_A__ZEROS P_id_A__U91 P_id_A__U33 P_id_A__U83 
         P_id_A__U23 P_id_A__U11 P_id_A__LENGTH P_id_A__U44 
         P_id_A__ISNATILIST P_id_A__U13 P_id_A__U71 P_id_A__U86 P_id_A__U32 
         P_id_A__U82 P_id_A__ISNATKIND P_id_A__U94 P_id_A__U43 P_id_A__U52 
         P_id_A__U21 P_id_A__U92 P_id_A__U41 P_id_A__U84 P_id_A__ISNAT 
         P_id_A__U12 P_id_MARK P_id_A__U45.
      
      Lemma marked_measure_equation :
       forall t, 
        marked_measure t = match t with
                             | (algebra.Alg.Term algebra.F.id_a__U46 
                                (x11::nil)) =>
                              P_id_A__U46 (measure x11)
                             | (algebra.Alg.Term 
                                algebra.F.id_a__isNatIListKind (x11::nil)) =>
                              P_id_A__ISNATILISTKIND (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U61 
                                (x11::nil)) =>
                              P_id_A__U61 (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U85 (x12::
                                x11::nil)) =>
                              P_id_A__U85 (measure x12) (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U31 (x12::
                                x11::nil)) =>
                              P_id_A__U31 (measure x12) (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U81 (x13::
                                x12::x11::nil)) =>
                              P_id_A__U81 (measure x13) (measure x12) 
                               (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U22 (x12::
                                x11::nil)) =>
                              P_id_A__U22 (measure x12) (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U93 (x13::
                                x12::x11::nil)) =>
                              P_id_A__U93 (measure x13) (measure x12) 
                               (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U42 (x13::
                                x12::x11::nil)) =>
                              P_id_A__U42 (measure x13) (measure x12) 
                               (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U51 (x12::
                                x11::nil)) =>
                              P_id_A__U51 (measure x12) (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__isNatList 
                                (x11::nil)) =>
                              P_id_A__ISNATLIST (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__zeros nil) =>
                              P_id_A__ZEROS 
                             | (algebra.Alg.Term algebra.F.id_a__U91 (x13::
                                x12::x11::nil)) =>
                              P_id_A__U91 (measure x13) (measure x12) 
                               (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U33 
                                (x11::nil)) =>
                              P_id_A__U33 (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U83 (x13::
                                x12::x11::nil)) =>
                              P_id_A__U83 (measure x13) (measure x12) 
                               (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U23 
                                (x11::nil)) =>
                              P_id_A__U23 (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U11 (x12::
                                x11::nil)) =>
                              P_id_A__U11 (measure x12) (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__length 
                                (x11::nil)) =>
                              P_id_A__LENGTH (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U44 (x13::
                                x12::x11::nil)) =>
                              P_id_A__U44 (measure x13) (measure x12) 
                               (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__isNatIList 
                                (x11::nil)) =>
                              P_id_A__ISNATILIST (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U13 
                                (x11::nil)) =>
                              P_id_A__U13 (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U71 
                                (x11::nil)) =>
                              P_id_A__U71 (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U86 
                                (x11::nil)) =>
                              P_id_A__U86 (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U32 (x12::
                                x11::nil)) =>
                              P_id_A__U32 (measure x12) (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U82 (x13::
                                x12::x11::nil)) =>
                              P_id_A__U82 (measure x13) (measure x12) 
                               (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__isNatKind 
                                (x11::nil)) =>
                              P_id_A__ISNATKIND (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U94 (x12::
                                x11::nil)) =>
                              P_id_A__U94 (measure x12) (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U43 (x13::
                                x12::x11::nil)) =>
                              P_id_A__U43 (measure x13) (measure x12) 
                               (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U52 
                                (x11::nil)) =>
                              P_id_A__U52 (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U21 (x12::
                                x11::nil)) =>
                              P_id_A__U21 (measure x12) (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U92 (x13::
                                x12::x11::nil)) =>
                              P_id_A__U92 (measure x13) (measure x12) 
                               (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U41 (x13::
                                x12::x11::nil)) =>
                              P_id_A__U41 (measure x13) (measure x12) 
                               (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U84 (x13::
                                x12::x11::nil)) =>
                              P_id_A__U84 (measure x13) (measure x12) 
                               (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__isNat 
                                (x11::nil)) =>
                              P_id_A__ISNAT (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U12 (x12::
                                x11::nil)) =>
                              P_id_A__U12 (measure x12) (measure x11)
                             | (algebra.Alg.Term algebra.F.id_mark 
                                (x11::nil)) =>
                              P_id_MARK (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U45 (x12::
                                x11::nil)) =>
                              P_id_A__U45 (measure x12) (measure x11)
                             | _ => measure t
                             end.
      Proof.
        reflexivity .
      Qed.
      
      Lemma marked_measure_star_monotonic :
       forall f l1 l2, 
        (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step 
                                                          R_xml_0_deep_rew.R_xml_0_rules)
                                                         ) l1 l2) ->
         marked_measure (algebra.Alg.Term f l1) <= marked_measure (algebra.Alg.Term 
                                                                    f 
                                                                    l2).
      Proof.
        unfold marked_measure in *.
        apply InterpZ.marked_measure_star_monotonic.
        intros ;apply P_id_U51_monotonic;assumption.
        intros ;apply P_id_a__U84_monotonic;assumption.
        intros ;apply P_id_a__U32_monotonic;assumption.
        intros ;apply P_id_isNatList_monotonic;assumption.
        intros ;apply P_id_a__U13_monotonic;assumption.
        intros ;apply P_id_U85_monotonic;assumption.
        intros ;apply P_id_a__length_monotonic;assumption.
        intros ;apply P_id_a__isNatIList_monotonic;assumption.
        intros ;apply P_id_U33_monotonic;assumption.
        intros ;apply P_id_a__U11_monotonic;assumption.
        intros ;apply P_id_U81_monotonic;assumption.
        intros ;apply P_id_a__U92_monotonic;assumption.
        intros ;apply P_id_a__U43_monotonic;assumption.
        intros ;apply P_id_U23_monotonic;assumption.
        intros ;apply P_id_a__isNatKind_monotonic;assumption.
        intros ;apply P_id_U93_monotonic;assumption.
        intros ;apply P_id_U11_monotonic;assumption.
        intros ;apply P_id_a__U71_monotonic;assumption.
        intros ;apply P_id_U44_monotonic;assumption.
        intros ;apply P_id_U61_monotonic;assumption.
        intros ;apply P_id_a__U86_monotonic;assumption.
        intros ;apply P_id_a__U41_monotonic;assumption.
        intros ;apply P_id_U22_monotonic;assumption.
        intros ;apply P_id_a__U21_monotonic;assumption.
        intros ;apply P_id_U91_monotonic;assumption.
        intros ;apply P_id_length_monotonic;assumption.
        intros ;apply P_id_a__U52_monotonic;assumption.
        intros ;apply P_id_U42_monotonic;assumption.
        intros ;apply P_id_a__U12_monotonic;assumption.
        intros ;apply P_id_U83_monotonic;assumption.
        intros ;apply P_id_a__U94_monotonic;assumption.
        intros ;apply P_id_a__U45_monotonic;assumption.
        intros ;apply P_id_U31_monotonic;assumption.
        intros ;apply P_id_a__isNat_monotonic;assumption.
        intros ;apply P_id_isNatIListKind_monotonic;assumption.
        intros ;apply P_id_a__U82_monotonic;assumption.
        intros ;apply P_id_U46_monotonic;assumption.
        intros ;apply P_id_cons_monotonic;assumption.
        intros ;apply P_id_U52_monotonic;assumption.
        intros ;apply P_id_a__U85_monotonic;assumption.
        intros ;apply P_id_a__U33_monotonic;assumption.
        intros ;apply P_id_U21_monotonic;assumption.
        intros ;apply P_id_a__isNatList_monotonic;assumption.
        intros ;apply P_id_U86_monotonic;assumption.
        intros ;apply P_id_mark_monotonic;assumption.
        intros ;apply P_id_a__U51_monotonic;assumption.
        intros ;apply P_id_U41_monotonic;assumption.
        intros ;apply P_id_U82_monotonic;assumption.
        intros ;apply P_id_a__U93_monotonic;assumption.
        intros ;apply P_id_a__U44_monotonic;assumption.
        intros ;apply P_id_isNat_monotonic;assumption.
        intros ;apply P_id_a__U23_monotonic;assumption.
        intros ;apply P_id_U94_monotonic;assumption.
        intros ;apply P_id_U12_monotonic;assumption.
        intros ;apply P_id_a__U81_monotonic;assumption.
        intros ;apply P_id_U45_monotonic;assumption.
        intros ;apply P_id_U71_monotonic;assumption.
        intros ;apply P_id_a__U91_monotonic;assumption.
        intros ;apply P_id_a__U42_monotonic;assumption.
        intros ;apply P_id_isNatKind_monotonic;assumption.
        intros ;apply P_id_a__U22_monotonic;assumption.
        intros ;apply P_id_U92_monotonic;assumption.
        intros ;apply P_id_a__U61_monotonic;assumption.
        intros ;apply P_id_U43_monotonic;assumption.
        intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
        intros ;apply P_id_U84_monotonic;assumption.
        intros ;apply P_id_s_monotonic;assumption.
        intros ;apply P_id_a__U46_monotonic;assumption.
        intros ;apply P_id_U32_monotonic;assumption.
        intros ;apply P_id_a__U31_monotonic;assumption.
        intros ;apply P_id_U13_monotonic;assumption.
        intros ;apply P_id_a__U83_monotonic;assumption.
        intros ;apply P_id_isNatIList_monotonic;assumption.
        intros ;apply P_id_a__zeros_bounded;assumption.
        intros ;apply P_id_U51_bounded;assumption.
        intros ;apply P_id_a__U84_bounded;assumption.
        intros ;apply P_id_a__U32_bounded;assumption.
        intros ;apply P_id_isNatList_bounded;assumption.
        intros ;apply P_id_a__U13_bounded;assumption.
        intros ;apply P_id_U85_bounded;assumption.
        intros ;apply P_id_a__length_bounded;assumption.
        intros ;apply P_id_a__isNatIList_bounded;assumption.
        intros ;apply P_id_U33_bounded;assumption.
        intros ;apply P_id_a__U11_bounded;assumption.
        intros ;apply P_id_U81_bounded;assumption.
        intros ;apply P_id_a__U92_bounded;assumption.
        intros ;apply P_id_a__U43_bounded;assumption.
        intros ;apply P_id_U23_bounded;assumption.
        intros ;apply P_id_a__isNatKind_bounded;assumption.
        intros ;apply P_id_U93_bounded;assumption.
        intros ;apply P_id_U11_bounded;assumption.
        intros ;apply P_id_a__U71_bounded;assumption.
        intros ;apply P_id_U44_bounded;assumption.
        intros ;apply P_id_0_bounded;assumption.
        intros ;apply P_id_U61_bounded;assumption.
        intros ;apply P_id_a__U86_bounded;assumption.
        intros ;apply P_id_a__U41_bounded;assumption.
        intros ;apply P_id_U22_bounded;assumption.
        intros ;apply P_id_a__U21_bounded;assumption.
        intros ;apply P_id_U91_bounded;assumption.
        intros ;apply P_id_length_bounded;assumption.
        intros ;apply P_id_a__U52_bounded;assumption.
        intros ;apply P_id_U42_bounded;assumption.
        intros ;apply P_id_a__U12_bounded;assumption.
        intros ;apply P_id_U83_bounded;assumption.
        intros ;apply P_id_a__U94_bounded;assumption.
        intros ;apply P_id_a__U45_bounded;assumption.
        intros ;apply P_id_U31_bounded;assumption.
        intros ;apply P_id_a__isNat_bounded;assumption.
        intros ;apply P_id_isNatIListKind_bounded;assumption.
        intros ;apply P_id_a__U82_bounded;assumption.
        intros ;apply P_id_U46_bounded;assumption.
        intros ;apply P_id_cons_bounded;assumption.
        intros ;apply P_id_U52_bounded;assumption.
        intros ;apply P_id_a__U85_bounded;assumption.
        intros ;apply P_id_a__U33_bounded;assumption.
        intros ;apply P_id_U21_bounded;assumption.
        intros ;apply P_id_a__isNatList_bounded;assumption.
        intros ;apply P_id_U86_bounded;assumption.
        intros ;apply P_id_mark_bounded;assumption.
        intros ;apply P_id_a__U51_bounded;assumption.
        intros ;apply P_id_U41_bounded;assumption.
        intros ;apply P_id_tt_bounded;assumption.
        intros ;apply P_id_U82_bounded;assumption.
        intros ;apply P_id_a__U93_bounded;assumption.
        intros ;apply P_id_a__U44_bounded;assumption.
        intros ;apply P_id_isNat_bounded;assumption.
        intros ;apply P_id_a__U23_bounded;assumption.
        intros ;apply P_id_U94_bounded;assumption.
        intros ;apply P_id_U12_bounded;assumption.
        intros ;apply P_id_a__U81_bounded;assumption.
        intros ;apply P_id_U45_bounded;assumption.
        intros ;apply P_id_zeros_bounded;assumption.
        intros ;apply P_id_U71_bounded;assumption.
        intros ;apply P_id_a__U91_bounded;assumption.
        intros ;apply P_id_a__U42_bounded;assumption.
        intros ;apply P_id_isNatKind_bounded;assumption.
        intros ;apply P_id_a__U22_bounded;assumption.
        intros ;apply P_id_U92_bounded;assumption.
        intros ;apply P_id_nil_bounded;assumption.
        intros ;apply P_id_a__U61_bounded;assumption.
        intros ;apply P_id_U43_bounded;assumption.
        intros ;apply P_id_a__isNatIListKind_bounded;assumption.
        intros ;apply P_id_U84_bounded;assumption.
        intros ;apply P_id_s_bounded;assumption.
        intros ;apply P_id_a__U46_bounded;assumption.
        intros ;apply P_id_U32_bounded;assumption.
        intros ;apply P_id_a__U31_bounded;assumption.
        intros ;apply P_id_U13_bounded;assumption.
        intros ;apply P_id_a__U83_bounded;assumption.
        intros ;apply P_id_isNatIList_bounded;assumption.
        apply rules_monotonic.
        intros ;apply P_id_A__U46_monotonic;assumption.
        intros ;apply P_id_A__ISNATILISTKIND_monotonic;assumption.
        intros ;apply P_id_A__U61_monotonic;assumption.
        intros ;apply P_id_A__U85_monotonic;assumption.
        intros ;apply P_id_A__U31_monotonic;assumption.
        intros ;apply P_id_A__U81_monotonic;assumption.
        intros ;apply P_id_A__U22_monotonic;assumption.
        intros ;apply P_id_A__U93_monotonic;assumption.
        intros ;apply P_id_A__U42_monotonic;assumption.
        intros ;apply P_id_A__U51_monotonic;assumption.
        intros ;apply P_id_A__ISNATLIST_monotonic;assumption.
        intros ;apply P_id_A__U91_monotonic;assumption.
        intros ;apply P_id_A__U33_monotonic;assumption.
        intros ;apply P_id_A__U83_monotonic;assumption.
        intros ;apply P_id_A__U23_monotonic;assumption.
        intros ;apply P_id_A__U11_monotonic;assumption.
        intros ;apply P_id_A__LENGTH_monotonic;assumption.
        intros ;apply P_id_A__U44_monotonic;assumption.
        intros ;apply P_id_A__ISNATILIST_monotonic;assumption.
        intros ;apply P_id_A__U13_monotonic;assumption.
        intros ;apply P_id_A__U71_monotonic;assumption.
        intros ;apply P_id_A__U86_monotonic;assumption.
        intros ;apply P_id_A__U32_monotonic;assumption.
        intros ;apply P_id_A__U82_monotonic;assumption.
        intros ;apply P_id_A__ISNATKIND_monotonic;assumption.
        intros ;apply P_id_A__U94_monotonic;assumption.
        intros ;apply P_id_A__U43_monotonic;assumption.
        intros ;apply P_id_A__U52_monotonic;assumption.
        intros ;apply P_id_A__U21_monotonic;assumption.
        intros ;apply P_id_A__U92_monotonic;assumption.
        intros ;apply P_id_A__U41_monotonic;assumption.
        intros ;apply P_id_A__U84_monotonic;assumption.
        intros ;apply P_id_A__ISNAT_monotonic;assumption.
        intros ;apply P_id_A__U12_monotonic;assumption.
        intros ;apply P_id_MARK_monotonic;assumption.
        intros ;apply P_id_A__U45_monotonic;assumption.
      Qed.
      
      Ltac rewrite_and_unfold  :=
       do 2 (rewrite marked_measure_equation);
        repeat (
        match goal with
          |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
           rewrite (measure_equation (algebra.Alg.Term f t))
          | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ =>
           rewrite (measure_equation (algebra.Alg.Term f t)) in H|-
          end
        ).
      
      Definition lt a b := (Zwf.Zwf 0) (marked_measure a) (marked_measure b).
      
      Definition le a b := marked_measure a <= marked_measure b.
      
      Lemma lt_le_compat : forall a b c, (lt a b) ->(le b c) ->lt a c.
      Proof.
        unfold lt, le in *.
        intros a b c.
        apply (interp.le_lt_compat_right (interp.o_Z 0)).
      Qed.
      
      Lemma wf_lt : well_founded lt.
      Proof.
        unfold lt in *.
        apply Inverse_Image.wf_inverse_image with  (B:=Z).
        apply Zwf.Zwf_well_founded.
      Qed.
      
      Lemma DP_R_xml_0_scc_65_large_scc_1_large_strict_in_lt :
       Relation_Definitions.inclusion _ 
        DP_R_xml_0_scc_65_large_scc_1_large_strict lt.
      Proof.
        unfold Relation_Definitions.inclusion, lt in *.
        
        intros a b H;destruct H;
         match goal with
           |  |- (Zwf.Zwf 0) _ (marked_measure (algebra.Alg.Term ?f ?l)) =>
            let l'' := algebra.Alg_ext.find_replacement l  in 
             ((apply (interp.le_lt_compat_right (interp.o_Z 0)) with
                (marked_measure (algebra.Alg.Term f l''));[idtac|
               apply marked_measure_star_monotonic;
                repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos);
                (assumption)||(constructor 1)]))
           end
         ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma DP_R_xml_0_scc_65_large_scc_1_large_large_in_le :
       Relation_Definitions.inclusion _ 
        DP_R_xml_0_scc_65_large_scc_1_large_large le.
      Proof.
        unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *.
        
        intros a b H;destruct H;
         match goal with
           |  |- _ <= marked_measure (algebra.Alg.Term ?f ?l) =>
            let l'' := algebra.Alg_ext.find_replacement l  in 
             ((apply (interp.le_trans (interp.o_Z 0)) with
                (marked_measure (algebra.Alg.Term f l''));[idtac|
               apply marked_measure_star_monotonic;
                repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos);
                (assumption)||(constructor 1)]))
           end
         ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Definition wf_DP_R_xml_0_scc_65_large_scc_1_large_large  := 
        WF_DP_R_xml_0_scc_65_large_scc_1_large_large.wf.
      
      
      Lemma wf :
       well_founded WF_DP_R_xml_0_scc_65_large_scc_1.DP_R_xml_0_scc_65_large_scc_1_large
        .
      Proof.
        intros x.
        apply (well_founded_ind wf_lt).
        clear x.
        intros x.
        pattern x.
        apply (@Acc_ind _ DP_R_xml_0_scc_65_large_scc_1_large_large).
        clear x.
        intros x _ IHx IHx'.
        constructor.
        intros y H.
        
        destruct H;
         (apply IHx';apply DP_R_xml_0_scc_65_large_scc_1_large_strict_in_lt;
           econstructor eassumption)||
         ((apply IHx;[econstructor eassumption|
           intros y' Hlt;apply IHx';apply lt_le_compat with (1:=Hlt) ;
            apply DP_R_xml_0_scc_65_large_scc_1_large_large_in_le;
            econstructor eassumption])).
        apply wf_DP_R_xml_0_scc_65_large_scc_1_large_large.
      Qed.
     End WF_DP_R_xml_0_scc_65_large_scc_1_large.
     
     Open Scope Z_scope.
     
     Import ring_extention.
     
     Notation Local "a <= b" := (Zle a b).
     
     Notation Local "a < b" := (Zlt a b).
     
     Definition P_id_a__zeros  := 1.
     
     Definition P_id_U51 (x11:Z) (x12:Z) := 2* x11.
     
     Definition P_id_a__U84 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
     
     Definition P_id_a__U32 (x11:Z) (x12:Z) := 1* x11.
     
     Definition P_id_isNatList (x11:Z) := 0.
     
     Definition P_id_a__U13 (x11:Z) := 1* x11.
     
     Definition P_id_U85 (x11:Z) (x12:Z) := 1* x11.
     
     Definition P_id_a__length (x11:Z) := 0.
     
     Definition P_id_a__isNatIList (x11:Z) := 0.
     
     Definition P_id_U33 (x11:Z) := 2* x11.
     
     Definition P_id_a__U11 (x11:Z) (x12:Z) := 1* x11.
     
     Definition P_id_U81 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
     
     Definition P_id_a__U92 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_a__U43 (x11:Z) (x12:Z) (x13:Z) := 2* x11.
     
     Definition P_id_U23 (x11:Z) := 1* x11.
     
     Definition P_id_a__isNatKind (x11:Z) := 0.
     
     Definition P_id_U93 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_U11 (x11:Z) (x12:Z) := 1* x11.
     
     Definition P_id_a__U71 (x11:Z) := 1* x11.
     
     Definition P_id_U44 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
     
     Definition P_id_0  := 0.
     
     Definition P_id_U61 (x11:Z) := 1* x11.
     
     Definition P_id_a__U86 (x11:Z) := 1* x11.
     
     Definition P_id_a__U41 (x11:Z) (x12:Z) (x13:Z) := 2* x11.
     
     Definition P_id_U22 (x11:Z) (x12:Z) := 1* x11.
     
     Definition P_id_a__U21 (x11:Z) (x12:Z) := 1* x11.
     
     Definition P_id_U91 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_length (x11:Z) := 0.
     
     Definition P_id_a__U52 (x11:Z) := 2* x11.
     
     Definition P_id_U42 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
     
     Definition P_id_a__U12 (x11:Z) (x12:Z) := 1* x11.
     
     Definition P_id_U83 (x11:Z) (x12:Z) (x13:Z) := 2* x11.
     
     Definition P_id_a__U94 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_a__U45 (x11:Z) (x12:Z) := 1* x11.
     
     Definition P_id_U31 (x11:Z) (x12:Z) := 2* x11.
     
     Definition P_id_a__isNat (x11:Z) := 0.
     
     Definition P_id_isNatIListKind (x11:Z) := 0.
     
     Definition P_id_a__U82 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
     
     Definition P_id_U46 (x11:Z) := 2* x11.
     
     Definition P_id_cons (x11:Z) (x12:Z) := 1 + 2* x11.
     
     Definition P_id_U52 (x11:Z) := 2* x11.
     
     Definition P_id_a__U85 (x11:Z) (x12:Z) := 1* x11.
     
     Definition P_id_a__U33 (x11:Z) := 2* x11.
     
     Definition P_id_U21 (x11:Z) (x12:Z) := 1* x11.
     
     Definition P_id_a__isNatList (x11:Z) := 0.
     
     Definition P_id_U86 (x11:Z) := 1* x11.
     
     Definition P_id_mark (x11:Z) := 2* x11.
     
     Definition P_id_a__U51 (x11:Z) (x12:Z) := 2* x11.
     
     Definition P_id_U41 (x11:Z) (x12:Z) (x13:Z) := 2* x11.
     
     Definition P_id_tt  := 0.
     
     Definition P_id_U82 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
     
     Definition P_id_a__U93 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_a__U44 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
     
     Definition P_id_isNat (x11:Z) := 0.
     
     Definition P_id_a__U23 (x11:Z) := 1* x11.
     
     Definition P_id_U94 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_U12 (x11:Z) (x12:Z) := 1* x11.
     
     Definition P_id_a__U81 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
     
     Definition P_id_U45 (x11:Z) (x12:Z) := 1* x11.
     
     Definition P_id_zeros  := 1.
     
     Definition P_id_U71 (x11:Z) := 1* x11.
     
     Definition P_id_a__U91 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_a__U42 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
     
     Definition P_id_isNatKind (x11:Z) := 0.
     
     Definition P_id_a__U22 (x11:Z) (x12:Z) := 1* x11.
     
     Definition P_id_U92 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_nil  := 2.
     
     Definition P_id_a__U61 (x11:Z) := 1* x11.
     
     Definition P_id_U43 (x11:Z) (x12:Z) (x13:Z) := 2* x11.
     
     Definition P_id_a__isNatIListKind (x11:Z) := 0.
     
     Definition P_id_U84 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
     
     Definition P_id_s (x11:Z) := 1* x11.
     
     Definition P_id_a__U46 (x11:Z) := 2* x11.
     
     Definition P_id_U32 (x11:Z) (x12:Z) := 1* x11.
     
     Definition P_id_a__U31 (x11:Z) (x12:Z) := 2* x11.
     
     Definition P_id_U13 (x11:Z) := 1* x11.
     
     Definition P_id_a__U83 (x11:Z) (x12:Z) (x13:Z) := 2* x11.
     
     Definition P_id_isNatIList (x11:Z) := 0.
     
     Lemma P_id_U51_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U51 x12 x14 <= P_id_U51 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U84_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U84 x12 x14 x16 <= P_id_a__U84 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U32_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U32 x12 x14 <= P_id_a__U32 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNatList_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_isNatList x12 <= P_id_isNatList x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U13_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U13 x12 <= P_id_a__U13 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U85_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U85 x12 x14 <= P_id_U85 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__length_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__length x12 <= P_id_a__length x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNatIList_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__isNatIList x12 <= P_id_a__isNatIList x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U33_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U33 x12 <= P_id_U33 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U11_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U11 x12 x14 <= P_id_a__U11 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U81_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U81 x12 x14 x16 <= P_id_U81 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U92_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U92 x12 x14 x16 <= P_id_a__U92 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U43_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U43 x12 x14 x16 <= P_id_a__U43 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U23_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U23 x12 <= P_id_U23 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNatKind_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__isNatKind x12 <= P_id_a__isNatKind x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U93_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U93 x12 x14 x16 <= P_id_U93 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U11_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U11 x12 x14 <= P_id_U11 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U71_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U71 x12 <= P_id_a__U71 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U44_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U44 x12 x14 x16 <= P_id_U44 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U61_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U61 x12 <= P_id_U61 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U86_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U86 x12 <= P_id_a__U86 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U41_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U41 x12 x14 x16 <= P_id_a__U41 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U22_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U22 x12 x14 <= P_id_U22 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U21_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U21 x12 x14 <= P_id_a__U21 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U91_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U91 x12 x14 x16 <= P_id_U91 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_length_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_length x12 <= P_id_length x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U52_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U52 x12 <= P_id_a__U52 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U42_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U42 x12 x14 x16 <= P_id_U42 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U12_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U12 x12 x14 <= P_id_a__U12 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U83_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U83 x12 x14 x16 <= P_id_U83 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U94_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U94 x12 x14 <= P_id_a__U94 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U45_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U45 x12 x14 <= P_id_a__U45 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U31_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U31 x12 x14 <= P_id_U31 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNat_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__isNat x12 <= P_id_a__isNat x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNatIListKind_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_isNatIListKind x12 <= P_id_isNatIListKind x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U82_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U82 x12 x14 x16 <= P_id_a__U82 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U46_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U46 x12 <= P_id_U46 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_cons_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_cons x12 x14 <= P_id_cons x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U52_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U52 x12 <= P_id_U52 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U85_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U85 x12 x14 <= P_id_a__U85 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U33_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U33 x12 <= P_id_a__U33 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U21_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U21 x12 x14 <= P_id_U21 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNatList_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__isNatList x12 <= P_id_a__isNatList x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U86_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U86 x12 <= P_id_U86 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_mark_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_mark x12 <= P_id_mark x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U51_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U51 x12 x14 <= P_id_a__U51 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U41_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U41 x12 x14 x16 <= P_id_U41 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U82_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U82 x12 x14 x16 <= P_id_U82 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U93_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U93 x12 x14 x16 <= P_id_a__U93 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U44_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U44 x12 x14 x16 <= P_id_a__U44 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNat_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_isNat x12 <= P_id_isNat x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U23_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U23 x12 <= P_id_a__U23 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U94_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U94 x12 x14 <= P_id_U94 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U12_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U12 x12 x14 <= P_id_U12 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U81_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U81 x12 x14 x16 <= P_id_a__U81 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U45_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U45 x12 x14 <= P_id_U45 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U71_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U71 x12 <= P_id_U71 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U91_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U91 x12 x14 x16 <= P_id_a__U91 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U42_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U42 x12 x14 x16 <= P_id_a__U42 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNatKind_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_isNatKind x12 <= P_id_isNatKind x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U22_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U22 x12 x14 <= P_id_a__U22 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U92_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U92 x12 x14 x16 <= P_id_U92 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U61_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U61 x12 <= P_id_a__U61 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U43_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U43 x12 x14 x16 <= P_id_U43 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNatIListKind_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__isNatIListKind x12 <= P_id_a__isNatIListKind x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U84_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U84 x12 x14 x16 <= P_id_U84 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_s_monotonic :
      forall x12 x11, (0 <= x12)/\ (x12 <= x11) ->P_id_s x12 <= P_id_s x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U46_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U46 x12 <= P_id_a__U46 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U32_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U32 x12 x14 <= P_id_U32 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U31_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U31 x12 x14 <= P_id_a__U31 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U13_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U13 x12 <= P_id_U13 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U83_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U83 x12 x14 x16 <= P_id_a__U83 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNatIList_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_isNatIList x12 <= P_id_isNatIList x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__zeros_bounded : 0 <= P_id_a__zeros .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U51_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U51 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U84_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U84 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U32_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U32 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNatList_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_isNatList x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U13_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U13 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U85_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U85 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__length_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__length x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNatIList_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__isNatIList x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U33_bounded : forall x11, (0 <= x11) ->0 <= P_id_U33 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U11_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U11 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U81_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U81 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U92_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U92 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U43_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U43 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U23_bounded : forall x11, (0 <= x11) ->0 <= P_id_U23 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNatKind_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__isNatKind x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U93_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U93 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U11_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U11 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U71_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U71 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U44_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U44 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_0_bounded : 0 <= P_id_0 .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U61_bounded : forall x11, (0 <= x11) ->0 <= P_id_U61 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U86_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U86 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U41_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U41 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U22_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U22 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U21_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U21 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U91_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U91 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_length_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_length x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U52_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U52 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U42_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U42 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U12_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U12 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U83_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U83 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U94_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U94 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U45_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U45 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U31_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U31 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNat_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__isNat x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNatIListKind_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_isNatIListKind x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U82_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U82 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U46_bounded : forall x11, (0 <= x11) ->0 <= P_id_U46 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_cons_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_cons x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U52_bounded : forall x11, (0 <= x11) ->0 <= P_id_U52 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U85_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U85 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U33_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U33 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U21_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U21 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNatList_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__isNatList x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U86_bounded : forall x11, (0 <= x11) ->0 <= P_id_U86 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_mark_bounded : forall x11, (0 <= x11) ->0 <= P_id_mark x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U51_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U51 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U41_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U41 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_tt_bounded : 0 <= P_id_tt .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U82_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U82 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U93_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U93 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U44_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U44 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNat_bounded : forall x11, (0 <= x11) ->0 <= P_id_isNat x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U23_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U23 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U94_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U94 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U12_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U12 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U81_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U81 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U45_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U45 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_zeros_bounded : 0 <= P_id_zeros .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U71_bounded : forall x11, (0 <= x11) ->0 <= P_id_U71 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U91_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U91 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U42_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U42 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNatKind_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_isNatKind x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U22_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U22 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U92_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U92 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_nil_bounded : 0 <= P_id_nil .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U61_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U61 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U43_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U43 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNatIListKind_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__isNatIListKind x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U84_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U84 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_s_bounded : forall x11, (0 <= x11) ->0 <= P_id_s x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U46_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U46 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U32_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U32 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U31_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U31 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U13_bounded : forall x11, (0 <= x11) ->0 <= P_id_U13 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U83_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U83 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNatIList_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_isNatIList x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Definition measure  := 
       InterpZ.measure 0 P_id_a__zeros P_id_U51 P_id_a__U84 P_id_a__U32 
        P_id_isNatList P_id_a__U13 P_id_U85 P_id_a__length 
        P_id_a__isNatIList P_id_U33 P_id_a__U11 P_id_U81 P_id_a__U92 
        P_id_a__U43 P_id_U23 P_id_a__isNatKind P_id_U93 P_id_U11 P_id_a__U71 
        P_id_U44 P_id_0 P_id_U61 P_id_a__U86 P_id_a__U41 P_id_U22 
        P_id_a__U21 P_id_U91 P_id_length P_id_a__U52 P_id_U42 P_id_a__U12 
        P_id_U83 P_id_a__U94 P_id_a__U45 P_id_U31 P_id_a__isNat 
        P_id_isNatIListKind P_id_a__U82 P_id_U46 P_id_cons P_id_U52 
        P_id_a__U85 P_id_a__U33 P_id_U21 P_id_a__isNatList P_id_U86 
        P_id_mark P_id_a__U51 P_id_U41 P_id_tt P_id_U82 P_id_a__U93 
        P_id_a__U44 P_id_isNat P_id_a__U23 P_id_U94 P_id_U12 P_id_a__U81 
        P_id_U45 P_id_zeros P_id_U71 P_id_a__U91 P_id_a__U42 P_id_isNatKind 
        P_id_a__U22 P_id_U92 P_id_nil P_id_a__U61 P_id_U43 
        P_id_a__isNatIListKind P_id_U84 P_id_s P_id_a__U46 P_id_U32 
        P_id_a__U31 P_id_U13 P_id_a__U83 P_id_isNatIList.
     
     Lemma measure_equation :
      forall t, 
       measure t = match t with
                     | (algebra.Alg.Term algebra.F.id_a__zeros nil) =>
                      P_id_a__zeros 
                     | (algebra.Alg.Term algebra.F.id_U51 (x12::x11::nil)) =>
                      P_id_U51 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U84 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U84 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U32 (x12::x11::nil)) =>
                      P_id_a__U32 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_isNatList (x11::nil)) =>
                      P_id_isNatList (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U13 (x11::nil)) =>
                      P_id_a__U13 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U85 (x12::x11::nil)) =>
                      P_id_U85 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__length (x11::nil)) =>
                      P_id_a__length (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__isNatIList 
                        (x11::nil)) =>
                      P_id_a__isNatIList (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U33 (x11::nil)) =>
                      P_id_U33 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U11 (x12::x11::nil)) =>
                      P_id_a__U11 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U81 (x13::x12::
                        x11::nil)) =>
                      P_id_U81 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U92 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U92 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U43 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U43 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U23 (x11::nil)) =>
                      P_id_U23 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__isNatKind 
                        (x11::nil)) =>
                      P_id_a__isNatKind (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U93 (x13::x12::
                        x11::nil)) =>
                      P_id_U93 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U11 (x12::x11::nil)) =>
                      P_id_U11 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U71 (x11::nil)) =>
                      P_id_a__U71 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U44 (x13::x12::
                        x11::nil)) =>
                      P_id_U44 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_0 nil) => P_id_0 
                     | (algebra.Alg.Term algebra.F.id_U61 (x11::nil)) =>
                      P_id_U61 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U86 (x11::nil)) =>
                      P_id_a__U86 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U41 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U41 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U22 (x12::x11::nil)) =>
                      P_id_U22 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U21 (x12::x11::nil)) =>
                      P_id_a__U21 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U91 (x13::x12::
                        x11::nil)) =>
                      P_id_U91 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_length (x11::nil)) =>
                      P_id_length (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U52 (x11::nil)) =>
                      P_id_a__U52 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U42 (x13::x12::
                        x11::nil)) =>
                      P_id_U42 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U12 (x12::x11::nil)) =>
                      P_id_a__U12 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U83 (x13::x12::
                        x11::nil)) =>
                      P_id_U83 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U94 (x12::x11::nil)) =>
                      P_id_a__U94 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U45 (x12::x11::nil)) =>
                      P_id_a__U45 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U31 (x12::x11::nil)) =>
                      P_id_U31 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__isNat (x11::nil)) =>
                      P_id_a__isNat (measure x11)
                     | (algebra.Alg.Term algebra.F.id_isNatIListKind 
                        (x11::nil)) =>
                      P_id_isNatIListKind (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U82 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U82 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U46 (x11::nil)) =>
                      P_id_U46 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_cons (x12::x11::nil)) =>
                      P_id_cons (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U52 (x11::nil)) =>
                      P_id_U52 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U85 (x12::x11::nil)) =>
                      P_id_a__U85 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U33 (x11::nil)) =>
                      P_id_a__U33 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U21 (x12::x11::nil)) =>
                      P_id_U21 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__isNatList 
                        (x11::nil)) =>
                      P_id_a__isNatList (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U86 (x11::nil)) =>
                      P_id_U86 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_mark (x11::nil)) =>
                      P_id_mark (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U51 (x12::x11::nil)) =>
                      P_id_a__U51 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U41 (x13::x12::
                        x11::nil)) =>
                      P_id_U41 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt 
                     | (algebra.Alg.Term algebra.F.id_U82 (x13::x12::
                        x11::nil)) =>
                      P_id_U82 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U93 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U93 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U44 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U44 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_isNat (x11::nil)) =>
                      P_id_isNat (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U23 (x11::nil)) =>
                      P_id_a__U23 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U94 (x12::x11::nil)) =>
                      P_id_U94 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U12 (x12::x11::nil)) =>
                      P_id_U12 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U81 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U81 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U45 (x12::x11::nil)) =>
                      P_id_U45 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_zeros nil) =>
                      P_id_zeros 
                     | (algebra.Alg.Term algebra.F.id_U71 (x11::nil)) =>
                      P_id_U71 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U91 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U91 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U42 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U42 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_isNatKind (x11::nil)) =>
                      P_id_isNatKind (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U22 (x12::x11::nil)) =>
                      P_id_a__U22 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U92 (x13::x12::
                        x11::nil)) =>
                      P_id_U92 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil 
                     | (algebra.Alg.Term algebra.F.id_a__U61 (x11::nil)) =>
                      P_id_a__U61 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U43 (x13::x12::
                        x11::nil)) =>
                      P_id_U43 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
                        (x11::nil)) =>
                      P_id_a__isNatIListKind (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U84 (x13::x12::
                        x11::nil)) =>
                      P_id_U84 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_s (x11::nil)) =>
                      P_id_s (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U46 (x11::nil)) =>
                      P_id_a__U46 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U32 (x12::x11::nil)) =>
                      P_id_U32 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U31 (x12::x11::nil)) =>
                      P_id_a__U31 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U13 (x11::nil)) =>
                      P_id_U13 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U83 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U83 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_isNatIList (x11::nil)) =>
                      P_id_isNatIList (measure x11)
                     | _ => 0
                     end.
     Proof.
       intros t;case t;intros ;apply refl_equal.
     Qed.
     
     Lemma measure_bounded : forall t, 0 <= measure t.
     Proof.
       unfold measure in |-*.
       
       apply InterpZ.measure_bounded;
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Ltac generate_pos_hyp  :=
      match goal with
        | H:context [measure ?x] |- _ =>
         let v := fresh "v" in 
          (let H := fresh "h" in 
            (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
              clearbody H;clearbody v))
        |  |- context [measure ?x] =>
         let v := fresh "v" in 
          (let H := fresh "h" in 
            (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
              clearbody H;clearbody v))
        end
      .
     
     Lemma rules_monotonic :
      forall l r, 
       (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) ->
        measure r <= measure l.
     Proof.
       intros l r H.
       fold measure in |-*.
       
       inversion H;clear H;subst;inversion H0;clear H0;subst;
        simpl algebra.EQT.T.apply_subst in |-*;
        repeat (
        match goal with
          |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
           rewrite (measure_equation (algebra.Alg.Term f t))
          end
        );repeat (generate_pos_hyp );
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma measure_star_monotonic :
      forall l r, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  r l) ->measure r <= measure l.
     Proof.
       unfold measure in *.
       apply InterpZ.measure_star_monotonic.
       intros ;apply P_id_U51_monotonic;assumption.
       intros ;apply P_id_a__U84_monotonic;assumption.
       intros ;apply P_id_a__U32_monotonic;assumption.
       intros ;apply P_id_isNatList_monotonic;assumption.
       intros ;apply P_id_a__U13_monotonic;assumption.
       intros ;apply P_id_U85_monotonic;assumption.
       intros ;apply P_id_a__length_monotonic;assumption.
       intros ;apply P_id_a__isNatIList_monotonic;assumption.
       intros ;apply P_id_U33_monotonic;assumption.
       intros ;apply P_id_a__U11_monotonic;assumption.
       intros ;apply P_id_U81_monotonic;assumption.
       intros ;apply P_id_a__U92_monotonic;assumption.
       intros ;apply P_id_a__U43_monotonic;assumption.
       intros ;apply P_id_U23_monotonic;assumption.
       intros ;apply P_id_a__isNatKind_monotonic;assumption.
       intros ;apply P_id_U93_monotonic;assumption.
       intros ;apply P_id_U11_monotonic;assumption.
       intros ;apply P_id_a__U71_monotonic;assumption.
       intros ;apply P_id_U44_monotonic;assumption.
       intros ;apply P_id_U61_monotonic;assumption.
       intros ;apply P_id_a__U86_monotonic;assumption.
       intros ;apply P_id_a__U41_monotonic;assumption.
       intros ;apply P_id_U22_monotonic;assumption.
       intros ;apply P_id_a__U21_monotonic;assumption.
       intros ;apply P_id_U91_monotonic;assumption.
       intros ;apply P_id_length_monotonic;assumption.
       intros ;apply P_id_a__U52_monotonic;assumption.
       intros ;apply P_id_U42_monotonic;assumption.
       intros ;apply P_id_a__U12_monotonic;assumption.
       intros ;apply P_id_U83_monotonic;assumption.
       intros ;apply P_id_a__U94_monotonic;assumption.
       intros ;apply P_id_a__U45_monotonic;assumption.
       intros ;apply P_id_U31_monotonic;assumption.
       intros ;apply P_id_a__isNat_monotonic;assumption.
       intros ;apply P_id_isNatIListKind_monotonic;assumption.
       intros ;apply P_id_a__U82_monotonic;assumption.
       intros ;apply P_id_U46_monotonic;assumption.
       intros ;apply P_id_cons_monotonic;assumption.
       intros ;apply P_id_U52_monotonic;assumption.
       intros ;apply P_id_a__U85_monotonic;assumption.
       intros ;apply P_id_a__U33_monotonic;assumption.
       intros ;apply P_id_U21_monotonic;assumption.
       intros ;apply P_id_a__isNatList_monotonic;assumption.
       intros ;apply P_id_U86_monotonic;assumption.
       intros ;apply P_id_mark_monotonic;assumption.
       intros ;apply P_id_a__U51_monotonic;assumption.
       intros ;apply P_id_U41_monotonic;assumption.
       intros ;apply P_id_U82_monotonic;assumption.
       intros ;apply P_id_a__U93_monotonic;assumption.
       intros ;apply P_id_a__U44_monotonic;assumption.
       intros ;apply P_id_isNat_monotonic;assumption.
       intros ;apply P_id_a__U23_monotonic;assumption.
       intros ;apply P_id_U94_monotonic;assumption.
       intros ;apply P_id_U12_monotonic;assumption.
       intros ;apply P_id_a__U81_monotonic;assumption.
       intros ;apply P_id_U45_monotonic;assumption.
       intros ;apply P_id_U71_monotonic;assumption.
       intros ;apply P_id_a__U91_monotonic;assumption.
       intros ;apply P_id_a__U42_monotonic;assumption.
       intros ;apply P_id_isNatKind_monotonic;assumption.
       intros ;apply P_id_a__U22_monotonic;assumption.
       intros ;apply P_id_U92_monotonic;assumption.
       intros ;apply P_id_a__U61_monotonic;assumption.
       intros ;apply P_id_U43_monotonic;assumption.
       intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
       intros ;apply P_id_U84_monotonic;assumption.
       intros ;apply P_id_s_monotonic;assumption.
       intros ;apply P_id_a__U46_monotonic;assumption.
       intros ;apply P_id_U32_monotonic;assumption.
       intros ;apply P_id_a__U31_monotonic;assumption.
       intros ;apply P_id_U13_monotonic;assumption.
       intros ;apply P_id_a__U83_monotonic;assumption.
       intros ;apply P_id_isNatIList_monotonic;assumption.
       intros ;apply P_id_a__zeros_bounded;assumption.
       intros ;apply P_id_U51_bounded;assumption.
       intros ;apply P_id_a__U84_bounded;assumption.
       intros ;apply P_id_a__U32_bounded;assumption.
       intros ;apply P_id_isNatList_bounded;assumption.
       intros ;apply P_id_a__U13_bounded;assumption.
       intros ;apply P_id_U85_bounded;assumption.
       intros ;apply P_id_a__length_bounded;assumption.
       intros ;apply P_id_a__isNatIList_bounded;assumption.
       intros ;apply P_id_U33_bounded;assumption.
       intros ;apply P_id_a__U11_bounded;assumption.
       intros ;apply P_id_U81_bounded;assumption.
       intros ;apply P_id_a__U92_bounded;assumption.
       intros ;apply P_id_a__U43_bounded;assumption.
       intros ;apply P_id_U23_bounded;assumption.
       intros ;apply P_id_a__isNatKind_bounded;assumption.
       intros ;apply P_id_U93_bounded;assumption.
       intros ;apply P_id_U11_bounded;assumption.
       intros ;apply P_id_a__U71_bounded;assumption.
       intros ;apply P_id_U44_bounded;assumption.
       intros ;apply P_id_0_bounded;assumption.
       intros ;apply P_id_U61_bounded;assumption.
       intros ;apply P_id_a__U86_bounded;assumption.
       intros ;apply P_id_a__U41_bounded;assumption.
       intros ;apply P_id_U22_bounded;assumption.
       intros ;apply P_id_a__U21_bounded;assumption.
       intros ;apply P_id_U91_bounded;assumption.
       intros ;apply P_id_length_bounded;assumption.
       intros ;apply P_id_a__U52_bounded;assumption.
       intros ;apply P_id_U42_bounded;assumption.
       intros ;apply P_id_a__U12_bounded;assumption.
       intros ;apply P_id_U83_bounded;assumption.
       intros ;apply P_id_a__U94_bounded;assumption.
       intros ;apply P_id_a__U45_bounded;assumption.
       intros ;apply P_id_U31_bounded;assumption.
       intros ;apply P_id_a__isNat_bounded;assumption.
       intros ;apply P_id_isNatIListKind_bounded;assumption.
       intros ;apply P_id_a__U82_bounded;assumption.
       intros ;apply P_id_U46_bounded;assumption.
       intros ;apply P_id_cons_bounded;assumption.
       intros ;apply P_id_U52_bounded;assumption.
       intros ;apply P_id_a__U85_bounded;assumption.
       intros ;apply P_id_a__U33_bounded;assumption.
       intros ;apply P_id_U21_bounded;assumption.
       intros ;apply P_id_a__isNatList_bounded;assumption.
       intros ;apply P_id_U86_bounded;assumption.
       intros ;apply P_id_mark_bounded;assumption.
       intros ;apply P_id_a__U51_bounded;assumption.
       intros ;apply P_id_U41_bounded;assumption.
       intros ;apply P_id_tt_bounded;assumption.
       intros ;apply P_id_U82_bounded;assumption.
       intros ;apply P_id_a__U93_bounded;assumption.
       intros ;apply P_id_a__U44_bounded;assumption.
       intros ;apply P_id_isNat_bounded;assumption.
       intros ;apply P_id_a__U23_bounded;assumption.
       intros ;apply P_id_U94_bounded;assumption.
       intros ;apply P_id_U12_bounded;assumption.
       intros ;apply P_id_a__U81_bounded;assumption.
       intros ;apply P_id_U45_bounded;assumption.
       intros ;apply P_id_zeros_bounded;assumption.
       intros ;apply P_id_U71_bounded;assumption.
       intros ;apply P_id_a__U91_bounded;assumption.
       intros ;apply P_id_a__U42_bounded;assumption.
       intros ;apply P_id_isNatKind_bounded;assumption.
       intros ;apply P_id_a__U22_bounded;assumption.
       intros ;apply P_id_U92_bounded;assumption.
       intros ;apply P_id_nil_bounded;assumption.
       intros ;apply P_id_a__U61_bounded;assumption.
       intros ;apply P_id_U43_bounded;assumption.
       intros ;apply P_id_a__isNatIListKind_bounded;assumption.
       intros ;apply P_id_U84_bounded;assumption.
       intros ;apply P_id_s_bounded;assumption.
       intros ;apply P_id_a__U46_bounded;assumption.
       intros ;apply P_id_U32_bounded;assumption.
       intros ;apply P_id_a__U31_bounded;assumption.
       intros ;apply P_id_U13_bounded;assumption.
       intros ;apply P_id_a__U83_bounded;assumption.
       intros ;apply P_id_isNatIList_bounded;assumption.
       apply rules_monotonic.
     Qed.
     
     Definition P_id_A__U46 (x11:Z) := 0.
     
     Definition P_id_A__ISNATILISTKIND (x11:Z) := 0.
     
     Definition P_id_A__U61 (x11:Z) := 0.
     
     Definition P_id_A__U85 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__U31 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__U81 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__U22 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__U93 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__U42 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__U51 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__ISNATLIST (x11:Z) := 0.
     
     Definition P_id_A__ZEROS  := 0.
     
     Definition P_id_A__U91 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__U33 (x11:Z) := 0.
     
     Definition P_id_A__U83 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__U23 (x11:Z) := 0.
     
     Definition P_id_A__U11 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__LENGTH (x11:Z) := 0.
     
     Definition P_id_A__U44 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__ISNATILIST (x11:Z) := 0.
     
     Definition P_id_A__U13 (x11:Z) := 0.
     
     Definition P_id_A__U71 (x11:Z) := 0.
     
     Definition P_id_A__U86 (x11:Z) := 0.
     
     Definition P_id_A__U32 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__U82 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__ISNATKIND (x11:Z) := 0.
     
     Definition P_id_A__U94 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__U43 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__U52 (x11:Z) := 0.
     
     Definition P_id_A__U21 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__U92 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__U41 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__U84 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__ISNAT (x11:Z) := 0.
     
     Definition P_id_A__U12 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_MARK (x11:Z) := 2* x11.
     
     Definition P_id_A__U45 (x11:Z) (x12:Z) := 0.
     
     Lemma P_id_A__U46_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U46 x12 <= P_id_A__U46 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__ISNATILISTKIND_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__ISNATILISTKIND x12 <= P_id_A__ISNATILISTKIND x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U61_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U61 x12 <= P_id_A__U61 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U85_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U85 x12 x14 <= P_id_A__U85 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U31_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U31 x12 x14 <= P_id_A__U31 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U81_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U81 x12 x14 x16 <= P_id_A__U81 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U22_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U22 x12 x14 <= P_id_A__U22 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U93_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U93 x12 x14 x16 <= P_id_A__U93 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U42_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U42 x12 x14 x16 <= P_id_A__U42 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U51_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U51 x12 x14 <= P_id_A__U51 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__ISNATLIST_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__ISNATLIST x12 <= P_id_A__ISNATLIST x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U91_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U91 x12 x14 x16 <= P_id_A__U91 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U33_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U33 x12 <= P_id_A__U33 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U83_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U83 x12 x14 x16 <= P_id_A__U83 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U23_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U23 x12 <= P_id_A__U23 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U11_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U11 x12 x14 <= P_id_A__U11 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__LENGTH_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__LENGTH x12 <= P_id_A__LENGTH x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U44_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U44 x12 x14 x16 <= P_id_A__U44 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__ISNATILIST_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__ISNATILIST x12 <= P_id_A__ISNATILIST x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U13_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U13 x12 <= P_id_A__U13 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U71_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U71 x12 <= P_id_A__U71 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U86_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U86 x12 <= P_id_A__U86 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U32_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U32 x12 x14 <= P_id_A__U32 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U82_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U82 x12 x14 x16 <= P_id_A__U82 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__ISNATKIND_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__ISNATKIND x12 <= P_id_A__ISNATKIND x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U94_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U94 x12 x14 <= P_id_A__U94 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U43_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U43 x12 x14 x16 <= P_id_A__U43 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U52_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U52 x12 <= P_id_A__U52 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U21_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U21 x12 x14 <= P_id_A__U21 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U92_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U92 x12 x14 x16 <= P_id_A__U92 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U41_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U41 x12 x14 x16 <= P_id_A__U41 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U84_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U84 x12 x14 x16 <= P_id_A__U84 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__ISNAT_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__ISNAT x12 <= P_id_A__ISNAT x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U12_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U12 x12 x14 <= P_id_A__U12 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_MARK_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_MARK x12 <= P_id_MARK x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U45_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U45 x12 x14 <= P_id_A__U45 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Definition marked_measure  := 
       InterpZ.marked_measure 0 P_id_a__zeros P_id_U51 P_id_a__U84 
        P_id_a__U32 P_id_isNatList P_id_a__U13 P_id_U85 P_id_a__length 
        P_id_a__isNatIList P_id_U33 P_id_a__U11 P_id_U81 P_id_a__U92 
        P_id_a__U43 P_id_U23 P_id_a__isNatKind P_id_U93 P_id_U11 P_id_a__U71 
        P_id_U44 P_id_0 P_id_U61 P_id_a__U86 P_id_a__U41 P_id_U22 
        P_id_a__U21 P_id_U91 P_id_length P_id_a__U52 P_id_U42 P_id_a__U12 
        P_id_U83 P_id_a__U94 P_id_a__U45 P_id_U31 P_id_a__isNat 
        P_id_isNatIListKind P_id_a__U82 P_id_U46 P_id_cons P_id_U52 
        P_id_a__U85 P_id_a__U33 P_id_U21 P_id_a__isNatList P_id_U86 
        P_id_mark P_id_a__U51 P_id_U41 P_id_tt P_id_U82 P_id_a__U93 
        P_id_a__U44 P_id_isNat P_id_a__U23 P_id_U94 P_id_U12 P_id_a__U81 
        P_id_U45 P_id_zeros P_id_U71 P_id_a__U91 P_id_a__U42 P_id_isNatKind 
        P_id_a__U22 P_id_U92 P_id_nil P_id_a__U61 P_id_U43 
        P_id_a__isNatIListKind P_id_U84 P_id_s P_id_a__U46 P_id_U32 
        P_id_a__U31 P_id_U13 P_id_a__U83 P_id_isNatIList P_id_A__U46 
        P_id_A__ISNATILISTKIND P_id_A__U61 P_id_A__U85 P_id_A__U31 
        P_id_A__U81 P_id_A__U22 P_id_A__U93 P_id_A__U42 P_id_A__U51 
        P_id_A__ISNATLIST P_id_A__ZEROS P_id_A__U91 P_id_A__U33 P_id_A__U83 
        P_id_A__U23 P_id_A__U11 P_id_A__LENGTH P_id_A__U44 
        P_id_A__ISNATILIST P_id_A__U13 P_id_A__U71 P_id_A__U86 P_id_A__U32 
        P_id_A__U82 P_id_A__ISNATKIND P_id_A__U94 P_id_A__U43 P_id_A__U52 
        P_id_A__U21 P_id_A__U92 P_id_A__U41 P_id_A__U84 P_id_A__ISNAT 
        P_id_A__U12 P_id_MARK P_id_A__U45.
     
     Lemma marked_measure_equation :
      forall t, 
       marked_measure t = match t with
                            | (algebra.Alg.Term algebra.F.id_a__U46 
                               (x11::nil)) =>
                             P_id_A__U46 (measure x11)
                            | (algebra.Alg.Term 
                               algebra.F.id_a__isNatIListKind (x11::nil)) =>
                             P_id_A__ISNATILISTKIND (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U61 
                               (x11::nil)) =>
                             P_id_A__U61 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U85 (x12::
                               x11::nil)) =>
                             P_id_A__U85 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U31 (x12::
                               x11::nil)) =>
                             P_id_A__U31 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U81 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U81 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U22 (x12::
                               x11::nil)) =>
                             P_id_A__U22 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U93 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U93 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U42 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U42 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U51 (x12::
                               x11::nil)) =>
                             P_id_A__U51 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__isNatList 
                               (x11::nil)) =>
                             P_id_A__ISNATLIST (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__zeros nil) =>
                             P_id_A__ZEROS 
                            | (algebra.Alg.Term algebra.F.id_a__U91 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U91 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U33 
                               (x11::nil)) =>
                             P_id_A__U33 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U83 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U83 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U23 
                               (x11::nil)) =>
                             P_id_A__U23 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U11 (x12::
                               x11::nil)) =>
                             P_id_A__U11 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__length 
                               (x11::nil)) =>
                             P_id_A__LENGTH (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U44 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U44 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__isNatIList 
                               (x11::nil)) =>
                             P_id_A__ISNATILIST (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U13 
                               (x11::nil)) =>
                             P_id_A__U13 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U71 
                               (x11::nil)) =>
                             P_id_A__U71 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U86 
                               (x11::nil)) =>
                             P_id_A__U86 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U32 (x12::
                               x11::nil)) =>
                             P_id_A__U32 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U82 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U82 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__isNatKind 
                               (x11::nil)) =>
                             P_id_A__ISNATKIND (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U94 (x12::
                               x11::nil)) =>
                             P_id_A__U94 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U43 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U43 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U52 
                               (x11::nil)) =>
                             P_id_A__U52 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U21 (x12::
                               x11::nil)) =>
                             P_id_A__U21 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U92 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U92 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U41 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U41 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U84 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U84 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__isNat 
                               (x11::nil)) =>
                             P_id_A__ISNAT (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U12 (x12::
                               x11::nil)) =>
                             P_id_A__U12 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_mark (x11::nil)) =>
                             P_id_MARK (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U45 (x12::
                               x11::nil)) =>
                             P_id_A__U45 (measure x12) (measure x11)
                            | _ => measure t
                            end.
     Proof.
       reflexivity .
     Qed.
     
     Lemma marked_measure_star_monotonic :
      forall f l1 l2, 
       (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step 
                                                         R_xml_0_deep_rew.R_xml_0_rules)
                                                        ) l1 l2) ->
        marked_measure (algebra.Alg.Term f l1) <= marked_measure (algebra.Alg.Term 
                                                                   f 
                                                                   l2).
     Proof.
       unfold marked_measure in *.
       apply InterpZ.marked_measure_star_monotonic.
       intros ;apply P_id_U51_monotonic;assumption.
       intros ;apply P_id_a__U84_monotonic;assumption.
       intros ;apply P_id_a__U32_monotonic;assumption.
       intros ;apply P_id_isNatList_monotonic;assumption.
       intros ;apply P_id_a__U13_monotonic;assumption.
       intros ;apply P_id_U85_monotonic;assumption.
       intros ;apply P_id_a__length_monotonic;assumption.
       intros ;apply P_id_a__isNatIList_monotonic;assumption.
       intros ;apply P_id_U33_monotonic;assumption.
       intros ;apply P_id_a__U11_monotonic;assumption.
       intros ;apply P_id_U81_monotonic;assumption.
       intros ;apply P_id_a__U92_monotonic;assumption.
       intros ;apply P_id_a__U43_monotonic;assumption.
       intros ;apply P_id_U23_monotonic;assumption.
       intros ;apply P_id_a__isNatKind_monotonic;assumption.
       intros ;apply P_id_U93_monotonic;assumption.
       intros ;apply P_id_U11_monotonic;assumption.
       intros ;apply P_id_a__U71_monotonic;assumption.
       intros ;apply P_id_U44_monotonic;assumption.
       intros ;apply P_id_U61_monotonic;assumption.
       intros ;apply P_id_a__U86_monotonic;assumption.
       intros ;apply P_id_a__U41_monotonic;assumption.
       intros ;apply P_id_U22_monotonic;assumption.
       intros ;apply P_id_a__U21_monotonic;assumption.
       intros ;apply P_id_U91_monotonic;assumption.
       intros ;apply P_id_length_monotonic;assumption.
       intros ;apply P_id_a__U52_monotonic;assumption.
       intros ;apply P_id_U42_monotonic;assumption.
       intros ;apply P_id_a__U12_monotonic;assumption.
       intros ;apply P_id_U83_monotonic;assumption.
       intros ;apply P_id_a__U94_monotonic;assumption.
       intros ;apply P_id_a__U45_monotonic;assumption.
       intros ;apply P_id_U31_monotonic;assumption.
       intros ;apply P_id_a__isNat_monotonic;assumption.
       intros ;apply P_id_isNatIListKind_monotonic;assumption.
       intros ;apply P_id_a__U82_monotonic;assumption.
       intros ;apply P_id_U46_monotonic;assumption.
       intros ;apply P_id_cons_monotonic;assumption.
       intros ;apply P_id_U52_monotonic;assumption.
       intros ;apply P_id_a__U85_monotonic;assumption.
       intros ;apply P_id_a__U33_monotonic;assumption.
       intros ;apply P_id_U21_monotonic;assumption.
       intros ;apply P_id_a__isNatList_monotonic;assumption.
       intros ;apply P_id_U86_monotonic;assumption.
       intros ;apply P_id_mark_monotonic;assumption.
       intros ;apply P_id_a__U51_monotonic;assumption.
       intros ;apply P_id_U41_monotonic;assumption.
       intros ;apply P_id_U82_monotonic;assumption.
       intros ;apply P_id_a__U93_monotonic;assumption.
       intros ;apply P_id_a__U44_monotonic;assumption.
       intros ;apply P_id_isNat_monotonic;assumption.
       intros ;apply P_id_a__U23_monotonic;assumption.
       intros ;apply P_id_U94_monotonic;assumption.
       intros ;apply P_id_U12_monotonic;assumption.
       intros ;apply P_id_a__U81_monotonic;assumption.
       intros ;apply P_id_U45_monotonic;assumption.
       intros ;apply P_id_U71_monotonic;assumption.
       intros ;apply P_id_a__U91_monotonic;assumption.
       intros ;apply P_id_a__U42_monotonic;assumption.
       intros ;apply P_id_isNatKind_monotonic;assumption.
       intros ;apply P_id_a__U22_monotonic;assumption.
       intros ;apply P_id_U92_monotonic;assumption.
       intros ;apply P_id_a__U61_monotonic;assumption.
       intros ;apply P_id_U43_monotonic;assumption.
       intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
       intros ;apply P_id_U84_monotonic;assumption.
       intros ;apply P_id_s_monotonic;assumption.
       intros ;apply P_id_a__U46_monotonic;assumption.
       intros ;apply P_id_U32_monotonic;assumption.
       intros ;apply P_id_a__U31_monotonic;assumption.
       intros ;apply P_id_U13_monotonic;assumption.
       intros ;apply P_id_a__U83_monotonic;assumption.
       intros ;apply P_id_isNatIList_monotonic;assumption.
       intros ;apply P_id_a__zeros_bounded;assumption.
       intros ;apply P_id_U51_bounded;assumption.
       intros ;apply P_id_a__U84_bounded;assumption.
       intros ;apply P_id_a__U32_bounded;assumption.
       intros ;apply P_id_isNatList_bounded;assumption.
       intros ;apply P_id_a__U13_bounded;assumption.
       intros ;apply P_id_U85_bounded;assumption.
       intros ;apply P_id_a__length_bounded;assumption.
       intros ;apply P_id_a__isNatIList_bounded;assumption.
       intros ;apply P_id_U33_bounded;assumption.
       intros ;apply P_id_a__U11_bounded;assumption.
       intros ;apply P_id_U81_bounded;assumption.
       intros ;apply P_id_a__U92_bounded;assumption.
       intros ;apply P_id_a__U43_bounded;assumption.
       intros ;apply P_id_U23_bounded;assumption.
       intros ;apply P_id_a__isNatKind_bounded;assumption.
       intros ;apply P_id_U93_bounded;assumption.
       intros ;apply P_id_U11_bounded;assumption.
       intros ;apply P_id_a__U71_bounded;assumption.
       intros ;apply P_id_U44_bounded;assumption.
       intros ;apply P_id_0_bounded;assumption.
       intros ;apply P_id_U61_bounded;assumption.
       intros ;apply P_id_a__U86_bounded;assumption.
       intros ;apply P_id_a__U41_bounded;assumption.
       intros ;apply P_id_U22_bounded;assumption.
       intros ;apply P_id_a__U21_bounded;assumption.
       intros ;apply P_id_U91_bounded;assumption.
       intros ;apply P_id_length_bounded;assumption.
       intros ;apply P_id_a__U52_bounded;assumption.
       intros ;apply P_id_U42_bounded;assumption.
       intros ;apply P_id_a__U12_bounded;assumption.
       intros ;apply P_id_U83_bounded;assumption.
       intros ;apply P_id_a__U94_bounded;assumption.
       intros ;apply P_id_a__U45_bounded;assumption.
       intros ;apply P_id_U31_bounded;assumption.
       intros ;apply P_id_a__isNat_bounded;assumption.
       intros ;apply P_id_isNatIListKind_bounded;assumption.
       intros ;apply P_id_a__U82_bounded;assumption.
       intros ;apply P_id_U46_bounded;assumption.
       intros ;apply P_id_cons_bounded;assumption.
       intros ;apply P_id_U52_bounded;assumption.
       intros ;apply P_id_a__U85_bounded;assumption.
       intros ;apply P_id_a__U33_bounded;assumption.
       intros ;apply P_id_U21_bounded;assumption.
       intros ;apply P_id_a__isNatList_bounded;assumption.
       intros ;apply P_id_U86_bounded;assumption.
       intros ;apply P_id_mark_bounded;assumption.
       intros ;apply P_id_a__U51_bounded;assumption.
       intros ;apply P_id_U41_bounded;assumption.
       intros ;apply P_id_tt_bounded;assumption.
       intros ;apply P_id_U82_bounded;assumption.
       intros ;apply P_id_a__U93_bounded;assumption.
       intros ;apply P_id_a__U44_bounded;assumption.
       intros ;apply P_id_isNat_bounded;assumption.
       intros ;apply P_id_a__U23_bounded;assumption.
       intros ;apply P_id_U94_bounded;assumption.
       intros ;apply P_id_U12_bounded;assumption.
       intros ;apply P_id_a__U81_bounded;assumption.
       intros ;apply P_id_U45_bounded;assumption.
       intros ;apply P_id_zeros_bounded;assumption.
       intros ;apply P_id_U71_bounded;assumption.
       intros ;apply P_id_a__U91_bounded;assumption.
       intros ;apply P_id_a__U42_bounded;assumption.
       intros ;apply P_id_isNatKind_bounded;assumption.
       intros ;apply P_id_a__U22_bounded;assumption.
       intros ;apply P_id_U92_bounded;assumption.
       intros ;apply P_id_nil_bounded;assumption.
       intros ;apply P_id_a__U61_bounded;assumption.
       intros ;apply P_id_U43_bounded;assumption.
       intros ;apply P_id_a__isNatIListKind_bounded;assumption.
       intros ;apply P_id_U84_bounded;assumption.
       intros ;apply P_id_s_bounded;assumption.
       intros ;apply P_id_a__U46_bounded;assumption.
       intros ;apply P_id_U32_bounded;assumption.
       intros ;apply P_id_a__U31_bounded;assumption.
       intros ;apply P_id_U13_bounded;assumption.
       intros ;apply P_id_a__U83_bounded;assumption.
       intros ;apply P_id_isNatIList_bounded;assumption.
       apply rules_monotonic.
       intros ;apply P_id_A__U46_monotonic;assumption.
       intros ;apply P_id_A__ISNATILISTKIND_monotonic;assumption.
       intros ;apply P_id_A__U61_monotonic;assumption.
       intros ;apply P_id_A__U85_monotonic;assumption.
       intros ;apply P_id_A__U31_monotonic;assumption.
       intros ;apply P_id_A__U81_monotonic;assumption.
       intros ;apply P_id_A__U22_monotonic;assumption.
       intros ;apply P_id_A__U93_monotonic;assumption.
       intros ;apply P_id_A__U42_monotonic;assumption.
       intros ;apply P_id_A__U51_monotonic;assumption.
       intros ;apply P_id_A__ISNATLIST_monotonic;assumption.
       intros ;apply P_id_A__U91_monotonic;assumption.
       intros ;apply P_id_A__U33_monotonic;assumption.
       intros ;apply P_id_A__U83_monotonic;assumption.
       intros ;apply P_id_A__U23_monotonic;assumption.
       intros ;apply P_id_A__U11_monotonic;assumption.
       intros ;apply P_id_A__LENGTH_monotonic;assumption.
       intros ;apply P_id_A__U44_monotonic;assumption.
       intros ;apply P_id_A__ISNATILIST_monotonic;assumption.
       intros ;apply P_id_A__U13_monotonic;assumption.
       intros ;apply P_id_A__U71_monotonic;assumption.
       intros ;apply P_id_A__U86_monotonic;assumption.
       intros ;apply P_id_A__U32_monotonic;assumption.
       intros ;apply P_id_A__U82_monotonic;assumption.
       intros ;apply P_id_A__ISNATKIND_monotonic;assumption.
       intros ;apply P_id_A__U94_monotonic;assumption.
       intros ;apply P_id_A__U43_monotonic;assumption.
       intros ;apply P_id_A__U52_monotonic;assumption.
       intros ;apply P_id_A__U21_monotonic;assumption.
       intros ;apply P_id_A__U92_monotonic;assumption.
       intros ;apply P_id_A__U41_monotonic;assumption.
       intros ;apply P_id_A__U84_monotonic;assumption.
       intros ;apply P_id_A__ISNAT_monotonic;assumption.
       intros ;apply P_id_A__U12_monotonic;assumption.
       intros ;apply P_id_MARK_monotonic;assumption.
       intros ;apply P_id_A__U45_monotonic;assumption.
     Qed.
     
     Ltac rewrite_and_unfold  :=
      do 2 (rewrite marked_measure_equation);
       repeat (
       match goal with
         |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
          rewrite (measure_equation (algebra.Alg.Term f t))
         | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ =>
          rewrite (measure_equation (algebra.Alg.Term f t)) in H|-
         end
       ).
     
     Definition lt a b := (Zwf.Zwf 0) (marked_measure a) (marked_measure b).
     
     Definition le a b := marked_measure a <= marked_measure b.
     
     Lemma lt_le_compat : forall a b c, (lt a b) ->(le b c) ->lt a c.
     Proof.
       unfold lt, le in *.
       intros a b c.
       apply (interp.le_lt_compat_right (interp.o_Z 0)).
     Qed.
     
     Lemma wf_lt : well_founded lt.
     Proof.
       unfold lt in *.
       apply Inverse_Image.wf_inverse_image with  (B:=Z).
       apply Zwf.Zwf_well_founded.
     Qed.
     
     Lemma DP_R_xml_0_scc_65_large_scc_1_strict_in_lt :
      Relation_Definitions.inclusion _ DP_R_xml_0_scc_65_large_scc_1_strict 
       lt.
     Proof.
       unfold Relation_Definitions.inclusion, lt in *.
       
       intros a b H;destruct H;
        match goal with
          |  |- (Zwf.Zwf 0) _ (marked_measure (algebra.Alg.Term ?f ?l)) =>
           let l'' := algebra.Alg_ext.find_replacement l  in 
            ((apply (interp.le_lt_compat_right (interp.o_Z 0)) with
               (marked_measure (algebra.Alg.Term f l''));[idtac|
              apply marked_measure_star_monotonic;
               repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos);
               (assumption)||(constructor 1)]))
          end
        ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma DP_R_xml_0_scc_65_large_scc_1_large_in_le :
      Relation_Definitions.inclusion _ DP_R_xml_0_scc_65_large_scc_1_large le.
     Proof.
       unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *.
       
       intros a b H;destruct H;
        match goal with
          |  |- _ <= marked_measure (algebra.Alg.Term ?f ?l) =>
           let l'' := algebra.Alg_ext.find_replacement l  in 
            ((apply (interp.le_trans (interp.o_Z 0)) with
               (marked_measure (algebra.Alg.Term f l''));[idtac|
              apply marked_measure_star_monotonic;
               repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos);
               (assumption)||(constructor 1)]))
          end
        ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Definition wf_DP_R_xml_0_scc_65_large_scc_1_large  := 
       WF_DP_R_xml_0_scc_65_large_scc_1_large.wf.
     
     
     Lemma wf :
      well_founded WF_DP_R_xml_0_scc_65_large.DP_R_xml_0_scc_65_large_scc_1.
     Proof.
       intros x.
       apply (well_founded_ind wf_lt).
       clear x.
       intros x.
       pattern x.
       apply (@Acc_ind _ DP_R_xml_0_scc_65_large_scc_1_large).
       clear x.
       intros x _ IHx IHx'.
       constructor.
       intros y H.
       
       destruct H;
        (apply IHx';apply DP_R_xml_0_scc_65_large_scc_1_strict_in_lt;
          econstructor eassumption)||
        ((apply IHx;[econstructor eassumption|
          intros y' Hlt;apply IHx';apply lt_le_compat with (1:=Hlt) ;
           apply DP_R_xml_0_scc_65_large_scc_1_large_in_le;
           econstructor eassumption])).
       apply wf_DP_R_xml_0_scc_65_large_scc_1_large.
     Qed.
    End WF_DP_R_xml_0_scc_65_large_scc_1.
    
    Definition wf_DP_R_xml_0_scc_65_large_scc_1  := 
      WF_DP_R_xml_0_scc_65_large_scc_1.wf.
    
    
    Lemma acc_DP_R_xml_0_scc_65_large_scc_1 :
     forall x y, 
      (DP_R_xml_0_scc_65_large_scc_1 x y) ->
       Acc WF_DP_R_xml_0_scc_65.DP_R_xml_0_scc_65_large x.
    Proof.
      intros x.
      pattern x.
      apply (@Acc_ind _ DP_R_xml_0_scc_65_large_scc_1).
      intros x' _ Hrec y h.
      
      inversion h;clear h;subst;
       constructor;intros _y _h;inversion _h;clear _h;subst;
        (eapply Hrec;econstructor eassumption)||
        ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
         (eapply Hrec;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))).
      apply wf_DP_R_xml_0_scc_65_large_scc_1.
    Qed.
    
    
    Inductive DP_R_xml_0_scc_65_large_non_scc_2  :
     algebra.Alg.term ->algebra.Alg.term ->Prop := 
       (* <a__U94(tt,L_),mark(L_)> *)
      | DP_R_xml_0_scc_65_large_non_scc_2_0 :
       forall x4 x12 x11, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   (algebra.Alg.Term algebra.F.id_tt nil) 
          x12) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x4 x11) ->
          DP_R_xml_0_scc_65_large_non_scc_2 (algebra.Alg.Term 
                                             algebra.F.id_mark (x4::nil)) 
           (algebra.Alg.Term algebra.F.id_a__U94 (x12::x11::nil))
    .
    
    
    Lemma acc_DP_R_xml_0_scc_65_large_non_scc_2 :
     forall x y, 
      (DP_R_xml_0_scc_65_large_non_scc_2 x y) ->
       Acc WF_DP_R_xml_0_scc_65.DP_R_xml_0_scc_65_large x.
    Proof.
      intros x y h.
      
      inversion h;clear h;subst;
       constructor;intros _y _h;inversion _h;clear _h;subst;
        (eapply acc_DP_R_xml_0_scc_65_large_scc_1;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
         (eapply Hrec;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))).
    Qed.
    
    
    Inductive DP_R_xml_0_scc_65_large_scc_3  :
     algebra.Alg.term ->algebra.Alg.term ->Prop := 
       (* <a__U94(tt,L_),a__length(mark(L_))> *)
      | DP_R_xml_0_scc_65_large_scc_3_0 :
       forall x4 x12 x11, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   (algebra.Alg.Term algebra.F.id_tt nil) 
          x12) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x4 x11) ->
          DP_R_xml_0_scc_65_large_scc_3 (algebra.Alg.Term 
                                         algebra.F.id_a__length 
                                         ((algebra.Alg.Term 
                                         algebra.F.id_mark (x4::nil))::nil)) 
           (algebra.Alg.Term algebra.F.id_a__U94 (x12::x11::nil))
      
       (* <a__length(cons(N_,L_)),a__U91(a__isNatList(L_),L_,N_)> *)
      | DP_R_xml_0_scc_65_large_scc_3_1 :
       forall x4 x5 x11, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_cons (x5::x4::nil)) x11) ->
         DP_R_xml_0_scc_65_large_scc_3 (algebra.Alg.Term algebra.F.id_a__U91 
                                        ((algebra.Alg.Term 
                                        algebra.F.id_a__isNatList 
                                        (x4::nil))::x4::x5::nil)) 
          (algebra.Alg.Term algebra.F.id_a__length (x11::nil))
      
       (* <a__U91(tt,L_,N_),a__U92(a__isNatIListKind(L_),L_,N_)> *)
      | DP_R_xml_0_scc_65_large_scc_3_2 :
       forall x4 x12 x5 x13 x11, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   (algebra.Alg.Term algebra.F.id_tt nil) 
          x13) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x4 x12) ->
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     x5 x11) ->
           DP_R_xml_0_scc_65_large_scc_3 (algebra.Alg.Term 
                                          algebra.F.id_a__U92 
                                          ((algebra.Alg.Term 
                                          algebra.F.id_a__isNatIListKind 
                                          (x4::nil))::x4::x5::nil)) 
            (algebra.Alg.Term algebra.F.id_a__U91 (x13::x12::x11::nil))
      
       (* <a__U92(tt,L_,N_),a__U93(a__isNat(N_),L_,N_)> *)
      | DP_R_xml_0_scc_65_large_scc_3_3 :
       forall x4 x12 x5 x13 x11, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   (algebra.Alg.Term algebra.F.id_tt nil) 
          x13) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x4 x12) ->
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     x5 x11) ->
           DP_R_xml_0_scc_65_large_scc_3 (algebra.Alg.Term 
                                          algebra.F.id_a__U93 
                                          ((algebra.Alg.Term 
                                          algebra.F.id_a__isNat (x5::nil))::
                                          x4::x5::nil)) 
            (algebra.Alg.Term algebra.F.id_a__U92 (x13::x12::x11::nil))
      
       (* <a__U93(tt,L_,N_),a__U94(a__isNatKind(N_),L_)> *)
      | DP_R_xml_0_scc_65_large_scc_3_4 :
       forall x4 x12 x5 x13 x11, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   (algebra.Alg.Term algebra.F.id_tt nil) 
          x13) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x4 x12) ->
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     x5 x11) ->
           DP_R_xml_0_scc_65_large_scc_3 (algebra.Alg.Term 
                                          algebra.F.id_a__U94 
                                          ((algebra.Alg.Term 
                                          algebra.F.id_a__isNatKind 
                                          (x5::nil))::x4::nil)) 
            (algebra.Alg.Term algebra.F.id_a__U93 (x13::x12::x11::nil))
    .
    
    
    Module WF_DP_R_xml_0_scc_65_large_scc_3.
     Inductive DP_R_xml_0_scc_65_large_scc_3_large  :
      algebra.Alg.term ->algebra.Alg.term ->Prop := 
        (* <a__U94(tt,L_),a__length(mark(L_))> *)
       | DP_R_xml_0_scc_65_large_scc_3_large_0 :
        forall x4 x12 x11, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    (algebra.Alg.Term algebra.F.id_tt nil) 
           x12) ->
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     x4 x11) ->
           DP_R_xml_0_scc_65_large_scc_3_large (algebra.Alg.Term 
                                                algebra.F.id_a__length 
                                                ((algebra.Alg.Term 
                                                algebra.F.id_mark 
                                                (x4::nil))::nil)) 
            (algebra.Alg.Term algebra.F.id_a__U94 (x12::x11::nil))
       
        (* <a__length(cons(N_,L_)),a__U91(a__isNatList(L_),L_,N_)> *)
       | DP_R_xml_0_scc_65_large_scc_3_large_1 :
        forall x4 x5 x11, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    
           (algebra.Alg.Term algebra.F.id_cons (x5::x4::nil)) x11) ->
          DP_R_xml_0_scc_65_large_scc_3_large (algebra.Alg.Term 
                                               algebra.F.id_a__U91 
                                               ((algebra.Alg.Term 
                                               algebra.F.id_a__isNatList 
                                               (x4::nil))::x4::x5::nil)) 
           (algebra.Alg.Term algebra.F.id_a__length (x11::nil))
       
        (* <a__U93(tt,L_,N_),a__U94(a__isNatKind(N_),L_)> *)
       | DP_R_xml_0_scc_65_large_scc_3_large_2 :
        forall x4 x12 x5 x13 x11, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    (algebra.Alg.Term algebra.F.id_tt nil) 
           x13) ->
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     x4 x12) ->
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                      x5 x11) ->
            DP_R_xml_0_scc_65_large_scc_3_large (algebra.Alg.Term 
                                                 algebra.F.id_a__U94 
                                                 ((algebra.Alg.Term 
                                                 algebra.F.id_a__isNatKind 
                                                 (x5::nil))::x4::nil)) 
             (algebra.Alg.Term algebra.F.id_a__U93 (x13::x12::x11::nil))
     .
     
     
     Inductive DP_R_xml_0_scc_65_large_scc_3_strict  :
      algebra.Alg.term ->algebra.Alg.term ->Prop := 
        (* <a__U91(tt,L_,N_),a__U92(a__isNatIListKind(L_),L_,N_)> *)
       | DP_R_xml_0_scc_65_large_scc_3_strict_0 :
        forall x4 x12 x5 x13 x11, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    (algebra.Alg.Term algebra.F.id_tt nil) 
           x13) ->
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     x4 x12) ->
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                      x5 x11) ->
            DP_R_xml_0_scc_65_large_scc_3_strict (algebra.Alg.Term 
                                                  algebra.F.id_a__U92 
                                                  ((algebra.Alg.Term 
                                                  algebra.F.id_a__isNatIListKind 
                                                  (x4::nil))::x4::x5::nil)) 
             (algebra.Alg.Term algebra.F.id_a__U91 (x13::x12::x11::nil))
       
        (* <a__U92(tt,L_,N_),a__U93(a__isNat(N_),L_,N_)> *)
       | DP_R_xml_0_scc_65_large_scc_3_strict_1 :
        forall x4 x12 x5 x13 x11, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    (algebra.Alg.Term algebra.F.id_tt nil) 
           x13) ->
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     x4 x12) ->
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                      x5 x11) ->
            DP_R_xml_0_scc_65_large_scc_3_strict (algebra.Alg.Term 
                                                  algebra.F.id_a__U93 
                                                  ((algebra.Alg.Term 
                                                  algebra.F.id_a__isNat 
                                                  (x5::nil))::x4::x5::nil)) 
             (algebra.Alg.Term algebra.F.id_a__U92 (x13::x12::x11::nil))
     .
     
     
     Module WF_DP_R_xml_0_scc_65_large_scc_3_large.
      Inductive DP_R_xml_0_scc_65_large_scc_3_large_non_scc_1  :
       algebra.Alg.term ->algebra.Alg.term ->Prop := 
         (* <a__length(cons(N_,L_)),a__U91(a__isNatList(L_),L_,N_)> *)
        | DP_R_xml_0_scc_65_large_scc_3_large_non_scc_1_0 :
         forall x4 x5 x11, 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     
            (algebra.Alg.Term algebra.F.id_cons (x5::x4::nil)) x11) ->
           DP_R_xml_0_scc_65_large_scc_3_large_non_scc_1 (algebra.Alg.Term 
                                                          algebra.F.id_a__U91 
                                                          ((algebra.Alg.Term 
                                                          algebra.F.id_a__isNatList 
                                                          (x4::nil))::x4::
                                                          x5::nil)) 
            (algebra.Alg.Term algebra.F.id_a__length (x11::nil))
      .
      
      
      Lemma acc_DP_R_xml_0_scc_65_large_scc_3_large_non_scc_1 :
       forall x y, 
        (DP_R_xml_0_scc_65_large_scc_3_large_non_scc_1 x y) ->
         Acc WF_DP_R_xml_0_scc_65_large_scc_3.DP_R_xml_0_scc_65_large_scc_3_large 
          x.
      Proof.
        intros x y h.
        
        inversion h;clear h;subst;
         constructor;intros _y _h;inversion _h;clear _h;subst;
          (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
          (eapply Hrec;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' )).
      Qed.
      
      
      Inductive DP_R_xml_0_scc_65_large_scc_3_large_non_scc_2  :
       algebra.Alg.term ->algebra.Alg.term ->Prop := 
         (* <a__U94(tt,L_),a__length(mark(L_))> *)
        | DP_R_xml_0_scc_65_large_scc_3_large_non_scc_2_0 :
         forall x4 x12 x11, 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     (algebra.Alg.Term algebra.F.id_tt nil) 
            x12) ->
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                      x4 x11) ->
            DP_R_xml_0_scc_65_large_scc_3_large_non_scc_2 (algebra.Alg.Term 
                                                           algebra.F.id_a__length 
                                                           ((algebra.Alg.Term 
                                                           algebra.F.id_mark 
                                                           (x4::nil))::nil)) 
             (algebra.Alg.Term algebra.F.id_a__U94 (x12::x11::nil))
      .
      
      
      Lemma acc_DP_R_xml_0_scc_65_large_scc_3_large_non_scc_2 :
       forall x y, 
        (DP_R_xml_0_scc_65_large_scc_3_large_non_scc_2 x y) ->
         Acc WF_DP_R_xml_0_scc_65_large_scc_3.DP_R_xml_0_scc_65_large_scc_3_large 
          x.
      Proof.
        intros x y h.
        
        inversion h;clear h;subst;
         constructor;intros _y _h;inversion _h;clear _h;subst;
          (eapply acc_DP_R_xml_0_scc_65_large_scc_3_large_non_scc_1;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
          ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
           (eapply Hrec;
             econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))).
      Qed.
      
      
      Inductive DP_R_xml_0_scc_65_large_scc_3_large_non_scc_3  :
       algebra.Alg.term ->algebra.Alg.term ->Prop := 
         (* <a__U93(tt,L_,N_),a__U94(a__isNatKind(N_),L_)> *)
        | DP_R_xml_0_scc_65_large_scc_3_large_non_scc_3_0 :
         forall x4 x12 x5 x13 x11, 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     (algebra.Alg.Term algebra.F.id_tt nil) 
            x13) ->
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                      x4 x12) ->
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                       x5 x11) ->
             DP_R_xml_0_scc_65_large_scc_3_large_non_scc_3 (algebra.Alg.Term 
                                                            algebra.F.id_a__U94 
                                                            ((algebra.Alg.Term 
                                                            algebra.F.id_a__isNatKind 
                                                            (x5::nil))::
                                                            x4::nil)) 
              (algebra.Alg.Term algebra.F.id_a__U93 (x13::x12::x11::nil))
      .
      
      
      Lemma acc_DP_R_xml_0_scc_65_large_scc_3_large_non_scc_3 :
       forall x y, 
        (DP_R_xml_0_scc_65_large_scc_3_large_non_scc_3 x y) ->
         Acc WF_DP_R_xml_0_scc_65_large_scc_3.DP_R_xml_0_scc_65_large_scc_3_large 
          x.
      Proof.
        intros x y h.
        
        inversion h;clear h;subst;
         constructor;intros _y _h;inversion _h;clear _h;subst;
          (eapply acc_DP_R_xml_0_scc_65_large_scc_3_large_non_scc_2;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
          ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
           (eapply Hrec;
             econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))).
      Qed.
      
      
      Lemma wf :
       well_founded WF_DP_R_xml_0_scc_65_large_scc_3.DP_R_xml_0_scc_65_large_scc_3_large
        .
      Proof.
        constructor;intros _y _h;inversion _h;clear _h;subst;
         (eapply acc_DP_R_xml_0_scc_65_large_scc_3_large_non_scc_3;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
         ((eapply acc_DP_R_xml_0_scc_65_large_scc_3_large_non_scc_2;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
          ((eapply acc_DP_R_xml_0_scc_65_large_scc_3_large_non_scc_1;
             econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
           ((eapply acc_DP_R_xml_0_scc_65_large_scc_3_large_non_scc_0;
              econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
            ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||(fail))))).
      Qed.
     End WF_DP_R_xml_0_scc_65_large_scc_3_large.
     
     Open Scope Z_scope.
     
     Import ring_extention.
     
     Notation Local "a <= b" := (Zle a b).
     
     Notation Local "a < b" := (Zlt a b).
     
     Definition P_id_a__zeros  := 0.
     
     Definition P_id_U51 (x11:Z) (x12:Z) := 2.
     
     Definition P_id_a__U84 (x11:Z) (x12:Z) (x13:Z) := 2* x13.
     
     Definition P_id_a__U32 (x11:Z) (x12:Z) := 2* x12.
     
     Definition P_id_isNatList (x11:Z) := 1* x11.
     
     Definition P_id_a__U13 (x11:Z) := 2* x11.
     
     Definition P_id_U85 (x11:Z) (x12:Z) := 1* x12.
     
     Definition P_id_a__length (x11:Z) := 2 + 2* x11.
     
     Definition P_id_a__isNatIList (x11:Z) := 2 + 3* x11.
     
     Definition P_id_U33 (x11:Z) := 1* x11.
     
     Definition P_id_a__U11 (x11:Z) (x12:Z) := 3* x12.
     
     Definition P_id_U81 (x11:Z) (x12:Z) (x13:Z) := 2* x13.
     
     Definition P_id_a__U92 (x11:Z) (x12:Z) (x13:Z) := 2* x11.
     
     Definition P_id_a__U43 (x11:Z) (x12:Z) (x13:Z) := 2.
     
     Definition P_id_U23 (x11:Z) := 2.
     
     Definition P_id_a__isNatKind (x11:Z) := 2* x11.
     
     Definition P_id_U93 (x11:Z) (x12:Z) (x13:Z) := 2.
     
     Definition P_id_U11 (x11:Z) (x12:Z) := 2* x12.
     
     Definition P_id_a__U71 (x11:Z) := 2.
     
     Definition P_id_U44 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
     
     Definition P_id_0  := 2.
     
     Definition P_id_U61 (x11:Z) := 1.
     
     Definition P_id_a__U86 (x11:Z) := 1* x11.
     
     Definition P_id_a__U41 (x11:Z) (x12:Z) (x13:Z) := 2.
     
     Definition P_id_U22 (x11:Z) (x12:Z) := 2.
     
     Definition P_id_a__U21 (x11:Z) (x12:Z) := 2.
     
     Definition P_id_U91 (x11:Z) (x12:Z) (x13:Z) := 2* x11 + 1* x12.
     
     Definition P_id_length (x11:Z) := 2 + 2* x11.
     
     Definition P_id_a__U52 (x11:Z) := 2.
     
     Definition P_id_U42 (x11:Z) (x12:Z) (x13:Z) := 1.
     
     Definition P_id_a__U12 (x11:Z) (x12:Z) := 2* x12.
     
     Definition P_id_U83 (x11:Z) (x12:Z) (x13:Z) := 1* x13.
     
     Definition P_id_a__U94 (x11:Z) (x12:Z) := 3.
     
     Definition P_id_a__U45 (x11:Z) (x12:Z) := 2.
     
     Definition P_id_U31 (x11:Z) (x12:Z) := 1* x11 + 3* x12.
     
     Definition P_id_a__isNat (x11:Z) := 2* x11.
     
     Definition P_id_isNatIListKind (x11:Z) := 1.
     
     Definition P_id_a__U82 (x11:Z) (x12:Z) (x13:Z) := 2* x13.
     
     Definition P_id_U46 (x11:Z) := 2.
     
     Definition P_id_cons (x11:Z) (x12:Z) := 3* x12.
     
     Definition P_id_U52 (x11:Z) := 1.
     
     Definition P_id_a__U85 (x11:Z) (x12:Z) := 2* x12.
     
     Definition P_id_a__U33 (x11:Z) := 1* x11.
     
     Definition P_id_U21 (x11:Z) (x12:Z) := 2.
     
     Definition P_id_a__isNatList (x11:Z) := 1* x11.
     
     Definition P_id_U86 (x11:Z) := 1* x11.
     
     Definition P_id_mark (x11:Z) := 2* x11.
     
     Definition P_id_a__U51 (x11:Z) (x12:Z) := 2.
     
     Definition P_id_U41 (x11:Z) (x12:Z) (x13:Z) := 1.
     
     Definition P_id_tt  := 2.
     
     Definition P_id_U82 (x11:Z) (x12:Z) (x13:Z) := 2* x13.
     
     Definition P_id_a__U93 (x11:Z) (x12:Z) (x13:Z) := 3.
     
     Definition P_id_a__U44 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
     
     Definition P_id_isNat (x11:Z) := 2* x11.
     
     Definition P_id_a__U23 (x11:Z) := 2.
     
     Definition P_id_U94 (x11:Z) (x12:Z) := 2.
     
     Definition P_id_U12 (x11:Z) (x12:Z) := 1* x12.
     
     Definition P_id_a__U81 (x11:Z) (x12:Z) (x13:Z) := 3* x13.
     
     Definition P_id_U45 (x11:Z) (x12:Z) := 1.
     
     Definition P_id_zeros  := 0.
     
     Definition P_id_U71 (x11:Z) := 1.
     
     Definition P_id_a__U91 (x11:Z) (x12:Z) (x13:Z) := 2* x11 + 1* x12.
     
     Definition P_id_a__U42 (x11:Z) (x12:Z) (x13:Z) := 2.
     
     Definition P_id_isNatKind (x11:Z) := 2* x11.
     
     Definition P_id_a__U22 (x11:Z) (x12:Z) := 2.
     
     Definition P_id_U92 (x11:Z) (x12:Z) (x13:Z) := 2* x11.
     
     Definition P_id_nil  := 2.
     
     Definition P_id_a__U61 (x11:Z) := 2.
     
     Definition P_id_U43 (x11:Z) (x12:Z) (x13:Z) := 2.
     
     Definition P_id_a__isNatIListKind (x11:Z) := 2.
     
     Definition P_id_U84 (x11:Z) (x12:Z) (x13:Z) := 2* x13.
     
     Definition P_id_s (x11:Z) := 2.
     
     Definition P_id_a__U46 (x11:Z) := 2.
     
     Definition P_id_U32 (x11:Z) (x12:Z) := 1* x12.
     
     Definition P_id_a__U31 (x11:Z) (x12:Z) := 1* x11 + 3* x12.
     
     Definition P_id_U13 (x11:Z) := 2* x11.
     
     Definition P_id_a__U83 (x11:Z) (x12:Z) (x13:Z) := 2* x13.
     
     Definition P_id_isNatIList (x11:Z) := 1 + 2* x11.
     
     Lemma P_id_U51_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U51 x12 x14 <= P_id_U51 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U84_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U84 x12 x14 x16 <= P_id_a__U84 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U32_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U32 x12 x14 <= P_id_a__U32 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNatList_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_isNatList x12 <= P_id_isNatList x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U13_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U13 x12 <= P_id_a__U13 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U85_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U85 x12 x14 <= P_id_U85 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__length_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__length x12 <= P_id_a__length x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNatIList_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__isNatIList x12 <= P_id_a__isNatIList x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U33_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U33 x12 <= P_id_U33 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U11_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U11 x12 x14 <= P_id_a__U11 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U81_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U81 x12 x14 x16 <= P_id_U81 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U92_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U92 x12 x14 x16 <= P_id_a__U92 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U43_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U43 x12 x14 x16 <= P_id_a__U43 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U23_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U23 x12 <= P_id_U23 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNatKind_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__isNatKind x12 <= P_id_a__isNatKind x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U93_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U93 x12 x14 x16 <= P_id_U93 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U11_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U11 x12 x14 <= P_id_U11 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U71_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U71 x12 <= P_id_a__U71 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U44_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U44 x12 x14 x16 <= P_id_U44 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U61_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U61 x12 <= P_id_U61 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U86_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U86 x12 <= P_id_a__U86 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U41_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U41 x12 x14 x16 <= P_id_a__U41 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U22_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U22 x12 x14 <= P_id_U22 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U21_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U21 x12 x14 <= P_id_a__U21 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U91_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U91 x12 x14 x16 <= P_id_U91 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_length_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_length x12 <= P_id_length x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U52_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U52 x12 <= P_id_a__U52 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U42_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U42 x12 x14 x16 <= P_id_U42 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U12_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U12 x12 x14 <= P_id_a__U12 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U83_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U83 x12 x14 x16 <= P_id_U83 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U94_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U94 x12 x14 <= P_id_a__U94 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U45_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U45 x12 x14 <= P_id_a__U45 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U31_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U31 x12 x14 <= P_id_U31 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNat_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__isNat x12 <= P_id_a__isNat x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNatIListKind_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_isNatIListKind x12 <= P_id_isNatIListKind x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U82_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U82 x12 x14 x16 <= P_id_a__U82 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U46_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U46 x12 <= P_id_U46 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_cons_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_cons x12 x14 <= P_id_cons x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U52_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U52 x12 <= P_id_U52 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U85_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U85 x12 x14 <= P_id_a__U85 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U33_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U33 x12 <= P_id_a__U33 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U21_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U21 x12 x14 <= P_id_U21 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNatList_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__isNatList x12 <= P_id_a__isNatList x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U86_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U86 x12 <= P_id_U86 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_mark_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_mark x12 <= P_id_mark x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U51_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U51 x12 x14 <= P_id_a__U51 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U41_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U41 x12 x14 x16 <= P_id_U41 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U82_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U82 x12 x14 x16 <= P_id_U82 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U93_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U93 x12 x14 x16 <= P_id_a__U93 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U44_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U44 x12 x14 x16 <= P_id_a__U44 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNat_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_isNat x12 <= P_id_isNat x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U23_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U23 x12 <= P_id_a__U23 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U94_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U94 x12 x14 <= P_id_U94 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U12_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U12 x12 x14 <= P_id_U12 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U81_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U81 x12 x14 x16 <= P_id_a__U81 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U45_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U45 x12 x14 <= P_id_U45 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U71_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U71 x12 <= P_id_U71 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U91_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U91 x12 x14 x16 <= P_id_a__U91 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U42_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U42 x12 x14 x16 <= P_id_a__U42 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNatKind_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_isNatKind x12 <= P_id_isNatKind x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U22_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U22 x12 x14 <= P_id_a__U22 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U92_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U92 x12 x14 x16 <= P_id_U92 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U61_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U61 x12 <= P_id_a__U61 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U43_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U43 x12 x14 x16 <= P_id_U43 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNatIListKind_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__isNatIListKind x12 <= P_id_a__isNatIListKind x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U84_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U84 x12 x14 x16 <= P_id_U84 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_s_monotonic :
      forall x12 x11, (0 <= x12)/\ (x12 <= x11) ->P_id_s x12 <= P_id_s x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U46_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U46 x12 <= P_id_a__U46 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U32_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U32 x12 x14 <= P_id_U32 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U31_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U31 x12 x14 <= P_id_a__U31 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U13_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U13 x12 <= P_id_U13 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U83_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U83 x12 x14 x16 <= P_id_a__U83 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNatIList_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_isNatIList x12 <= P_id_isNatIList x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__zeros_bounded : 0 <= P_id_a__zeros .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U51_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U51 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U84_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U84 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U32_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U32 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNatList_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_isNatList x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U13_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U13 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U85_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U85 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__length_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__length x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNatIList_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__isNatIList x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U33_bounded : forall x11, (0 <= x11) ->0 <= P_id_U33 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U11_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U11 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U81_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U81 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U92_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U92 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U43_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U43 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U23_bounded : forall x11, (0 <= x11) ->0 <= P_id_U23 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNatKind_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__isNatKind x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U93_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U93 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U11_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U11 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U71_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U71 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U44_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U44 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_0_bounded : 0 <= P_id_0 .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U61_bounded : forall x11, (0 <= x11) ->0 <= P_id_U61 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U86_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U86 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U41_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U41 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U22_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U22 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U21_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U21 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U91_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U91 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_length_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_length x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U52_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U52 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U42_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U42 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U12_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U12 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U83_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U83 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U94_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U94 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U45_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U45 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U31_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U31 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNat_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__isNat x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNatIListKind_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_isNatIListKind x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U82_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U82 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U46_bounded : forall x11, (0 <= x11) ->0 <= P_id_U46 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_cons_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_cons x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U52_bounded : forall x11, (0 <= x11) ->0 <= P_id_U52 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U85_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U85 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U33_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U33 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U21_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U21 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNatList_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__isNatList x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U86_bounded : forall x11, (0 <= x11) ->0 <= P_id_U86 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_mark_bounded : forall x11, (0 <= x11) ->0 <= P_id_mark x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U51_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U51 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U41_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U41 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_tt_bounded : 0 <= P_id_tt .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U82_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U82 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U93_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U93 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U44_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U44 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNat_bounded : forall x11, (0 <= x11) ->0 <= P_id_isNat x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U23_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U23 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U94_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U94 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U12_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U12 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U81_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U81 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U45_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U45 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_zeros_bounded : 0 <= P_id_zeros .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U71_bounded : forall x11, (0 <= x11) ->0 <= P_id_U71 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U91_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U91 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U42_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U42 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNatKind_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_isNatKind x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U22_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U22 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U92_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U92 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_nil_bounded : 0 <= P_id_nil .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U61_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U61 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U43_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U43 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNatIListKind_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__isNatIListKind x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U84_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U84 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_s_bounded : forall x11, (0 <= x11) ->0 <= P_id_s x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U46_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U46 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U32_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U32 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U31_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U31 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U13_bounded : forall x11, (0 <= x11) ->0 <= P_id_U13 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__U83_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U83 x13 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNatIList_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_isNatIList x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Definition measure  := 
       InterpZ.measure 0 P_id_a__zeros P_id_U51 P_id_a__U84 P_id_a__U32 
        P_id_isNatList P_id_a__U13 P_id_U85 P_id_a__length 
        P_id_a__isNatIList P_id_U33 P_id_a__U11 P_id_U81 P_id_a__U92 
        P_id_a__U43 P_id_U23 P_id_a__isNatKind P_id_U93 P_id_U11 P_id_a__U71 
        P_id_U44 P_id_0 P_id_U61 P_id_a__U86 P_id_a__U41 P_id_U22 
        P_id_a__U21 P_id_U91 P_id_length P_id_a__U52 P_id_U42 P_id_a__U12 
        P_id_U83 P_id_a__U94 P_id_a__U45 P_id_U31 P_id_a__isNat 
        P_id_isNatIListKind P_id_a__U82 P_id_U46 P_id_cons P_id_U52 
        P_id_a__U85 P_id_a__U33 P_id_U21 P_id_a__isNatList P_id_U86 
        P_id_mark P_id_a__U51 P_id_U41 P_id_tt P_id_U82 P_id_a__U93 
        P_id_a__U44 P_id_isNat P_id_a__U23 P_id_U94 P_id_U12 P_id_a__U81 
        P_id_U45 P_id_zeros P_id_U71 P_id_a__U91 P_id_a__U42 P_id_isNatKind 
        P_id_a__U22 P_id_U92 P_id_nil P_id_a__U61 P_id_U43 
        P_id_a__isNatIListKind P_id_U84 P_id_s P_id_a__U46 P_id_U32 
        P_id_a__U31 P_id_U13 P_id_a__U83 P_id_isNatIList.
     
     Lemma measure_equation :
      forall t, 
       measure t = match t with
                     | (algebra.Alg.Term algebra.F.id_a__zeros nil) =>
                      P_id_a__zeros 
                     | (algebra.Alg.Term algebra.F.id_U51 (x12::x11::nil)) =>
                      P_id_U51 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U84 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U84 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U32 (x12::x11::nil)) =>
                      P_id_a__U32 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_isNatList (x11::nil)) =>
                      P_id_isNatList (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U13 (x11::nil)) =>
                      P_id_a__U13 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U85 (x12::x11::nil)) =>
                      P_id_U85 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__length (x11::nil)) =>
                      P_id_a__length (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__isNatIList 
                        (x11::nil)) =>
                      P_id_a__isNatIList (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U33 (x11::nil)) =>
                      P_id_U33 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U11 (x12::x11::nil)) =>
                      P_id_a__U11 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U81 (x13::x12::
                        x11::nil)) =>
                      P_id_U81 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U92 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U92 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U43 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U43 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U23 (x11::nil)) =>
                      P_id_U23 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__isNatKind 
                        (x11::nil)) =>
                      P_id_a__isNatKind (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U93 (x13::x12::
                        x11::nil)) =>
                      P_id_U93 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U11 (x12::x11::nil)) =>
                      P_id_U11 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U71 (x11::nil)) =>
                      P_id_a__U71 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U44 (x13::x12::
                        x11::nil)) =>
                      P_id_U44 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_0 nil) => P_id_0 
                     | (algebra.Alg.Term algebra.F.id_U61 (x11::nil)) =>
                      P_id_U61 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U86 (x11::nil)) =>
                      P_id_a__U86 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U41 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U41 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U22 (x12::x11::nil)) =>
                      P_id_U22 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U21 (x12::x11::nil)) =>
                      P_id_a__U21 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U91 (x13::x12::
                        x11::nil)) =>
                      P_id_U91 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_length (x11::nil)) =>
                      P_id_length (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U52 (x11::nil)) =>
                      P_id_a__U52 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U42 (x13::x12::
                        x11::nil)) =>
                      P_id_U42 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U12 (x12::x11::nil)) =>
                      P_id_a__U12 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U83 (x13::x12::
                        x11::nil)) =>
                      P_id_U83 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U94 (x12::x11::nil)) =>
                      P_id_a__U94 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U45 (x12::x11::nil)) =>
                      P_id_a__U45 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U31 (x12::x11::nil)) =>
                      P_id_U31 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__isNat (x11::nil)) =>
                      P_id_a__isNat (measure x11)
                     | (algebra.Alg.Term algebra.F.id_isNatIListKind 
                        (x11::nil)) =>
                      P_id_isNatIListKind (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U82 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U82 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U46 (x11::nil)) =>
                      P_id_U46 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_cons (x12::x11::nil)) =>
                      P_id_cons (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U52 (x11::nil)) =>
                      P_id_U52 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U85 (x12::x11::nil)) =>
                      P_id_a__U85 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U33 (x11::nil)) =>
                      P_id_a__U33 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U21 (x12::x11::nil)) =>
                      P_id_U21 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__isNatList 
                        (x11::nil)) =>
                      P_id_a__isNatList (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U86 (x11::nil)) =>
                      P_id_U86 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_mark (x11::nil)) =>
                      P_id_mark (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U51 (x12::x11::nil)) =>
                      P_id_a__U51 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U41 (x13::x12::
                        x11::nil)) =>
                      P_id_U41 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt 
                     | (algebra.Alg.Term algebra.F.id_U82 (x13::x12::
                        x11::nil)) =>
                      P_id_U82 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U93 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U93 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U44 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U44 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_isNat (x11::nil)) =>
                      P_id_isNat (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U23 (x11::nil)) =>
                      P_id_a__U23 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U94 (x12::x11::nil)) =>
                      P_id_U94 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U12 (x12::x11::nil)) =>
                      P_id_U12 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U81 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U81 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U45 (x12::x11::nil)) =>
                      P_id_U45 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_zeros nil) =>
                      P_id_zeros 
                     | (algebra.Alg.Term algebra.F.id_U71 (x11::nil)) =>
                      P_id_U71 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U91 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U91 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U42 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U42 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_isNatKind (x11::nil)) =>
                      P_id_isNatKind (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U22 (x12::x11::nil)) =>
                      P_id_a__U22 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U92 (x13::x12::
                        x11::nil)) =>
                      P_id_U92 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil 
                     | (algebra.Alg.Term algebra.F.id_a__U61 (x11::nil)) =>
                      P_id_a__U61 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U43 (x13::x12::
                        x11::nil)) =>
                      P_id_U43 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
                        (x11::nil)) =>
                      P_id_a__isNatIListKind (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U84 (x13::x12::
                        x11::nil)) =>
                      P_id_U84 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_s (x11::nil)) =>
                      P_id_s (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U46 (x11::nil)) =>
                      P_id_a__U46 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U32 (x12::x11::nil)) =>
                      P_id_U32 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U31 (x12::x11::nil)) =>
                      P_id_a__U31 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U13 (x11::nil)) =>
                      P_id_U13 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U83 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U83 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_isNatIList (x11::nil)) =>
                      P_id_isNatIList (measure x11)
                     | _ => 0
                     end.
     Proof.
       intros t;case t;intros ;apply refl_equal.
     Qed.
     
     Lemma measure_bounded : forall t, 0 <= measure t.
     Proof.
       unfold measure in |-*.
       
       apply InterpZ.measure_bounded;
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Ltac generate_pos_hyp  :=
      match goal with
        | H:context [measure ?x] |- _ =>
         let v := fresh "v" in 
          (let H := fresh "h" in 
            (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
              clearbody H;clearbody v))
        |  |- context [measure ?x] =>
         let v := fresh "v" in 
          (let H := fresh "h" in 
            (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
              clearbody H;clearbody v))
        end
      .
     
     Lemma rules_monotonic :
      forall l r, 
       (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) ->
        measure r <= measure l.
     Proof.
       intros l r H.
       fold measure in |-*.
       
       inversion H;clear H;subst;inversion H0;clear H0;subst;
        simpl algebra.EQT.T.apply_subst in |-*;
        repeat (
        match goal with
          |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
           rewrite (measure_equation (algebra.Alg.Term f t))
          end
        );repeat (generate_pos_hyp );
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma measure_star_monotonic :
      forall l r, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  r l) ->measure r <= measure l.
     Proof.
       unfold measure in *.
       apply InterpZ.measure_star_monotonic.
       intros ;apply P_id_U51_monotonic;assumption.
       intros ;apply P_id_a__U84_monotonic;assumption.
       intros ;apply P_id_a__U32_monotonic;assumption.
       intros ;apply P_id_isNatList_monotonic;assumption.
       intros ;apply P_id_a__U13_monotonic;assumption.
       intros ;apply P_id_U85_monotonic;assumption.
       intros ;apply P_id_a__length_monotonic;assumption.
       intros ;apply P_id_a__isNatIList_monotonic;assumption.
       intros ;apply P_id_U33_monotonic;assumption.
       intros ;apply P_id_a__U11_monotonic;assumption.
       intros ;apply P_id_U81_monotonic;assumption.
       intros ;apply P_id_a__U92_monotonic;assumption.
       intros ;apply P_id_a__U43_monotonic;assumption.
       intros ;apply P_id_U23_monotonic;assumption.
       intros ;apply P_id_a__isNatKind_monotonic;assumption.
       intros ;apply P_id_U93_monotonic;assumption.
       intros ;apply P_id_U11_monotonic;assumption.
       intros ;apply P_id_a__U71_monotonic;assumption.
       intros ;apply P_id_U44_monotonic;assumption.
       intros ;apply P_id_U61_monotonic;assumption.
       intros ;apply P_id_a__U86_monotonic;assumption.
       intros ;apply P_id_a__U41_monotonic;assumption.
       intros ;apply P_id_U22_monotonic;assumption.
       intros ;apply P_id_a__U21_monotonic;assumption.
       intros ;apply P_id_U91_monotonic;assumption.
       intros ;apply P_id_length_monotonic;assumption.
       intros ;apply P_id_a__U52_monotonic;assumption.
       intros ;apply P_id_U42_monotonic;assumption.
       intros ;apply P_id_a__U12_monotonic;assumption.
       intros ;apply P_id_U83_monotonic;assumption.
       intros ;apply P_id_a__U94_monotonic;assumption.
       intros ;apply P_id_a__U45_monotonic;assumption.
       intros ;apply P_id_U31_monotonic;assumption.
       intros ;apply P_id_a__isNat_monotonic;assumption.
       intros ;apply P_id_isNatIListKind_monotonic;assumption.
       intros ;apply P_id_a__U82_monotonic;assumption.
       intros ;apply P_id_U46_monotonic;assumption.
       intros ;apply P_id_cons_monotonic;assumption.
       intros ;apply P_id_U52_monotonic;assumption.
       intros ;apply P_id_a__U85_monotonic;assumption.
       intros ;apply P_id_a__U33_monotonic;assumption.
       intros ;apply P_id_U21_monotonic;assumption.
       intros ;apply P_id_a__isNatList_monotonic;assumption.
       intros ;apply P_id_U86_monotonic;assumption.
       intros ;apply P_id_mark_monotonic;assumption.
       intros ;apply P_id_a__U51_monotonic;assumption.
       intros ;apply P_id_U41_monotonic;assumption.
       intros ;apply P_id_U82_monotonic;assumption.
       intros ;apply P_id_a__U93_monotonic;assumption.
       intros ;apply P_id_a__U44_monotonic;assumption.
       intros ;apply P_id_isNat_monotonic;assumption.
       intros ;apply P_id_a__U23_monotonic;assumption.
       intros ;apply P_id_U94_monotonic;assumption.
       intros ;apply P_id_U12_monotonic;assumption.
       intros ;apply P_id_a__U81_monotonic;assumption.
       intros ;apply P_id_U45_monotonic;assumption.
       intros ;apply P_id_U71_monotonic;assumption.
       intros ;apply P_id_a__U91_monotonic;assumption.
       intros ;apply P_id_a__U42_monotonic;assumption.
       intros ;apply P_id_isNatKind_monotonic;assumption.
       intros ;apply P_id_a__U22_monotonic;assumption.
       intros ;apply P_id_U92_monotonic;assumption.
       intros ;apply P_id_a__U61_monotonic;assumption.
       intros ;apply P_id_U43_monotonic;assumption.
       intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
       intros ;apply P_id_U84_monotonic;assumption.
       intros ;apply P_id_s_monotonic;assumption.
       intros ;apply P_id_a__U46_monotonic;assumption.
       intros ;apply P_id_U32_monotonic;assumption.
       intros ;apply P_id_a__U31_monotonic;assumption.
       intros ;apply P_id_U13_monotonic;assumption.
       intros ;apply P_id_a__U83_monotonic;assumption.
       intros ;apply P_id_isNatIList_monotonic;assumption.
       intros ;apply P_id_a__zeros_bounded;assumption.
       intros ;apply P_id_U51_bounded;assumption.
       intros ;apply P_id_a__U84_bounded;assumption.
       intros ;apply P_id_a__U32_bounded;assumption.
       intros ;apply P_id_isNatList_bounded;assumption.
       intros ;apply P_id_a__U13_bounded;assumption.
       intros ;apply P_id_U85_bounded;assumption.
       intros ;apply P_id_a__length_bounded;assumption.
       intros ;apply P_id_a__isNatIList_bounded;assumption.
       intros ;apply P_id_U33_bounded;assumption.
       intros ;apply P_id_a__U11_bounded;assumption.
       intros ;apply P_id_U81_bounded;assumption.
       intros ;apply P_id_a__U92_bounded;assumption.
       intros ;apply P_id_a__U43_bounded;assumption.
       intros ;apply P_id_U23_bounded;assumption.
       intros ;apply P_id_a__isNatKind_bounded;assumption.
       intros ;apply P_id_U93_bounded;assumption.
       intros ;apply P_id_U11_bounded;assumption.
       intros ;apply P_id_a__U71_bounded;assumption.
       intros ;apply P_id_U44_bounded;assumption.
       intros ;apply P_id_0_bounded;assumption.
       intros ;apply P_id_U61_bounded;assumption.
       intros ;apply P_id_a__U86_bounded;assumption.
       intros ;apply P_id_a__U41_bounded;assumption.
       intros ;apply P_id_U22_bounded;assumption.
       intros ;apply P_id_a__U21_bounded;assumption.
       intros ;apply P_id_U91_bounded;assumption.
       intros ;apply P_id_length_bounded;assumption.
       intros ;apply P_id_a__U52_bounded;assumption.
       intros ;apply P_id_U42_bounded;assumption.
       intros ;apply P_id_a__U12_bounded;assumption.
       intros ;apply P_id_U83_bounded;assumption.
       intros ;apply P_id_a__U94_bounded;assumption.
       intros ;apply P_id_a__U45_bounded;assumption.
       intros ;apply P_id_U31_bounded;assumption.
       intros ;apply P_id_a__isNat_bounded;assumption.
       intros ;apply P_id_isNatIListKind_bounded;assumption.
       intros ;apply P_id_a__U82_bounded;assumption.
       intros ;apply P_id_U46_bounded;assumption.
       intros ;apply P_id_cons_bounded;assumption.
       intros ;apply P_id_U52_bounded;assumption.
       intros ;apply P_id_a__U85_bounded;assumption.
       intros ;apply P_id_a__U33_bounded;assumption.
       intros ;apply P_id_U21_bounded;assumption.
       intros ;apply P_id_a__isNatList_bounded;assumption.
       intros ;apply P_id_U86_bounded;assumption.
       intros ;apply P_id_mark_bounded;assumption.
       intros ;apply P_id_a__U51_bounded;assumption.
       intros ;apply P_id_U41_bounded;assumption.
       intros ;apply P_id_tt_bounded;assumption.
       intros ;apply P_id_U82_bounded;assumption.
       intros ;apply P_id_a__U93_bounded;assumption.
       intros ;apply P_id_a__U44_bounded;assumption.
       intros ;apply P_id_isNat_bounded;assumption.
       intros ;apply P_id_a__U23_bounded;assumption.
       intros ;apply P_id_U94_bounded;assumption.
       intros ;apply P_id_U12_bounded;assumption.
       intros ;apply P_id_a__U81_bounded;assumption.
       intros ;apply P_id_U45_bounded;assumption.
       intros ;apply P_id_zeros_bounded;assumption.
       intros ;apply P_id_U71_bounded;assumption.
       intros ;apply P_id_a__U91_bounded;assumption.
       intros ;apply P_id_a__U42_bounded;assumption.
       intros ;apply P_id_isNatKind_bounded;assumption.
       intros ;apply P_id_a__U22_bounded;assumption.
       intros ;apply P_id_U92_bounded;assumption.
       intros ;apply P_id_nil_bounded;assumption.
       intros ;apply P_id_a__U61_bounded;assumption.
       intros ;apply P_id_U43_bounded;assumption.
       intros ;apply P_id_a__isNatIListKind_bounded;assumption.
       intros ;apply P_id_U84_bounded;assumption.
       intros ;apply P_id_s_bounded;assumption.
       intros ;apply P_id_a__U46_bounded;assumption.
       intros ;apply P_id_U32_bounded;assumption.
       intros ;apply P_id_a__U31_bounded;assumption.
       intros ;apply P_id_U13_bounded;assumption.
       intros ;apply P_id_a__U83_bounded;assumption.
       intros ;apply P_id_isNatIList_bounded;assumption.
       apply rules_monotonic.
     Qed.
     
     Definition P_id_A__U46 (x11:Z) := 0.
     
     Definition P_id_A__ISNATILISTKIND (x11:Z) := 0.
     
     Definition P_id_A__U61 (x11:Z) := 0.
     
     Definition P_id_A__U85 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__U31 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__U81 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__U22 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__U93 (x11:Z) (x12:Z) (x13:Z) := 2* x12.
     
     Definition P_id_A__U42 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__U51 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__ISNATLIST (x11:Z) := 0.
     
     Definition P_id_A__ZEROS  := 0.
     
     Definition P_id_A__U91 (x11:Z) (x12:Z) (x13:Z) := 1* x11 + 2* x12.
     
     Definition P_id_A__U33 (x11:Z) := 0.
     
     Definition P_id_A__U83 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__U23 (x11:Z) := 0.
     
     Definition P_id_A__U11 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__LENGTH (x11:Z) := 1* x11.
     
     Definition P_id_A__U44 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__ISNATILIST (x11:Z) := 0.
     
     Definition P_id_A__U13 (x11:Z) := 0.
     
     Definition P_id_A__U71 (x11:Z) := 0.
     
     Definition P_id_A__U86 (x11:Z) := 0.
     
     Definition P_id_A__U32 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__U82 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__ISNATKIND (x11:Z) := 0.
     
     Definition P_id_A__U94 (x11:Z) (x12:Z) := 2* x12.
     
     Definition P_id_A__U43 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__U52 (x11:Z) := 0.
     
     Definition P_id_A__U21 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__U92 (x11:Z) (x12:Z) (x13:Z) := 1 + 2* x12.
     
     Definition P_id_A__U41 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__U84 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__ISNAT (x11:Z) := 0.
     
     Definition P_id_A__U12 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_MARK (x11:Z) := 0.
     
     Definition P_id_A__U45 (x11:Z) (x12:Z) := 0.
     
     Lemma P_id_A__U46_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U46 x12 <= P_id_A__U46 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__ISNATILISTKIND_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__ISNATILISTKIND x12 <= P_id_A__ISNATILISTKIND x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U61_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U61 x12 <= P_id_A__U61 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U85_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U85 x12 x14 <= P_id_A__U85 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U31_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U31 x12 x14 <= P_id_A__U31 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U81_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U81 x12 x14 x16 <= P_id_A__U81 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U22_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U22 x12 x14 <= P_id_A__U22 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U93_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U93 x12 x14 x16 <= P_id_A__U93 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U42_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U42 x12 x14 x16 <= P_id_A__U42 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U51_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U51 x12 x14 <= P_id_A__U51 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__ISNATLIST_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__ISNATLIST x12 <= P_id_A__ISNATLIST x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U91_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U91 x12 x14 x16 <= P_id_A__U91 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U33_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U33 x12 <= P_id_A__U33 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U83_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U83 x12 x14 x16 <= P_id_A__U83 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U23_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U23 x12 <= P_id_A__U23 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U11_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U11 x12 x14 <= P_id_A__U11 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__LENGTH_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__LENGTH x12 <= P_id_A__LENGTH x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U44_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U44 x12 x14 x16 <= P_id_A__U44 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__ISNATILIST_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__ISNATILIST x12 <= P_id_A__ISNATILIST x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U13_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U13 x12 <= P_id_A__U13 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U71_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U71 x12 <= P_id_A__U71 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U86_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U86 x12 <= P_id_A__U86 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U32_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U32 x12 x14 <= P_id_A__U32 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U82_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U82 x12 x14 x16 <= P_id_A__U82 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__ISNATKIND_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__ISNATKIND x12 <= P_id_A__ISNATKIND x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U94_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U94 x12 x14 <= P_id_A__U94 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U43_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U43 x12 x14 x16 <= P_id_A__U43 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U52_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U52 x12 <= P_id_A__U52 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U21_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U21 x12 x14 <= P_id_A__U21 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U92_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U92 x12 x14 x16 <= P_id_A__U92 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U41_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U41 x12 x14 x16 <= P_id_A__U41 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U84_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U84 x12 x14 x16 <= P_id_A__U84 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__ISNAT_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__ISNAT x12 <= P_id_A__ISNAT x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U12_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U12 x12 x14 <= P_id_A__U12 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_MARK_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_MARK x12 <= P_id_MARK x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__U45_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U45 x12 x14 <= P_id_A__U45 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Definition marked_measure  := 
       InterpZ.marked_measure 0 P_id_a__zeros P_id_U51 P_id_a__U84 
        P_id_a__U32 P_id_isNatList P_id_a__U13 P_id_U85 P_id_a__length 
        P_id_a__isNatIList P_id_U33 P_id_a__U11 P_id_U81 P_id_a__U92 
        P_id_a__U43 P_id_U23 P_id_a__isNatKind P_id_U93 P_id_U11 P_id_a__U71 
        P_id_U44 P_id_0 P_id_U61 P_id_a__U86 P_id_a__U41 P_id_U22 
        P_id_a__U21 P_id_U91 P_id_length P_id_a__U52 P_id_U42 P_id_a__U12 
        P_id_U83 P_id_a__U94 P_id_a__U45 P_id_U31 P_id_a__isNat 
        P_id_isNatIListKind P_id_a__U82 P_id_U46 P_id_cons P_id_U52 
        P_id_a__U85 P_id_a__U33 P_id_U21 P_id_a__isNatList P_id_U86 
        P_id_mark P_id_a__U51 P_id_U41 P_id_tt P_id_U82 P_id_a__U93 
        P_id_a__U44 P_id_isNat P_id_a__U23 P_id_U94 P_id_U12 P_id_a__U81 
        P_id_U45 P_id_zeros P_id_U71 P_id_a__U91 P_id_a__U42 P_id_isNatKind 
        P_id_a__U22 P_id_U92 P_id_nil P_id_a__U61 P_id_U43 
        P_id_a__isNatIListKind P_id_U84 P_id_s P_id_a__U46 P_id_U32 
        P_id_a__U31 P_id_U13 P_id_a__U83 P_id_isNatIList P_id_A__U46 
        P_id_A__ISNATILISTKIND P_id_A__U61 P_id_A__U85 P_id_A__U31 
        P_id_A__U81 P_id_A__U22 P_id_A__U93 P_id_A__U42 P_id_A__U51 
        P_id_A__ISNATLIST P_id_A__ZEROS P_id_A__U91 P_id_A__U33 P_id_A__U83 
        P_id_A__U23 P_id_A__U11 P_id_A__LENGTH P_id_A__U44 
        P_id_A__ISNATILIST P_id_A__U13 P_id_A__U71 P_id_A__U86 P_id_A__U32 
        P_id_A__U82 P_id_A__ISNATKIND P_id_A__U94 P_id_A__U43 P_id_A__U52 
        P_id_A__U21 P_id_A__U92 P_id_A__U41 P_id_A__U84 P_id_A__ISNAT 
        P_id_A__U12 P_id_MARK P_id_A__U45.
     
     Lemma marked_measure_equation :
      forall t, 
       marked_measure t = match t with
                            | (algebra.Alg.Term algebra.F.id_a__U46 
                               (x11::nil)) =>
                             P_id_A__U46 (measure x11)
                            | (algebra.Alg.Term 
                               algebra.F.id_a__isNatIListKind (x11::nil)) =>
                             P_id_A__ISNATILISTKIND (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U61 
                               (x11::nil)) =>
                             P_id_A__U61 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U85 (x12::
                               x11::nil)) =>
                             P_id_A__U85 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U31 (x12::
                               x11::nil)) =>
                             P_id_A__U31 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U81 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U81 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U22 (x12::
                               x11::nil)) =>
                             P_id_A__U22 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U93 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U93 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U42 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U42 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U51 (x12::
                               x11::nil)) =>
                             P_id_A__U51 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__isNatList 
                               (x11::nil)) =>
                             P_id_A__ISNATLIST (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__zeros nil) =>
                             P_id_A__ZEROS 
                            | (algebra.Alg.Term algebra.F.id_a__U91 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U91 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U33 
                               (x11::nil)) =>
                             P_id_A__U33 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U83 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U83 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U23 
                               (x11::nil)) =>
                             P_id_A__U23 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U11 (x12::
                               x11::nil)) =>
                             P_id_A__U11 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__length 
                               (x11::nil)) =>
                             P_id_A__LENGTH (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U44 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U44 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__isNatIList 
                               (x11::nil)) =>
                             P_id_A__ISNATILIST (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U13 
                               (x11::nil)) =>
                             P_id_A__U13 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U71 
                               (x11::nil)) =>
                             P_id_A__U71 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U86 
                               (x11::nil)) =>
                             P_id_A__U86 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U32 (x12::
                               x11::nil)) =>
                             P_id_A__U32 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U82 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U82 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__isNatKind 
                               (x11::nil)) =>
                             P_id_A__ISNATKIND (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U94 (x12::
                               x11::nil)) =>
                             P_id_A__U94 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U43 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U43 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U52 
                               (x11::nil)) =>
                             P_id_A__U52 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U21 (x12::
                               x11::nil)) =>
                             P_id_A__U21 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U92 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U92 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U41 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U41 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U84 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U84 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__isNat 
                               (x11::nil)) =>
                             P_id_A__ISNAT (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U12 (x12::
                               x11::nil)) =>
                             P_id_A__U12 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_mark (x11::nil)) =>
                             P_id_MARK (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U45 (x12::
                               x11::nil)) =>
                             P_id_A__U45 (measure x12) (measure x11)
                            | _ => measure t
                            end.
     Proof.
       reflexivity .
     Qed.
     
     Lemma marked_measure_star_monotonic :
      forall f l1 l2, 
       (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step 
                                                         R_xml_0_deep_rew.R_xml_0_rules)
                                                        ) l1 l2) ->
        marked_measure (algebra.Alg.Term f l1) <= marked_measure (algebra.Alg.Term 
                                                                   f 
                                                                   l2).
     Proof.
       unfold marked_measure in *.
       apply InterpZ.marked_measure_star_monotonic.
       intros ;apply P_id_U51_monotonic;assumption.
       intros ;apply P_id_a__U84_monotonic;assumption.
       intros ;apply P_id_a__U32_monotonic;assumption.
       intros ;apply P_id_isNatList_monotonic;assumption.
       intros ;apply P_id_a__U13_monotonic;assumption.
       intros ;apply P_id_U85_monotonic;assumption.
       intros ;apply P_id_a__length_monotonic;assumption.
       intros ;apply P_id_a__isNatIList_monotonic;assumption.
       intros ;apply P_id_U33_monotonic;assumption.
       intros ;apply P_id_a__U11_monotonic;assumption.
       intros ;apply P_id_U81_monotonic;assumption.
       intros ;apply P_id_a__U92_monotonic;assumption.
       intros ;apply P_id_a__U43_monotonic;assumption.
       intros ;apply P_id_U23_monotonic;assumption.
       intros ;apply P_id_a__isNatKind_monotonic;assumption.
       intros ;apply P_id_U93_monotonic;assumption.
       intros ;apply P_id_U11_monotonic;assumption.
       intros ;apply P_id_a__U71_monotonic;assumption.
       intros ;apply P_id_U44_monotonic;assumption.
       intros ;apply P_id_U61_monotonic;assumption.
       intros ;apply P_id_a__U86_monotonic;assumption.
       intros ;apply P_id_a__U41_monotonic;assumption.
       intros ;apply P_id_U22_monotonic;assumption.
       intros ;apply P_id_a__U21_monotonic;assumption.
       intros ;apply P_id_U91_monotonic;assumption.
       intros ;apply P_id_length_monotonic;assumption.
       intros ;apply P_id_a__U52_monotonic;assumption.
       intros ;apply P_id_U42_monotonic;assumption.
       intros ;apply P_id_a__U12_monotonic;assumption.
       intros ;apply P_id_U83_monotonic;assumption.
       intros ;apply P_id_a__U94_monotonic;assumption.
       intros ;apply P_id_a__U45_monotonic;assumption.
       intros ;apply P_id_U31_monotonic;assumption.
       intros ;apply P_id_a__isNat_monotonic;assumption.
       intros ;apply P_id_isNatIListKind_monotonic;assumption.
       intros ;apply P_id_a__U82_monotonic;assumption.
       intros ;apply P_id_U46_monotonic;assumption.
       intros ;apply P_id_cons_monotonic;assumption.
       intros ;apply P_id_U52_monotonic;assumption.
       intros ;apply P_id_a__U85_monotonic;assumption.
       intros ;apply P_id_a__U33_monotonic;assumption.
       intros ;apply P_id_U21_monotonic;assumption.
       intros ;apply P_id_a__isNatList_monotonic;assumption.
       intros ;apply P_id_U86_monotonic;assumption.
       intros ;apply P_id_mark_monotonic;assumption.
       intros ;apply P_id_a__U51_monotonic;assumption.
       intros ;apply P_id_U41_monotonic;assumption.
       intros ;apply P_id_U82_monotonic;assumption.
       intros ;apply P_id_a__U93_monotonic;assumption.
       intros ;apply P_id_a__U44_monotonic;assumption.
       intros ;apply P_id_isNat_monotonic;assumption.
       intros ;apply P_id_a__U23_monotonic;assumption.
       intros ;apply P_id_U94_monotonic;assumption.
       intros ;apply P_id_U12_monotonic;assumption.
       intros ;apply P_id_a__U81_monotonic;assumption.
       intros ;apply P_id_U45_monotonic;assumption.
       intros ;apply P_id_U71_monotonic;assumption.
       intros ;apply P_id_a__U91_monotonic;assumption.
       intros ;apply P_id_a__U42_monotonic;assumption.
       intros ;apply P_id_isNatKind_monotonic;assumption.
       intros ;apply P_id_a__U22_monotonic;assumption.
       intros ;apply P_id_U92_monotonic;assumption.
       intros ;apply P_id_a__U61_monotonic;assumption.
       intros ;apply P_id_U43_monotonic;assumption.
       intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
       intros ;apply P_id_U84_monotonic;assumption.
       intros ;apply P_id_s_monotonic;assumption.
       intros ;apply P_id_a__U46_monotonic;assumption.
       intros ;apply P_id_U32_monotonic;assumption.
       intros ;apply P_id_a__U31_monotonic;assumption.
       intros ;apply P_id_U13_monotonic;assumption.
       intros ;apply P_id_a__U83_monotonic;assumption.
       intros ;apply P_id_isNatIList_monotonic;assumption.
       intros ;apply P_id_a__zeros_bounded;assumption.
       intros ;apply P_id_U51_bounded;assumption.
       intros ;apply P_id_a__U84_bounded;assumption.
       intros ;apply P_id_a__U32_bounded;assumption.
       intros ;apply P_id_isNatList_bounded;assumption.
       intros ;apply P_id_a__U13_bounded;assumption.
       intros ;apply P_id_U85_bounded;assumption.
       intros ;apply P_id_a__length_bounded;assumption.
       intros ;apply P_id_a__isNatIList_bounded;assumption.
       intros ;apply P_id_U33_bounded;assumption.
       intros ;apply P_id_a__U11_bounded;assumption.
       intros ;apply P_id_U81_bounded;assumption.
       intros ;apply P_id_a__U92_bounded;assumption.
       intros ;apply P_id_a__U43_bounded;assumption.
       intros ;apply P_id_U23_bounded;assumption.
       intros ;apply P_id_a__isNatKind_bounded;assumption.
       intros ;apply P_id_U93_bounded;assumption.
       intros ;apply P_id_U11_bounded;assumption.
       intros ;apply P_id_a__U71_bounded;assumption.
       intros ;apply P_id_U44_bounded;assumption.
       intros ;apply P_id_0_bounded;assumption.
       intros ;apply P_id_U61_bounded;assumption.
       intros ;apply P_id_a__U86_bounded;assumption.
       intros ;apply P_id_a__U41_bounded;assumption.
       intros ;apply P_id_U22_bounded;assumption.
       intros ;apply P_id_a__U21_bounded;assumption.
       intros ;apply P_id_U91_bounded;assumption.
       intros ;apply P_id_length_bounded;assumption.
       intros ;apply P_id_a__U52_bounded;assumption.
       intros ;apply P_id_U42_bounded;assumption.
       intros ;apply P_id_a__U12_bounded;assumption.
       intros ;apply P_id_U83_bounded;assumption.
       intros ;apply P_id_a__U94_bounded;assumption.
       intros ;apply P_id_a__U45_bounded;assumption.
       intros ;apply P_id_U31_bounded;assumption.
       intros ;apply P_id_a__isNat_bounded;assumption.
       intros ;apply P_id_isNatIListKind_bounded;assumption.
       intros ;apply P_id_a__U82_bounded;assumption.
       intros ;apply P_id_U46_bounded;assumption.
       intros ;apply P_id_cons_bounded;assumption.
       intros ;apply P_id_U52_bounded;assumption.
       intros ;apply P_id_a__U85_bounded;assumption.
       intros ;apply P_id_a__U33_bounded;assumption.
       intros ;apply P_id_U21_bounded;assumption.
       intros ;apply P_id_a__isNatList_bounded;assumption.
       intros ;apply P_id_U86_bounded;assumption.
       intros ;apply P_id_mark_bounded;assumption.
       intros ;apply P_id_a__U51_bounded;assumption.
       intros ;apply P_id_U41_bounded;assumption.
       intros ;apply P_id_tt_bounded;assumption.
       intros ;apply P_id_U82_bounded;assumption.
       intros ;apply P_id_a__U93_bounded;assumption.
       intros ;apply P_id_a__U44_bounded;assumption.
       intros ;apply P_id_isNat_bounded;assumption.
       intros ;apply P_id_a__U23_bounded;assumption.
       intros ;apply P_id_U94_bounded;assumption.
       intros ;apply P_id_U12_bounded;assumption.
       intros ;apply P_id_a__U81_bounded;assumption.
       intros ;apply P_id_U45_bounded;assumption.
       intros ;apply P_id_zeros_bounded;assumption.
       intros ;apply P_id_U71_bounded;assumption.
       intros ;apply P_id_a__U91_bounded;assumption.
       intros ;apply P_id_a__U42_bounded;assumption.
       intros ;apply P_id_isNatKind_bounded;assumption.
       intros ;apply P_id_a__U22_bounded;assumption.
       intros ;apply P_id_U92_bounded;assumption.
       intros ;apply P_id_nil_bounded;assumption.
       intros ;apply P_id_a__U61_bounded;assumption.
       intros ;apply P_id_U43_bounded;assumption.
       intros ;apply P_id_a__isNatIListKind_bounded;assumption.
       intros ;apply P_id_U84_bounded;assumption.
       intros ;apply P_id_s_bounded;assumption.
       intros ;apply P_id_a__U46_bounded;assumption.
       intros ;apply P_id_U32_bounded;assumption.
       intros ;apply P_id_a__U31_bounded;assumption.
       intros ;apply P_id_U13_bounded;assumption.
       intros ;apply P_id_a__U83_bounded;assumption.
       intros ;apply P_id_isNatIList_bounded;assumption.
       apply rules_monotonic.
       intros ;apply P_id_A__U46_monotonic;assumption.
       intros ;apply P_id_A__ISNATILISTKIND_monotonic;assumption.
       intros ;apply P_id_A__U61_monotonic;assumption.
       intros ;apply P_id_A__U85_monotonic;assumption.
       intros ;apply P_id_A__U31_monotonic;assumption.
       intros ;apply P_id_A__U81_monotonic;assumption.
       intros ;apply P_id_A__U22_monotonic;assumption.
       intros ;apply P_id_A__U93_monotonic;assumption.
       intros ;apply P_id_A__U42_monotonic;assumption.
       intros ;apply P_id_A__U51_monotonic;assumption.
       intros ;apply P_id_A__ISNATLIST_monotonic;assumption.
       intros ;apply P_id_A__U91_monotonic;assumption.
       intros ;apply P_id_A__U33_monotonic;assumption.
       intros ;apply P_id_A__U83_monotonic;assumption.
       intros ;apply P_id_A__U23_monotonic;assumption.
       intros ;apply P_id_A__U11_monotonic;assumption.
       intros ;apply P_id_A__LENGTH_monotonic;assumption.
       intros ;apply P_id_A__U44_monotonic;assumption.
       intros ;apply P_id_A__ISNATILIST_monotonic;assumption.
       intros ;apply P_id_A__U13_monotonic;assumption.
       intros ;apply P_id_A__U71_monotonic;assumption.
       intros ;apply P_id_A__U86_monotonic;assumption.
       intros ;apply P_id_A__U32_monotonic;assumption.
       intros ;apply P_id_A__U82_monotonic;assumption.
       intros ;apply P_id_A__ISNATKIND_monotonic;assumption.
       intros ;apply P_id_A__U94_monotonic;assumption.
       intros ;apply P_id_A__U43_monotonic;assumption.
       intros ;apply P_id_A__U52_monotonic;assumption.
       intros ;apply P_id_A__U21_monotonic;assumption.
       intros ;apply P_id_A__U92_monotonic;assumption.
       intros ;apply P_id_A__U41_monotonic;assumption.
       intros ;apply P_id_A__U84_monotonic;assumption.
       intros ;apply P_id_A__ISNAT_monotonic;assumption.
       intros ;apply P_id_A__U12_monotonic;assumption.
       intros ;apply P_id_MARK_monotonic;assumption.
       intros ;apply P_id_A__U45_monotonic;assumption.
     Qed.
     
     Ltac rewrite_and_unfold  :=
      do 2 (rewrite marked_measure_equation);
       repeat (
       match goal with
         |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
          rewrite (measure_equation (algebra.Alg.Term f t))
         | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ =>
          rewrite (measure_equation (algebra.Alg.Term f t)) in H|-
         end
       ).
     
     Definition lt a b := (Zwf.Zwf 0) (marked_measure a) (marked_measure b).
     
     Definition le a b := marked_measure a <= marked_measure b.
     
     Lemma lt_le_compat : forall a b c, (lt a b) ->(le b c) ->lt a c.
     Proof.
       unfold lt, le in *.
       intros a b c.
       apply (interp.le_lt_compat_right (interp.o_Z 0)).
     Qed.
     
     Lemma wf_lt : well_founded lt.
     Proof.
       unfold lt in *.
       apply Inverse_Image.wf_inverse_image with  (B:=Z).
       apply Zwf.Zwf_well_founded.
     Qed.
     
     Lemma DP_R_xml_0_scc_65_large_scc_3_strict_in_lt :
      Relation_Definitions.inclusion _ DP_R_xml_0_scc_65_large_scc_3_strict 
       lt.
     Proof.
       unfold Relation_Definitions.inclusion, lt in *.
       
       intros a b H;destruct H;
        match goal with
          |  |- (Zwf.Zwf 0) _ (marked_measure (algebra.Alg.Term ?f ?l)) =>
           let l'' := algebra.Alg_ext.find_replacement l  in 
            ((apply (interp.le_lt_compat_right (interp.o_Z 0)) with
               (marked_measure (algebra.Alg.Term f l''));[idtac|
              apply marked_measure_star_monotonic;
               repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos);
               (assumption)||(constructor 1)]))
          end
        ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma DP_R_xml_0_scc_65_large_scc_3_large_in_le :
      Relation_Definitions.inclusion _ DP_R_xml_0_scc_65_large_scc_3_large le.
     Proof.
       unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *.
       
       intros a b H;destruct H;
        match goal with
          |  |- _ <= marked_measure (algebra.Alg.Term ?f ?l) =>
           let l'' := algebra.Alg_ext.find_replacement l  in 
            ((apply (interp.le_trans (interp.o_Z 0)) with
               (marked_measure (algebra.Alg.Term f l''));[idtac|
              apply marked_measure_star_monotonic;
               repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos);
               (assumption)||(constructor 1)]))
          end
        ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Definition wf_DP_R_xml_0_scc_65_large_scc_3_large  := 
       WF_DP_R_xml_0_scc_65_large_scc_3_large.wf.
     
     
     Lemma wf :
      well_founded WF_DP_R_xml_0_scc_65_large.DP_R_xml_0_scc_65_large_scc_3.
     Proof.
       intros x.
       apply (well_founded_ind wf_lt).
       clear x.
       intros x.
       pattern x.
       apply (@Acc_ind _ DP_R_xml_0_scc_65_large_scc_3_large).
       clear x.
       intros x _ IHx IHx'.
       constructor.
       intros y H.
       
       destruct H;
        (apply IHx';apply DP_R_xml_0_scc_65_large_scc_3_strict_in_lt;
          econstructor eassumption)||
        ((apply IHx;[econstructor eassumption|
          intros y' Hlt;apply IHx';apply lt_le_compat with (1:=Hlt) ;
           apply DP_R_xml_0_scc_65_large_scc_3_large_in_le;
           econstructor eassumption])).
       apply wf_DP_R_xml_0_scc_65_large_scc_3_large.
     Qed.
    End WF_DP_R_xml_0_scc_65_large_scc_3.
    
    Definition wf_DP_R_xml_0_scc_65_large_scc_3  := 
      WF_DP_R_xml_0_scc_65_large_scc_3.wf.
    
    
    Lemma acc_DP_R_xml_0_scc_65_large_scc_3 :
     forall x y, 
      (DP_R_xml_0_scc_65_large_scc_3 x y) ->
       Acc WF_DP_R_xml_0_scc_65.DP_R_xml_0_scc_65_large x.
    Proof.
      intros x.
      pattern x.
      apply (@Acc_ind _ DP_R_xml_0_scc_65_large_scc_3).
      intros x' _ Hrec y h.
      
      inversion h;clear h;subst;
       constructor;intros _y _h;inversion _h;clear _h;subst;
        (eapply Hrec;econstructor eassumption)||
        ((eapply acc_DP_R_xml_0_scc_65_large_non_scc_2;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
         ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
          (eapply Hrec;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))).
      apply wf_DP_R_xml_0_scc_65_large_scc_3.
    Qed.
    
    
    Lemma wf : well_founded WF_DP_R_xml_0_scc_65.DP_R_xml_0_scc_65_large.
    Proof.
      constructor;intros _y _h;inversion _h;clear _h;subst;
       (eapply acc_DP_R_xml_0_scc_65_large_non_scc_2;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_scc_65_large_non_scc_1;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((eapply acc_DP_R_xml_0_scc_65_large_non_scc_0;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
         ((eapply acc_DP_R_xml_0_scc_65_large_scc_3;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
          ((eapply acc_DP_R_xml_0_scc_65_large_scc_2;
             econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
           ((eapply acc_DP_R_xml_0_scc_65_large_scc_1;
              econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
            ((eapply acc_DP_R_xml_0_scc_65_large_scc_0;
               econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
             ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||(fail)))))))).
    Qed.
   End WF_DP_R_xml_0_scc_65_large.
   
   Open Scope Z_scope.
   
   Import ring_extention.
   
   Notation Local "a <= b" := (Zle a b).
   
   Notation Local "a < b" := (Zlt a b).
   
   Definition P_id_a__zeros  := 0.
   
   Definition P_id_U51 (x11:Z) (x12:Z) := 1* x11.
   
   Definition P_id_a__U84 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
   
   Definition P_id_a__U32 (x11:Z) (x12:Z) := 1* x11.
   
   Definition P_id_isNatList (x11:Z) := 0.
   
   Definition P_id_a__U13 (x11:Z) := 1* x11.
   
   Definition P_id_U85 (x11:Z) (x12:Z) := 2* x11.
   
   Definition P_id_a__length (x11:Z) := 1 + 1* x11.
   
   Definition P_id_a__isNatIList (x11:Z) := 0.
   
   Definition P_id_U33 (x11:Z) := 1* x11.
   
   Definition P_id_a__U11 (x11:Z) (x12:Z) := 2* x11.
   
   Definition P_id_U81 (x11:Z) (x12:Z) (x13:Z) := 2* x11.
   
   Definition P_id_a__U92 (x11:Z) (x12:Z) (x13:Z) := 1 + 1* x11 + 3* x12.
   
   Definition P_id_a__U43 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
   
   Definition P_id_U23 (x11:Z) := 1* x11.
   
   Definition P_id_a__isNatKind (x11:Z) := 0.
   
   Definition P_id_U93 (x11:Z) (x12:Z) (x13:Z) := 1 + 2* x11 + 2* x12.
   
   Definition P_id_U11 (x11:Z) (x12:Z) := 2* x11.
   
   Definition P_id_a__U71 (x11:Z) := 1* x11.
   
   Definition P_id_U44 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
   
   Definition P_id_0  := 0.
   
   Definition P_id_U61 (x11:Z) := 2* x11.
   
   Definition P_id_a__U86 (x11:Z) := 1* x11.
   
   Definition P_id_a__U41 (x11:Z) (x12:Z) (x13:Z) := 2* x11.
   
   Definition P_id_U22 (x11:Z) (x12:Z) := 2* x11.
   
   Definition P_id_a__U21 (x11:Z) (x12:Z) := 2* x11.
   
   Definition P_id_U91 (x11:Z) (x12:Z) (x13:Z) := 1 + 1* x11 + 2* x12.
   
   Definition P_id_length (x11:Z) := 1 + 1* x11.
   
   Definition P_id_a__U52 (x11:Z) := 2* x11.
   
   Definition P_id_U42 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
   
   Definition P_id_a__U12 (x11:Z) (x12:Z) := 1* x11.
   
   Definition P_id_U83 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
   
   Definition P_id_a__U94 (x11:Z) (x12:Z) := 1 + 1* x11 + 2* x12.
   
   Definition P_id_a__U45 (x11:Z) (x12:Z) := 1* x11.
   
   Definition P_id_U31 (x11:Z) (x12:Z) := 2* x11.
   
   Definition P_id_a__isNat (x11:Z) := 0.
   
   Definition P_id_isNatIListKind (x11:Z) := 0.
   
   Definition P_id_a__U82 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
   
   Definition P_id_U46 (x11:Z) := 1* x11.
   
   Definition P_id_cons (x11:Z) (x12:Z) := 2* x11 + 3* x12.
   
   Definition P_id_U52 (x11:Z) := 2* x11.
   
   Definition P_id_a__U85 (x11:Z) (x12:Z) := 2* x11.
   
   Definition P_id_a__U33 (x11:Z) := 1* x11.
   
   Definition P_id_U21 (x11:Z) (x12:Z) := 2* x11.
   
   Definition P_id_a__isNatList (x11:Z) := 0.
   
   Definition P_id_U86 (x11:Z) := 1* x11.
   
   Definition P_id_mark (x11:Z) := 2* x11.
   
   Definition P_id_a__U51 (x11:Z) (x12:Z) := 1* x11.
   
   Definition P_id_U41 (x11:Z) (x12:Z) (x13:Z) := 2* x11.
   
   Definition P_id_tt  := 0.
   
   Definition P_id_U82 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
   
   Definition P_id_a__U93 (x11:Z) (x12:Z) (x13:Z) := 1 + 2* x11 + 3* x12.
   
   Definition P_id_a__U44 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
   
   Definition P_id_isNat (x11:Z) := 0.
   
   Definition P_id_a__U23 (x11:Z) := 1* x11.
   
   Definition P_id_U94 (x11:Z) (x12:Z) := 1 + 1* x11 + 2* x12.
   
   Definition P_id_U12 (x11:Z) (x12:Z) := 1* x11.
   
   Definition P_id_a__U81 (x11:Z) (x12:Z) (x13:Z) := 2* x11.
   
   Definition P_id_U45 (x11:Z) (x12:Z) := 1* x11.
   
   Definition P_id_zeros  := 0.
   
   Definition P_id_U71 (x11:Z) := 1* x11.
   
   Definition P_id_a__U91 (x11:Z) (x12:Z) (x13:Z) := 1 + 1* x11 + 3* x12.
   
   Definition P_id_a__U42 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
   
   Definition P_id_isNatKind (x11:Z) := 0.
   
   Definition P_id_a__U22 (x11:Z) (x12:Z) := 2* x11.
   
   Definition P_id_U92 (x11:Z) (x12:Z) (x13:Z) := 1 + 1* x11 + 2* x12.
   
   Definition P_id_nil  := 2.
   
   Definition P_id_a__U61 (x11:Z) := 2* x11.
   
   Definition P_id_U43 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
   
   Definition P_id_a__isNatIListKind (x11:Z) := 0.
   
   Definition P_id_U84 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
   
   Definition P_id_s (x11:Z) := 1* x11.
   
   Definition P_id_a__U46 (x11:Z) := 1* x11.
   
   Definition P_id_U32 (x11:Z) (x12:Z) := 1* x11.
   
   Definition P_id_a__U31 (x11:Z) (x12:Z) := 2* x11.
   
   Definition P_id_U13 (x11:Z) := 1* x11.
   
   Definition P_id_a__U83 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
   
   Definition P_id_isNatIList (x11:Z) := 0.
   
   Lemma P_id_U51_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_U51 x12 x14 <= P_id_U51 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U84_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U84 x12 x14 x16 <= P_id_a__U84 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U32_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_a__U32 x12 x14 <= P_id_a__U32 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_isNatList_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_isNatList x12 <= P_id_isNatList x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U13_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_a__U13 x12 <= P_id_a__U13 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U85_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_U85 x12 x14 <= P_id_U85 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__length_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_a__length x12 <= P_id_a__length x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__isNatIList_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->
      P_id_a__isNatIList x12 <= P_id_a__isNatIList x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U33_monotonic :
    forall x12 x11, (0 <= x12)/\ (x12 <= x11) ->P_id_U33 x12 <= P_id_U33 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U11_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_a__U11 x12 x14 <= P_id_a__U11 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U81_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_U81 x12 x14 x16 <= P_id_U81 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U92_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U92 x12 x14 x16 <= P_id_a__U92 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U43_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U43 x12 x14 x16 <= P_id_a__U43 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U23_monotonic :
    forall x12 x11, (0 <= x12)/\ (x12 <= x11) ->P_id_U23 x12 <= P_id_U23 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__isNatKind_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->
      P_id_a__isNatKind x12 <= P_id_a__isNatKind x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U93_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_U93 x12 x14 x16 <= P_id_U93 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U11_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_U11 x12 x14 <= P_id_U11 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U71_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_a__U71 x12 <= P_id_a__U71 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U44_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_U44 x12 x14 x16 <= P_id_U44 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U61_monotonic :
    forall x12 x11, (0 <= x12)/\ (x12 <= x11) ->P_id_U61 x12 <= P_id_U61 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U86_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_a__U86 x12 <= P_id_a__U86 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U41_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U41 x12 x14 x16 <= P_id_a__U41 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U22_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_U22 x12 x14 <= P_id_U22 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U21_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_a__U21 x12 x14 <= P_id_a__U21 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U91_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_U91 x12 x14 x16 <= P_id_U91 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_length_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_length x12 <= P_id_length x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U52_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_a__U52 x12 <= P_id_a__U52 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U42_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_U42 x12 x14 x16 <= P_id_U42 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U12_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_a__U12 x12 x14 <= P_id_a__U12 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U83_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_U83 x12 x14 x16 <= P_id_U83 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U94_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_a__U94 x12 x14 <= P_id_a__U94 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U45_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_a__U45 x12 x14 <= P_id_a__U45 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U31_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_U31 x12 x14 <= P_id_U31 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__isNat_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_a__isNat x12 <= P_id_a__isNat x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_isNatIListKind_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->
      P_id_isNatIListKind x12 <= P_id_isNatIListKind x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U82_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U82 x12 x14 x16 <= P_id_a__U82 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U46_monotonic :
    forall x12 x11, (0 <= x12)/\ (x12 <= x11) ->P_id_U46 x12 <= P_id_U46 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_cons_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_cons x12 x14 <= P_id_cons x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U52_monotonic :
    forall x12 x11, (0 <= x12)/\ (x12 <= x11) ->P_id_U52 x12 <= P_id_U52 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U85_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_a__U85 x12 x14 <= P_id_a__U85 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U33_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_a__U33 x12 <= P_id_a__U33 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U21_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_U21 x12 x14 <= P_id_U21 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__isNatList_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->
      P_id_a__isNatList x12 <= P_id_a__isNatList x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U86_monotonic :
    forall x12 x11, (0 <= x12)/\ (x12 <= x11) ->P_id_U86 x12 <= P_id_U86 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_mark_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_mark x12 <= P_id_mark x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U51_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_a__U51 x12 x14 <= P_id_a__U51 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U41_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_U41 x12 x14 x16 <= P_id_U41 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U82_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_U82 x12 x14 x16 <= P_id_U82 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U93_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U93 x12 x14 x16 <= P_id_a__U93 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U44_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U44 x12 x14 x16 <= P_id_a__U44 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_isNat_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_isNat x12 <= P_id_isNat x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U23_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_a__U23 x12 <= P_id_a__U23 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U94_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_U94 x12 x14 <= P_id_U94 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U12_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_U12 x12 x14 <= P_id_U12 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U81_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U81 x12 x14 x16 <= P_id_a__U81 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U45_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_U45 x12 x14 <= P_id_U45 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U71_monotonic :
    forall x12 x11, (0 <= x12)/\ (x12 <= x11) ->P_id_U71 x12 <= P_id_U71 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U91_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U91 x12 x14 x16 <= P_id_a__U91 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U42_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U42 x12 x14 x16 <= P_id_a__U42 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_isNatKind_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_isNatKind x12 <= P_id_isNatKind x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U22_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_a__U22 x12 x14 <= P_id_a__U22 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U92_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_U92 x12 x14 x16 <= P_id_U92 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U61_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_a__U61 x12 <= P_id_a__U61 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U43_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_U43 x12 x14 x16 <= P_id_U43 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__isNatIListKind_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->
      P_id_a__isNatIListKind x12 <= P_id_a__isNatIListKind x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U84_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_U84 x12 x14 x16 <= P_id_U84 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_s_monotonic :
    forall x12 x11, (0 <= x12)/\ (x12 <= x11) ->P_id_s x12 <= P_id_s x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U46_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_a__U46 x12 <= P_id_a__U46 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U32_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_U32 x12 x14 <= P_id_U32 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U31_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_a__U31 x12 x14 <= P_id_a__U31 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U13_monotonic :
    forall x12 x11, (0 <= x12)/\ (x12 <= x11) ->P_id_U13 x12 <= P_id_U13 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U83_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U83 x12 x14 x16 <= P_id_a__U83 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_isNatIList_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_isNatIList x12 <= P_id_isNatIList x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__zeros_bounded : 0 <= P_id_a__zeros .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U51_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U51 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U84_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U84 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U32_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U32 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_isNatList_bounded :
    forall x11, (0 <= x11) ->0 <= P_id_isNatList x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U13_bounded : forall x11, (0 <= x11) ->0 <= P_id_a__U13 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U85_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U85 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__length_bounded :
    forall x11, (0 <= x11) ->0 <= P_id_a__length x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__isNatIList_bounded :
    forall x11, (0 <= x11) ->0 <= P_id_a__isNatIList x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U33_bounded : forall x11, (0 <= x11) ->0 <= P_id_U33 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U11_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U11 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U81_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U81 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U92_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U92 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U43_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U43 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U23_bounded : forall x11, (0 <= x11) ->0 <= P_id_U23 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__isNatKind_bounded :
    forall x11, (0 <= x11) ->0 <= P_id_a__isNatKind x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U93_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U93 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U11_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U11 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U71_bounded : forall x11, (0 <= x11) ->0 <= P_id_a__U71 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U44_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U44 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_0_bounded : 0 <= P_id_0 .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U61_bounded : forall x11, (0 <= x11) ->0 <= P_id_U61 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U86_bounded : forall x11, (0 <= x11) ->0 <= P_id_a__U86 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U41_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U41 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U22_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U22 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U21_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U21 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U91_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U91 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_length_bounded : forall x11, (0 <= x11) ->0 <= P_id_length x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U52_bounded : forall x11, (0 <= x11) ->0 <= P_id_a__U52 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U42_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U42 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U12_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U12 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U83_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U83 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U94_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U94 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U45_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U45 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U31_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U31 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__isNat_bounded :
    forall x11, (0 <= x11) ->0 <= P_id_a__isNat x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_isNatIListKind_bounded :
    forall x11, (0 <= x11) ->0 <= P_id_isNatIListKind x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U82_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U82 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U46_bounded : forall x11, (0 <= x11) ->0 <= P_id_U46 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_cons_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_cons x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U52_bounded : forall x11, (0 <= x11) ->0 <= P_id_U52 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U85_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U85 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U33_bounded : forall x11, (0 <= x11) ->0 <= P_id_a__U33 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U21_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U21 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__isNatList_bounded :
    forall x11, (0 <= x11) ->0 <= P_id_a__isNatList x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U86_bounded : forall x11, (0 <= x11) ->0 <= P_id_U86 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_mark_bounded : forall x11, (0 <= x11) ->0 <= P_id_mark x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U51_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U51 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U41_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U41 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_tt_bounded : 0 <= P_id_tt .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U82_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U82 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U93_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U93 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U44_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U44 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_isNat_bounded : forall x11, (0 <= x11) ->0 <= P_id_isNat x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U23_bounded : forall x11, (0 <= x11) ->0 <= P_id_a__U23 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U94_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U94 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U12_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U12 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U81_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U81 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U45_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U45 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_zeros_bounded : 0 <= P_id_zeros .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U71_bounded : forall x11, (0 <= x11) ->0 <= P_id_U71 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U91_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U91 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U42_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U42 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_isNatKind_bounded :
    forall x11, (0 <= x11) ->0 <= P_id_isNatKind x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U22_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U22 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U92_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U92 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_nil_bounded : 0 <= P_id_nil .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U61_bounded : forall x11, (0 <= x11) ->0 <= P_id_a__U61 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U43_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U43 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__isNatIListKind_bounded :
    forall x11, (0 <= x11) ->0 <= P_id_a__isNatIListKind x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U84_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U84 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_s_bounded : forall x11, (0 <= x11) ->0 <= P_id_s x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U46_bounded : forall x11, (0 <= x11) ->0 <= P_id_a__U46 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U32_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U32 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U31_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U31 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U13_bounded : forall x11, (0 <= x11) ->0 <= P_id_U13 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__U83_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U83 x13 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_isNatIList_bounded :
    forall x11, (0 <= x11) ->0 <= P_id_isNatIList x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Definition measure  := 
     InterpZ.measure 0 P_id_a__zeros P_id_U51 P_id_a__U84 P_id_a__U32 
      P_id_isNatList P_id_a__U13 P_id_U85 P_id_a__length P_id_a__isNatIList 
      P_id_U33 P_id_a__U11 P_id_U81 P_id_a__U92 P_id_a__U43 P_id_U23 
      P_id_a__isNatKind P_id_U93 P_id_U11 P_id_a__U71 P_id_U44 P_id_0 
      P_id_U61 P_id_a__U86 P_id_a__U41 P_id_U22 P_id_a__U21 P_id_U91 
      P_id_length P_id_a__U52 P_id_U42 P_id_a__U12 P_id_U83 P_id_a__U94 
      P_id_a__U45 P_id_U31 P_id_a__isNat P_id_isNatIListKind P_id_a__U82 
      P_id_U46 P_id_cons P_id_U52 P_id_a__U85 P_id_a__U33 P_id_U21 
      P_id_a__isNatList P_id_U86 P_id_mark P_id_a__U51 P_id_U41 P_id_tt 
      P_id_U82 P_id_a__U93 P_id_a__U44 P_id_isNat P_id_a__U23 P_id_U94 
      P_id_U12 P_id_a__U81 P_id_U45 P_id_zeros P_id_U71 P_id_a__U91 
      P_id_a__U42 P_id_isNatKind P_id_a__U22 P_id_U92 P_id_nil P_id_a__U61 
      P_id_U43 P_id_a__isNatIListKind P_id_U84 P_id_s P_id_a__U46 P_id_U32 
      P_id_a__U31 P_id_U13 P_id_a__U83 P_id_isNatIList.
   
   Lemma measure_equation :
    forall t, 
     measure t = match t with
                   | (algebra.Alg.Term algebra.F.id_a__zeros nil) =>
                    P_id_a__zeros 
                   | (algebra.Alg.Term algebra.F.id_U51 (x12::x11::nil)) =>
                    P_id_U51 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U84 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U84 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U32 (x12::x11::nil)) =>
                    P_id_a__U32 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_isNatList (x11::nil)) =>
                    P_id_isNatList (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U13 (x11::nil)) =>
                    P_id_a__U13 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U85 (x12::x11::nil)) =>
                    P_id_U85 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__length (x11::nil)) =>
                    P_id_a__length (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__isNatIList (x11::nil)) =>
                    P_id_a__isNatIList (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U33 (x11::nil)) =>
                    P_id_U33 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U11 (x12::x11::nil)) =>
                    P_id_a__U11 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U81 (x13::x12::x11::nil)) =>
                    P_id_U81 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U92 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U92 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U43 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U43 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U23 (x11::nil)) =>
                    P_id_U23 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__isNatKind (x11::nil)) =>
                    P_id_a__isNatKind (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U93 (x13::x12::x11::nil)) =>
                    P_id_U93 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U11 (x12::x11::nil)) =>
                    P_id_U11 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U71 (x11::nil)) =>
                    P_id_a__U71 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U44 (x13::x12::x11::nil)) =>
                    P_id_U44 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_0 nil) => P_id_0 
                   | (algebra.Alg.Term algebra.F.id_U61 (x11::nil)) =>
                    P_id_U61 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U86 (x11::nil)) =>
                    P_id_a__U86 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U41 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U41 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U22 (x12::x11::nil)) =>
                    P_id_U22 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U21 (x12::x11::nil)) =>
                    P_id_a__U21 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U91 (x13::x12::x11::nil)) =>
                    P_id_U91 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_length (x11::nil)) =>
                    P_id_length (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U52 (x11::nil)) =>
                    P_id_a__U52 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U42 (x13::x12::x11::nil)) =>
                    P_id_U42 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U12 (x12::x11::nil)) =>
                    P_id_a__U12 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U83 (x13::x12::x11::nil)) =>
                    P_id_U83 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U94 (x12::x11::nil)) =>
                    P_id_a__U94 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U45 (x12::x11::nil)) =>
                    P_id_a__U45 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U31 (x12::x11::nil)) =>
                    P_id_U31 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__isNat (x11::nil)) =>
                    P_id_a__isNat (measure x11)
                   | (algebra.Alg.Term algebra.F.id_isNatIListKind 
                      (x11::nil)) =>
                    P_id_isNatIListKind (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U82 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U82 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U46 (x11::nil)) =>
                    P_id_U46 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_cons (x12::x11::nil)) =>
                    P_id_cons (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U52 (x11::nil)) =>
                    P_id_U52 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U85 (x12::x11::nil)) =>
                    P_id_a__U85 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U33 (x11::nil)) =>
                    P_id_a__U33 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U21 (x12::x11::nil)) =>
                    P_id_U21 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__isNatList (x11::nil)) =>
                    P_id_a__isNatList (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U86 (x11::nil)) =>
                    P_id_U86 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_mark (x11::nil)) =>
                    P_id_mark (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U51 (x12::x11::nil)) =>
                    P_id_a__U51 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U41 (x13::x12::x11::nil)) =>
                    P_id_U41 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt 
                   | (algebra.Alg.Term algebra.F.id_U82 (x13::x12::x11::nil)) =>
                    P_id_U82 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U93 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U93 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U44 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U44 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_isNat (x11::nil)) =>
                    P_id_isNat (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U23 (x11::nil)) =>
                    P_id_a__U23 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U94 (x12::x11::nil)) =>
                    P_id_U94 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U12 (x12::x11::nil)) =>
                    P_id_U12 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U81 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U81 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U45 (x12::x11::nil)) =>
                    P_id_U45 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_zeros nil) => P_id_zeros 
                   | (algebra.Alg.Term algebra.F.id_U71 (x11::nil)) =>
                    P_id_U71 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U91 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U91 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U42 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U42 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_isNatKind (x11::nil)) =>
                    P_id_isNatKind (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U22 (x12::x11::nil)) =>
                    P_id_a__U22 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U92 (x13::x12::x11::nil)) =>
                    P_id_U92 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil 
                   | (algebra.Alg.Term algebra.F.id_a__U61 (x11::nil)) =>
                    P_id_a__U61 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U43 (x13::x12::x11::nil)) =>
                    P_id_U43 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
                      (x11::nil)) =>
                    P_id_a__isNatIListKind (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U84 (x13::x12::x11::nil)) =>
                    P_id_U84 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_s (x11::nil)) =>
                    P_id_s (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U46 (x11::nil)) =>
                    P_id_a__U46 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U32 (x12::x11::nil)) =>
                    P_id_U32 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U31 (x12::x11::nil)) =>
                    P_id_a__U31 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U13 (x11::nil)) =>
                    P_id_U13 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U83 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U83 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_isNatIList (x11::nil)) =>
                    P_id_isNatIList (measure x11)
                   | _ => 0
                   end.
   Proof.
     intros t;case t;intros ;apply refl_equal.
   Qed.
   
   Lemma measure_bounded : forall t, 0 <= measure t.
   Proof.
     unfold measure in |-*.
     
     apply InterpZ.measure_bounded;
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Ltac generate_pos_hyp  :=
    match goal with
      | H:context [measure ?x] |- _ =>
       let v := fresh "v" in 
        (let H := fresh "h" in 
          (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
            clearbody H;clearbody v))
      |  |- context [measure ?x] =>
       let v := fresh "v" in 
        (let H := fresh "h" in 
          (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
            clearbody H;clearbody v))
      end
    .
   
   Lemma rules_monotonic :
    forall l r, 
     (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) ->
      measure r <= measure l.
   Proof.
     intros l r H.
     fold measure in |-*.
     
     inversion H;clear H;subst;inversion H0;clear H0;subst;
      simpl algebra.EQT.T.apply_subst in |-*;
      repeat (
      match goal with
        |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
         rewrite (measure_equation (algebra.Alg.Term f t))
        end
      );repeat (generate_pos_hyp );
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma measure_star_monotonic :
    forall l r, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                r l) ->measure r <= measure l.
   Proof.
     unfold measure in *.
     apply InterpZ.measure_star_monotonic.
     intros ;apply P_id_U51_monotonic;assumption.
     intros ;apply P_id_a__U84_monotonic;assumption.
     intros ;apply P_id_a__U32_monotonic;assumption.
     intros ;apply P_id_isNatList_monotonic;assumption.
     intros ;apply P_id_a__U13_monotonic;assumption.
     intros ;apply P_id_U85_monotonic;assumption.
     intros ;apply P_id_a__length_monotonic;assumption.
     intros ;apply P_id_a__isNatIList_monotonic;assumption.
     intros ;apply P_id_U33_monotonic;assumption.
     intros ;apply P_id_a__U11_monotonic;assumption.
     intros ;apply P_id_U81_monotonic;assumption.
     intros ;apply P_id_a__U92_monotonic;assumption.
     intros ;apply P_id_a__U43_monotonic;assumption.
     intros ;apply P_id_U23_monotonic;assumption.
     intros ;apply P_id_a__isNatKind_monotonic;assumption.
     intros ;apply P_id_U93_monotonic;assumption.
     intros ;apply P_id_U11_monotonic;assumption.
     intros ;apply P_id_a__U71_monotonic;assumption.
     intros ;apply P_id_U44_monotonic;assumption.
     intros ;apply P_id_U61_monotonic;assumption.
     intros ;apply P_id_a__U86_monotonic;assumption.
     intros ;apply P_id_a__U41_monotonic;assumption.
     intros ;apply P_id_U22_monotonic;assumption.
     intros ;apply P_id_a__U21_monotonic;assumption.
     intros ;apply P_id_U91_monotonic;assumption.
     intros ;apply P_id_length_monotonic;assumption.
     intros ;apply P_id_a__U52_monotonic;assumption.
     intros ;apply P_id_U42_monotonic;assumption.
     intros ;apply P_id_a__U12_monotonic;assumption.
     intros ;apply P_id_U83_monotonic;assumption.
     intros ;apply P_id_a__U94_monotonic;assumption.
     intros ;apply P_id_a__U45_monotonic;assumption.
     intros ;apply P_id_U31_monotonic;assumption.
     intros ;apply P_id_a__isNat_monotonic;assumption.
     intros ;apply P_id_isNatIListKind_monotonic;assumption.
     intros ;apply P_id_a__U82_monotonic;assumption.
     intros ;apply P_id_U46_monotonic;assumption.
     intros ;apply P_id_cons_monotonic;assumption.
     intros ;apply P_id_U52_monotonic;assumption.
     intros ;apply P_id_a__U85_monotonic;assumption.
     intros ;apply P_id_a__U33_monotonic;assumption.
     intros ;apply P_id_U21_monotonic;assumption.
     intros ;apply P_id_a__isNatList_monotonic;assumption.
     intros ;apply P_id_U86_monotonic;assumption.
     intros ;apply P_id_mark_monotonic;assumption.
     intros ;apply P_id_a__U51_monotonic;assumption.
     intros ;apply P_id_U41_monotonic;assumption.
     intros ;apply P_id_U82_monotonic;assumption.
     intros ;apply P_id_a__U93_monotonic;assumption.
     intros ;apply P_id_a__U44_monotonic;assumption.
     intros ;apply P_id_isNat_monotonic;assumption.
     intros ;apply P_id_a__U23_monotonic;assumption.
     intros ;apply P_id_U94_monotonic;assumption.
     intros ;apply P_id_U12_monotonic;assumption.
     intros ;apply P_id_a__U81_monotonic;assumption.
     intros ;apply P_id_U45_monotonic;assumption.
     intros ;apply P_id_U71_monotonic;assumption.
     intros ;apply P_id_a__U91_monotonic;assumption.
     intros ;apply P_id_a__U42_monotonic;assumption.
     intros ;apply P_id_isNatKind_monotonic;assumption.
     intros ;apply P_id_a__U22_monotonic;assumption.
     intros ;apply P_id_U92_monotonic;assumption.
     intros ;apply P_id_a__U61_monotonic;assumption.
     intros ;apply P_id_U43_monotonic;assumption.
     intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
     intros ;apply P_id_U84_monotonic;assumption.
     intros ;apply P_id_s_monotonic;assumption.
     intros ;apply P_id_a__U46_monotonic;assumption.
     intros ;apply P_id_U32_monotonic;assumption.
     intros ;apply P_id_a__U31_monotonic;assumption.
     intros ;apply P_id_U13_monotonic;assumption.
     intros ;apply P_id_a__U83_monotonic;assumption.
     intros ;apply P_id_isNatIList_monotonic;assumption.
     intros ;apply P_id_a__zeros_bounded;assumption.
     intros ;apply P_id_U51_bounded;assumption.
     intros ;apply P_id_a__U84_bounded;assumption.
     intros ;apply P_id_a__U32_bounded;assumption.
     intros ;apply P_id_isNatList_bounded;assumption.
     intros ;apply P_id_a__U13_bounded;assumption.
     intros ;apply P_id_U85_bounded;assumption.
     intros ;apply P_id_a__length_bounded;assumption.
     intros ;apply P_id_a__isNatIList_bounded;assumption.
     intros ;apply P_id_U33_bounded;assumption.
     intros ;apply P_id_a__U11_bounded;assumption.
     intros ;apply P_id_U81_bounded;assumption.
     intros ;apply P_id_a__U92_bounded;assumption.
     intros ;apply P_id_a__U43_bounded;assumption.
     intros ;apply P_id_U23_bounded;assumption.
     intros ;apply P_id_a__isNatKind_bounded;assumption.
     intros ;apply P_id_U93_bounded;assumption.
     intros ;apply P_id_U11_bounded;assumption.
     intros ;apply P_id_a__U71_bounded;assumption.
     intros ;apply P_id_U44_bounded;assumption.
     intros ;apply P_id_0_bounded;assumption.
     intros ;apply P_id_U61_bounded;assumption.
     intros ;apply P_id_a__U86_bounded;assumption.
     intros ;apply P_id_a__U41_bounded;assumption.
     intros ;apply P_id_U22_bounded;assumption.
     intros ;apply P_id_a__U21_bounded;assumption.
     intros ;apply P_id_U91_bounded;assumption.
     intros ;apply P_id_length_bounded;assumption.
     intros ;apply P_id_a__U52_bounded;assumption.
     intros ;apply P_id_U42_bounded;assumption.
     intros ;apply P_id_a__U12_bounded;assumption.
     intros ;apply P_id_U83_bounded;assumption.
     intros ;apply P_id_a__U94_bounded;assumption.
     intros ;apply P_id_a__U45_bounded;assumption.
     intros ;apply P_id_U31_bounded;assumption.
     intros ;apply P_id_a__isNat_bounded;assumption.
     intros ;apply P_id_isNatIListKind_bounded;assumption.
     intros ;apply P_id_a__U82_bounded;assumption.
     intros ;apply P_id_U46_bounded;assumption.
     intros ;apply P_id_cons_bounded;assumption.
     intros ;apply P_id_U52_bounded;assumption.
     intros ;apply P_id_a__U85_bounded;assumption.
     intros ;apply P_id_a__U33_bounded;assumption.
     intros ;apply P_id_U21_bounded;assumption.
     intros ;apply P_id_a__isNatList_bounded;assumption.
     intros ;apply P_id_U86_bounded;assumption.
     intros ;apply P_id_mark_bounded;assumption.
     intros ;apply P_id_a__U51_bounded;assumption.
     intros ;apply P_id_U41_bounded;assumption.
     intros ;apply P_id_tt_bounded;assumption.
     intros ;apply P_id_U82_bounded;assumption.
     intros ;apply P_id_a__U93_bounded;assumption.
     intros ;apply P_id_a__U44_bounded;assumption.
     intros ;apply P_id_isNat_bounded;assumption.
     intros ;apply P_id_a__U23_bounded;assumption.
     intros ;apply P_id_U94_bounded;assumption.
     intros ;apply P_id_U12_bounded;assumption.
     intros ;apply P_id_a__U81_bounded;assumption.
     intros ;apply P_id_U45_bounded;assumption.
     intros ;apply P_id_zeros_bounded;assumption.
     intros ;apply P_id_U71_bounded;assumption.
     intros ;apply P_id_a__U91_bounded;assumption.
     intros ;apply P_id_a__U42_bounded;assumption.
     intros ;apply P_id_isNatKind_bounded;assumption.
     intros ;apply P_id_a__U22_bounded;assumption.
     intros ;apply P_id_U92_bounded;assumption.
     intros ;apply P_id_nil_bounded;assumption.
     intros ;apply P_id_a__U61_bounded;assumption.
     intros ;apply P_id_U43_bounded;assumption.
     intros ;apply P_id_a__isNatIListKind_bounded;assumption.
     intros ;apply P_id_U84_bounded;assumption.
     intros ;apply P_id_s_bounded;assumption.
     intros ;apply P_id_a__U46_bounded;assumption.
     intros ;apply P_id_U32_bounded;assumption.
     intros ;apply P_id_a__U31_bounded;assumption.
     intros ;apply P_id_U13_bounded;assumption.
     intros ;apply P_id_a__U83_bounded;assumption.
     intros ;apply P_id_isNatIList_bounded;assumption.
     apply rules_monotonic.
   Qed.
   
   Definition P_id_A__U46 (x11:Z) := 0.
   
   Definition P_id_A__ISNATILISTKIND (x11:Z) := 0.
   
   Definition P_id_A__U61 (x11:Z) := 0.
   
   Definition P_id_A__U85 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_A__U31 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_A__U81 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_A__U22 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_A__U93 (x11:Z) (x12:Z) (x13:Z) := 2* x11 + 2* x12.
   
   Definition P_id_A__U42 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_A__U51 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_A__ISNATLIST (x11:Z) := 0.
   
   Definition P_id_A__ZEROS  := 0.
   
   Definition P_id_A__U91 (x11:Z) (x12:Z) (x13:Z) := 2* x12.
   
   Definition P_id_A__U33 (x11:Z) := 0.
   
   Definition P_id_A__U83 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_A__U23 (x11:Z) := 0.
   
   Definition P_id_A__U11 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_A__LENGTH (x11:Z) := 1* x11.
   
   Definition P_id_A__U44 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_A__ISNATILIST (x11:Z) := 0.
   
   Definition P_id_A__U13 (x11:Z) := 0.
   
   Definition P_id_A__U71 (x11:Z) := 0.
   
   Definition P_id_A__U86 (x11:Z) := 0.
   
   Definition P_id_A__U32 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_A__U82 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_A__ISNATKIND (x11:Z) := 0.
   
   Definition P_id_A__U94 (x11:Z) (x12:Z) := 2* x12.
   
   Definition P_id_A__U43 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_A__U52 (x11:Z) := 0.
   
   Definition P_id_A__U21 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_A__U92 (x11:Z) (x12:Z) (x13:Z) := 2* x12.
   
   Definition P_id_A__U41 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_A__U84 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_A__ISNAT (x11:Z) := 0.
   
   Definition P_id_A__U12 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_MARK (x11:Z) := 2* x11.
   
   Definition P_id_A__U45 (x11:Z) (x12:Z) := 0.
   
   Lemma P_id_A__U46_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_A__U46 x12 <= P_id_A__U46 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__ISNATILISTKIND_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->
      P_id_A__ISNATILISTKIND x12 <= P_id_A__ISNATILISTKIND x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U61_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_A__U61 x12 <= P_id_A__U61 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U85_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_A__U85 x12 x14 <= P_id_A__U85 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U31_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_A__U31 x12 x14 <= P_id_A__U31 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U81_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U81 x12 x14 x16 <= P_id_A__U81 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U22_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_A__U22 x12 x14 <= P_id_A__U22 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U93_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U93 x12 x14 x16 <= P_id_A__U93 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U42_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U42 x12 x14 x16 <= P_id_A__U42 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U51_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_A__U51 x12 x14 <= P_id_A__U51 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__ISNATLIST_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->
      P_id_A__ISNATLIST x12 <= P_id_A__ISNATLIST x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U91_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U91 x12 x14 x16 <= P_id_A__U91 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U33_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_A__U33 x12 <= P_id_A__U33 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U83_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U83 x12 x14 x16 <= P_id_A__U83 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U23_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_A__U23 x12 <= P_id_A__U23 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U11_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_A__U11 x12 x14 <= P_id_A__U11 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__LENGTH_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_A__LENGTH x12 <= P_id_A__LENGTH x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U44_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U44 x12 x14 x16 <= P_id_A__U44 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__ISNATILIST_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->
      P_id_A__ISNATILIST x12 <= P_id_A__ISNATILIST x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U13_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_A__U13 x12 <= P_id_A__U13 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U71_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_A__U71 x12 <= P_id_A__U71 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U86_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_A__U86 x12 <= P_id_A__U86 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U32_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_A__U32 x12 x14 <= P_id_A__U32 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U82_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U82 x12 x14 x16 <= P_id_A__U82 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__ISNATKIND_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->
      P_id_A__ISNATKIND x12 <= P_id_A__ISNATKIND x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U94_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_A__U94 x12 x14 <= P_id_A__U94 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U43_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U43 x12 x14 x16 <= P_id_A__U43 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U52_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_A__U52 x12 <= P_id_A__U52 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U21_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_A__U21 x12 x14 <= P_id_A__U21 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U92_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U92 x12 x14 x16 <= P_id_A__U92 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U41_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U41 x12 x14 x16 <= P_id_A__U41 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U84_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U84 x12 x14 x16 <= P_id_A__U84 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__ISNAT_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_A__ISNAT x12 <= P_id_A__ISNAT x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U12_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_A__U12 x12 x14 <= P_id_A__U12 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_MARK_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_MARK x12 <= P_id_MARK x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__U45_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_A__U45 x12 x14 <= P_id_A__U45 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Definition marked_measure  := 
     InterpZ.marked_measure 0 P_id_a__zeros P_id_U51 P_id_a__U84 P_id_a__U32 
      P_id_isNatList P_id_a__U13 P_id_U85 P_id_a__length P_id_a__isNatIList 
      P_id_U33 P_id_a__U11 P_id_U81 P_id_a__U92 P_id_a__U43 P_id_U23 
      P_id_a__isNatKind P_id_U93 P_id_U11 P_id_a__U71 P_id_U44 P_id_0 
      P_id_U61 P_id_a__U86 P_id_a__U41 P_id_U22 P_id_a__U21 P_id_U91 
      P_id_length P_id_a__U52 P_id_U42 P_id_a__U12 P_id_U83 P_id_a__U94 
      P_id_a__U45 P_id_U31 P_id_a__isNat P_id_isNatIListKind P_id_a__U82 
      P_id_U46 P_id_cons P_id_U52 P_id_a__U85 P_id_a__U33 P_id_U21 
      P_id_a__isNatList P_id_U86 P_id_mark P_id_a__U51 P_id_U41 P_id_tt 
      P_id_U82 P_id_a__U93 P_id_a__U44 P_id_isNat P_id_a__U23 P_id_U94 
      P_id_U12 P_id_a__U81 P_id_U45 P_id_zeros P_id_U71 P_id_a__U91 
      P_id_a__U42 P_id_isNatKind P_id_a__U22 P_id_U92 P_id_nil P_id_a__U61 
      P_id_U43 P_id_a__isNatIListKind P_id_U84 P_id_s P_id_a__U46 P_id_U32 
      P_id_a__U31 P_id_U13 P_id_a__U83 P_id_isNatIList P_id_A__U46 
      P_id_A__ISNATILISTKIND P_id_A__U61 P_id_A__U85 P_id_A__U31 P_id_A__U81 
      P_id_A__U22 P_id_A__U93 P_id_A__U42 P_id_A__U51 P_id_A__ISNATLIST 
      P_id_A__ZEROS P_id_A__U91 P_id_A__U33 P_id_A__U83 P_id_A__U23 
      P_id_A__U11 P_id_A__LENGTH P_id_A__U44 P_id_A__ISNATILIST P_id_A__U13 
      P_id_A__U71 P_id_A__U86 P_id_A__U32 P_id_A__U82 P_id_A__ISNATKIND 
      P_id_A__U94 P_id_A__U43 P_id_A__U52 P_id_A__U21 P_id_A__U92 
      P_id_A__U41 P_id_A__U84 P_id_A__ISNAT P_id_A__U12 P_id_MARK P_id_A__U45
     .
   
   Lemma marked_measure_equation :
    forall t, 
     marked_measure t = match t with
                          | (algebra.Alg.Term algebra.F.id_a__U46 (x11::nil)) =>
                           P_id_A__U46 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
                             (x11::nil)) =>
                           P_id_A__ISNATILISTKIND (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U61 (x11::nil)) =>
                           P_id_A__U61 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U85 (x12::
                             x11::nil)) =>
                           P_id_A__U85 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U31 (x12::
                             x11::nil)) =>
                           P_id_A__U31 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U81 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U81 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U22 (x12::
                             x11::nil)) =>
                           P_id_A__U22 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U93 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U93 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U42 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U42 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U51 (x12::
                             x11::nil)) =>
                           P_id_A__U51 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__isNatList 
                             (x11::nil)) =>
                           P_id_A__ISNATLIST (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__zeros nil) =>
                           P_id_A__ZEROS 
                          | (algebra.Alg.Term algebra.F.id_a__U91 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U91 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U33 (x11::nil)) =>
                           P_id_A__U33 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U83 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U83 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U23 (x11::nil)) =>
                           P_id_A__U23 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U11 (x12::
                             x11::nil)) =>
                           P_id_A__U11 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__length 
                             (x11::nil)) =>
                           P_id_A__LENGTH (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U44 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U44 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__isNatIList 
                             (x11::nil)) =>
                           P_id_A__ISNATILIST (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U13 (x11::nil)) =>
                           P_id_A__U13 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U71 (x11::nil)) =>
                           P_id_A__U71 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U86 (x11::nil)) =>
                           P_id_A__U86 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U32 (x12::
                             x11::nil)) =>
                           P_id_A__U32 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U82 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U82 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__isNatKind 
                             (x11::nil)) =>
                           P_id_A__ISNATKIND (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U94 (x12::
                             x11::nil)) =>
                           P_id_A__U94 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U43 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U43 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U52 (x11::nil)) =>
                           P_id_A__U52 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U21 (x12::
                             x11::nil)) =>
                           P_id_A__U21 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U92 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U92 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U41 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U41 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U84 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U84 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__isNat 
                             (x11::nil)) =>
                           P_id_A__ISNAT (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U12 (x12::
                             x11::nil)) =>
                           P_id_A__U12 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_mark (x11::nil)) =>
                           P_id_MARK (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U45 (x12::
                             x11::nil)) =>
                           P_id_A__U45 (measure x12) (measure x11)
                          | _ => measure t
                          end.
   Proof.
     reflexivity .
   Qed.
   
   Lemma marked_measure_star_monotonic :
    forall f l1 l2, 
     (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step 
                                                       R_xml_0_deep_rew.R_xml_0_rules)
                                                      ) l1 l2) ->
      marked_measure (algebra.Alg.Term f l1) <= marked_measure (algebra.Alg.Term 
                                                                 f l2).
   Proof.
     unfold marked_measure in *.
     apply InterpZ.marked_measure_star_monotonic.
     intros ;apply P_id_U51_monotonic;assumption.
     intros ;apply P_id_a__U84_monotonic;assumption.
     intros ;apply P_id_a__U32_monotonic;assumption.
     intros ;apply P_id_isNatList_monotonic;assumption.
     intros ;apply P_id_a__U13_monotonic;assumption.
     intros ;apply P_id_U85_monotonic;assumption.
     intros ;apply P_id_a__length_monotonic;assumption.
     intros ;apply P_id_a__isNatIList_monotonic;assumption.
     intros ;apply P_id_U33_monotonic;assumption.
     intros ;apply P_id_a__U11_monotonic;assumption.
     intros ;apply P_id_U81_monotonic;assumption.
     intros ;apply P_id_a__U92_monotonic;assumption.
     intros ;apply P_id_a__U43_monotonic;assumption.
     intros ;apply P_id_U23_monotonic;assumption.
     intros ;apply P_id_a__isNatKind_monotonic;assumption.
     intros ;apply P_id_U93_monotonic;assumption.
     intros ;apply P_id_U11_monotonic;assumption.
     intros ;apply P_id_a__U71_monotonic;assumption.
     intros ;apply P_id_U44_monotonic;assumption.
     intros ;apply P_id_U61_monotonic;assumption.
     intros ;apply P_id_a__U86_monotonic;assumption.
     intros ;apply P_id_a__U41_monotonic;assumption.
     intros ;apply P_id_U22_monotonic;assumption.
     intros ;apply P_id_a__U21_monotonic;assumption.
     intros ;apply P_id_U91_monotonic;assumption.
     intros ;apply P_id_length_monotonic;assumption.
     intros ;apply P_id_a__U52_monotonic;assumption.
     intros ;apply P_id_U42_monotonic;assumption.
     intros ;apply P_id_a__U12_monotonic;assumption.
     intros ;apply P_id_U83_monotonic;assumption.
     intros ;apply P_id_a__U94_monotonic;assumption.
     intros ;apply P_id_a__U45_monotonic;assumption.
     intros ;apply P_id_U31_monotonic;assumption.
     intros ;apply P_id_a__isNat_monotonic;assumption.
     intros ;apply P_id_isNatIListKind_monotonic;assumption.
     intros ;apply P_id_a__U82_monotonic;assumption.
     intros ;apply P_id_U46_monotonic;assumption.
     intros ;apply P_id_cons_monotonic;assumption.
     intros ;apply P_id_U52_monotonic;assumption.
     intros ;apply P_id_a__U85_monotonic;assumption.
     intros ;apply P_id_a__U33_monotonic;assumption.
     intros ;apply P_id_U21_monotonic;assumption.
     intros ;apply P_id_a__isNatList_monotonic;assumption.
     intros ;apply P_id_U86_monotonic;assumption.
     intros ;apply P_id_mark_monotonic;assumption.
     intros ;apply P_id_a__U51_monotonic;assumption.
     intros ;apply P_id_U41_monotonic;assumption.
     intros ;apply P_id_U82_monotonic;assumption.
     intros ;apply P_id_a__U93_monotonic;assumption.
     intros ;apply P_id_a__U44_monotonic;assumption.
     intros ;apply P_id_isNat_monotonic;assumption.
     intros ;apply P_id_a__U23_monotonic;assumption.
     intros ;apply P_id_U94_monotonic;assumption.
     intros ;apply P_id_U12_monotonic;assumption.
     intros ;apply P_id_a__U81_monotonic;assumption.
     intros ;apply P_id_U45_monotonic;assumption.
     intros ;apply P_id_U71_monotonic;assumption.
     intros ;apply P_id_a__U91_monotonic;assumption.
     intros ;apply P_id_a__U42_monotonic;assumption.
     intros ;apply P_id_isNatKind_monotonic;assumption.
     intros ;apply P_id_a__U22_monotonic;assumption.
     intros ;apply P_id_U92_monotonic;assumption.
     intros ;apply P_id_a__U61_monotonic;assumption.
     intros ;apply P_id_U43_monotonic;assumption.
     intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
     intros ;apply P_id_U84_monotonic;assumption.
     intros ;apply P_id_s_monotonic;assumption.
     intros ;apply P_id_a__U46_monotonic;assumption.
     intros ;apply P_id_U32_monotonic;assumption.
     intros ;apply P_id_a__U31_monotonic;assumption.
     intros ;apply P_id_U13_monotonic;assumption.
     intros ;apply P_id_a__U83_monotonic;assumption.
     intros ;apply P_id_isNatIList_monotonic;assumption.
     intros ;apply P_id_a__zeros_bounded;assumption.
     intros ;apply P_id_U51_bounded;assumption.
     intros ;apply P_id_a__U84_bounded;assumption.
     intros ;apply P_id_a__U32_bounded;assumption.
     intros ;apply P_id_isNatList_bounded;assumption.
     intros ;apply P_id_a__U13_bounded;assumption.
     intros ;apply P_id_U85_bounded;assumption.
     intros ;apply P_id_a__length_bounded;assumption.
     intros ;apply P_id_a__isNatIList_bounded;assumption.
     intros ;apply P_id_U33_bounded;assumption.
     intros ;apply P_id_a__U11_bounded;assumption.
     intros ;apply P_id_U81_bounded;assumption.
     intros ;apply P_id_a__U92_bounded;assumption.
     intros ;apply P_id_a__U43_bounded;assumption.
     intros ;apply P_id_U23_bounded;assumption.
     intros ;apply P_id_a__isNatKind_bounded;assumption.
     intros ;apply P_id_U93_bounded;assumption.
     intros ;apply P_id_U11_bounded;assumption.
     intros ;apply P_id_a__U71_bounded;assumption.
     intros ;apply P_id_U44_bounded;assumption.
     intros ;apply P_id_0_bounded;assumption.
     intros ;apply P_id_U61_bounded;assumption.
     intros ;apply P_id_a__U86_bounded;assumption.
     intros ;apply P_id_a__U41_bounded;assumption.
     intros ;apply P_id_U22_bounded;assumption.
     intros ;apply P_id_a__U21_bounded;assumption.
     intros ;apply P_id_U91_bounded;assumption.
     intros ;apply P_id_length_bounded;assumption.
     intros ;apply P_id_a__U52_bounded;assumption.
     intros ;apply P_id_U42_bounded;assumption.
     intros ;apply P_id_a__U12_bounded;assumption.
     intros ;apply P_id_U83_bounded;assumption.
     intros ;apply P_id_a__U94_bounded;assumption.
     intros ;apply P_id_a__U45_bounded;assumption.
     intros ;apply P_id_U31_bounded;assumption.
     intros ;apply P_id_a__isNat_bounded;assumption.
     intros ;apply P_id_isNatIListKind_bounded;assumption.
     intros ;apply P_id_a__U82_bounded;assumption.
     intros ;apply P_id_U46_bounded;assumption.
     intros ;apply P_id_cons_bounded;assumption.
     intros ;apply P_id_U52_bounded;assumption.
     intros ;apply P_id_a__U85_bounded;assumption.
     intros ;apply P_id_a__U33_bounded;assumption.
     intros ;apply P_id_U21_bounded;assumption.
     intros ;apply P_id_a__isNatList_bounded;assumption.
     intros ;apply P_id_U86_bounded;assumption.
     intros ;apply P_id_mark_bounded;assumption.
     intros ;apply P_id_a__U51_bounded;assumption.
     intros ;apply P_id_U41_bounded;assumption.
     intros ;apply P_id_tt_bounded;assumption.
     intros ;apply P_id_U82_bounded;assumption.
     intros ;apply P_id_a__U93_bounded;assumption.
     intros ;apply P_id_a__U44_bounded;assumption.
     intros ;apply P_id_isNat_bounded;assumption.
     intros ;apply P_id_a__U23_bounded;assumption.
     intros ;apply P_id_U94_bounded;assumption.
     intros ;apply P_id_U12_bounded;assumption.
     intros ;apply P_id_a__U81_bounded;assumption.
     intros ;apply P_id_U45_bounded;assumption.
     intros ;apply P_id_zeros_bounded;assumption.
     intros ;apply P_id_U71_bounded;assumption.
     intros ;apply P_id_a__U91_bounded;assumption.
     intros ;apply P_id_a__U42_bounded;assumption.
     intros ;apply P_id_isNatKind_bounded;assumption.
     intros ;apply P_id_a__U22_bounded;assumption.
     intros ;apply P_id_U92_bounded;assumption.
     intros ;apply P_id_nil_bounded;assumption.
     intros ;apply P_id_a__U61_bounded;assumption.
     intros ;apply P_id_U43_bounded;assumption.
     intros ;apply P_id_a__isNatIListKind_bounded;assumption.
     intros ;apply P_id_U84_bounded;assumption.
     intros ;apply P_id_s_bounded;assumption.
     intros ;apply P_id_a__U46_bounded;assumption.
     intros ;apply P_id_U32_bounded;assumption.
     intros ;apply P_id_a__U31_bounded;assumption.
     intros ;apply P_id_U13_bounded;assumption.
     intros ;apply P_id_a__U83_bounded;assumption.
     intros ;apply P_id_isNatIList_bounded;assumption.
     apply rules_monotonic.
     intros ;apply P_id_A__U46_monotonic;assumption.
     intros ;apply P_id_A__ISNATILISTKIND_monotonic;assumption.
     intros ;apply P_id_A__U61_monotonic;assumption.
     intros ;apply P_id_A__U85_monotonic;assumption.
     intros ;apply P_id_A__U31_monotonic;assumption.
     intros ;apply P_id_A__U81_monotonic;assumption.
     intros ;apply P_id_A__U22_monotonic;assumption.
     intros ;apply P_id_A__U93_monotonic;assumption.
     intros ;apply P_id_A__U42_monotonic;assumption.
     intros ;apply P_id_A__U51_monotonic;assumption.
     intros ;apply P_id_A__ISNATLIST_monotonic;assumption.
     intros ;apply P_id_A__U91_monotonic;assumption.
     intros ;apply P_id_A__U33_monotonic;assumption.
     intros ;apply P_id_A__U83_monotonic;assumption.
     intros ;apply P_id_A__U23_monotonic;assumption.
     intros ;apply P_id_A__U11_monotonic;assumption.
     intros ;apply P_id_A__LENGTH_monotonic;assumption.
     intros ;apply P_id_A__U44_monotonic;assumption.
     intros ;apply P_id_A__ISNATILIST_monotonic;assumption.
     intros ;apply P_id_A__U13_monotonic;assumption.
     intros ;apply P_id_A__U71_monotonic;assumption.
     intros ;apply P_id_A__U86_monotonic;assumption.
     intros ;apply P_id_A__U32_monotonic;assumption.
     intros ;apply P_id_A__U82_monotonic;assumption.
     intros ;apply P_id_A__ISNATKIND_monotonic;assumption.
     intros ;apply P_id_A__U94_monotonic;assumption.
     intros ;apply P_id_A__U43_monotonic;assumption.
     intros ;apply P_id_A__U52_monotonic;assumption.
     intros ;apply P_id_A__U21_monotonic;assumption.
     intros ;apply P_id_A__U92_monotonic;assumption.
     intros ;apply P_id_A__U41_monotonic;assumption.
     intros ;apply P_id_A__U84_monotonic;assumption.
     intros ;apply P_id_A__ISNAT_monotonic;assumption.
     intros ;apply P_id_A__U12_monotonic;assumption.
     intros ;apply P_id_MARK_monotonic;assumption.
     intros ;apply P_id_A__U45_monotonic;assumption.
   Qed.
   
   Ltac rewrite_and_unfold  :=
    do 2 (rewrite marked_measure_equation);
     repeat (
     match goal with
       |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
        rewrite (measure_equation (algebra.Alg.Term f t))
       | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ =>
        rewrite (measure_equation (algebra.Alg.Term f t)) in H|-
       end
     ).
   
   Definition lt a b := (Zwf.Zwf 0) (marked_measure a) (marked_measure b).
   
   Definition le a b := marked_measure a <= marked_measure b.
   
   Lemma lt_le_compat : forall a b c, (lt a b) ->(le b c) ->lt a c.
   Proof.
     unfold lt, le in *.
     intros a b c.
     apply (interp.le_lt_compat_right (interp.o_Z 0)).
   Qed.
   
   Lemma wf_lt : well_founded lt.
   Proof.
     unfold lt in *.
     apply Inverse_Image.wf_inverse_image with  (B:=Z).
     apply Zwf.Zwf_well_founded.
   Qed.
   
   Lemma DP_R_xml_0_scc_65_strict_in_lt :
    Relation_Definitions.inclusion _ DP_R_xml_0_scc_65_strict lt.
   Proof.
     unfold Relation_Definitions.inclusion, lt in *.
     
     intros a b H;destruct H;
      match goal with
        |  |- (Zwf.Zwf 0) _ (marked_measure (algebra.Alg.Term ?f ?l)) =>
         let l'' := algebra.Alg_ext.find_replacement l  in 
          ((apply (interp.le_lt_compat_right (interp.o_Z 0)) with
             (marked_measure (algebra.Alg.Term f l''));[idtac|
            apply marked_measure_star_monotonic;
             repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos);
             (assumption)||(constructor 1)]))
        end
      ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma DP_R_xml_0_scc_65_large_in_le :
    Relation_Definitions.inclusion _ DP_R_xml_0_scc_65_large le.
   Proof.
     unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *.
     
     intros a b H;destruct H;
      match goal with
        |  |- _ <= marked_measure (algebra.Alg.Term ?f ?l) =>
         let l'' := algebra.Alg_ext.find_replacement l  in 
          ((apply (interp.le_trans (interp.o_Z 0)) with
             (marked_measure (algebra.Alg.Term f l''));[idtac|
            apply marked_measure_star_monotonic;
             repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos);
             (assumption)||(constructor 1)]))
        end
      ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Definition wf_DP_R_xml_0_scc_65_large  := WF_DP_R_xml_0_scc_65_large.wf.
   
   
   Lemma wf : well_founded WF_DP_R_xml_0.DP_R_xml_0_scc_65.
   Proof.
     intros x.
     apply (well_founded_ind wf_lt).
     clear x.
     intros x.
     pattern x.
     apply (@Acc_ind _ DP_R_xml_0_scc_65_large).
     clear x.
     intros x _ IHx IHx'.
     constructor.
     intros y H.
     
     destruct H;
      (apply IHx';apply DP_R_xml_0_scc_65_strict_in_lt;
        econstructor eassumption)||
      ((apply IHx;[econstructor eassumption|
        intros y' Hlt;apply IHx';apply lt_le_compat with (1:=Hlt) ;
         apply DP_R_xml_0_scc_65_large_in_le;econstructor eassumption])).
     apply wf_DP_R_xml_0_scc_65_large.
   Qed.
  End WF_DP_R_xml_0_scc_65.
  
  Definition wf_DP_R_xml_0_scc_65  := WF_DP_R_xml_0_scc_65.wf.
  
  
  Lemma acc_DP_R_xml_0_scc_65 :
   forall x y, (DP_R_xml_0_scc_65 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x.
    pattern x.
    apply (@Acc_ind _ DP_R_xml_0_scc_65).
    intros x' _ Hrec y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply Hrec;econstructor eassumption)||
      ((eapply acc_DP_R_xml_0_non_scc_64;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_63;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((eapply acc_DP_R_xml_0_non_scc_62;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
         ((eapply acc_DP_R_xml_0_non_scc_61;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
          ((eapply acc_DP_R_xml_0_non_scc_60;
             econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
           ((eapply acc_DP_R_xml_0_non_scc_59;
              econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
            ((eapply acc_DP_R_xml_0_non_scc_56;
               econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
             ((eapply acc_DP_R_xml_0_non_scc_54;
                econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
              ((eapply acc_DP_R_xml_0_non_scc_52;
                 econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
               ((eapply acc_DP_R_xml_0_non_scc_50;
                  econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
                ((eapply acc_DP_R_xml_0_non_scc_49;
                   econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
                 ((eapply acc_DP_R_xml_0_non_scc_48;
                    econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
                  ((eapply acc_DP_R_xml_0_non_scc_47;
                     econstructor 
                     (eassumption)||(algebra.Alg_ext.star_refl' ))||
                   ((eapply acc_DP_R_xml_0_non_scc_46;
                      econstructor 
                      (eassumption)||(algebra.Alg_ext.star_refl' ))||
                    ((eapply acc_DP_R_xml_0_non_scc_45;
                       econstructor 
                       (eassumption)||(algebra.Alg_ext.star_refl' ))||
                     ((eapply acc_DP_R_xml_0_non_scc_44;
                        econstructor 
                        (eassumption)||(algebra.Alg_ext.star_refl' ))||
                      ((eapply acc_DP_R_xml_0_non_scc_43;
                         econstructor 
                         (eassumption)||(algebra.Alg_ext.star_refl' ))||
                       ((eapply acc_DP_R_xml_0_non_scc_42;
                          econstructor 
                          (eassumption)||(algebra.Alg_ext.star_refl' ))||
                        ((eapply acc_DP_R_xml_0_non_scc_41;
                           econstructor 
                           (eassumption)||(algebra.Alg_ext.star_refl' ))||
                         ((eapply acc_DP_R_xml_0_non_scc_40;
                            econstructor 
                            (eassumption)||(algebra.Alg_ext.star_refl' ))||
                          ((eapply acc_DP_R_xml_0_non_scc_39;
                             econstructor 
                             (eassumption)||(algebra.Alg_ext.star_refl' ))||
                           ((eapply acc_DP_R_xml_0_non_scc_27;
                              econstructor 
                              (eassumption)||(algebra.Alg_ext.star_refl' ))||
                            ((eapply acc_DP_R_xml_0_non_scc_24;
                               econstructor 
                               (eassumption)||(algebra.Alg_ext.star_refl' ))||
                             ((eapply acc_DP_R_xml_0_non_scc_20;
                                econstructor 
                                (eassumption)||(algebra.Alg_ext.star_refl' ))||
                              ((eapply acc_DP_R_xml_0_non_scc_17;
                                 econstructor 
                                 (eassumption)||(algebra.Alg_ext.star_refl' ))||
                               ((eapply acc_DP_R_xml_0_non_scc_15;
                                  econstructor 
                                  (eassumption)||
                                  (algebra.Alg_ext.star_refl' ))||
                                ((eapply acc_DP_R_xml_0_non_scc_9;
                                   econstructor 
                                   (eassumption)||
                                   (algebra.Alg_ext.star_refl' ))||
                                 ((eapply acc_DP_R_xml_0_non_scc_8;
                                    econstructor 
                                    (eassumption)||
                                    (algebra.Alg_ext.star_refl' ))||
                                  ((eapply acc_DP_R_xml_0_non_scc_7;
                                     econstructor 
                                     (eassumption)||
                                     (algebra.Alg_ext.star_refl' ))||
                                   ((eapply acc_DP_R_xml_0_non_scc_6;
                                      econstructor 
                                      (eassumption)||
                                      (algebra.Alg_ext.star_refl' ))||
                                    ((eapply acc_DP_R_xml_0_non_scc_5;
                                       econstructor 
                                       (eassumption)||
                                       (algebra.Alg_ext.star_refl' ))||
                                     ((eapply acc_DP_R_xml_0_non_scc_4;
                                        econstructor 
                                        (eassumption)||
                                        (algebra.Alg_ext.star_refl' ))||
                                      ((eapply acc_DP_R_xml_0_non_scc_3;
                                         econstructor 
                                         (eassumption)||
                                         (algebra.Alg_ext.star_refl' ))||
                                       ((eapply acc_DP_R_xml_0_non_scc_2;
                                          econstructor 
                                          (eassumption)||
                                          (algebra.Alg_ext.star_refl' ))||
                                        ((eapply acc_DP_R_xml_0_non_scc_1;
                                           econstructor 
                                           (eassumption)||
                                           (algebra.Alg_ext.star_refl' ))||
                                         ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
                                          (eapply Hrec;
                                            econstructor 
                                            (eassumption)||
                                            (algebra.Alg_ext.star_refl' )))))))))))))))))))))))))))))))))))))).
    apply wf_DP_R_xml_0_scc_65.
  Qed.
  
  
  Lemma wf : well_founded WF_R_xml_0_deep_rew.DP_R_xml_0.
  Proof.
    constructor;intros _y _h;inversion _h;clear _h;subst;
     (eapply acc_DP_R_xml_0_non_scc_64;
       econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
     ((eapply acc_DP_R_xml_0_non_scc_63;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_62;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_61;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((eapply acc_DP_R_xml_0_non_scc_60;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
         ((eapply acc_DP_R_xml_0_non_scc_59;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
          ((eapply acc_DP_R_xml_0_non_scc_58;
             econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
           ((eapply acc_DP_R_xml_0_non_scc_57;
              econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
            ((eapply acc_DP_R_xml_0_non_scc_56;
               econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
             ((eapply acc_DP_R_xml_0_non_scc_55;
                econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
              ((eapply acc_DP_R_xml_0_non_scc_54;
                 econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
               ((eapply acc_DP_R_xml_0_non_scc_53;
                  econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
                ((eapply acc_DP_R_xml_0_non_scc_52;
                   econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
                 ((eapply acc_DP_R_xml_0_non_scc_51;
                    econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
                  ((eapply acc_DP_R_xml_0_non_scc_50;
                     econstructor 
                     (eassumption)||(algebra.Alg_ext.star_refl' ))||
                   ((eapply acc_DP_R_xml_0_non_scc_49;
                      econstructor 
                      (eassumption)||(algebra.Alg_ext.star_refl' ))||
                    ((eapply acc_DP_R_xml_0_non_scc_48;
                       econstructor 
                       (eassumption)||(algebra.Alg_ext.star_refl' ))||
                     ((eapply acc_DP_R_xml_0_non_scc_47;
                        econstructor 
                        (eassumption)||(algebra.Alg_ext.star_refl' ))||
                      ((eapply acc_DP_R_xml_0_non_scc_46;
                         econstructor 
                         (eassumption)||(algebra.Alg_ext.star_refl' ))||
                       ((eapply acc_DP_R_xml_0_non_scc_45;
                          econstructor 
                          (eassumption)||(algebra.Alg_ext.star_refl' ))||
                        ((eapply acc_DP_R_xml_0_non_scc_44;
                           econstructor 
                           (eassumption)||(algebra.Alg_ext.star_refl' ))||
                         ((eapply acc_DP_R_xml_0_non_scc_43;
                            econstructor 
                            (eassumption)||(algebra.Alg_ext.star_refl' ))||
                          ((eapply acc_DP_R_xml_0_non_scc_42;
                             econstructor 
                             (eassumption)||(algebra.Alg_ext.star_refl' ))||
                           ((eapply acc_DP_R_xml_0_non_scc_41;
                              econstructor 
                              (eassumption)||(algebra.Alg_ext.star_refl' ))||
                            ((eapply acc_DP_R_xml_0_non_scc_40;
                               econstructor 
                               (eassumption)||(algebra.Alg_ext.star_refl' ))||
                             ((eapply acc_DP_R_xml_0_non_scc_39;
                                econstructor 
                                (eassumption)||(algebra.Alg_ext.star_refl' ))||
                              ((eapply acc_DP_R_xml_0_non_scc_38;
                                 econstructor 
                                 (eassumption)||(algebra.Alg_ext.star_refl' ))||
                               ((eapply acc_DP_R_xml_0_non_scc_37;
                                  econstructor 
                                  (eassumption)||
                                  (algebra.Alg_ext.star_refl' ))||
                                ((eapply acc_DP_R_xml_0_non_scc_36;
                                   econstructor 
                                   (eassumption)||
                                   (algebra.Alg_ext.star_refl' ))||
                                 ((eapply acc_DP_R_xml_0_non_scc_35;
                                    econstructor 
                                    (eassumption)||
                                    (algebra.Alg_ext.star_refl' ))||
                                  ((eapply acc_DP_R_xml_0_non_scc_34;
                                     econstructor 
                                     (eassumption)||
                                     (algebra.Alg_ext.star_refl' ))||
                                   ((eapply acc_DP_R_xml_0_non_scc_33;
                                      econstructor 
                                      (eassumption)||
                                      (algebra.Alg_ext.star_refl' ))||
                                    ((eapply acc_DP_R_xml_0_non_scc_32;
                                       econstructor 
                                       (eassumption)||
                                       (algebra.Alg_ext.star_refl' ))||
                                     ((eapply acc_DP_R_xml_0_non_scc_31;
                                        econstructor 
                                        (eassumption)||
                                        (algebra.Alg_ext.star_refl' ))||
                                      ((eapply acc_DP_R_xml_0_non_scc_30;
                                         econstructor 
                                         (eassumption)||
                                         (algebra.Alg_ext.star_refl' ))||
                                       ((eapply acc_DP_R_xml_0_non_scc_29;
                                          econstructor 
                                          (eassumption)||
                                          (algebra.Alg_ext.star_refl' ))||
                                        ((eapply acc_DP_R_xml_0_non_scc_28;
                                           econstructor 
                                           (eassumption)||
                                           (algebra.Alg_ext.star_refl' ))||
                                         ((eapply acc_DP_R_xml_0_non_scc_27;
                                            econstructor 
                                            (eassumption)||
                                            (algebra.Alg_ext.star_refl' ))||
                                          ((eapply acc_DP_R_xml_0_non_scc_26;
                                             econstructor 
                                             (eassumption)||
                                             (algebra.Alg_ext.star_refl' ))||
                                           ((eapply acc_DP_R_xml_0_non_scc_25;
                                              econstructor 
                                              (eassumption)||
                                              (algebra.Alg_ext.star_refl' ))||
                                            ((eapply acc_DP_R_xml_0_non_scc_24;
                                               
                                               econstructor 
                                               (eassumption)||
                                               (algebra.Alg_ext.star_refl' ))||
                                             ((eapply acc_DP_R_xml_0_non_scc_23;
                                                
                                                econstructor 
                                                (eassumption)||
                                                (algebra.Alg_ext.star_refl' ))||
                                              ((eapply acc_DP_R_xml_0_non_scc_22;
                                                 
                                                 econstructor 
                                                 (eassumption)||
                                                 (algebra.Alg_ext.star_refl' ))||
                                               ((eapply acc_DP_R_xml_0_non_scc_21;
                                                  
                                                  econstructor 
                                                  (eassumption)||
                                                  (algebra.Alg_ext.star_refl' ))||
                                                ((eapply acc_DP_R_xml_0_non_scc_20;
                                                   
                                                   econstructor 
                                                   (eassumption)||
                                                   (algebra.Alg_ext.star_refl' ))||
                                                 ((eapply acc_DP_R_xml_0_non_scc_19;
                                                    
                                                    econstructor 
                                                    (eassumption)||
                                                    (algebra.Alg_ext.star_refl' ))||
                                                  ((eapply acc_DP_R_xml_0_non_scc_18;
                                                     
                                                     econstructor 
                                                     (eassumption)||
                                                     (algebra.Alg_ext.star_refl' ))||
                                                   ((eapply acc_DP_R_xml_0_non_scc_17;
                                                      
                                                      econstructor 
                                                      (eassumption)||
                                                      (algebra.Alg_ext.star_refl' ))||
                                                    ((eapply acc_DP_R_xml_0_non_scc_16;
                                                       
                                                       econstructor 
                                                       (eassumption)||
                                                       (algebra.Alg_ext.star_refl' ))||
                                                     ((eapply acc_DP_R_xml_0_non_scc_15;
                                                        
                                                        econstructor 
                                                        (eassumption)||
                                                        (algebra.Alg_ext.star_refl' ))||
                                                      ((eapply acc_DP_R_xml_0_non_scc_14;
                                                         
                                                         econstructor 
                                                         (eassumption)||
                                                         (algebra.Alg_ext.star_refl' ))||
                                                       ((eapply acc_DP_R_xml_0_non_scc_13;
                                                          
                                                          econstructor 
                                                          (eassumption)||
                                                          (algebra.Alg_ext.star_refl' ))||
                                                        ((eapply acc_DP_R_xml_0_non_scc_12;
                                                           
                                                           econstructor 
                                                           (eassumption)||
                                                           (algebra.Alg_ext.star_refl' ))||
                                                         ((eapply acc_DP_R_xml_0_non_scc_11;
                                                            
                                                            econstructor 
                                                            (eassumption)||
                                                            (algebra.Alg_ext.star_refl' ))||
                                                          ((eapply acc_DP_R_xml_0_non_scc_10;
                                                             
                                                             econstructor 
                                                             (eassumption)||
                                                             (algebra.Alg_ext.star_refl' ))||
                                                           ((eapply acc_DP_R_xml_0_non_scc_9;
                                                              
                                                              econstructor 
                                                              (eassumption)||
                                                              (algebra.Alg_ext.star_refl' ))||
                                                            ((eapply 
                                                              acc_DP_R_xml_0_non_scc_8;
                                                               
                                                               econstructor 
                                                               (eassumption)||
                                                               (algebra.Alg_ext.star_refl' ))||
                                                             ((eapply 
                                                               acc_DP_R_xml_0_non_scc_7;
                                                                
                                                                econstructor 
                                                                (eassumption)||
                                                                (algebra.Alg_ext.star_refl' ))||
                                                              ((eapply 
                                                                acc_DP_R_xml_0_non_scc_6;
                                                                 
                                                                 econstructor 
                                                                 (eassumption)||
                                                                 (algebra.Alg_ext.star_refl' ))||
                                                               ((eapply 
                                                                 acc_DP_R_xml_0_non_scc_5;
                                                                  
                                                                  econstructor 
                                                                  (eassumption)||
                                                                  (algebra.Alg_ext.star_refl' ))||
                                                                ((eapply 
                                                                  acc_DP_R_xml_0_non_scc_4;
                                                                   
                                                                   econstructor 
                                                                   (eassumption)||
                                                                   (algebra.Alg_ext.star_refl' ))||
                                                                 ((eapply 
                                                                   acc_DP_R_xml_0_non_scc_3;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                  ((eapply 
                                                                    acc_DP_R_xml_0_non_scc_2;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                   ((
                                                                    eapply 
                                                                    acc_DP_R_xml_0_non_scc_1;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_non_scc_0;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_65;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_64;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_63;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_62;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_61;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_60;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_59;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_58;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_57;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_56;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_55;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_54;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_53;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_52;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_51;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_50;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_49;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_48;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_47;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_46;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_45;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_44;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_43;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_42;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_41;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_40;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_39;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_38;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_37;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_36;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_35;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_34;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_33;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_32;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_31;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_30;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_29;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_28;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_27;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_26;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_25;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_24;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_23;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_22;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_21;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_20;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_19;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_18;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_17;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_16;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_15;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_14;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_13;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_12;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_11;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_10;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_9;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_8;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_7;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_6;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_5;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_4;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_3;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_2;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_1;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_0;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
                                                                    (
                                                                    fail)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))).
  Qed.
 End WF_DP_R_xml_0.
 
 Definition wf_H  := WF_DP_R_xml_0.wf.
 
 Lemma wf :
  well_founded (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules).
 Proof.
   apply ddp.dp_criterion.
   apply R_xml_0_deep_rew.R_xml_0_non_var.
   apply R_xml_0_deep_rew.R_xml_0_reg.
   
   intros ;
    apply (ddp.constructor_defined_dec _ _ 
            R_xml_0_deep_rew.R_xml_0_rules_included).
   refine (Inclusion.wf_incl _ _ _ _ wf_H).
   intros x y H.
   destruct (R_xml_0_dp_step_spec H) as [f [l1 [l2 [H1 [H2 H3]]]]].
   
   destruct (ddp.dp_list_complete _ _ 
              R_xml_0_deep_rew.R_xml_0_rules_included _ _ H3)
    as [x' [y' [sigma [h1 [h2 h3]]]]].
   clear H3.
   subst.
   vm_compute in h3|-.
   let e := type of h3 in (dp_concl_tac h2 h3 ltac:(fun _ => idtac) e).
 Qed.
End WF_R_xml_0_deep_rew.


(* 
*** Local Variables: ***
*** coq-prog-name: "coqtop" ***
*** coq-prog-args: ("-emacs-U" "-I" "$COCCINELLE/examples" "-I" "$COCCINELLE/term_algebra" "-I" "$COCCINELLE/term_orderings" "-I" "$COCCINELLE/basis" "-I" "$COCCINELLE/list_extensions" "-I" "$COCCINELLE/examples/cime_trace/") ***
*** compile-command: "coqc -I $COCCINELLE/term_algebra -I $COCCINELLE/term_orderings -I $COCCINELLE/basis -I $COCCINELLE/list_extensions -I $COCCINELLE/examples/cime_trace/ -I $COCCINELLE/examples/  c_output/strat/tpdb-5.0___TRS___TRCSR___LengthOfFiniteLists_complete_noand_GM.trs/a3pat.v" ***
*** End: ***
 *)