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_isNatKind *)
    | id_isNatKind : symb
     (* id_a__isNatIList *)
    | id_a__isNatIList : symb
     (* id_a__U21 *)
    | id_a__U21 : symb
     (* id_U41 *)
    | id_U41 : symb
     (* id_a__and *)
    | id_a__and : symb
     (* id_a__U11 *)
    | id_a__U11 : symb
     (* id_U21 *)
    | id_U21 : symb
     (* id_a__U61 *)
    | id_a__U61 : symb
     (* id_a__U32 *)
    | id_a__U32 : symb
     (* id_U51 *)
    | id_U51 : symb
     (* id_isNatIListKind *)
    | id_isNatIListKind : symb
     (* id_0 *)
    | id_0 : symb
     (* id_U12 *)
    | id_U12 : symb
     (* id_a__U52 *)
    | id_a__U52 : symb
     (* id_a__isNat *)
    | id_a__isNat : symb
     (* id_U43 *)
    | id_U43 : symb
     (* id_a__isNatIListKind *)
    | id_a__isNatIListKind : symb
     (* id_a__U12 *)
    | id_a__U12 : symb
     (* id_U31 *)
    | id_U31 : symb
     (* id_a__length *)
    | id_a__length : symb
     (* id_a__U42 *)
    | id_a__U42 : symb
     (* id_U53 *)
    | id_U53 : symb
     (* id_and *)
    | id_and : symb
     (* id_cons *)
    | id_cons : symb
     (* id_U11 *)
    | id_U11 : symb
     (* id_a__U51 *)
    | id_a__U51 : symb
     (* id_a__U22 *)
    | id_a__U22 : symb
     (* id_U42 *)
    | id_U42 : symb
     (* id_length *)
    | id_length : symb
     (* id_tt *)
    | id_tt : symb
     (* id_U22 *)
    | id_U22 : symb
     (* id_s *)
    | id_s : symb
     (* id_a__U41 *)
    | id_a__U41 : symb
     (* id_U52 *)
    | id_U52 : symb
     (* id_nil *)
    | id_nil : symb
     (* id_zeros *)
    | id_zeros : symb
     (* id_isNatList *)
    | id_isNatList : symb
     (* id_a__U53 *)
    | id_a__U53 : symb
     (* id_a__U31 *)
    | id_a__U31 : symb
     (* id_isNatIList *)
    | id_isNatIList : symb
     (* id_a__isNatKind *)
    | id_a__isNatKind : symb
     (* id_a__isNatList *)
    | id_a__isNatList : symb
     (* id_U32 *)
    | id_U32 : symb
     (* id_mark *)
    | id_mark : symb
     (* id_a__U43 *)
    | id_a__U43 : symb
     (* id_U61 *)
    | id_U61 : symb
     (* id_isNat *)
    | id_isNat : symb
  .
  
  
  Definition symb_eq_bool (f1 f2:symb) : bool := 
    match f1,f2 with
      | id_a__zeros,id_a__zeros => true
      | id_isNatKind,id_isNatKind => true
      | id_a__isNatIList,id_a__isNatIList => true
      | id_a__U21,id_a__U21 => true
      | id_U41,id_U41 => true
      | id_a__and,id_a__and => true
      | id_a__U11,id_a__U11 => true
      | id_U21,id_U21 => true
      | id_a__U61,id_a__U61 => true
      | id_a__U32,id_a__U32 => true
      | id_U51,id_U51 => true
      | id_isNatIListKind,id_isNatIListKind => true
      | id_0,id_0 => true
      | id_U12,id_U12 => true
      | id_a__U52,id_a__U52 => true
      | id_a__isNat,id_a__isNat => true
      | id_U43,id_U43 => true
      | id_a__isNatIListKind,id_a__isNatIListKind => true
      | id_a__U12,id_a__U12 => true
      | id_U31,id_U31 => true
      | id_a__length,id_a__length => true
      | id_a__U42,id_a__U42 => true
      | id_U53,id_U53 => true
      | id_and,id_and => true
      | id_cons,id_cons => true
      | id_U11,id_U11 => true
      | id_a__U51,id_a__U51 => true
      | id_a__U22,id_a__U22 => true
      | id_U42,id_U42 => true
      | id_length,id_length => true
      | id_tt,id_tt => true
      | id_U22,id_U22 => true
      | id_s,id_s => true
      | id_a__U41,id_a__U41 => true
      | id_U52,id_U52 => true
      | id_nil,id_nil => true
      | id_zeros,id_zeros => true
      | id_isNatList,id_isNatList => true
      | id_a__U53,id_a__U53 => true
      | id_a__U31,id_a__U31 => true
      | id_isNatIList,id_isNatIList => true
      | id_a__isNatKind,id_a__isNatKind => true
      | id_a__isNatList,id_a__isNatList => true
      | id_U32,id_U32 => true
      | id_mark,id_mark => true
      | id_a__U43,id_a__U43 => true
      | id_U61,id_U61 => true
      | id_isNat,id_isNat => 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_isNatKind,id_isNatKind => refl_equal _
             | id_a__isNatIList,id_a__isNatIList => refl_equal _
             | id_a__U21,id_a__U21 => refl_equal _
             | id_U41,id_U41 => refl_equal _
             | id_a__and,id_a__and => refl_equal _
             | id_a__U11,id_a__U11 => refl_equal _
             | id_U21,id_U21 => refl_equal _
             | id_a__U61,id_a__U61 => refl_equal _
             | id_a__U32,id_a__U32 => refl_equal _
             | id_U51,id_U51 => refl_equal _
             | id_isNatIListKind,id_isNatIListKind => refl_equal _
             | id_0,id_0 => refl_equal _
             | id_U12,id_U12 => refl_equal _
             | id_a__U52,id_a__U52 => refl_equal _
             | id_a__isNat,id_a__isNat => refl_equal _
             | id_U43,id_U43 => refl_equal _
             | id_a__isNatIListKind,id_a__isNatIListKind => refl_equal _
             | id_a__U12,id_a__U12 => refl_equal _
             | id_U31,id_U31 => refl_equal _
             | id_a__length,id_a__length => refl_equal _
             | id_a__U42,id_a__U42 => refl_equal _
             | id_U53,id_U53 => refl_equal _
             | id_and,id_and => refl_equal _
             | id_cons,id_cons => refl_equal _
             | id_U11,id_U11 => refl_equal _
             | id_a__U51,id_a__U51 => refl_equal _
             | id_a__U22,id_a__U22 => refl_equal _
             | id_U42,id_U42 => refl_equal _
             | id_length,id_length => refl_equal _
             | id_tt,id_tt => refl_equal _
             | id_U22,id_U22 => refl_equal _
             | id_s,id_s => refl_equal _
             | id_a__U41,id_a__U41 => refl_equal _
             | id_U52,id_U52 => refl_equal _
             | id_nil,id_nil => refl_equal _
             | id_zeros,id_zeros => refl_equal _
             | id_isNatList,id_isNatList => refl_equal _
             | id_a__U53,id_a__U53 => refl_equal _
             | id_a__U31,id_a__U31 => refl_equal _
             | id_isNatIList,id_isNatIList => refl_equal _
             | id_a__isNatKind,id_a__isNatKind => refl_equal _
             | id_a__isNatList,id_a__isNatList => refl_equal _
             | id_U32,id_U32 => refl_equal _
             | id_mark,id_mark => refl_equal _
             | id_a__U43,id_a__U43 => refl_equal _
             | id_U61,id_U61 => refl_equal _
             | id_isNat,id_isNat => refl_equal _
             | _,_ => _
             end;intros abs;discriminate.
  Defined.
  
  
  Definition arity (f:symb) := 
    match f with
      | id_a__zeros => term.Free 0
      | id_isNatKind => term.Free 1
      | id_a__isNatIList => term.Free 1
      | id_a__U21 => term.Free 2
      | id_U41 => term.Free 3
      | id_a__and => term.Free 2
      | id_a__U11 => term.Free 2
      | id_U21 => term.Free 2
      | id_a__U61 => term.Free 2
      | id_a__U32 => term.Free 1
      | id_U51 => term.Free 3
      | id_isNatIListKind => term.Free 1
      | id_0 => term.Free 0
      | id_U12 => term.Free 1
      | id_a__U52 => term.Free 2
      | id_a__isNat => term.Free 1
      | id_U43 => term.Free 1
      | id_a__isNatIListKind => term.Free 1
      | id_a__U12 => term.Free 1
      | id_U31 => term.Free 2
      | id_a__length => term.Free 1
      | id_a__U42 => term.Free 2
      | id_U53 => term.Free 1
      | id_and => term.Free 2
      | id_cons => term.Free 2
      | id_U11 => term.Free 2
      | id_a__U51 => term.Free 3
      | id_a__U22 => term.Free 1
      | id_U42 => term.Free 2
      | id_length => term.Free 1
      | id_tt => term.Free 0
      | id_U22 => term.Free 1
      | id_s => term.Free 1
      | id_a__U41 => term.Free 3
      | id_U52 => term.Free 2
      | id_nil => term.Free 0
      | id_zeros => term.Free 0
      | id_isNatList => term.Free 1
      | id_a__U53 => term.Free 1
      | id_a__U31 => term.Free 2
      | id_isNatIList => term.Free 1
      | id_a__isNatKind => term.Free 1
      | id_a__isNatList => term.Free 1
      | id_U32 => term.Free 1
      | id_mark => term.Free 1
      | id_a__U43 => term.Free 1
      | id_U61 => term.Free 2
      | id_isNat => 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_isNatKind => false
      | id_a__zeros,id_a__isNatIList => false
      | id_a__zeros,id_a__U21 => false
      | id_a__zeros,id_U41 => false
      | id_a__zeros,id_a__and => false
      | id_a__zeros,id_a__U11 => false
      | id_a__zeros,id_U21 => false
      | id_a__zeros,id_a__U61 => false
      | id_a__zeros,id_a__U32 => false
      | id_a__zeros,id_U51 => false
      | id_a__zeros,id_isNatIListKind => false
      | id_a__zeros,id_0 => false
      | id_a__zeros,id_U12 => false
      | id_a__zeros,id_a__U52 => false
      | id_a__zeros,id_a__isNat => false
      | id_a__zeros,id_U43 => false
      | id_a__zeros,id_a__isNatIListKind => false
      | id_a__zeros,id_a__U12 => false
      | id_a__zeros,id_U31 => false
      | id_a__zeros,id_a__length => false
      | id_a__zeros,id_a__U42 => false
      | id_a__zeros,id_U53 => false
      | id_a__zeros,id_and => false
      | id_a__zeros,id_cons => false
      | id_a__zeros,id_U11 => false
      | id_a__zeros,id_a__U51 => false
      | id_a__zeros,id_a__U22 => false
      | id_a__zeros,id_U42 => false
      | id_a__zeros,id_length => false
      | id_a__zeros,id_tt => false
      | id_a__zeros,id_U22 => false
      | id_a__zeros,id_s => false
      | id_a__zeros,id_a__U41 => false
      | id_a__zeros,id_U52 => false
      | id_a__zeros,id_nil => false
      | id_a__zeros,id_zeros => false
      | id_a__zeros,id_isNatList => false
      | id_a__zeros,id_a__U53 => false
      | id_a__zeros,id_a__U31 => false
      | id_a__zeros,id_isNatIList => false
      | id_a__zeros,id_a__isNatKind => false
      | id_a__zeros,id_a__isNatList => false
      | id_a__zeros,id_U32 => false
      | id_a__zeros,id_mark => false
      | id_a__zeros,id_a__U43 => false
      | id_a__zeros,id_U61 => false
      | id_a__zeros,id_isNat => false
      | id_isNatKind,id_a__zeros => true
      | id_isNatKind,id_isNatKind => true
      | id_isNatKind,id_a__isNatIList => false
      | id_isNatKind,id_a__U21 => false
      | id_isNatKind,id_U41 => false
      | id_isNatKind,id_a__and => false
      | id_isNatKind,id_a__U11 => false
      | id_isNatKind,id_U21 => false
      | id_isNatKind,id_a__U61 => false
      | id_isNatKind,id_a__U32 => false
      | id_isNatKind,id_U51 => false
      | id_isNatKind,id_isNatIListKind => false
      | id_isNatKind,id_0 => false
      | id_isNatKind,id_U12 => false
      | id_isNatKind,id_a__U52 => false
      | id_isNatKind,id_a__isNat => false
      | id_isNatKind,id_U43 => false
      | id_isNatKind,id_a__isNatIListKind => false
      | id_isNatKind,id_a__U12 => false
      | id_isNatKind,id_U31 => false
      | id_isNatKind,id_a__length => false
      | id_isNatKind,id_a__U42 => false
      | id_isNatKind,id_U53 => false
      | id_isNatKind,id_and => false
      | id_isNatKind,id_cons => false
      | id_isNatKind,id_U11 => false
      | id_isNatKind,id_a__U51 => false
      | id_isNatKind,id_a__U22 => false
      | id_isNatKind,id_U42 => false
      | id_isNatKind,id_length => false
      | id_isNatKind,id_tt => false
      | id_isNatKind,id_U22 => false
      | id_isNatKind,id_s => false
      | id_isNatKind,id_a__U41 => false
      | id_isNatKind,id_U52 => false
      | id_isNatKind,id_nil => false
      | id_isNatKind,id_zeros => false
      | id_isNatKind,id_isNatList => false
      | id_isNatKind,id_a__U53 => false
      | id_isNatKind,id_a__U31 => false
      | id_isNatKind,id_isNatIList => false
      | id_isNatKind,id_a__isNatKind => false
      | id_isNatKind,id_a__isNatList => false
      | id_isNatKind,id_U32 => false
      | id_isNatKind,id_mark => false
      | id_isNatKind,id_a__U43 => false
      | id_isNatKind,id_U61 => false
      | id_isNatKind,id_isNat => false
      | id_a__isNatIList,id_a__zeros => true
      | id_a__isNatIList,id_isNatKind => true
      | id_a__isNatIList,id_a__isNatIList => true
      | id_a__isNatIList,id_a__U21 => false
      | id_a__isNatIList,id_U41 => false
      | id_a__isNatIList,id_a__and => false
      | id_a__isNatIList,id_a__U11 => false
      | id_a__isNatIList,id_U21 => false
      | id_a__isNatIList,id_a__U61 => false
      | id_a__isNatIList,id_a__U32 => false
      | id_a__isNatIList,id_U51 => false
      | id_a__isNatIList,id_isNatIListKind => false
      | id_a__isNatIList,id_0 => false
      | id_a__isNatIList,id_U12 => false
      | id_a__isNatIList,id_a__U52 => false
      | id_a__isNatIList,id_a__isNat => false
      | id_a__isNatIList,id_U43 => false
      | id_a__isNatIList,id_a__isNatIListKind => false
      | id_a__isNatIList,id_a__U12 => false
      | id_a__isNatIList,id_U31 => false
      | id_a__isNatIList,id_a__length => false
      | id_a__isNatIList,id_a__U42 => false
      | id_a__isNatIList,id_U53 => false
      | id_a__isNatIList,id_and => false
      | id_a__isNatIList,id_cons => false
      | id_a__isNatIList,id_U11 => false
      | id_a__isNatIList,id_a__U51 => false
      | id_a__isNatIList,id_a__U22 => false
      | id_a__isNatIList,id_U42 => false
      | id_a__isNatIList,id_length => false
      | id_a__isNatIList,id_tt => false
      | id_a__isNatIList,id_U22 => false
      | id_a__isNatIList,id_s => false
      | id_a__isNatIList,id_a__U41 => false
      | id_a__isNatIList,id_U52 => false
      | id_a__isNatIList,id_nil => false
      | id_a__isNatIList,id_zeros => false
      | id_a__isNatIList,id_isNatList => false
      | id_a__isNatIList,id_a__U53 => false
      | id_a__isNatIList,id_a__U31 => false
      | id_a__isNatIList,id_isNatIList => false
      | id_a__isNatIList,id_a__isNatKind => false
      | id_a__isNatIList,id_a__isNatList => false
      | id_a__isNatIList,id_U32 => false
      | id_a__isNatIList,id_mark => false
      | id_a__isNatIList,id_a__U43 => false
      | id_a__isNatIList,id_U61 => false
      | id_a__isNatIList,id_isNat => false
      | id_a__U21,id_a__zeros => true
      | id_a__U21,id_isNatKind => true
      | id_a__U21,id_a__isNatIList => true
      | id_a__U21,id_a__U21 => true
      | id_a__U21,id_U41 => false
      | id_a__U21,id_a__and => false
      | id_a__U21,id_a__U11 => false
      | id_a__U21,id_U21 => false
      | id_a__U21,id_a__U61 => false
      | id_a__U21,id_a__U32 => false
      | id_a__U21,id_U51 => false
      | id_a__U21,id_isNatIListKind => false
      | id_a__U21,id_0 => false
      | id_a__U21,id_U12 => false
      | id_a__U21,id_a__U52 => false
      | id_a__U21,id_a__isNat => false
      | id_a__U21,id_U43 => false
      | id_a__U21,id_a__isNatIListKind => false
      | id_a__U21,id_a__U12 => false
      | id_a__U21,id_U31 => false
      | id_a__U21,id_a__length => false
      | id_a__U21,id_a__U42 => false
      | id_a__U21,id_U53 => false
      | id_a__U21,id_and => false
      | id_a__U21,id_cons => false
      | id_a__U21,id_U11 => false
      | id_a__U21,id_a__U51 => false
      | id_a__U21,id_a__U22 => false
      | id_a__U21,id_U42 => false
      | id_a__U21,id_length => false
      | id_a__U21,id_tt => false
      | id_a__U21,id_U22 => false
      | id_a__U21,id_s => false
      | id_a__U21,id_a__U41 => false
      | id_a__U21,id_U52 => false
      | id_a__U21,id_nil => false
      | id_a__U21,id_zeros => false
      | id_a__U21,id_isNatList => false
      | id_a__U21,id_a__U53 => false
      | id_a__U21,id_a__U31 => false
      | id_a__U21,id_isNatIList => false
      | id_a__U21,id_a__isNatKind => false
      | id_a__U21,id_a__isNatList => false
      | id_a__U21,id_U32 => false
      | id_a__U21,id_mark => false
      | id_a__U21,id_a__U43 => false
      | id_a__U21,id_U61 => false
      | id_a__U21,id_isNat => false
      | id_U41,id_a__zeros => true
      | id_U41,id_isNatKind => true
      | id_U41,id_a__isNatIList => true
      | id_U41,id_a__U21 => true
      | id_U41,id_U41 => true
      | id_U41,id_a__and => false
      | id_U41,id_a__U11 => false
      | id_U41,id_U21 => false
      | id_U41,id_a__U61 => false
      | id_U41,id_a__U32 => false
      | id_U41,id_U51 => false
      | id_U41,id_isNatIListKind => false
      | id_U41,id_0 => false
      | id_U41,id_U12 => false
      | id_U41,id_a__U52 => false
      | id_U41,id_a__isNat => false
      | id_U41,id_U43 => false
      | id_U41,id_a__isNatIListKind => false
      | id_U41,id_a__U12 => false
      | id_U41,id_U31 => false
      | id_U41,id_a__length => false
      | id_U41,id_a__U42 => false
      | id_U41,id_U53 => false
      | id_U41,id_and => false
      | id_U41,id_cons => false
      | id_U41,id_U11 => false
      | id_U41,id_a__U51 => false
      | id_U41,id_a__U22 => false
      | id_U41,id_U42 => false
      | id_U41,id_length => false
      | id_U41,id_tt => false
      | id_U41,id_U22 => false
      | id_U41,id_s => false
      | id_U41,id_a__U41 => false
      | id_U41,id_U52 => false
      | id_U41,id_nil => false
      | id_U41,id_zeros => false
      | id_U41,id_isNatList => false
      | id_U41,id_a__U53 => false
      | id_U41,id_a__U31 => false
      | id_U41,id_isNatIList => false
      | id_U41,id_a__isNatKind => false
      | id_U41,id_a__isNatList => false
      | id_U41,id_U32 => false
      | id_U41,id_mark => false
      | id_U41,id_a__U43 => false
      | id_U41,id_U61 => false
      | id_U41,id_isNat => false
      | id_a__and,id_a__zeros => true
      | id_a__and,id_isNatKind => true
      | id_a__and,id_a__isNatIList => true
      | id_a__and,id_a__U21 => true
      | id_a__and,id_U41 => true
      | id_a__and,id_a__and => true
      | id_a__and,id_a__U11 => false
      | id_a__and,id_U21 => false
      | id_a__and,id_a__U61 => false
      | id_a__and,id_a__U32 => false
      | id_a__and,id_U51 => false
      | id_a__and,id_isNatIListKind => false
      | id_a__and,id_0 => false
      | id_a__and,id_U12 => false
      | id_a__and,id_a__U52 => false
      | id_a__and,id_a__isNat => false
      | id_a__and,id_U43 => false
      | id_a__and,id_a__isNatIListKind => false
      | id_a__and,id_a__U12 => false
      | id_a__and,id_U31 => false
      | id_a__and,id_a__length => false
      | id_a__and,id_a__U42 => false
      | id_a__and,id_U53 => false
      | id_a__and,id_and => false
      | id_a__and,id_cons => false
      | id_a__and,id_U11 => false
      | id_a__and,id_a__U51 => false
      | id_a__and,id_a__U22 => false
      | id_a__and,id_U42 => false
      | id_a__and,id_length => false
      | id_a__and,id_tt => false
      | id_a__and,id_U22 => false
      | id_a__and,id_s => false
      | id_a__and,id_a__U41 => false
      | id_a__and,id_U52 => false
      | id_a__and,id_nil => false
      | id_a__and,id_zeros => false
      | id_a__and,id_isNatList => false
      | id_a__and,id_a__U53 => false
      | id_a__and,id_a__U31 => false
      | id_a__and,id_isNatIList => false
      | id_a__and,id_a__isNatKind => false
      | id_a__and,id_a__isNatList => false
      | id_a__and,id_U32 => false
      | id_a__and,id_mark => false
      | id_a__and,id_a__U43 => false
      | id_a__and,id_U61 => false
      | id_a__and,id_isNat => false
      | id_a__U11,id_a__zeros => true
      | id_a__U11,id_isNatKind => true
      | id_a__U11,id_a__isNatIList => true
      | id_a__U11,id_a__U21 => true
      | id_a__U11,id_U41 => true
      | id_a__U11,id_a__and => true
      | id_a__U11,id_a__U11 => true
      | id_a__U11,id_U21 => false
      | id_a__U11,id_a__U61 => false
      | id_a__U11,id_a__U32 => false
      | id_a__U11,id_U51 => false
      | id_a__U11,id_isNatIListKind => false
      | id_a__U11,id_0 => false
      | id_a__U11,id_U12 => false
      | id_a__U11,id_a__U52 => false
      | id_a__U11,id_a__isNat => false
      | id_a__U11,id_U43 => false
      | id_a__U11,id_a__isNatIListKind => false
      | id_a__U11,id_a__U12 => false
      | id_a__U11,id_U31 => false
      | id_a__U11,id_a__length => false
      | id_a__U11,id_a__U42 => false
      | id_a__U11,id_U53 => false
      | id_a__U11,id_and => false
      | id_a__U11,id_cons => false
      | id_a__U11,id_U11 => false
      | id_a__U11,id_a__U51 => false
      | id_a__U11,id_a__U22 => false
      | id_a__U11,id_U42 => false
      | id_a__U11,id_length => false
      | id_a__U11,id_tt => false
      | id_a__U11,id_U22 => false
      | id_a__U11,id_s => false
      | id_a__U11,id_a__U41 => false
      | id_a__U11,id_U52 => false
      | id_a__U11,id_nil => false
      | id_a__U11,id_zeros => false
      | id_a__U11,id_isNatList => false
      | id_a__U11,id_a__U53 => false
      | id_a__U11,id_a__U31 => false
      | id_a__U11,id_isNatIList => false
      | id_a__U11,id_a__isNatKind => false
      | id_a__U11,id_a__isNatList => false
      | id_a__U11,id_U32 => false
      | id_a__U11,id_mark => false
      | id_a__U11,id_a__U43 => false
      | id_a__U11,id_U61 => false
      | id_a__U11,id_isNat => false
      | id_U21,id_a__zeros => true
      | id_U21,id_isNatKind => true
      | id_U21,id_a__isNatIList => true
      | id_U21,id_a__U21 => true
      | id_U21,id_U41 => true
      | id_U21,id_a__and => true
      | id_U21,id_a__U11 => true
      | id_U21,id_U21 => true
      | id_U21,id_a__U61 => false
      | id_U21,id_a__U32 => false
      | id_U21,id_U51 => false
      | id_U21,id_isNatIListKind => false
      | id_U21,id_0 => false
      | id_U21,id_U12 => false
      | id_U21,id_a__U52 => false
      | id_U21,id_a__isNat => false
      | id_U21,id_U43 => false
      | id_U21,id_a__isNatIListKind => false
      | id_U21,id_a__U12 => false
      | id_U21,id_U31 => false
      | id_U21,id_a__length => false
      | id_U21,id_a__U42 => false
      | id_U21,id_U53 => false
      | id_U21,id_and => false
      | id_U21,id_cons => false
      | id_U21,id_U11 => false
      | id_U21,id_a__U51 => false
      | id_U21,id_a__U22 => false
      | id_U21,id_U42 => false
      | id_U21,id_length => false
      | id_U21,id_tt => false
      | id_U21,id_U22 => false
      | id_U21,id_s => false
      | id_U21,id_a__U41 => false
      | id_U21,id_U52 => false
      | id_U21,id_nil => false
      | id_U21,id_zeros => false
      | id_U21,id_isNatList => false
      | id_U21,id_a__U53 => false
      | id_U21,id_a__U31 => false
      | id_U21,id_isNatIList => false
      | id_U21,id_a__isNatKind => false
      | id_U21,id_a__isNatList => false
      | id_U21,id_U32 => false
      | id_U21,id_mark => false
      | id_U21,id_a__U43 => false
      | id_U21,id_U61 => false
      | id_U21,id_isNat => false
      | id_a__U61,id_a__zeros => true
      | id_a__U61,id_isNatKind => true
      | id_a__U61,id_a__isNatIList => true
      | id_a__U61,id_a__U21 => true
      | id_a__U61,id_U41 => true
      | id_a__U61,id_a__and => true
      | id_a__U61,id_a__U11 => true
      | id_a__U61,id_U21 => true
      | id_a__U61,id_a__U61 => true
      | id_a__U61,id_a__U32 => false
      | id_a__U61,id_U51 => false
      | id_a__U61,id_isNatIListKind => false
      | id_a__U61,id_0 => false
      | id_a__U61,id_U12 => false
      | id_a__U61,id_a__U52 => false
      | id_a__U61,id_a__isNat => false
      | id_a__U61,id_U43 => false
      | id_a__U61,id_a__isNatIListKind => false
      | id_a__U61,id_a__U12 => false
      | id_a__U61,id_U31 => false
      | id_a__U61,id_a__length => false
      | id_a__U61,id_a__U42 => false
      | id_a__U61,id_U53 => false
      | id_a__U61,id_and => false
      | id_a__U61,id_cons => false
      | id_a__U61,id_U11 => false
      | id_a__U61,id_a__U51 => false
      | id_a__U61,id_a__U22 => false
      | id_a__U61,id_U42 => false
      | id_a__U61,id_length => false
      | id_a__U61,id_tt => false
      | id_a__U61,id_U22 => false
      | id_a__U61,id_s => false
      | id_a__U61,id_a__U41 => false
      | id_a__U61,id_U52 => false
      | id_a__U61,id_nil => false
      | id_a__U61,id_zeros => false
      | id_a__U61,id_isNatList => false
      | id_a__U61,id_a__U53 => false
      | id_a__U61,id_a__U31 => false
      | id_a__U61,id_isNatIList => false
      | id_a__U61,id_a__isNatKind => false
      | id_a__U61,id_a__isNatList => false
      | id_a__U61,id_U32 => false
      | id_a__U61,id_mark => false
      | id_a__U61,id_a__U43 => false
      | id_a__U61,id_U61 => false
      | id_a__U61,id_isNat => false
      | id_a__U32,id_a__zeros => true
      | id_a__U32,id_isNatKind => true
      | id_a__U32,id_a__isNatIList => true
      | id_a__U32,id_a__U21 => true
      | id_a__U32,id_U41 => true
      | id_a__U32,id_a__and => true
      | id_a__U32,id_a__U11 => true
      | id_a__U32,id_U21 => true
      | id_a__U32,id_a__U61 => true
      | id_a__U32,id_a__U32 => true
      | id_a__U32,id_U51 => false
      | id_a__U32,id_isNatIListKind => false
      | id_a__U32,id_0 => false
      | id_a__U32,id_U12 => false
      | id_a__U32,id_a__U52 => false
      | id_a__U32,id_a__isNat => false
      | id_a__U32,id_U43 => false
      | id_a__U32,id_a__isNatIListKind => false
      | id_a__U32,id_a__U12 => false
      | id_a__U32,id_U31 => false
      | id_a__U32,id_a__length => false
      | id_a__U32,id_a__U42 => false
      | id_a__U32,id_U53 => false
      | id_a__U32,id_and => false
      | id_a__U32,id_cons => false
      | id_a__U32,id_U11 => false
      | id_a__U32,id_a__U51 => false
      | id_a__U32,id_a__U22 => false
      | id_a__U32,id_U42 => false
      | id_a__U32,id_length => false
      | id_a__U32,id_tt => false
      | id_a__U32,id_U22 => false
      | id_a__U32,id_s => false
      | id_a__U32,id_a__U41 => false
      | id_a__U32,id_U52 => false
      | id_a__U32,id_nil => false
      | id_a__U32,id_zeros => false
      | id_a__U32,id_isNatList => false
      | id_a__U32,id_a__U53 => false
      | id_a__U32,id_a__U31 => false
      | id_a__U32,id_isNatIList => false
      | id_a__U32,id_a__isNatKind => false
      | id_a__U32,id_a__isNatList => false
      | id_a__U32,id_U32 => false
      | id_a__U32,id_mark => false
      | id_a__U32,id_a__U43 => false
      | id_a__U32,id_U61 => false
      | id_a__U32,id_isNat => false
      | id_U51,id_a__zeros => true
      | id_U51,id_isNatKind => true
      | id_U51,id_a__isNatIList => true
      | id_U51,id_a__U21 => true
      | id_U51,id_U41 => true
      | id_U51,id_a__and => true
      | id_U51,id_a__U11 => true
      | id_U51,id_U21 => true
      | id_U51,id_a__U61 => true
      | id_U51,id_a__U32 => true
      | id_U51,id_U51 => true
      | id_U51,id_isNatIListKind => false
      | id_U51,id_0 => false
      | id_U51,id_U12 => false
      | id_U51,id_a__U52 => false
      | id_U51,id_a__isNat => false
      | id_U51,id_U43 => false
      | id_U51,id_a__isNatIListKind => false
      | id_U51,id_a__U12 => false
      | id_U51,id_U31 => false
      | id_U51,id_a__length => false
      | id_U51,id_a__U42 => false
      | id_U51,id_U53 => false
      | id_U51,id_and => false
      | id_U51,id_cons => false
      | id_U51,id_U11 => false
      | id_U51,id_a__U51 => false
      | id_U51,id_a__U22 => false
      | id_U51,id_U42 => false
      | id_U51,id_length => false
      | id_U51,id_tt => false
      | id_U51,id_U22 => false
      | id_U51,id_s => false
      | id_U51,id_a__U41 => false
      | id_U51,id_U52 => false
      | id_U51,id_nil => false
      | id_U51,id_zeros => false
      | id_U51,id_isNatList => false
      | id_U51,id_a__U53 => false
      | id_U51,id_a__U31 => false
      | id_U51,id_isNatIList => false
      | id_U51,id_a__isNatKind => false
      | id_U51,id_a__isNatList => false
      | id_U51,id_U32 => false
      | id_U51,id_mark => false
      | id_U51,id_a__U43 => false
      | id_U51,id_U61 => false
      | id_U51,id_isNat => false
      | id_isNatIListKind,id_a__zeros => true
      | id_isNatIListKind,id_isNatKind => true
      | id_isNatIListKind,id_a__isNatIList => true
      | id_isNatIListKind,id_a__U21 => true
      | id_isNatIListKind,id_U41 => true
      | id_isNatIListKind,id_a__and => true
      | id_isNatIListKind,id_a__U11 => true
      | id_isNatIListKind,id_U21 => true
      | id_isNatIListKind,id_a__U61 => true
      | id_isNatIListKind,id_a__U32 => true
      | id_isNatIListKind,id_U51 => true
      | id_isNatIListKind,id_isNatIListKind => true
      | id_isNatIListKind,id_0 => false
      | id_isNatIListKind,id_U12 => false
      | id_isNatIListKind,id_a__U52 => false
      | id_isNatIListKind,id_a__isNat => false
      | id_isNatIListKind,id_U43 => false
      | id_isNatIListKind,id_a__isNatIListKind => false
      | id_isNatIListKind,id_a__U12 => false
      | id_isNatIListKind,id_U31 => false
      | id_isNatIListKind,id_a__length => false
      | id_isNatIListKind,id_a__U42 => false
      | id_isNatIListKind,id_U53 => false
      | id_isNatIListKind,id_and => false
      | id_isNatIListKind,id_cons => false
      | id_isNatIListKind,id_U11 => false
      | id_isNatIListKind,id_a__U51 => false
      | id_isNatIListKind,id_a__U22 => false
      | id_isNatIListKind,id_U42 => false
      | id_isNatIListKind,id_length => false
      | id_isNatIListKind,id_tt => false
      | id_isNatIListKind,id_U22 => false
      | id_isNatIListKind,id_s => false
      | id_isNatIListKind,id_a__U41 => false
      | id_isNatIListKind,id_U52 => false
      | id_isNatIListKind,id_nil => false
      | id_isNatIListKind,id_zeros => false
      | id_isNatIListKind,id_isNatList => false
      | id_isNatIListKind,id_a__U53 => false
      | id_isNatIListKind,id_a__U31 => false
      | id_isNatIListKind,id_isNatIList => false
      | id_isNatIListKind,id_a__isNatKind => false
      | id_isNatIListKind,id_a__isNatList => false
      | id_isNatIListKind,id_U32 => false
      | id_isNatIListKind,id_mark => false
      | id_isNatIListKind,id_a__U43 => false
      | id_isNatIListKind,id_U61 => false
      | id_isNatIListKind,id_isNat => false
      | id_0,id_a__zeros => true
      | id_0,id_isNatKind => true
      | id_0,id_a__isNatIList => true
      | id_0,id_a__U21 => true
      | id_0,id_U41 => true
      | id_0,id_a__and => true
      | id_0,id_a__U11 => true
      | id_0,id_U21 => true
      | id_0,id_a__U61 => true
      | id_0,id_a__U32 => true
      | id_0,id_U51 => true
      | id_0,id_isNatIListKind => true
      | id_0,id_0 => true
      | id_0,id_U12 => false
      | id_0,id_a__U52 => false
      | id_0,id_a__isNat => false
      | id_0,id_U43 => false
      | id_0,id_a__isNatIListKind => false
      | id_0,id_a__U12 => false
      | id_0,id_U31 => false
      | id_0,id_a__length => false
      | id_0,id_a__U42 => false
      | id_0,id_U53 => false
      | id_0,id_and => false
      | id_0,id_cons => false
      | id_0,id_U11 => false
      | id_0,id_a__U51 => false
      | id_0,id_a__U22 => false
      | id_0,id_U42 => false
      | id_0,id_length => false
      | id_0,id_tt => false
      | id_0,id_U22 => false
      | id_0,id_s => false
      | id_0,id_a__U41 => false
      | id_0,id_U52 => false
      | id_0,id_nil => false
      | id_0,id_zeros => false
      | id_0,id_isNatList => false
      | id_0,id_a__U53 => false
      | id_0,id_a__U31 => false
      | id_0,id_isNatIList => false
      | id_0,id_a__isNatKind => false
      | id_0,id_a__isNatList => false
      | id_0,id_U32 => false
      | id_0,id_mark => false
      | id_0,id_a__U43 => false
      | id_0,id_U61 => false
      | id_0,id_isNat => false
      | id_U12,id_a__zeros => true
      | id_U12,id_isNatKind => true
      | id_U12,id_a__isNatIList => true
      | id_U12,id_a__U21 => true
      | id_U12,id_U41 => true
      | id_U12,id_a__and => true
      | id_U12,id_a__U11 => true
      | id_U12,id_U21 => true
      | id_U12,id_a__U61 => true
      | id_U12,id_a__U32 => true
      | id_U12,id_U51 => true
      | id_U12,id_isNatIListKind => true
      | id_U12,id_0 => true
      | id_U12,id_U12 => true
      | id_U12,id_a__U52 => false
      | id_U12,id_a__isNat => false
      | id_U12,id_U43 => false
      | id_U12,id_a__isNatIListKind => false
      | id_U12,id_a__U12 => false
      | id_U12,id_U31 => false
      | id_U12,id_a__length => false
      | id_U12,id_a__U42 => false
      | id_U12,id_U53 => false
      | id_U12,id_and => false
      | id_U12,id_cons => false
      | id_U12,id_U11 => false
      | id_U12,id_a__U51 => false
      | id_U12,id_a__U22 => false
      | id_U12,id_U42 => false
      | id_U12,id_length => false
      | id_U12,id_tt => false
      | id_U12,id_U22 => false
      | id_U12,id_s => false
      | id_U12,id_a__U41 => false
      | id_U12,id_U52 => false
      | id_U12,id_nil => false
      | id_U12,id_zeros => false
      | id_U12,id_isNatList => false
      | id_U12,id_a__U53 => false
      | id_U12,id_a__U31 => false
      | id_U12,id_isNatIList => false
      | id_U12,id_a__isNatKind => false
      | id_U12,id_a__isNatList => false
      | id_U12,id_U32 => false
      | id_U12,id_mark => false
      | id_U12,id_a__U43 => false
      | id_U12,id_U61 => false
      | id_U12,id_isNat => false
      | id_a__U52,id_a__zeros => true
      | id_a__U52,id_isNatKind => true
      | id_a__U52,id_a__isNatIList => true
      | id_a__U52,id_a__U21 => true
      | id_a__U52,id_U41 => true
      | id_a__U52,id_a__and => true
      | id_a__U52,id_a__U11 => true
      | id_a__U52,id_U21 => true
      | id_a__U52,id_a__U61 => true
      | id_a__U52,id_a__U32 => true
      | id_a__U52,id_U51 => true
      | id_a__U52,id_isNatIListKind => true
      | id_a__U52,id_0 => true
      | id_a__U52,id_U12 => true
      | id_a__U52,id_a__U52 => true
      | id_a__U52,id_a__isNat => false
      | id_a__U52,id_U43 => false
      | id_a__U52,id_a__isNatIListKind => false
      | id_a__U52,id_a__U12 => false
      | id_a__U52,id_U31 => false
      | id_a__U52,id_a__length => false
      | id_a__U52,id_a__U42 => false
      | id_a__U52,id_U53 => false
      | id_a__U52,id_and => false
      | id_a__U52,id_cons => false
      | id_a__U52,id_U11 => false
      | id_a__U52,id_a__U51 => false
      | id_a__U52,id_a__U22 => false
      | id_a__U52,id_U42 => false
      | id_a__U52,id_length => false
      | id_a__U52,id_tt => false
      | id_a__U52,id_U22 => false
      | id_a__U52,id_s => false
      | id_a__U52,id_a__U41 => false
      | id_a__U52,id_U52 => false
      | id_a__U52,id_nil => false
      | id_a__U52,id_zeros => false
      | id_a__U52,id_isNatList => false
      | id_a__U52,id_a__U53 => false
      | id_a__U52,id_a__U31 => false
      | id_a__U52,id_isNatIList => false
      | id_a__U52,id_a__isNatKind => false
      | id_a__U52,id_a__isNatList => false
      | id_a__U52,id_U32 => false
      | id_a__U52,id_mark => false
      | id_a__U52,id_a__U43 => false
      | id_a__U52,id_U61 => false
      | id_a__U52,id_isNat => false
      | id_a__isNat,id_a__zeros => true
      | id_a__isNat,id_isNatKind => true
      | id_a__isNat,id_a__isNatIList => true
      | id_a__isNat,id_a__U21 => true
      | id_a__isNat,id_U41 => true
      | id_a__isNat,id_a__and => true
      | id_a__isNat,id_a__U11 => true
      | id_a__isNat,id_U21 => true
      | id_a__isNat,id_a__U61 => true
      | id_a__isNat,id_a__U32 => true
      | id_a__isNat,id_U51 => true
      | id_a__isNat,id_isNatIListKind => true
      | id_a__isNat,id_0 => true
      | id_a__isNat,id_U12 => true
      | id_a__isNat,id_a__U52 => true
      | id_a__isNat,id_a__isNat => true
      | id_a__isNat,id_U43 => false
      | id_a__isNat,id_a__isNatIListKind => false
      | id_a__isNat,id_a__U12 => false
      | id_a__isNat,id_U31 => false
      | id_a__isNat,id_a__length => false
      | id_a__isNat,id_a__U42 => false
      | id_a__isNat,id_U53 => false
      | id_a__isNat,id_and => false
      | id_a__isNat,id_cons => false
      | id_a__isNat,id_U11 => false
      | id_a__isNat,id_a__U51 => false
      | id_a__isNat,id_a__U22 => false
      | id_a__isNat,id_U42 => false
      | id_a__isNat,id_length => false
      | id_a__isNat,id_tt => false
      | id_a__isNat,id_U22 => false
      | id_a__isNat,id_s => false
      | id_a__isNat,id_a__U41 => false
      | id_a__isNat,id_U52 => false
      | id_a__isNat,id_nil => false
      | id_a__isNat,id_zeros => false
      | id_a__isNat,id_isNatList => false
      | id_a__isNat,id_a__U53 => false
      | id_a__isNat,id_a__U31 => false
      | id_a__isNat,id_isNatIList => false
      | id_a__isNat,id_a__isNatKind => false
      | id_a__isNat,id_a__isNatList => false
      | id_a__isNat,id_U32 => false
      | id_a__isNat,id_mark => false
      | id_a__isNat,id_a__U43 => false
      | id_a__isNat,id_U61 => false
      | id_a__isNat,id_isNat => false
      | id_U43,id_a__zeros => true
      | id_U43,id_isNatKind => true
      | id_U43,id_a__isNatIList => true
      | id_U43,id_a__U21 => true
      | id_U43,id_U41 => true
      | id_U43,id_a__and => true
      | id_U43,id_a__U11 => true
      | id_U43,id_U21 => true
      | id_U43,id_a__U61 => true
      | id_U43,id_a__U32 => true
      | id_U43,id_U51 => true
      | id_U43,id_isNatIListKind => true
      | id_U43,id_0 => true
      | id_U43,id_U12 => true
      | id_U43,id_a__U52 => true
      | id_U43,id_a__isNat => true
      | id_U43,id_U43 => true
      | id_U43,id_a__isNatIListKind => false
      | id_U43,id_a__U12 => false
      | id_U43,id_U31 => false
      | id_U43,id_a__length => false
      | id_U43,id_a__U42 => false
      | id_U43,id_U53 => false
      | id_U43,id_and => false
      | id_U43,id_cons => false
      | id_U43,id_U11 => false
      | id_U43,id_a__U51 => false
      | id_U43,id_a__U22 => false
      | id_U43,id_U42 => false
      | id_U43,id_length => false
      | id_U43,id_tt => false
      | id_U43,id_U22 => false
      | id_U43,id_s => false
      | id_U43,id_a__U41 => false
      | id_U43,id_U52 => false
      | id_U43,id_nil => false
      | id_U43,id_zeros => false
      | id_U43,id_isNatList => false
      | id_U43,id_a__U53 => false
      | id_U43,id_a__U31 => false
      | id_U43,id_isNatIList => false
      | id_U43,id_a__isNatKind => false
      | id_U43,id_a__isNatList => false
      | id_U43,id_U32 => false
      | id_U43,id_mark => false
      | id_U43,id_a__U43 => false
      | id_U43,id_U61 => false
      | id_U43,id_isNat => false
      | id_a__isNatIListKind,id_a__zeros => true
      | id_a__isNatIListKind,id_isNatKind => true
      | id_a__isNatIListKind,id_a__isNatIList => true
      | id_a__isNatIListKind,id_a__U21 => true
      | id_a__isNatIListKind,id_U41 => true
      | id_a__isNatIListKind,id_a__and => true
      | id_a__isNatIListKind,id_a__U11 => true
      | id_a__isNatIListKind,id_U21 => true
      | id_a__isNatIListKind,id_a__U61 => true
      | id_a__isNatIListKind,id_a__U32 => true
      | id_a__isNatIListKind,id_U51 => true
      | id_a__isNatIListKind,id_isNatIListKind => true
      | id_a__isNatIListKind,id_0 => true
      | id_a__isNatIListKind,id_U12 => true
      | id_a__isNatIListKind,id_a__U52 => true
      | id_a__isNatIListKind,id_a__isNat => true
      | id_a__isNatIListKind,id_U43 => true
      | id_a__isNatIListKind,id_a__isNatIListKind => true
      | id_a__isNatIListKind,id_a__U12 => false
      | id_a__isNatIListKind,id_U31 => false
      | id_a__isNatIListKind,id_a__length => false
      | id_a__isNatIListKind,id_a__U42 => false
      | id_a__isNatIListKind,id_U53 => false
      | id_a__isNatIListKind,id_and => false
      | id_a__isNatIListKind,id_cons => false
      | id_a__isNatIListKind,id_U11 => false
      | id_a__isNatIListKind,id_a__U51 => false
      | id_a__isNatIListKind,id_a__U22 => false
      | id_a__isNatIListKind,id_U42 => false
      | id_a__isNatIListKind,id_length => false
      | id_a__isNatIListKind,id_tt => false
      | id_a__isNatIListKind,id_U22 => false
      | id_a__isNatIListKind,id_s => false
      | id_a__isNatIListKind,id_a__U41 => false
      | id_a__isNatIListKind,id_U52 => false
      | id_a__isNatIListKind,id_nil => false
      | id_a__isNatIListKind,id_zeros => false
      | id_a__isNatIListKind,id_isNatList => false
      | id_a__isNatIListKind,id_a__U53 => false
      | id_a__isNatIListKind,id_a__U31 => false
      | id_a__isNatIListKind,id_isNatIList => false
      | id_a__isNatIListKind,id_a__isNatKind => false
      | id_a__isNatIListKind,id_a__isNatList => false
      | id_a__isNatIListKind,id_U32 => false
      | id_a__isNatIListKind,id_mark => false
      | id_a__isNatIListKind,id_a__U43 => false
      | id_a__isNatIListKind,id_U61 => false
      | id_a__isNatIListKind,id_isNat => false
      | id_a__U12,id_a__zeros => true
      | id_a__U12,id_isNatKind => true
      | id_a__U12,id_a__isNatIList => true
      | id_a__U12,id_a__U21 => true
      | id_a__U12,id_U41 => true
      | id_a__U12,id_a__and => true
      | id_a__U12,id_a__U11 => true
      | id_a__U12,id_U21 => true
      | id_a__U12,id_a__U61 => true
      | id_a__U12,id_a__U32 => true
      | id_a__U12,id_U51 => true
      | id_a__U12,id_isNatIListKind => true
      | id_a__U12,id_0 => true
      | id_a__U12,id_U12 => true
      | id_a__U12,id_a__U52 => true
      | id_a__U12,id_a__isNat => true
      | id_a__U12,id_U43 => true
      | id_a__U12,id_a__isNatIListKind => true
      | id_a__U12,id_a__U12 => true
      | id_a__U12,id_U31 => false
      | id_a__U12,id_a__length => false
      | id_a__U12,id_a__U42 => false
      | id_a__U12,id_U53 => false
      | id_a__U12,id_and => false
      | id_a__U12,id_cons => false
      | id_a__U12,id_U11 => false
      | id_a__U12,id_a__U51 => false
      | id_a__U12,id_a__U22 => false
      | id_a__U12,id_U42 => false
      | id_a__U12,id_length => false
      | id_a__U12,id_tt => false
      | id_a__U12,id_U22 => false
      | id_a__U12,id_s => false
      | id_a__U12,id_a__U41 => false
      | id_a__U12,id_U52 => false
      | id_a__U12,id_nil => false
      | id_a__U12,id_zeros => false
      | id_a__U12,id_isNatList => false
      | id_a__U12,id_a__U53 => false
      | id_a__U12,id_a__U31 => false
      | id_a__U12,id_isNatIList => false
      | id_a__U12,id_a__isNatKind => false
      | id_a__U12,id_a__isNatList => false
      | id_a__U12,id_U32 => false
      | id_a__U12,id_mark => false
      | id_a__U12,id_a__U43 => false
      | id_a__U12,id_U61 => false
      | id_a__U12,id_isNat => false
      | id_U31,id_a__zeros => true
      | id_U31,id_isNatKind => true
      | id_U31,id_a__isNatIList => true
      | id_U31,id_a__U21 => true
      | id_U31,id_U41 => true
      | id_U31,id_a__and => true
      | id_U31,id_a__U11 => true
      | id_U31,id_U21 => true
      | id_U31,id_a__U61 => true
      | id_U31,id_a__U32 => true
      | id_U31,id_U51 => true
      | id_U31,id_isNatIListKind => true
      | id_U31,id_0 => true
      | id_U31,id_U12 => true
      | id_U31,id_a__U52 => true
      | id_U31,id_a__isNat => true
      | id_U31,id_U43 => true
      | id_U31,id_a__isNatIListKind => true
      | id_U31,id_a__U12 => true
      | id_U31,id_U31 => true
      | id_U31,id_a__length => false
      | id_U31,id_a__U42 => false
      | id_U31,id_U53 => false
      | id_U31,id_and => false
      | id_U31,id_cons => false
      | id_U31,id_U11 => false
      | id_U31,id_a__U51 => false
      | id_U31,id_a__U22 => false
      | id_U31,id_U42 => false
      | id_U31,id_length => false
      | id_U31,id_tt => false
      | id_U31,id_U22 => false
      | id_U31,id_s => false
      | id_U31,id_a__U41 => false
      | id_U31,id_U52 => false
      | id_U31,id_nil => false
      | id_U31,id_zeros => false
      | id_U31,id_isNatList => false
      | id_U31,id_a__U53 => false
      | id_U31,id_a__U31 => false
      | id_U31,id_isNatIList => false
      | id_U31,id_a__isNatKind => false
      | id_U31,id_a__isNatList => false
      | id_U31,id_U32 => false
      | id_U31,id_mark => false
      | id_U31,id_a__U43 => false
      | id_U31,id_U61 => false
      | id_U31,id_isNat => false
      | id_a__length,id_a__zeros => true
      | id_a__length,id_isNatKind => true
      | id_a__length,id_a__isNatIList => true
      | id_a__length,id_a__U21 => true
      | id_a__length,id_U41 => true
      | id_a__length,id_a__and => true
      | id_a__length,id_a__U11 => true
      | id_a__length,id_U21 => true
      | id_a__length,id_a__U61 => true
      | id_a__length,id_a__U32 => true
      | id_a__length,id_U51 => true
      | id_a__length,id_isNatIListKind => true
      | id_a__length,id_0 => true
      | id_a__length,id_U12 => true
      | id_a__length,id_a__U52 => true
      | id_a__length,id_a__isNat => true
      | id_a__length,id_U43 => true
      | id_a__length,id_a__isNatIListKind => true
      | id_a__length,id_a__U12 => true
      | id_a__length,id_U31 => true
      | id_a__length,id_a__length => true
      | id_a__length,id_a__U42 => false
      | id_a__length,id_U53 => false
      | id_a__length,id_and => false
      | id_a__length,id_cons => false
      | id_a__length,id_U11 => false
      | id_a__length,id_a__U51 => false
      | id_a__length,id_a__U22 => false
      | id_a__length,id_U42 => false
      | id_a__length,id_length => false
      | id_a__length,id_tt => false
      | id_a__length,id_U22 => false
      | id_a__length,id_s => false
      | id_a__length,id_a__U41 => false
      | id_a__length,id_U52 => false
      | id_a__length,id_nil => false
      | id_a__length,id_zeros => false
      | id_a__length,id_isNatList => false
      | id_a__length,id_a__U53 => false
      | id_a__length,id_a__U31 => false
      | id_a__length,id_isNatIList => false
      | id_a__length,id_a__isNatKind => false
      | id_a__length,id_a__isNatList => false
      | id_a__length,id_U32 => false
      | id_a__length,id_mark => false
      | id_a__length,id_a__U43 => false
      | id_a__length,id_U61 => false
      | id_a__length,id_isNat => false
      | id_a__U42,id_a__zeros => true
      | id_a__U42,id_isNatKind => true
      | id_a__U42,id_a__isNatIList => true
      | id_a__U42,id_a__U21 => true
      | id_a__U42,id_U41 => true
      | id_a__U42,id_a__and => true
      | id_a__U42,id_a__U11 => true
      | id_a__U42,id_U21 => true
      | id_a__U42,id_a__U61 => true
      | id_a__U42,id_a__U32 => true
      | id_a__U42,id_U51 => true
      | id_a__U42,id_isNatIListKind => true
      | id_a__U42,id_0 => true
      | id_a__U42,id_U12 => true
      | id_a__U42,id_a__U52 => true
      | id_a__U42,id_a__isNat => true
      | id_a__U42,id_U43 => true
      | id_a__U42,id_a__isNatIListKind => true
      | id_a__U42,id_a__U12 => true
      | id_a__U42,id_U31 => true
      | id_a__U42,id_a__length => true
      | id_a__U42,id_a__U42 => true
      | id_a__U42,id_U53 => false
      | id_a__U42,id_and => false
      | id_a__U42,id_cons => false
      | id_a__U42,id_U11 => false
      | id_a__U42,id_a__U51 => false
      | id_a__U42,id_a__U22 => false
      | id_a__U42,id_U42 => false
      | id_a__U42,id_length => false
      | id_a__U42,id_tt => false
      | id_a__U42,id_U22 => false
      | id_a__U42,id_s => false
      | id_a__U42,id_a__U41 => false
      | id_a__U42,id_U52 => false
      | id_a__U42,id_nil => false
      | id_a__U42,id_zeros => false
      | id_a__U42,id_isNatList => false
      | id_a__U42,id_a__U53 => false
      | id_a__U42,id_a__U31 => false
      | id_a__U42,id_isNatIList => false
      | id_a__U42,id_a__isNatKind => false
      | id_a__U42,id_a__isNatList => false
      | id_a__U42,id_U32 => false
      | id_a__U42,id_mark => false
      | id_a__U42,id_a__U43 => false
      | id_a__U42,id_U61 => false
      | id_a__U42,id_isNat => false
      | id_U53,id_a__zeros => true
      | id_U53,id_isNatKind => true
      | id_U53,id_a__isNatIList => true
      | id_U53,id_a__U21 => true
      | id_U53,id_U41 => true
      | id_U53,id_a__and => true
      | id_U53,id_a__U11 => true
      | id_U53,id_U21 => true
      | id_U53,id_a__U61 => true
      | id_U53,id_a__U32 => true
      | id_U53,id_U51 => true
      | id_U53,id_isNatIListKind => true
      | id_U53,id_0 => true
      | id_U53,id_U12 => true
      | id_U53,id_a__U52 => true
      | id_U53,id_a__isNat => true
      | id_U53,id_U43 => true
      | id_U53,id_a__isNatIListKind => true
      | id_U53,id_a__U12 => true
      | id_U53,id_U31 => true
      | id_U53,id_a__length => true
      | id_U53,id_a__U42 => true
      | id_U53,id_U53 => true
      | id_U53,id_and => false
      | id_U53,id_cons => false
      | id_U53,id_U11 => false
      | id_U53,id_a__U51 => false
      | id_U53,id_a__U22 => false
      | id_U53,id_U42 => false
      | id_U53,id_length => false
      | id_U53,id_tt => false
      | id_U53,id_U22 => false
      | id_U53,id_s => false
      | id_U53,id_a__U41 => false
      | id_U53,id_U52 => false
      | id_U53,id_nil => false
      | id_U53,id_zeros => false
      | id_U53,id_isNatList => false
      | id_U53,id_a__U53 => false
      | id_U53,id_a__U31 => false
      | id_U53,id_isNatIList => false
      | id_U53,id_a__isNatKind => false
      | id_U53,id_a__isNatList => false
      | id_U53,id_U32 => false
      | id_U53,id_mark => false
      | id_U53,id_a__U43 => false
      | id_U53,id_U61 => false
      | id_U53,id_isNat => false
      | id_and,id_a__zeros => true
      | id_and,id_isNatKind => true
      | id_and,id_a__isNatIList => true
      | id_and,id_a__U21 => true
      | id_and,id_U41 => true
      | id_and,id_a__and => true
      | id_and,id_a__U11 => true
      | id_and,id_U21 => true
      | id_and,id_a__U61 => true
      | id_and,id_a__U32 => true
      | id_and,id_U51 => true
      | id_and,id_isNatIListKind => true
      | id_and,id_0 => true
      | id_and,id_U12 => true
      | id_and,id_a__U52 => true
      | id_and,id_a__isNat => true
      | id_and,id_U43 => true
      | id_and,id_a__isNatIListKind => true
      | id_and,id_a__U12 => true
      | id_and,id_U31 => true
      | id_and,id_a__length => true
      | id_and,id_a__U42 => true
      | id_and,id_U53 => true
      | id_and,id_and => true
      | id_and,id_cons => false
      | id_and,id_U11 => false
      | id_and,id_a__U51 => false
      | id_and,id_a__U22 => false
      | id_and,id_U42 => false
      | id_and,id_length => false
      | id_and,id_tt => false
      | id_and,id_U22 => false
      | id_and,id_s => false
      | id_and,id_a__U41 => false
      | id_and,id_U52 => false
      | id_and,id_nil => false
      | id_and,id_zeros => false
      | id_and,id_isNatList => false
      | id_and,id_a__U53 => false
      | id_and,id_a__U31 => false
      | id_and,id_isNatIList => false
      | id_and,id_a__isNatKind => false
      | id_and,id_a__isNatList => false
      | id_and,id_U32 => false
      | id_and,id_mark => false
      | id_and,id_a__U43 => false
      | id_and,id_U61 => false
      | id_and,id_isNat => false
      | id_cons,id_a__zeros => true
      | id_cons,id_isNatKind => true
      | id_cons,id_a__isNatIList => true
      | id_cons,id_a__U21 => true
      | id_cons,id_U41 => true
      | id_cons,id_a__and => true
      | id_cons,id_a__U11 => true
      | id_cons,id_U21 => true
      | id_cons,id_a__U61 => true
      | id_cons,id_a__U32 => true
      | id_cons,id_U51 => true
      | id_cons,id_isNatIListKind => true
      | id_cons,id_0 => true
      | id_cons,id_U12 => true
      | id_cons,id_a__U52 => true
      | id_cons,id_a__isNat => true
      | id_cons,id_U43 => true
      | id_cons,id_a__isNatIListKind => true
      | id_cons,id_a__U12 => true
      | id_cons,id_U31 => true
      | id_cons,id_a__length => true
      | id_cons,id_a__U42 => true
      | id_cons,id_U53 => true
      | id_cons,id_and => true
      | id_cons,id_cons => true
      | id_cons,id_U11 => false
      | id_cons,id_a__U51 => false
      | id_cons,id_a__U22 => false
      | id_cons,id_U42 => false
      | id_cons,id_length => false
      | id_cons,id_tt => false
      | id_cons,id_U22 => false
      | id_cons,id_s => false
      | id_cons,id_a__U41 => false
      | id_cons,id_U52 => false
      | id_cons,id_nil => false
      | id_cons,id_zeros => false
      | id_cons,id_isNatList => false
      | id_cons,id_a__U53 => false
      | id_cons,id_a__U31 => false
      | id_cons,id_isNatIList => false
      | id_cons,id_a__isNatKind => false
      | id_cons,id_a__isNatList => false
      | id_cons,id_U32 => false
      | id_cons,id_mark => false
      | id_cons,id_a__U43 => false
      | id_cons,id_U61 => false
      | id_cons,id_isNat => false
      | id_U11,id_a__zeros => true
      | id_U11,id_isNatKind => true
      | id_U11,id_a__isNatIList => true
      | id_U11,id_a__U21 => true
      | id_U11,id_U41 => true
      | id_U11,id_a__and => true
      | id_U11,id_a__U11 => true
      | id_U11,id_U21 => true
      | id_U11,id_a__U61 => true
      | id_U11,id_a__U32 => true
      | id_U11,id_U51 => true
      | id_U11,id_isNatIListKind => true
      | id_U11,id_0 => true
      | id_U11,id_U12 => true
      | id_U11,id_a__U52 => true
      | id_U11,id_a__isNat => true
      | id_U11,id_U43 => true
      | id_U11,id_a__isNatIListKind => true
      | id_U11,id_a__U12 => true
      | id_U11,id_U31 => true
      | id_U11,id_a__length => true
      | id_U11,id_a__U42 => true
      | id_U11,id_U53 => true
      | id_U11,id_and => true
      | id_U11,id_cons => true
      | id_U11,id_U11 => true
      | id_U11,id_a__U51 => false
      | id_U11,id_a__U22 => false
      | id_U11,id_U42 => false
      | id_U11,id_length => false
      | id_U11,id_tt => false
      | id_U11,id_U22 => false
      | id_U11,id_s => false
      | id_U11,id_a__U41 => false
      | id_U11,id_U52 => false
      | id_U11,id_nil => false
      | id_U11,id_zeros => false
      | id_U11,id_isNatList => false
      | id_U11,id_a__U53 => false
      | id_U11,id_a__U31 => false
      | id_U11,id_isNatIList => false
      | id_U11,id_a__isNatKind => false
      | id_U11,id_a__isNatList => false
      | id_U11,id_U32 => false
      | id_U11,id_mark => false
      | id_U11,id_a__U43 => false
      | id_U11,id_U61 => false
      | id_U11,id_isNat => false
      | id_a__U51,id_a__zeros => true
      | id_a__U51,id_isNatKind => true
      | id_a__U51,id_a__isNatIList => true
      | id_a__U51,id_a__U21 => true
      | id_a__U51,id_U41 => true
      | id_a__U51,id_a__and => true
      | id_a__U51,id_a__U11 => true
      | id_a__U51,id_U21 => true
      | id_a__U51,id_a__U61 => true
      | id_a__U51,id_a__U32 => true
      | id_a__U51,id_U51 => true
      | id_a__U51,id_isNatIListKind => true
      | id_a__U51,id_0 => true
      | id_a__U51,id_U12 => true
      | id_a__U51,id_a__U52 => true
      | id_a__U51,id_a__isNat => true
      | id_a__U51,id_U43 => true
      | id_a__U51,id_a__isNatIListKind => true
      | id_a__U51,id_a__U12 => true
      | id_a__U51,id_U31 => true
      | id_a__U51,id_a__length => true
      | id_a__U51,id_a__U42 => true
      | id_a__U51,id_U53 => true
      | id_a__U51,id_and => true
      | id_a__U51,id_cons => true
      | id_a__U51,id_U11 => true
      | id_a__U51,id_a__U51 => true
      | id_a__U51,id_a__U22 => false
      | id_a__U51,id_U42 => false
      | id_a__U51,id_length => false
      | id_a__U51,id_tt => false
      | id_a__U51,id_U22 => false
      | id_a__U51,id_s => false
      | id_a__U51,id_a__U41 => false
      | id_a__U51,id_U52 => false
      | id_a__U51,id_nil => false
      | id_a__U51,id_zeros => false
      | id_a__U51,id_isNatList => false
      | id_a__U51,id_a__U53 => false
      | id_a__U51,id_a__U31 => false
      | id_a__U51,id_isNatIList => false
      | id_a__U51,id_a__isNatKind => false
      | id_a__U51,id_a__isNatList => false
      | id_a__U51,id_U32 => false
      | id_a__U51,id_mark => false
      | id_a__U51,id_a__U43 => false
      | id_a__U51,id_U61 => false
      | id_a__U51,id_isNat => false
      | id_a__U22,id_a__zeros => true
      | id_a__U22,id_isNatKind => true
      | id_a__U22,id_a__isNatIList => true
      | id_a__U22,id_a__U21 => true
      | id_a__U22,id_U41 => true
      | id_a__U22,id_a__and => true
      | id_a__U22,id_a__U11 => true
      | id_a__U22,id_U21 => true
      | id_a__U22,id_a__U61 => true
      | id_a__U22,id_a__U32 => true
      | id_a__U22,id_U51 => true
      | id_a__U22,id_isNatIListKind => true
      | id_a__U22,id_0 => true
      | id_a__U22,id_U12 => true
      | id_a__U22,id_a__U52 => true
      | id_a__U22,id_a__isNat => true
      | id_a__U22,id_U43 => true
      | id_a__U22,id_a__isNatIListKind => true
      | id_a__U22,id_a__U12 => true
      | id_a__U22,id_U31 => true
      | id_a__U22,id_a__length => true
      | id_a__U22,id_a__U42 => true
      | id_a__U22,id_U53 => true
      | id_a__U22,id_and => true
      | id_a__U22,id_cons => true
      | id_a__U22,id_U11 => true
      | id_a__U22,id_a__U51 => true
      | id_a__U22,id_a__U22 => true
      | id_a__U22,id_U42 => false
      | id_a__U22,id_length => false
      | id_a__U22,id_tt => false
      | id_a__U22,id_U22 => false
      | id_a__U22,id_s => false
      | id_a__U22,id_a__U41 => false
      | id_a__U22,id_U52 => false
      | id_a__U22,id_nil => false
      | id_a__U22,id_zeros => false
      | id_a__U22,id_isNatList => false
      | id_a__U22,id_a__U53 => false
      | id_a__U22,id_a__U31 => false
      | id_a__U22,id_isNatIList => false
      | id_a__U22,id_a__isNatKind => false
      | id_a__U22,id_a__isNatList => false
      | id_a__U22,id_U32 => false
      | id_a__U22,id_mark => false
      | id_a__U22,id_a__U43 => false
      | id_a__U22,id_U61 => false
      | id_a__U22,id_isNat => false
      | id_U42,id_a__zeros => true
      | id_U42,id_isNatKind => true
      | id_U42,id_a__isNatIList => true
      | id_U42,id_a__U21 => true
      | id_U42,id_U41 => true
      | id_U42,id_a__and => true
      | id_U42,id_a__U11 => true
      | id_U42,id_U21 => true
      | id_U42,id_a__U61 => true
      | id_U42,id_a__U32 => true
      | id_U42,id_U51 => true
      | id_U42,id_isNatIListKind => true
      | id_U42,id_0 => true
      | id_U42,id_U12 => true
      | id_U42,id_a__U52 => true
      | id_U42,id_a__isNat => true
      | id_U42,id_U43 => true
      | id_U42,id_a__isNatIListKind => true
      | id_U42,id_a__U12 => true
      | id_U42,id_U31 => true
      | id_U42,id_a__length => true
      | id_U42,id_a__U42 => true
      | id_U42,id_U53 => true
      | id_U42,id_and => true
      | id_U42,id_cons => true
      | id_U42,id_U11 => true
      | id_U42,id_a__U51 => true
      | id_U42,id_a__U22 => true
      | id_U42,id_U42 => true
      | id_U42,id_length => false
      | id_U42,id_tt => false
      | id_U42,id_U22 => false
      | id_U42,id_s => false
      | id_U42,id_a__U41 => false
      | id_U42,id_U52 => false
      | id_U42,id_nil => false
      | id_U42,id_zeros => false
      | id_U42,id_isNatList => false
      | id_U42,id_a__U53 => false
      | id_U42,id_a__U31 => false
      | id_U42,id_isNatIList => false
      | id_U42,id_a__isNatKind => false
      | id_U42,id_a__isNatList => false
      | id_U42,id_U32 => false
      | id_U42,id_mark => false
      | id_U42,id_a__U43 => false
      | id_U42,id_U61 => false
      | id_U42,id_isNat => false
      | id_length,id_a__zeros => true
      | id_length,id_isNatKind => true
      | id_length,id_a__isNatIList => true
      | id_length,id_a__U21 => true
      | id_length,id_U41 => true
      | id_length,id_a__and => true
      | id_length,id_a__U11 => true
      | id_length,id_U21 => true
      | id_length,id_a__U61 => true
      | id_length,id_a__U32 => true
      | id_length,id_U51 => true
      | id_length,id_isNatIListKind => true
      | id_length,id_0 => true
      | id_length,id_U12 => true
      | id_length,id_a__U52 => true
      | id_length,id_a__isNat => true
      | id_length,id_U43 => true
      | id_length,id_a__isNatIListKind => true
      | id_length,id_a__U12 => true
      | id_length,id_U31 => true
      | id_length,id_a__length => true
      | id_length,id_a__U42 => true
      | id_length,id_U53 => true
      | id_length,id_and => true
      | id_length,id_cons => true
      | id_length,id_U11 => true
      | id_length,id_a__U51 => true
      | id_length,id_a__U22 => true
      | id_length,id_U42 => true
      | id_length,id_length => true
      | id_length,id_tt => false
      | id_length,id_U22 => false
      | id_length,id_s => false
      | id_length,id_a__U41 => false
      | id_length,id_U52 => false
      | id_length,id_nil => false
      | id_length,id_zeros => false
      | id_length,id_isNatList => false
      | id_length,id_a__U53 => false
      | id_length,id_a__U31 => false
      | id_length,id_isNatIList => false
      | id_length,id_a__isNatKind => false
      | id_length,id_a__isNatList => false
      | id_length,id_U32 => false
      | id_length,id_mark => false
      | id_length,id_a__U43 => false
      | id_length,id_U61 => false
      | id_length,id_isNat => false
      | id_tt,id_a__zeros => true
      | id_tt,id_isNatKind => true
      | id_tt,id_a__isNatIList => true
      | id_tt,id_a__U21 => true
      | id_tt,id_U41 => true
      | id_tt,id_a__and => true
      | id_tt,id_a__U11 => true
      | id_tt,id_U21 => true
      | id_tt,id_a__U61 => true
      | id_tt,id_a__U32 => true
      | id_tt,id_U51 => true
      | id_tt,id_isNatIListKind => true
      | id_tt,id_0 => true
      | id_tt,id_U12 => true
      | id_tt,id_a__U52 => true
      | id_tt,id_a__isNat => true
      | id_tt,id_U43 => true
      | id_tt,id_a__isNatIListKind => true
      | id_tt,id_a__U12 => true
      | id_tt,id_U31 => true
      | id_tt,id_a__length => true
      | id_tt,id_a__U42 => true
      | id_tt,id_U53 => true
      | id_tt,id_and => true
      | id_tt,id_cons => true
      | id_tt,id_U11 => true
      | id_tt,id_a__U51 => true
      | id_tt,id_a__U22 => true
      | id_tt,id_U42 => true
      | id_tt,id_length => true
      | id_tt,id_tt => true
      | id_tt,id_U22 => false
      | id_tt,id_s => false
      | id_tt,id_a__U41 => false
      | id_tt,id_U52 => false
      | id_tt,id_nil => false
      | id_tt,id_zeros => false
      | id_tt,id_isNatList => false
      | id_tt,id_a__U53 => false
      | id_tt,id_a__U31 => false
      | id_tt,id_isNatIList => false
      | id_tt,id_a__isNatKind => false
      | id_tt,id_a__isNatList => false
      | id_tt,id_U32 => false
      | id_tt,id_mark => false
      | id_tt,id_a__U43 => false
      | id_tt,id_U61 => false
      | id_tt,id_isNat => false
      | id_U22,id_a__zeros => true
      | id_U22,id_isNatKind => true
      | id_U22,id_a__isNatIList => true
      | id_U22,id_a__U21 => true
      | id_U22,id_U41 => true
      | id_U22,id_a__and => true
      | id_U22,id_a__U11 => true
      | id_U22,id_U21 => true
      | id_U22,id_a__U61 => true
      | id_U22,id_a__U32 => true
      | id_U22,id_U51 => true
      | id_U22,id_isNatIListKind => true
      | id_U22,id_0 => true
      | id_U22,id_U12 => true
      | id_U22,id_a__U52 => true
      | id_U22,id_a__isNat => true
      | id_U22,id_U43 => true
      | id_U22,id_a__isNatIListKind => true
      | id_U22,id_a__U12 => true
      | id_U22,id_U31 => true
      | id_U22,id_a__length => true
      | id_U22,id_a__U42 => true
      | id_U22,id_U53 => true
      | id_U22,id_and => true
      | id_U22,id_cons => true
      | id_U22,id_U11 => true
      | id_U22,id_a__U51 => true
      | id_U22,id_a__U22 => true
      | id_U22,id_U42 => true
      | id_U22,id_length => true
      | id_U22,id_tt => true
      | id_U22,id_U22 => true
      | id_U22,id_s => false
      | id_U22,id_a__U41 => false
      | id_U22,id_U52 => false
      | id_U22,id_nil => false
      | id_U22,id_zeros => false
      | id_U22,id_isNatList => false
      | id_U22,id_a__U53 => false
      | id_U22,id_a__U31 => false
      | id_U22,id_isNatIList => false
      | id_U22,id_a__isNatKind => false
      | id_U22,id_a__isNatList => false
      | id_U22,id_U32 => false
      | id_U22,id_mark => false
      | id_U22,id_a__U43 => false
      | id_U22,id_U61 => false
      | id_U22,id_isNat => false
      | id_s,id_a__zeros => true
      | id_s,id_isNatKind => true
      | id_s,id_a__isNatIList => true
      | id_s,id_a__U21 => true
      | id_s,id_U41 => true
      | id_s,id_a__and => true
      | id_s,id_a__U11 => true
      | id_s,id_U21 => true
      | id_s,id_a__U61 => true
      | id_s,id_a__U32 => true
      | id_s,id_U51 => true
      | id_s,id_isNatIListKind => true
      | id_s,id_0 => true
      | id_s,id_U12 => true
      | id_s,id_a__U52 => true
      | id_s,id_a__isNat => true
      | id_s,id_U43 => true
      | id_s,id_a__isNatIListKind => true
      | id_s,id_a__U12 => true
      | id_s,id_U31 => true
      | id_s,id_a__length => true
      | id_s,id_a__U42 => true
      | id_s,id_U53 => true
      | id_s,id_and => true
      | id_s,id_cons => true
      | id_s,id_U11 => true
      | id_s,id_a__U51 => true
      | id_s,id_a__U22 => true
      | id_s,id_U42 => true
      | id_s,id_length => true
      | id_s,id_tt => true
      | id_s,id_U22 => true
      | id_s,id_s => true
      | id_s,id_a__U41 => false
      | id_s,id_U52 => false
      | id_s,id_nil => false
      | id_s,id_zeros => false
      | id_s,id_isNatList => false
      | id_s,id_a__U53 => false
      | id_s,id_a__U31 => false
      | id_s,id_isNatIList => false
      | id_s,id_a__isNatKind => false
      | id_s,id_a__isNatList => false
      | id_s,id_U32 => false
      | id_s,id_mark => false
      | id_s,id_a__U43 => false
      | id_s,id_U61 => false
      | id_s,id_isNat => false
      | id_a__U41,id_a__zeros => true
      | id_a__U41,id_isNatKind => true
      | id_a__U41,id_a__isNatIList => true
      | id_a__U41,id_a__U21 => true
      | id_a__U41,id_U41 => true
      | id_a__U41,id_a__and => true
      | id_a__U41,id_a__U11 => true
      | id_a__U41,id_U21 => true
      | id_a__U41,id_a__U61 => true
      | id_a__U41,id_a__U32 => true
      | id_a__U41,id_U51 => true
      | id_a__U41,id_isNatIListKind => true
      | id_a__U41,id_0 => true
      | id_a__U41,id_U12 => true
      | id_a__U41,id_a__U52 => true
      | id_a__U41,id_a__isNat => true
      | id_a__U41,id_U43 => true
      | id_a__U41,id_a__isNatIListKind => true
      | id_a__U41,id_a__U12 => true
      | id_a__U41,id_U31 => true
      | id_a__U41,id_a__length => true
      | id_a__U41,id_a__U42 => true
      | id_a__U41,id_U53 => true
      | id_a__U41,id_and => true
      | id_a__U41,id_cons => true
      | id_a__U41,id_U11 => true
      | id_a__U41,id_a__U51 => true
      | id_a__U41,id_a__U22 => true
      | id_a__U41,id_U42 => true
      | id_a__U41,id_length => true
      | id_a__U41,id_tt => true
      | id_a__U41,id_U22 => true
      | id_a__U41,id_s => true
      | id_a__U41,id_a__U41 => true
      | id_a__U41,id_U52 => false
      | id_a__U41,id_nil => false
      | id_a__U41,id_zeros => false
      | id_a__U41,id_isNatList => false
      | id_a__U41,id_a__U53 => false
      | id_a__U41,id_a__U31 => false
      | id_a__U41,id_isNatIList => false
      | id_a__U41,id_a__isNatKind => false
      | id_a__U41,id_a__isNatList => false
      | id_a__U41,id_U32 => false
      | id_a__U41,id_mark => false
      | id_a__U41,id_a__U43 => false
      | id_a__U41,id_U61 => false
      | id_a__U41,id_isNat => false
      | id_U52,id_a__zeros => true
      | id_U52,id_isNatKind => true
      | id_U52,id_a__isNatIList => true
      | id_U52,id_a__U21 => true
      | id_U52,id_U41 => true
      | id_U52,id_a__and => true
      | id_U52,id_a__U11 => true
      | id_U52,id_U21 => true
      | id_U52,id_a__U61 => true
      | id_U52,id_a__U32 => true
      | id_U52,id_U51 => true
      | id_U52,id_isNatIListKind => true
      | id_U52,id_0 => true
      | id_U52,id_U12 => true
      | id_U52,id_a__U52 => true
      | id_U52,id_a__isNat => true
      | id_U52,id_U43 => true
      | id_U52,id_a__isNatIListKind => true
      | id_U52,id_a__U12 => true
      | id_U52,id_U31 => true
      | id_U52,id_a__length => true
      | id_U52,id_a__U42 => true
      | id_U52,id_U53 => true
      | id_U52,id_and => true
      | id_U52,id_cons => true
      | id_U52,id_U11 => true
      | id_U52,id_a__U51 => true
      | id_U52,id_a__U22 => true
      | id_U52,id_U42 => true
      | id_U52,id_length => true
      | id_U52,id_tt => true
      | id_U52,id_U22 => true
      | id_U52,id_s => true
      | id_U52,id_a__U41 => true
      | id_U52,id_U52 => true
      | id_U52,id_nil => false
      | id_U52,id_zeros => false
      | id_U52,id_isNatList => false
      | id_U52,id_a__U53 => false
      | id_U52,id_a__U31 => false
      | id_U52,id_isNatIList => false
      | id_U52,id_a__isNatKind => false
      | id_U52,id_a__isNatList => false
      | id_U52,id_U32 => false
      | id_U52,id_mark => false
      | id_U52,id_a__U43 => false
      | id_U52,id_U61 => false
      | id_U52,id_isNat => false
      | id_nil,id_a__zeros => true
      | id_nil,id_isNatKind => true
      | id_nil,id_a__isNatIList => true
      | id_nil,id_a__U21 => true
      | id_nil,id_U41 => true
      | id_nil,id_a__and => true
      | id_nil,id_a__U11 => true
      | id_nil,id_U21 => true
      | id_nil,id_a__U61 => true
      | id_nil,id_a__U32 => true
      | id_nil,id_U51 => true
      | id_nil,id_isNatIListKind => true
      | id_nil,id_0 => true
      | id_nil,id_U12 => true
      | id_nil,id_a__U52 => true
      | id_nil,id_a__isNat => true
      | id_nil,id_U43 => true
      | id_nil,id_a__isNatIListKind => true
      | id_nil,id_a__U12 => true
      | id_nil,id_U31 => true
      | id_nil,id_a__length => true
      | id_nil,id_a__U42 => true
      | id_nil,id_U53 => true
      | id_nil,id_and => true
      | id_nil,id_cons => true
      | id_nil,id_U11 => true
      | id_nil,id_a__U51 => true
      | id_nil,id_a__U22 => true
      | id_nil,id_U42 => true
      | id_nil,id_length => true
      | id_nil,id_tt => true
      | id_nil,id_U22 => true
      | id_nil,id_s => true
      | id_nil,id_a__U41 => true
      | id_nil,id_U52 => true
      | id_nil,id_nil => true
      | id_nil,id_zeros => false
      | id_nil,id_isNatList => false
      | id_nil,id_a__U53 => false
      | id_nil,id_a__U31 => false
      | id_nil,id_isNatIList => false
      | id_nil,id_a__isNatKind => false
      | id_nil,id_a__isNatList => false
      | id_nil,id_U32 => false
      | id_nil,id_mark => false
      | id_nil,id_a__U43 => false
      | id_nil,id_U61 => false
      | id_nil,id_isNat => false
      | id_zeros,id_a__zeros => true
      | id_zeros,id_isNatKind => true
      | id_zeros,id_a__isNatIList => true
      | id_zeros,id_a__U21 => true
      | id_zeros,id_U41 => true
      | id_zeros,id_a__and => true
      | id_zeros,id_a__U11 => true
      | id_zeros,id_U21 => true
      | id_zeros,id_a__U61 => true
      | id_zeros,id_a__U32 => true
      | id_zeros,id_U51 => true
      | id_zeros,id_isNatIListKind => true
      | id_zeros,id_0 => true
      | id_zeros,id_U12 => true
      | id_zeros,id_a__U52 => true
      | id_zeros,id_a__isNat => true
      | id_zeros,id_U43 => true
      | id_zeros,id_a__isNatIListKind => true
      | id_zeros,id_a__U12 => true
      | id_zeros,id_U31 => true
      | id_zeros,id_a__length => true
      | id_zeros,id_a__U42 => true
      | id_zeros,id_U53 => true
      | id_zeros,id_and => true
      | id_zeros,id_cons => true
      | id_zeros,id_U11 => true
      | id_zeros,id_a__U51 => true
      | id_zeros,id_a__U22 => true
      | id_zeros,id_U42 => true
      | id_zeros,id_length => true
      | id_zeros,id_tt => true
      | id_zeros,id_U22 => true
      | id_zeros,id_s => true
      | id_zeros,id_a__U41 => true
      | id_zeros,id_U52 => true
      | id_zeros,id_nil => true
      | id_zeros,id_zeros => true
      | id_zeros,id_isNatList => false
      | id_zeros,id_a__U53 => false
      | id_zeros,id_a__U31 => false
      | id_zeros,id_isNatIList => false
      | id_zeros,id_a__isNatKind => false
      | id_zeros,id_a__isNatList => false
      | id_zeros,id_U32 => false
      | id_zeros,id_mark => false
      | id_zeros,id_a__U43 => false
      | id_zeros,id_U61 => false
      | id_zeros,id_isNat => false
      | id_isNatList,id_a__zeros => true
      | id_isNatList,id_isNatKind => true
      | id_isNatList,id_a__isNatIList => true
      | id_isNatList,id_a__U21 => true
      | id_isNatList,id_U41 => true
      | id_isNatList,id_a__and => true
      | id_isNatList,id_a__U11 => true
      | id_isNatList,id_U21 => true
      | id_isNatList,id_a__U61 => true
      | id_isNatList,id_a__U32 => true
      | id_isNatList,id_U51 => true
      | id_isNatList,id_isNatIListKind => true
      | id_isNatList,id_0 => true
      | id_isNatList,id_U12 => true
      | id_isNatList,id_a__U52 => true
      | id_isNatList,id_a__isNat => true
      | id_isNatList,id_U43 => true
      | id_isNatList,id_a__isNatIListKind => true
      | id_isNatList,id_a__U12 => true
      | id_isNatList,id_U31 => true
      | id_isNatList,id_a__length => true
      | id_isNatList,id_a__U42 => true
      | id_isNatList,id_U53 => true
      | id_isNatList,id_and => true
      | id_isNatList,id_cons => true
      | id_isNatList,id_U11 => true
      | id_isNatList,id_a__U51 => true
      | id_isNatList,id_a__U22 => true
      | id_isNatList,id_U42 => true
      | id_isNatList,id_length => true
      | id_isNatList,id_tt => true
      | id_isNatList,id_U22 => true
      | id_isNatList,id_s => true
      | id_isNatList,id_a__U41 => true
      | id_isNatList,id_U52 => true
      | id_isNatList,id_nil => true
      | id_isNatList,id_zeros => true
      | id_isNatList,id_isNatList => true
      | id_isNatList,id_a__U53 => false
      | id_isNatList,id_a__U31 => false
      | id_isNatList,id_isNatIList => false
      | id_isNatList,id_a__isNatKind => false
      | id_isNatList,id_a__isNatList => false
      | id_isNatList,id_U32 => false
      | id_isNatList,id_mark => false
      | id_isNatList,id_a__U43 => false
      | id_isNatList,id_U61 => false
      | id_isNatList,id_isNat => false
      | id_a__U53,id_a__zeros => true
      | id_a__U53,id_isNatKind => true
      | id_a__U53,id_a__isNatIList => true
      | id_a__U53,id_a__U21 => true
      | id_a__U53,id_U41 => true
      | id_a__U53,id_a__and => true
      | id_a__U53,id_a__U11 => true
      | id_a__U53,id_U21 => true
      | id_a__U53,id_a__U61 => true
      | id_a__U53,id_a__U32 => true
      | id_a__U53,id_U51 => true
      | id_a__U53,id_isNatIListKind => true
      | id_a__U53,id_0 => true
      | id_a__U53,id_U12 => true
      | id_a__U53,id_a__U52 => true
      | id_a__U53,id_a__isNat => true
      | id_a__U53,id_U43 => true
      | id_a__U53,id_a__isNatIListKind => true
      | id_a__U53,id_a__U12 => true
      | id_a__U53,id_U31 => true
      | id_a__U53,id_a__length => true
      | id_a__U53,id_a__U42 => true
      | id_a__U53,id_U53 => true
      | id_a__U53,id_and => true
      | id_a__U53,id_cons => true
      | id_a__U53,id_U11 => true
      | id_a__U53,id_a__U51 => true
      | id_a__U53,id_a__U22 => true
      | id_a__U53,id_U42 => true
      | id_a__U53,id_length => true
      | id_a__U53,id_tt => true
      | id_a__U53,id_U22 => true
      | id_a__U53,id_s => true
      | id_a__U53,id_a__U41 => true
      | id_a__U53,id_U52 => true
      | id_a__U53,id_nil => true
      | id_a__U53,id_zeros => true
      | id_a__U53,id_isNatList => true
      | id_a__U53,id_a__U53 => true
      | id_a__U53,id_a__U31 => false
      | id_a__U53,id_isNatIList => false
      | id_a__U53,id_a__isNatKind => false
      | id_a__U53,id_a__isNatList => false
      | id_a__U53,id_U32 => false
      | id_a__U53,id_mark => false
      | id_a__U53,id_a__U43 => false
      | id_a__U53,id_U61 => false
      | id_a__U53,id_isNat => false
      | id_a__U31,id_a__zeros => true
      | id_a__U31,id_isNatKind => true
      | id_a__U31,id_a__isNatIList => true
      | id_a__U31,id_a__U21 => true
      | id_a__U31,id_U41 => true
      | id_a__U31,id_a__and => true
      | id_a__U31,id_a__U11 => true
      | id_a__U31,id_U21 => true
      | id_a__U31,id_a__U61 => true
      | id_a__U31,id_a__U32 => true
      | id_a__U31,id_U51 => true
      | id_a__U31,id_isNatIListKind => true
      | id_a__U31,id_0 => true
      | id_a__U31,id_U12 => true
      | id_a__U31,id_a__U52 => true
      | id_a__U31,id_a__isNat => true
      | id_a__U31,id_U43 => true
      | id_a__U31,id_a__isNatIListKind => true
      | id_a__U31,id_a__U12 => true
      | id_a__U31,id_U31 => true
      | id_a__U31,id_a__length => true
      | id_a__U31,id_a__U42 => true
      | id_a__U31,id_U53 => true
      | id_a__U31,id_and => true
      | id_a__U31,id_cons => true
      | id_a__U31,id_U11 => true
      | id_a__U31,id_a__U51 => true
      | id_a__U31,id_a__U22 => true
      | id_a__U31,id_U42 => true
      | id_a__U31,id_length => true
      | id_a__U31,id_tt => true
      | id_a__U31,id_U22 => true
      | id_a__U31,id_s => true
      | id_a__U31,id_a__U41 => true
      | id_a__U31,id_U52 => true
      | id_a__U31,id_nil => true
      | id_a__U31,id_zeros => true
      | id_a__U31,id_isNatList => true
      | id_a__U31,id_a__U53 => true
      | id_a__U31,id_a__U31 => true
      | id_a__U31,id_isNatIList => false
      | id_a__U31,id_a__isNatKind => false
      | id_a__U31,id_a__isNatList => false
      | id_a__U31,id_U32 => false
      | id_a__U31,id_mark => false
      | id_a__U31,id_a__U43 => false
      | id_a__U31,id_U61 => false
      | id_a__U31,id_isNat => false
      | id_isNatIList,id_a__zeros => true
      | id_isNatIList,id_isNatKind => true
      | id_isNatIList,id_a__isNatIList => true
      | id_isNatIList,id_a__U21 => true
      | id_isNatIList,id_U41 => true
      | id_isNatIList,id_a__and => true
      | id_isNatIList,id_a__U11 => true
      | id_isNatIList,id_U21 => true
      | id_isNatIList,id_a__U61 => true
      | id_isNatIList,id_a__U32 => true
      | id_isNatIList,id_U51 => true
      | id_isNatIList,id_isNatIListKind => true
      | id_isNatIList,id_0 => true
      | id_isNatIList,id_U12 => true
      | id_isNatIList,id_a__U52 => true
      | id_isNatIList,id_a__isNat => true
      | id_isNatIList,id_U43 => true
      | id_isNatIList,id_a__isNatIListKind => true
      | id_isNatIList,id_a__U12 => true
      | id_isNatIList,id_U31 => true
      | id_isNatIList,id_a__length => true
      | id_isNatIList,id_a__U42 => true
      | id_isNatIList,id_U53 => true
      | id_isNatIList,id_and => true
      | id_isNatIList,id_cons => true
      | id_isNatIList,id_U11 => true
      | id_isNatIList,id_a__U51 => true
      | id_isNatIList,id_a__U22 => true
      | id_isNatIList,id_U42 => true
      | id_isNatIList,id_length => true
      | id_isNatIList,id_tt => true
      | id_isNatIList,id_U22 => true
      | id_isNatIList,id_s => true
      | id_isNatIList,id_a__U41 => true
      | id_isNatIList,id_U52 => true
      | id_isNatIList,id_nil => true
      | id_isNatIList,id_zeros => true
      | id_isNatIList,id_isNatList => true
      | id_isNatIList,id_a__U53 => true
      | id_isNatIList,id_a__U31 => true
      | id_isNatIList,id_isNatIList => true
      | id_isNatIList,id_a__isNatKind => false
      | id_isNatIList,id_a__isNatList => false
      | id_isNatIList,id_U32 => false
      | id_isNatIList,id_mark => false
      | id_isNatIList,id_a__U43 => false
      | id_isNatIList,id_U61 => false
      | id_isNatIList,id_isNat => false
      | id_a__isNatKind,id_a__zeros => true
      | id_a__isNatKind,id_isNatKind => true
      | id_a__isNatKind,id_a__isNatIList => true
      | id_a__isNatKind,id_a__U21 => true
      | id_a__isNatKind,id_U41 => true
      | id_a__isNatKind,id_a__and => true
      | id_a__isNatKind,id_a__U11 => true
      | id_a__isNatKind,id_U21 => true
      | id_a__isNatKind,id_a__U61 => true
      | id_a__isNatKind,id_a__U32 => true
      | id_a__isNatKind,id_U51 => true
      | id_a__isNatKind,id_isNatIListKind => true
      | id_a__isNatKind,id_0 => true
      | id_a__isNatKind,id_U12 => true
      | id_a__isNatKind,id_a__U52 => true
      | id_a__isNatKind,id_a__isNat => true
      | id_a__isNatKind,id_U43 => true
      | id_a__isNatKind,id_a__isNatIListKind => true
      | id_a__isNatKind,id_a__U12 => true
      | id_a__isNatKind,id_U31 => true
      | id_a__isNatKind,id_a__length => true
      | id_a__isNatKind,id_a__U42 => true
      | id_a__isNatKind,id_U53 => true
      | id_a__isNatKind,id_and => true
      | id_a__isNatKind,id_cons => true
      | id_a__isNatKind,id_U11 => true
      | id_a__isNatKind,id_a__U51 => true
      | id_a__isNatKind,id_a__U22 => true
      | id_a__isNatKind,id_U42 => true
      | id_a__isNatKind,id_length => true
      | id_a__isNatKind,id_tt => true
      | id_a__isNatKind,id_U22 => true
      | id_a__isNatKind,id_s => true
      | id_a__isNatKind,id_a__U41 => true
      | id_a__isNatKind,id_U52 => true
      | id_a__isNatKind,id_nil => true
      | id_a__isNatKind,id_zeros => true
      | id_a__isNatKind,id_isNatList => true
      | id_a__isNatKind,id_a__U53 => true
      | id_a__isNatKind,id_a__U31 => true
      | id_a__isNatKind,id_isNatIList => true
      | id_a__isNatKind,id_a__isNatKind => true
      | id_a__isNatKind,id_a__isNatList => false
      | id_a__isNatKind,id_U32 => false
      | id_a__isNatKind,id_mark => false
      | id_a__isNatKind,id_a__U43 => false
      | id_a__isNatKind,id_U61 => false
      | id_a__isNatKind,id_isNat => false
      | id_a__isNatList,id_a__zeros => true
      | id_a__isNatList,id_isNatKind => true
      | id_a__isNatList,id_a__isNatIList => true
      | id_a__isNatList,id_a__U21 => true
      | id_a__isNatList,id_U41 => true
      | id_a__isNatList,id_a__and => true
      | id_a__isNatList,id_a__U11 => true
      | id_a__isNatList,id_U21 => true
      | id_a__isNatList,id_a__U61 => true
      | id_a__isNatList,id_a__U32 => true
      | id_a__isNatList,id_U51 => true
      | id_a__isNatList,id_isNatIListKind => true
      | id_a__isNatList,id_0 => true
      | id_a__isNatList,id_U12 => true
      | id_a__isNatList,id_a__U52 => true
      | id_a__isNatList,id_a__isNat => true
      | id_a__isNatList,id_U43 => true
      | id_a__isNatList,id_a__isNatIListKind => true
      | id_a__isNatList,id_a__U12 => true
      | id_a__isNatList,id_U31 => true
      | id_a__isNatList,id_a__length => true
      | id_a__isNatList,id_a__U42 => true
      | id_a__isNatList,id_U53 => true
      | id_a__isNatList,id_and => true
      | id_a__isNatList,id_cons => true
      | id_a__isNatList,id_U11 => true
      | id_a__isNatList,id_a__U51 => true
      | id_a__isNatList,id_a__U22 => true
      | id_a__isNatList,id_U42 => true
      | id_a__isNatList,id_length => true
      | id_a__isNatList,id_tt => true
      | id_a__isNatList,id_U22 => true
      | id_a__isNatList,id_s => true
      | id_a__isNatList,id_a__U41 => true
      | id_a__isNatList,id_U52 => true
      | id_a__isNatList,id_nil => true
      | id_a__isNatList,id_zeros => true
      | id_a__isNatList,id_isNatList => true
      | id_a__isNatList,id_a__U53 => true
      | id_a__isNatList,id_a__U31 => true
      | id_a__isNatList,id_isNatIList => true
      | id_a__isNatList,id_a__isNatKind => true
      | id_a__isNatList,id_a__isNatList => true
      | id_a__isNatList,id_U32 => false
      | id_a__isNatList,id_mark => false
      | id_a__isNatList,id_a__U43 => false
      | id_a__isNatList,id_U61 => false
      | id_a__isNatList,id_isNat => false
      | id_U32,id_a__zeros => true
      | id_U32,id_isNatKind => true
      | id_U32,id_a__isNatIList => true
      | id_U32,id_a__U21 => true
      | id_U32,id_U41 => true
      | id_U32,id_a__and => true
      | id_U32,id_a__U11 => true
      | id_U32,id_U21 => true
      | id_U32,id_a__U61 => true
      | id_U32,id_a__U32 => true
      | id_U32,id_U51 => true
      | id_U32,id_isNatIListKind => true
      | id_U32,id_0 => true
      | id_U32,id_U12 => true
      | id_U32,id_a__U52 => true
      | id_U32,id_a__isNat => true
      | id_U32,id_U43 => true
      | id_U32,id_a__isNatIListKind => true
      | id_U32,id_a__U12 => true
      | id_U32,id_U31 => true
      | id_U32,id_a__length => true
      | id_U32,id_a__U42 => true
      | id_U32,id_U53 => true
      | id_U32,id_and => true
      | id_U32,id_cons => true
      | id_U32,id_U11 => true
      | id_U32,id_a__U51 => true
      | id_U32,id_a__U22 => true
      | id_U32,id_U42 => true
      | id_U32,id_length => true
      | id_U32,id_tt => true
      | id_U32,id_U22 => true
      | id_U32,id_s => true
      | id_U32,id_a__U41 => true
      | id_U32,id_U52 => true
      | id_U32,id_nil => true
      | id_U32,id_zeros => true
      | id_U32,id_isNatList => true
      | id_U32,id_a__U53 => true
      | id_U32,id_a__U31 => true
      | id_U32,id_isNatIList => true
      | id_U32,id_a__isNatKind => true
      | id_U32,id_a__isNatList => true
      | id_U32,id_U32 => true
      | id_U32,id_mark => false
      | id_U32,id_a__U43 => false
      | id_U32,id_U61 => false
      | id_U32,id_isNat => false
      | id_mark,id_a__zeros => true
      | id_mark,id_isNatKind => true
      | id_mark,id_a__isNatIList => true
      | id_mark,id_a__U21 => true
      | id_mark,id_U41 => true
      | id_mark,id_a__and => true
      | id_mark,id_a__U11 => true
      | id_mark,id_U21 => true
      | id_mark,id_a__U61 => true
      | id_mark,id_a__U32 => true
      | id_mark,id_U51 => true
      | id_mark,id_isNatIListKind => true
      | id_mark,id_0 => true
      | id_mark,id_U12 => true
      | id_mark,id_a__U52 => true
      | id_mark,id_a__isNat => true
      | id_mark,id_U43 => true
      | id_mark,id_a__isNatIListKind => true
      | id_mark,id_a__U12 => true
      | id_mark,id_U31 => true
      | id_mark,id_a__length => true
      | id_mark,id_a__U42 => true
      | id_mark,id_U53 => true
      | id_mark,id_and => true
      | id_mark,id_cons => true
      | id_mark,id_U11 => true
      | id_mark,id_a__U51 => true
      | id_mark,id_a__U22 => true
      | id_mark,id_U42 => true
      | id_mark,id_length => true
      | id_mark,id_tt => true
      | id_mark,id_U22 => true
      | id_mark,id_s => true
      | id_mark,id_a__U41 => true
      | id_mark,id_U52 => true
      | id_mark,id_nil => true
      | id_mark,id_zeros => true
      | id_mark,id_isNatList => true
      | id_mark,id_a__U53 => true
      | id_mark,id_a__U31 => true
      | id_mark,id_isNatIList => true
      | id_mark,id_a__isNatKind => true
      | id_mark,id_a__isNatList => true
      | id_mark,id_U32 => true
      | id_mark,id_mark => true
      | id_mark,id_a__U43 => false
      | id_mark,id_U61 => false
      | id_mark,id_isNat => false
      | id_a__U43,id_a__zeros => true
      | id_a__U43,id_isNatKind => true
      | id_a__U43,id_a__isNatIList => true
      | id_a__U43,id_a__U21 => true
      | id_a__U43,id_U41 => true
      | id_a__U43,id_a__and => true
      | id_a__U43,id_a__U11 => true
      | id_a__U43,id_U21 => true
      | id_a__U43,id_a__U61 => true
      | id_a__U43,id_a__U32 => true
      | id_a__U43,id_U51 => true
      | id_a__U43,id_isNatIListKind => true
      | id_a__U43,id_0 => true
      | id_a__U43,id_U12 => true
      | id_a__U43,id_a__U52 => true
      | id_a__U43,id_a__isNat => true
      | id_a__U43,id_U43 => true
      | id_a__U43,id_a__isNatIListKind => true
      | id_a__U43,id_a__U12 => true
      | id_a__U43,id_U31 => true
      | id_a__U43,id_a__length => true
      | id_a__U43,id_a__U42 => true
      | id_a__U43,id_U53 => true
      | id_a__U43,id_and => true
      | id_a__U43,id_cons => true
      | id_a__U43,id_U11 => true
      | id_a__U43,id_a__U51 => true
      | id_a__U43,id_a__U22 => true
      | id_a__U43,id_U42 => true
      | id_a__U43,id_length => true
      | id_a__U43,id_tt => true
      | id_a__U43,id_U22 => true
      | id_a__U43,id_s => true
      | id_a__U43,id_a__U41 => true
      | id_a__U43,id_U52 => true
      | id_a__U43,id_nil => true
      | id_a__U43,id_zeros => true
      | id_a__U43,id_isNatList => true
      | id_a__U43,id_a__U53 => true
      | id_a__U43,id_a__U31 => true
      | id_a__U43,id_isNatIList => true
      | id_a__U43,id_a__isNatKind => true
      | id_a__U43,id_a__isNatList => true
      | id_a__U43,id_U32 => true
      | id_a__U43,id_mark => true
      | id_a__U43,id_a__U43 => true
      | id_a__U43,id_U61 => false
      | id_a__U43,id_isNat => false
      | id_U61,id_a__zeros => true
      | id_U61,id_isNatKind => true
      | id_U61,id_a__isNatIList => true
      | id_U61,id_a__U21 => true
      | id_U61,id_U41 => true
      | id_U61,id_a__and => true
      | id_U61,id_a__U11 => true
      | id_U61,id_U21 => true
      | id_U61,id_a__U61 => true
      | id_U61,id_a__U32 => true
      | id_U61,id_U51 => true
      | id_U61,id_isNatIListKind => true
      | id_U61,id_0 => true
      | id_U61,id_U12 => true
      | id_U61,id_a__U52 => true
      | id_U61,id_a__isNat => true
      | id_U61,id_U43 => true
      | id_U61,id_a__isNatIListKind => true
      | id_U61,id_a__U12 => true
      | id_U61,id_U31 => true
      | id_U61,id_a__length => true
      | id_U61,id_a__U42 => true
      | id_U61,id_U53 => true
      | id_U61,id_and => true
      | id_U61,id_cons => true
      | id_U61,id_U11 => true
      | id_U61,id_a__U51 => true
      | id_U61,id_a__U22 => true
      | id_U61,id_U42 => true
      | id_U61,id_length => true
      | id_U61,id_tt => true
      | id_U61,id_U22 => true
      | id_U61,id_s => true
      | id_U61,id_a__U41 => true
      | id_U61,id_U52 => true
      | id_U61,id_nil => true
      | id_U61,id_zeros => true
      | id_U61,id_isNatList => true
      | id_U61,id_a__U53 => true
      | id_U61,id_a__U31 => true
      | id_U61,id_isNatIList => true
      | id_U61,id_a__isNatKind => true
      | id_U61,id_a__isNatList => true
      | id_U61,id_U32 => true
      | id_U61,id_mark => true
      | id_U61,id_a__U43 => true
      | id_U61,id_U61 => true
      | id_U61,id_isNat => false
      | id_isNat,id_a__zeros => true
      | id_isNat,id_isNatKind => true
      | id_isNat,id_a__isNatIList => true
      | id_isNat,id_a__U21 => true
      | id_isNat,id_U41 => true
      | id_isNat,id_a__and => true
      | id_isNat,id_a__U11 => true
      | id_isNat,id_U21 => true
      | id_isNat,id_a__U61 => true
      | id_isNat,id_a__U32 => true
      | id_isNat,id_U51 => true
      | id_isNat,id_isNatIListKind => true
      | id_isNat,id_0 => true
      | id_isNat,id_U12 => true
      | id_isNat,id_a__U52 => true
      | id_isNat,id_a__isNat => true
      | id_isNat,id_U43 => true
      | id_isNat,id_a__isNatIListKind => true
      | id_isNat,id_a__U12 => true
      | id_isNat,id_U31 => true
      | id_isNat,id_a__length => true
      | id_isNat,id_a__U42 => true
      | id_isNat,id_U53 => true
      | id_isNat,id_and => true
      | id_isNat,id_cons => true
      | id_isNat,id_U11 => true
      | id_isNat,id_a__U51 => true
      | id_isNat,id_a__U22 => true
      | id_isNat,id_U42 => true
      | id_isNat,id_length => true
      | id_isNat,id_tt => true
      | id_isNat,id_U22 => true
      | id_isNat,id_s => true
      | id_isNat,id_a__U41 => true
      | id_isNat,id_U52 => true
      | id_isNat,id_nil => true
      | id_isNat,id_zeros => true
      | id_isNat,id_isNatList => true
      | id_isNat,id_a__U53 => true
      | id_isNat,id_a__U31 => true
      | id_isNat,id_isNatIList => true
      | id_isNat,id_a__isNatKind => true
      | id_isNat,id_a__isNatList => true
      | id_isNat,id_U32 => true
      | id_isNat,id_mark => true
      | id_isNat,id_a__U43 => true
      | id_isNat,id_U61 => true
      | id_isNat,id_isNat => 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__isNatList(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__isNatList 
                   ((algebra.Alg.Var 1)::nil))::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) -> tt *)
   | R_xml_0_rule_2 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_tt nil) 
     (algebra.Alg.Term algebra.F.id_a__U12 ((algebra.Alg.Term 
      algebra.F.id_tt nil)::nil))
    (* a__U21(tt,V1_) -> a__U22(a__isNat(V1_)) *)
   | R_xml_0_rule_3 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U22 ((algebra.Alg.Term 
                   algebra.F.id_a__isNat ((algebra.Alg.Var 1)::nil))::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) -> tt *)
   | R_xml_0_rule_4 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_tt nil) 
     (algebra.Alg.Term algebra.F.id_a__U22 ((algebra.Alg.Term 
      algebra.F.id_tt nil)::nil))
    (* a__U31(tt,V_) -> a__U32(a__isNatList(V_)) *)
   | R_xml_0_rule_5 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U32 ((algebra.Alg.Term 
                   algebra.F.id_a__isNatList 
                   ((algebra.Alg.Var 2)::nil))::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) -> 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__U32 ((algebra.Alg.Term 
      algebra.F.id_tt nil)::nil))
    (* a__U41(tt,V1_,V2_) -> a__U42(a__isNat(V1_),V2_) *)
   | R_xml_0_rule_7 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U42 ((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__U41 ((algebra.Alg.Term 
      algebra.F.id_tt nil)::(algebra.Alg.Var 1)::(algebra.Alg.Var 3)::nil))
    (* a__U42(tt,V2_) -> a__U43(a__isNatIList(V2_)) *)
   | R_xml_0_rule_8 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U43 ((algebra.Alg.Term 
                   algebra.F.id_a__isNatIList 
                   ((algebra.Alg.Var 3)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U42 ((algebra.Alg.Term 
      algebra.F.id_tt nil)::(algebra.Alg.Var 3)::nil))
    (* a__U43(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__U43 ((algebra.Alg.Term 
      algebra.F.id_tt nil)::nil))
    (* a__U51(tt,V1_,V2_) -> a__U52(a__isNat(V1_),V2_) *)
   | R_xml_0_rule_10 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U52 ((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__U51 ((algebra.Alg.Term 
      algebra.F.id_tt nil)::(algebra.Alg.Var 1)::(algebra.Alg.Var 3)::nil))
    (* a__U52(tt,V2_) -> a__U53(a__isNatList(V2_)) *)
   | R_xml_0_rule_11 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U53 ((algebra.Alg.Term 
                   algebra.F.id_a__isNatList 
                   ((algebra.Alg.Var 3)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U52 ((algebra.Alg.Term 
      algebra.F.id_tt nil)::(algebra.Alg.Var 3)::nil))
    (* a__U53(tt) -> tt *)
   | R_xml_0_rule_12 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_tt nil) 
     (algebra.Alg.Term algebra.F.id_a__U53 ((algebra.Alg.Term 
      algebra.F.id_tt nil)::nil))
    (* a__U61(tt,L_) -> s(a__length(mark(L_))) *)
   | R_xml_0_rule_13 :
    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__U61 ((algebra.Alg.Term 
      algebra.F.id_tt nil)::(algebra.Alg.Var 4)::nil))
    (* a__and(tt,X_) -> mark(X_) *)
   | R_xml_0_rule_14 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_mark 
                   ((algebra.Alg.Var 5)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__and ((algebra.Alg.Term 
      algebra.F.id_tt nil)::(algebra.Alg.Var 5)::nil))
    (* a__isNat(0) -> 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__isNat ((algebra.Alg.Term 
      algebra.F.id_0 nil)::nil))
   
    (* a__isNat(length(V1_)) -> a__U11(a__isNatIListKind(V1_),V1_) *)
   | R_xml_0_rule_16 :
    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_17 :
    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_18 :
    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_19 :
    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__and(a__isNatKind(V1_),isNatIListKind(V2_)),V1_,V2_) *)
   | R_xml_0_rule_20 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U41 ((algebra.Alg.Term 
                   algebra.F.id_a__and ((algebra.Alg.Term 
                   algebra.F.id_a__isNatKind ((algebra.Alg.Var 1)::nil))::
                   (algebra.Alg.Term algebra.F.id_isNatIListKind 
                   ((algebra.Alg.Var 3)::nil))::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_21 :
    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_22 :
    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__and(a__isNatKind(V1_),isNatIListKind(V2_)) *)
   | R_xml_0_rule_23 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__and ((algebra.Alg.Term 
                   algebra.F.id_a__isNatKind ((algebra.Alg.Var 1)::nil))::
                   (algebra.Alg.Term algebra.F.id_isNatIListKind 
                   ((algebra.Alg.Var 3)::nil))::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_24 :
    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__isNatIListKind(V1_) *)
   | R_xml_0_rule_25 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
                   ((algebra.Alg.Var 1)::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__isNatKind(V1_) *)
   | R_xml_0_rule_26 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__isNatKind 
                   ((algebra.Alg.Var 1)::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_27 :
    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__U51(a__and(a__isNatKind(V1_),isNatIListKind(V2_)),V1_,V2_) *)
   | R_xml_0_rule_28 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U51 ((algebra.Alg.Term 
                   algebra.F.id_a__and ((algebra.Alg.Term 
                   algebra.F.id_a__isNatKind ((algebra.Alg.Var 1)::nil))::
                   (algebra.Alg.Term algebra.F.id_isNatIListKind 
                   ((algebra.Alg.Var 3)::nil))::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_29 :
    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__U61(a__and(a__and(a__isNatList(L_),isNatIListKind(L_)),
        and(isNat(N_),isNatKind(N_))),L_) *)
   | R_xml_0_rule_30 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U61 ((algebra.Alg.Term 
                   algebra.F.id_a__and ((algebra.Alg.Term 
                   algebra.F.id_a__and ((algebra.Alg.Term 
                   algebra.F.id_a__isNatList ((algebra.Alg.Var 4)::nil))::
                   (algebra.Alg.Term algebra.F.id_isNatIListKind 
                   ((algebra.Alg.Var 4)::nil))::nil))::(algebra.Alg.Term 
                   algebra.F.id_and ((algebra.Alg.Term algebra.F.id_isNat 
                   ((algebra.Alg.Var 6)::nil))::(algebra.Alg.Term 
                   algebra.F.id_isNatKind 
                   ((algebra.Alg.Var 6)::nil))::nil))::nil))::
                   (algebra.Alg.Var 4)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__length ((algebra.Alg.Term 
      algebra.F.id_cons ((algebra.Alg.Var 6)::
      (algebra.Alg.Var 4)::nil))::nil))
    (* mark(zeros) -> a__zeros *)
   | R_xml_0_rule_31 :
    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_32 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U11 ((algebra.Alg.Term 
                   algebra.F.id_mark ((algebra.Alg.Var 7)::nil))::
                   (algebra.Alg.Var 8)::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U11 
      ((algebra.Alg.Var 7)::(algebra.Alg.Var 8)::nil))::nil))
    (* mark(U12(X_)) -> a__U12(mark(X_)) *)
   | R_xml_0_rule_33 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U12 ((algebra.Alg.Term 
                   algebra.F.id_mark ((algebra.Alg.Var 5)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U12 
      ((algebra.Alg.Var 5)::nil))::nil))
    (* mark(isNatList(X_)) -> a__isNatList(X_) *)
   | R_xml_0_rule_34 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__isNatList 
                   ((algebra.Alg.Var 5)::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term 
      algebra.F.id_isNatList ((algebra.Alg.Var 5)::nil))::nil))
    (* mark(U21(X1_,X2_)) -> a__U21(mark(X1_),X2_) *)
   | R_xml_0_rule_35 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U21 ((algebra.Alg.Term 
                   algebra.F.id_mark ((algebra.Alg.Var 7)::nil))::
                   (algebra.Alg.Var 8)::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U21 
      ((algebra.Alg.Var 7)::(algebra.Alg.Var 8)::nil))::nil))
    (* mark(U22(X_)) -> a__U22(mark(X_)) *)
   | R_xml_0_rule_36 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U22 ((algebra.Alg.Term 
                   algebra.F.id_mark ((algebra.Alg.Var 5)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U22 
      ((algebra.Alg.Var 5)::nil))::nil))
    (* mark(isNat(X_)) -> a__isNat(X_) *)
   | R_xml_0_rule_37 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__isNat 
                   ((algebra.Alg.Var 5)::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term 
      algebra.F.id_isNat ((algebra.Alg.Var 5)::nil))::nil))
    (* mark(U31(X1_,X2_)) -> a__U31(mark(X1_),X2_) *)
   | R_xml_0_rule_38 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U31 ((algebra.Alg.Term 
                   algebra.F.id_mark ((algebra.Alg.Var 7)::nil))::
                   (algebra.Alg.Var 8)::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U31 
      ((algebra.Alg.Var 7)::(algebra.Alg.Var 8)::nil))::nil))
    (* mark(U32(X_)) -> a__U32(mark(X_)) *)
   | R_xml_0_rule_39 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U32 ((algebra.Alg.Term 
                   algebra.F.id_mark ((algebra.Alg.Var 5)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U32 
      ((algebra.Alg.Var 5)::nil))::nil))
   
    (* mark(U41(X1_,X2_,X3_)) -> a__U41(mark(X1_),X2_,X3_) *)
   | R_xml_0_rule_40 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U41 ((algebra.Alg.Term 
                   algebra.F.id_mark ((algebra.Alg.Var 7)::nil))::
                   (algebra.Alg.Var 8)::(algebra.Alg.Var 9)::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U41 
      ((algebra.Alg.Var 7)::(algebra.Alg.Var 8)::
      (algebra.Alg.Var 9)::nil))::nil))
    (* mark(U42(X1_,X2_)) -> a__U42(mark(X1_),X2_) *)
   | R_xml_0_rule_41 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U42 ((algebra.Alg.Term 
                   algebra.F.id_mark ((algebra.Alg.Var 7)::nil))::
                   (algebra.Alg.Var 8)::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U42 
      ((algebra.Alg.Var 7)::(algebra.Alg.Var 8)::nil))::nil))
    (* mark(U43(X_)) -> a__U43(mark(X_)) *)
   | R_xml_0_rule_42 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U43 ((algebra.Alg.Term 
                   algebra.F.id_mark ((algebra.Alg.Var 5)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U43 
      ((algebra.Alg.Var 5)::nil))::nil))
    (* mark(isNatIList(X_)) -> a__isNatIList(X_) *)
   | R_xml_0_rule_43 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__isNatIList 
                   ((algebra.Alg.Var 5)::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term 
      algebra.F.id_isNatIList ((algebra.Alg.Var 5)::nil))::nil))
   
    (* mark(U51(X1_,X2_,X3_)) -> a__U51(mark(X1_),X2_,X3_) *)
   | R_xml_0_rule_44 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U51 ((algebra.Alg.Term 
                   algebra.F.id_mark ((algebra.Alg.Var 7)::nil))::
                   (algebra.Alg.Var 8)::(algebra.Alg.Var 9)::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U51 
      ((algebra.Alg.Var 7)::(algebra.Alg.Var 8)::
      (algebra.Alg.Var 9)::nil))::nil))
    (* mark(U52(X1_,X2_)) -> a__U52(mark(X1_),X2_) *)
   | R_xml_0_rule_45 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U52 ((algebra.Alg.Term 
                   algebra.F.id_mark ((algebra.Alg.Var 7)::nil))::
                   (algebra.Alg.Var 8)::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U52 
      ((algebra.Alg.Var 7)::(algebra.Alg.Var 8)::nil))::nil))
    (* mark(U53(X_)) -> a__U53(mark(X_)) *)
   | R_xml_0_rule_46 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U53 ((algebra.Alg.Term 
                   algebra.F.id_mark ((algebra.Alg.Var 5)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U53 
      ((algebra.Alg.Var 5)::nil))::nil))
    (* mark(U61(X1_,X2_)) -> a__U61(mark(X1_),X2_) *)
   | R_xml_0_rule_47 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U61 ((algebra.Alg.Term 
                   algebra.F.id_mark ((algebra.Alg.Var 7)::nil))::
                   (algebra.Alg.Var 8)::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U61 
      ((algebra.Alg.Var 7)::(algebra.Alg.Var 8)::nil))::nil))
    (* mark(length(X_)) -> a__length(mark(X_)) *)
   | R_xml_0_rule_48 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__length 
                   ((algebra.Alg.Term algebra.F.id_mark 
                   ((algebra.Alg.Var 5)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term 
      algebra.F.id_length ((algebra.Alg.Var 5)::nil))::nil))
    (* mark(and(X1_,X2_)) -> a__and(mark(X1_),X2_) *)
   | R_xml_0_rule_49 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__and ((algebra.Alg.Term 
                   algebra.F.id_mark ((algebra.Alg.Var 7)::nil))::
                   (algebra.Alg.Var 8)::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_and 
      ((algebra.Alg.Var 7)::(algebra.Alg.Var 8)::nil))::nil))
   
    (* mark(isNatIListKind(X_)) -> a__isNatIListKind(X_) *)
   | R_xml_0_rule_50 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
                   ((algebra.Alg.Var 5)::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term 
      algebra.F.id_isNatIListKind ((algebra.Alg.Var 5)::nil))::nil))
    (* mark(isNatKind(X_)) -> a__isNatKind(X_) *)
   | R_xml_0_rule_51 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__isNatKind 
                   ((algebra.Alg.Var 5)::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term 
      algebra.F.id_isNatKind ((algebra.Alg.Var 5)::nil))::nil))
    (* mark(cons(X1_,X2_)) -> cons(mark(X1_),X2_) *)
   | R_xml_0_rule_52 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term 
                   algebra.F.id_mark ((algebra.Alg.Var 7)::nil))::
                   (algebra.Alg.Var 8)::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term 
      algebra.F.id_cons ((algebra.Alg.Var 7)::
      (algebra.Alg.Var 8)::nil))::nil))
    (* mark(0) -> 0 *)
   | R_xml_0_rule_53 :
    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_54 :
    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_55 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_s ((algebra.Alg.Term 
                   algebra.F.id_mark ((algebra.Alg.Var 5)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_s 
      ((algebra.Alg.Var 5)::nil))::nil))
    (* mark(nil) -> nil *)
   | R_xml_0_rule_56 :
    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_57 :
    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_58 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_U11 ((algebra.Alg.Var 7)::
                   (algebra.Alg.Var 8)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U11 ((algebra.Alg.Var 7)::
      (algebra.Alg.Var 8)::nil))
    (* a__U12(X_) -> U12(X_) *)
   | R_xml_0_rule_59 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_U12 
                   ((algebra.Alg.Var 5)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U12 ((algebra.Alg.Var 5)::nil))
    (* a__isNatList(X_) -> isNatList(X_) *)
   | R_xml_0_rule_60 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_isNatList 
                   ((algebra.Alg.Var 5)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__isNatList ((algebra.Alg.Var 5)::nil))
    (* a__U21(X1_,X2_) -> U21(X1_,X2_) *)
   | R_xml_0_rule_61 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_U21 ((algebra.Alg.Var 7)::
                   (algebra.Alg.Var 8)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U21 ((algebra.Alg.Var 7)::
      (algebra.Alg.Var 8)::nil))
    (* a__U22(X_) -> U22(X_) *)
   | R_xml_0_rule_62 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_U22 
                   ((algebra.Alg.Var 5)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U22 ((algebra.Alg.Var 5)::nil))
    (* a__isNat(X_) -> isNat(X_) *)
   | R_xml_0_rule_63 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_isNat 
                   ((algebra.Alg.Var 5)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__isNat ((algebra.Alg.Var 5)::nil))
    (* a__U31(X1_,X2_) -> U31(X1_,X2_) *)
   | R_xml_0_rule_64 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_U31 ((algebra.Alg.Var 7)::
                   (algebra.Alg.Var 8)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U31 ((algebra.Alg.Var 7)::
      (algebra.Alg.Var 8)::nil))
    (* a__U32(X_) -> U32(X_) *)
   | R_xml_0_rule_65 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_U32 
                   ((algebra.Alg.Var 5)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U32 ((algebra.Alg.Var 5)::nil))
    (* a__U41(X1_,X2_,X3_) -> U41(X1_,X2_,X3_) *)
   | R_xml_0_rule_66 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_U41 ((algebra.Alg.Var 7)::
                   (algebra.Alg.Var 8)::(algebra.Alg.Var 9)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U41 ((algebra.Alg.Var 7)::
      (algebra.Alg.Var 8)::(algebra.Alg.Var 9)::nil))
    (* a__U42(X1_,X2_) -> U42(X1_,X2_) *)
   | R_xml_0_rule_67 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_U42 ((algebra.Alg.Var 7)::
                   (algebra.Alg.Var 8)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U42 ((algebra.Alg.Var 7)::
      (algebra.Alg.Var 8)::nil))
    (* a__U43(X_) -> U43(X_) *)
   | R_xml_0_rule_68 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_U43 
                   ((algebra.Alg.Var 5)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U43 ((algebra.Alg.Var 5)::nil))
    (* a__isNatIList(X_) -> isNatIList(X_) *)
   | R_xml_0_rule_69 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_isNatIList 
                   ((algebra.Alg.Var 5)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__isNatIList ((algebra.Alg.Var 5)::nil))
    (* a__U51(X1_,X2_,X3_) -> U51(X1_,X2_,X3_) *)
   | R_xml_0_rule_70 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_U51 ((algebra.Alg.Var 7)::
                   (algebra.Alg.Var 8)::(algebra.Alg.Var 9)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U51 ((algebra.Alg.Var 7)::
      (algebra.Alg.Var 8)::(algebra.Alg.Var 9)::nil))
    (* a__U52(X1_,X2_) -> U52(X1_,X2_) *)
   | R_xml_0_rule_71 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_U52 ((algebra.Alg.Var 7)::
                   (algebra.Alg.Var 8)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U52 ((algebra.Alg.Var 7)::
      (algebra.Alg.Var 8)::nil))
    (* a__U53(X_) -> U53(X_) *)
   | R_xml_0_rule_72 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_U53 
                   ((algebra.Alg.Var 5)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U53 ((algebra.Alg.Var 5)::nil))
    (* a__U61(X1_,X2_) -> U61(X1_,X2_) *)
   | R_xml_0_rule_73 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_U61 ((algebra.Alg.Var 7)::
                   (algebra.Alg.Var 8)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__U61 ((algebra.Alg.Var 7)::
      (algebra.Alg.Var 8)::nil))
    (* a__length(X_) -> length(X_) *)
   | R_xml_0_rule_74 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_length 
                   ((algebra.Alg.Var 5)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__length ((algebra.Alg.Var 5)::nil))
    (* a__and(X1_,X2_) -> and(X1_,X2_) *)
   | R_xml_0_rule_75 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_and ((algebra.Alg.Var 7)::
                   (algebra.Alg.Var 8)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__and ((algebra.Alg.Var 7)::
      (algebra.Alg.Var 8)::nil))
    (* a__isNatIListKind(X_) -> isNatIListKind(X_) *)
   | R_xml_0_rule_76 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_isNatIListKind 
                   ((algebra.Alg.Var 5)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
      ((algebra.Alg.Var 5)::nil))
    (* a__isNatKind(X_) -> isNatKind(X_) *)
   | R_xml_0_rule_77 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_isNatKind 
                   ((algebra.Alg.Var 5)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__isNatKind ((algebra.Alg.Var 5)::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__isNatList ((algebra.Alg.Var 1)::nil))::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)::nil)),(algebra.Alg.Term algebra.F.id_tt nil))::
    R_xml_0_rule_as_list_1.
 
 
 Definition R_xml_0_rule_as_list_3  := 
   ((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__isNat ((algebra.Alg.Var 1)::nil))::nil)))::
    R_xml_0_rule_as_list_2.
 
 
 Definition R_xml_0_rule_as_list_4  := 
   ((algebra.Alg.Term algebra.F.id_a__U22 ((algebra.Alg.Term algebra.F.id_tt 
     nil)::nil)),(algebra.Alg.Term algebra.F.id_tt nil))::
    R_xml_0_rule_as_list_3.
 
 
 Definition R_xml_0_rule_as_list_5  := 
   ((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__isNatList ((algebra.Alg.Var 2)::nil))::nil)))::
    R_xml_0_rule_as_list_4.
 
 
 Definition R_xml_0_rule_as_list_6  := 
   ((algebra.Alg.Term algebra.F.id_a__U32 ((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__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__isNat ((algebra.Alg.Var 1)::nil))::
     (algebra.Alg.Var 3)::nil)))::R_xml_0_rule_as_list_6.
 
 
 Definition R_xml_0_rule_as_list_8  := 
   ((algebra.Alg.Term algebra.F.id_a__U42 ((algebra.Alg.Term algebra.F.id_tt 
     nil)::(algebra.Alg.Var 3)::nil)),
    (algebra.Alg.Term algebra.F.id_a__U43 ((algebra.Alg.Term 
     algebra.F.id_a__isNatIList ((algebra.Alg.Var 3)::nil))::nil)))::
    R_xml_0_rule_as_list_7.
 
 
 Definition R_xml_0_rule_as_list_9  := 
   ((algebra.Alg.Term algebra.F.id_a__U43 ((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__U51 ((algebra.Alg.Term algebra.F.id_tt 
     nil)::(algebra.Alg.Var 1)::(algebra.Alg.Var 3)::nil)),
    (algebra.Alg.Term algebra.F.id_a__U52 ((algebra.Alg.Term 
     algebra.F.id_a__isNat ((algebra.Alg.Var 1)::nil))::
     (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__U52 ((algebra.Alg.Term algebra.F.id_tt 
     nil)::(algebra.Alg.Var 3)::nil)),
    (algebra.Alg.Term algebra.F.id_a__U53 ((algebra.Alg.Term 
     algebra.F.id_a__isNatList ((algebra.Alg.Var 3)::nil))::nil)))::
    R_xml_0_rule_as_list_10.
 
 
 Definition R_xml_0_rule_as_list_12  := 
   ((algebra.Alg.Term algebra.F.id_a__U53 ((algebra.Alg.Term algebra.F.id_tt 
     nil)::nil)),(algebra.Alg.Term algebra.F.id_tt nil))::
    R_xml_0_rule_as_list_11.
 
 
 Definition R_xml_0_rule_as_list_13  := 
   ((algebra.Alg.Term algebra.F.id_a__U61 ((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_12.
 
 
 Definition R_xml_0_rule_as_list_14  := 
   ((algebra.Alg.Term algebra.F.id_a__and ((algebra.Alg.Term algebra.F.id_tt 
     nil)::(algebra.Alg.Var 5)::nil)),
    (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 5)::nil)))::
    R_xml_0_rule_as_list_13.
 
 
 Definition R_xml_0_rule_as_list_15  := 
   ((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_14.
 
 
 Definition R_xml_0_rule_as_list_16  := 
   ((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_15.
 
 
 Definition R_xml_0_rule_as_list_17  := 
   ((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_16.
 
 
 Definition R_xml_0_rule_as_list_18  := 
   ((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_17.
 
 
 Definition R_xml_0_rule_as_list_19  := 
   ((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_18.
 
 
 Definition R_xml_0_rule_as_list_20  := 
   ((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__and ((algebra.Alg.Term algebra.F.id_a__isNatKind 
     ((algebra.Alg.Var 1)::nil))::(algebra.Alg.Term 
     algebra.F.id_isNatIListKind ((algebra.Alg.Var 3)::nil))::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__isNatIListKind ((algebra.Alg.Term 
     algebra.F.id_nil nil)::nil)),(algebra.Alg.Term algebra.F.id_tt nil))::
    R_xml_0_rule_as_list_20.
 
 
 Definition R_xml_0_rule_as_list_22  := 
   ((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_21.
 
 
 Definition R_xml_0_rule_as_list_23  := 
   ((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__and ((algebra.Alg.Term 
     algebra.F.id_a__isNatKind ((algebra.Alg.Var 1)::nil))::
     (algebra.Alg.Term algebra.F.id_isNatIListKind 
     ((algebra.Alg.Var 3)::nil))::nil)))::R_xml_0_rule_as_list_22.
 
 
 Definition R_xml_0_rule_as_list_24  := 
   ((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_23.
 
 
 Definition R_xml_0_rule_as_list_25  := 
   ((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__isNatIListKind 
     ((algebra.Alg.Var 1)::nil)))::R_xml_0_rule_as_list_24.
 
 
 Definition R_xml_0_rule_as_list_26  := 
   ((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__isNatKind ((algebra.Alg.Var 1)::nil)))::
    R_xml_0_rule_as_list_25.
 
 
 Definition R_xml_0_rule_as_list_27  := 
   ((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_26.
 
 
 Definition R_xml_0_rule_as_list_28  := 
   ((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__U51 ((algebra.Alg.Term 
     algebra.F.id_a__and ((algebra.Alg.Term algebra.F.id_a__isNatKind 
     ((algebra.Alg.Var 1)::nil))::(algebra.Alg.Term 
     algebra.F.id_isNatIListKind ((algebra.Alg.Var 3)::nil))::nil))::
     (algebra.Alg.Var 1)::(algebra.Alg.Var 3)::nil)))::
    R_xml_0_rule_as_list_27.
 
 
 Definition R_xml_0_rule_as_list_29  := 
   ((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_28.
 
 
 Definition R_xml_0_rule_as_list_30  := 
   ((algebra.Alg.Term algebra.F.id_a__length ((algebra.Alg.Term 
     algebra.F.id_cons ((algebra.Alg.Var 6)::
     (algebra.Alg.Var 4)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__U61 ((algebra.Alg.Term 
     algebra.F.id_a__and ((algebra.Alg.Term algebra.F.id_a__and 
     ((algebra.Alg.Term algebra.F.id_a__isNatList 
     ((algebra.Alg.Var 4)::nil))::(algebra.Alg.Term 
     algebra.F.id_isNatIListKind ((algebra.Alg.Var 4)::nil))::nil))::
     (algebra.Alg.Term algebra.F.id_and ((algebra.Alg.Term 
     algebra.F.id_isNat ((algebra.Alg.Var 6)::nil))::(algebra.Alg.Term 
     algebra.F.id_isNatKind ((algebra.Alg.Var 6)::nil))::nil))::nil))::
     (algebra.Alg.Var 4)::nil)))::R_xml_0_rule_as_list_29.
 
 
 Definition R_xml_0_rule_as_list_31  := 
   ((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_30.
 
 
 Definition R_xml_0_rule_as_list_32  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U11 
     ((algebra.Alg.Var 7)::(algebra.Alg.Var 8)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__U11 ((algebra.Alg.Term 
     algebra.F.id_mark ((algebra.Alg.Var 7)::nil))::
     (algebra.Alg.Var 8)::nil)))::R_xml_0_rule_as_list_31.
 
 
 Definition R_xml_0_rule_as_list_33  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U12 
     ((algebra.Alg.Var 5)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__U12 ((algebra.Alg.Term 
     algebra.F.id_mark ((algebra.Alg.Var 5)::nil))::nil)))::
    R_xml_0_rule_as_list_32.
 
 
 Definition R_xml_0_rule_as_list_34  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term 
     algebra.F.id_isNatList ((algebra.Alg.Var 5)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__isNatList ((algebra.Alg.Var 5)::nil)))::
    R_xml_0_rule_as_list_33.
 
 
 Definition R_xml_0_rule_as_list_35  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U21 
     ((algebra.Alg.Var 7)::(algebra.Alg.Var 8)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__U21 ((algebra.Alg.Term 
     algebra.F.id_mark ((algebra.Alg.Var 7)::nil))::
     (algebra.Alg.Var 8)::nil)))::R_xml_0_rule_as_list_34.
 
 
 Definition R_xml_0_rule_as_list_36  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U22 
     ((algebra.Alg.Var 5)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__U22 ((algebra.Alg.Term 
     algebra.F.id_mark ((algebra.Alg.Var 5)::nil))::nil)))::
    R_xml_0_rule_as_list_35.
 
 
 Definition R_xml_0_rule_as_list_37  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term 
     algebra.F.id_isNat ((algebra.Alg.Var 5)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__isNat ((algebra.Alg.Var 5)::nil)))::
    R_xml_0_rule_as_list_36.
 
 
 Definition R_xml_0_rule_as_list_38  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U31 
     ((algebra.Alg.Var 7)::(algebra.Alg.Var 8)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__U31 ((algebra.Alg.Term 
     algebra.F.id_mark ((algebra.Alg.Var 7)::nil))::
     (algebra.Alg.Var 8)::nil)))::R_xml_0_rule_as_list_37.
 
 
 Definition R_xml_0_rule_as_list_39  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U32 
     ((algebra.Alg.Var 5)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__U32 ((algebra.Alg.Term 
     algebra.F.id_mark ((algebra.Alg.Var 5)::nil))::nil)))::
    R_xml_0_rule_as_list_38.
 
 
 Definition R_xml_0_rule_as_list_40  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U41 
     ((algebra.Alg.Var 7)::(algebra.Alg.Var 8)::
     (algebra.Alg.Var 9)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__U41 ((algebra.Alg.Term 
     algebra.F.id_mark ((algebra.Alg.Var 7)::nil))::(algebra.Alg.Var 8)::
     (algebra.Alg.Var 9)::nil)))::R_xml_0_rule_as_list_39.
 
 
 Definition R_xml_0_rule_as_list_41  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U42 
     ((algebra.Alg.Var 7)::(algebra.Alg.Var 8)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__U42 ((algebra.Alg.Term 
     algebra.F.id_mark ((algebra.Alg.Var 7)::nil))::
     (algebra.Alg.Var 8)::nil)))::R_xml_0_rule_as_list_40.
 
 
 Definition R_xml_0_rule_as_list_42  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U43 
     ((algebra.Alg.Var 5)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__U43 ((algebra.Alg.Term 
     algebra.F.id_mark ((algebra.Alg.Var 5)::nil))::nil)))::
    R_xml_0_rule_as_list_41.
 
 
 Definition R_xml_0_rule_as_list_43  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term 
     algebra.F.id_isNatIList ((algebra.Alg.Var 5)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__isNatIList ((algebra.Alg.Var 5)::nil)))::
    R_xml_0_rule_as_list_42.
 
 
 Definition R_xml_0_rule_as_list_44  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U51 
     ((algebra.Alg.Var 7)::(algebra.Alg.Var 8)::
     (algebra.Alg.Var 9)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__U51 ((algebra.Alg.Term 
     algebra.F.id_mark ((algebra.Alg.Var 7)::nil))::(algebra.Alg.Var 8)::
     (algebra.Alg.Var 9)::nil)))::R_xml_0_rule_as_list_43.
 
 
 Definition R_xml_0_rule_as_list_45  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U52 
     ((algebra.Alg.Var 7)::(algebra.Alg.Var 8)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__U52 ((algebra.Alg.Term 
     algebra.F.id_mark ((algebra.Alg.Var 7)::nil))::
     (algebra.Alg.Var 8)::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_U53 
     ((algebra.Alg.Var 5)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__U53 ((algebra.Alg.Term 
     algebra.F.id_mark ((algebra.Alg.Var 5)::nil))::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_U61 
     ((algebra.Alg.Var 7)::(algebra.Alg.Var 8)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__U61 ((algebra.Alg.Term 
     algebra.F.id_mark ((algebra.Alg.Var 7)::nil))::
     (algebra.Alg.Var 8)::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_length ((algebra.Alg.Var 5)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__length ((algebra.Alg.Term 
     algebra.F.id_mark ((algebra.Alg.Var 5)::nil))::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_and 
     ((algebra.Alg.Var 7)::(algebra.Alg.Var 8)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__and ((algebra.Alg.Term 
     algebra.F.id_mark ((algebra.Alg.Var 7)::nil))::
     (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_isNatIListKind ((algebra.Alg.Var 5)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
     ((algebra.Alg.Var 5)::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_isNatKind ((algebra.Alg.Var 5)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__isNatKind ((algebra.Alg.Var 5)::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_cons 
     ((algebra.Alg.Var 7)::(algebra.Alg.Var 8)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_mark 
     ((algebra.Alg.Var 7)::nil))::(algebra.Alg.Var 8)::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_0 
     nil)::nil)),(algebra.Alg.Term algebra.F.id_0 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_tt 
     nil)::nil)),(algebra.Alg.Term algebra.F.id_tt 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_s 
     ((algebra.Alg.Var 5)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_s ((algebra.Alg.Term algebra.F.id_mark 
     ((algebra.Alg.Var 5)::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_nil 
     nil)::nil)),(algebra.Alg.Term algebra.F.id_nil nil))::
    R_xml_0_rule_as_list_55.
 
 
 Definition R_xml_0_rule_as_list_57  := 
   ((algebra.Alg.Term algebra.F.id_a__zeros nil),
    (algebra.Alg.Term algebra.F.id_zeros nil))::R_xml_0_rule_as_list_56.
 
 
 Definition R_xml_0_rule_as_list_58  := 
   ((algebra.Alg.Term algebra.F.id_a__U11 ((algebra.Alg.Var 7)::
     (algebra.Alg.Var 8)::nil)),
    (algebra.Alg.Term algebra.F.id_U11 ((algebra.Alg.Var 7)::
     (algebra.Alg.Var 8)::nil)))::R_xml_0_rule_as_list_57.
 
 
 Definition R_xml_0_rule_as_list_59  := 
   ((algebra.Alg.Term algebra.F.id_a__U12 ((algebra.Alg.Var 5)::nil)),
    (algebra.Alg.Term algebra.F.id_U12 ((algebra.Alg.Var 5)::nil)))::
    R_xml_0_rule_as_list_58.
 
 
 Definition R_xml_0_rule_as_list_60  := 
   ((algebra.Alg.Term algebra.F.id_a__isNatList ((algebra.Alg.Var 5)::nil)),
    (algebra.Alg.Term algebra.F.id_isNatList ((algebra.Alg.Var 5)::nil)))::
    R_xml_0_rule_as_list_59.
 
 
 Definition R_xml_0_rule_as_list_61  := 
   ((algebra.Alg.Term algebra.F.id_a__U21 ((algebra.Alg.Var 7)::
     (algebra.Alg.Var 8)::nil)),
    (algebra.Alg.Term algebra.F.id_U21 ((algebra.Alg.Var 7)::
     (algebra.Alg.Var 8)::nil)))::R_xml_0_rule_as_list_60.
 
 
 Definition R_xml_0_rule_as_list_62  := 
   ((algebra.Alg.Term algebra.F.id_a__U22 ((algebra.Alg.Var 5)::nil)),
    (algebra.Alg.Term algebra.F.id_U22 ((algebra.Alg.Var 5)::nil)))::
    R_xml_0_rule_as_list_61.
 
 
 Definition R_xml_0_rule_as_list_63  := 
   ((algebra.Alg.Term algebra.F.id_a__isNat ((algebra.Alg.Var 5)::nil)),
    (algebra.Alg.Term algebra.F.id_isNat ((algebra.Alg.Var 5)::nil)))::
    R_xml_0_rule_as_list_62.
 
 
 Definition R_xml_0_rule_as_list_64  := 
   ((algebra.Alg.Term algebra.F.id_a__U31 ((algebra.Alg.Var 7)::
     (algebra.Alg.Var 8)::nil)),
    (algebra.Alg.Term algebra.F.id_U31 ((algebra.Alg.Var 7)::
     (algebra.Alg.Var 8)::nil)))::R_xml_0_rule_as_list_63.
 
 
 Definition R_xml_0_rule_as_list_65  := 
   ((algebra.Alg.Term algebra.F.id_a__U32 ((algebra.Alg.Var 5)::nil)),
    (algebra.Alg.Term algebra.F.id_U32 ((algebra.Alg.Var 5)::nil)))::
    R_xml_0_rule_as_list_64.
 
 
 Definition R_xml_0_rule_as_list_66  := 
   ((algebra.Alg.Term algebra.F.id_a__U41 ((algebra.Alg.Var 7)::
     (algebra.Alg.Var 8)::(algebra.Alg.Var 9)::nil)),
    (algebra.Alg.Term algebra.F.id_U41 ((algebra.Alg.Var 7)::
     (algebra.Alg.Var 8)::(algebra.Alg.Var 9)::nil)))::
    R_xml_0_rule_as_list_65.
 
 
 Definition R_xml_0_rule_as_list_67  := 
   ((algebra.Alg.Term algebra.F.id_a__U42 ((algebra.Alg.Var 7)::
     (algebra.Alg.Var 8)::nil)),
    (algebra.Alg.Term algebra.F.id_U42 ((algebra.Alg.Var 7)::
     (algebra.Alg.Var 8)::nil)))::R_xml_0_rule_as_list_66.
 
 
 Definition R_xml_0_rule_as_list_68  := 
   ((algebra.Alg.Term algebra.F.id_a__U43 ((algebra.Alg.Var 5)::nil)),
    (algebra.Alg.Term algebra.F.id_U43 ((algebra.Alg.Var 5)::nil)))::
    R_xml_0_rule_as_list_67.
 
 
 Definition R_xml_0_rule_as_list_69  := 
   ((algebra.Alg.Term algebra.F.id_a__isNatIList ((algebra.Alg.Var 5)::nil)),
    (algebra.Alg.Term algebra.F.id_isNatIList ((algebra.Alg.Var 5)::nil)))::
    R_xml_0_rule_as_list_68.
 
 
 Definition R_xml_0_rule_as_list_70  := 
   ((algebra.Alg.Term algebra.F.id_a__U51 ((algebra.Alg.Var 7)::
     (algebra.Alg.Var 8)::(algebra.Alg.Var 9)::nil)),
    (algebra.Alg.Term algebra.F.id_U51 ((algebra.Alg.Var 7)::
     (algebra.Alg.Var 8)::(algebra.Alg.Var 9)::nil)))::
    R_xml_0_rule_as_list_69.
 
 
 Definition R_xml_0_rule_as_list_71  := 
   ((algebra.Alg.Term algebra.F.id_a__U52 ((algebra.Alg.Var 7)::
     (algebra.Alg.Var 8)::nil)),
    (algebra.Alg.Term algebra.F.id_U52 ((algebra.Alg.Var 7)::
     (algebra.Alg.Var 8)::nil)))::R_xml_0_rule_as_list_70.
 
 
 Definition R_xml_0_rule_as_list_72  := 
   ((algebra.Alg.Term algebra.F.id_a__U53 ((algebra.Alg.Var 5)::nil)),
    (algebra.Alg.Term algebra.F.id_U53 ((algebra.Alg.Var 5)::nil)))::
    R_xml_0_rule_as_list_71.
 
 
 Definition R_xml_0_rule_as_list_73  := 
   ((algebra.Alg.Term algebra.F.id_a__U61 ((algebra.Alg.Var 7)::
     (algebra.Alg.Var 8)::nil)),
    (algebra.Alg.Term algebra.F.id_U61 ((algebra.Alg.Var 7)::
     (algebra.Alg.Var 8)::nil)))::R_xml_0_rule_as_list_72.
 
 
 Definition R_xml_0_rule_as_list_74  := 
   ((algebra.Alg.Term algebra.F.id_a__length ((algebra.Alg.Var 5)::nil)),
    (algebra.Alg.Term algebra.F.id_length ((algebra.Alg.Var 5)::nil)))::
    R_xml_0_rule_as_list_73.
 
 
 Definition R_xml_0_rule_as_list_75  := 
   ((algebra.Alg.Term algebra.F.id_a__and ((algebra.Alg.Var 7)::
     (algebra.Alg.Var 8)::nil)),
    (algebra.Alg.Term algebra.F.id_and ((algebra.Alg.Var 7)::
     (algebra.Alg.Var 8)::nil)))::R_xml_0_rule_as_list_74.
 
 
 Definition R_xml_0_rule_as_list_76  := 
   ((algebra.Alg.Term algebra.F.id_a__isNatIListKind 
     ((algebra.Alg.Var 5)::nil)),
    (algebra.Alg.Term algebra.F.id_isNatIListKind ((algebra.Alg.Var 5)::nil)))::
    R_xml_0_rule_as_list_75.
 
 
 Definition R_xml_0_rule_as_list_77  := 
   ((algebra.Alg.Term algebra.F.id_a__isNatKind ((algebra.Alg.Var 5)::nil)),
    (algebra.Alg.Term algebra.F.id_isNatKind ((algebra.Alg.Var 5)::nil)))::
    R_xml_0_rule_as_list_76.
 
 Definition R_xml_0_rule_as_list  := R_xml_0_rule_as_list_77.
 
 
 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 78.
   injection H;intros ;subst;constructor 77.
   injection H;intros ;subst;constructor 76.
   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.
   rewrite  <- or_ext_generated.or25_equiv in H|-.
   case H;clear H;intros H.
   injection H;intros ;subst;constructor 54.
   injection H;intros ;subst;constructor 53.
   injection H;intros ;subst;constructor 52.
   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.
   rewrite  <- or_ext_generated.or25_equiv in H|-.
   case H;clear H;intros H.
   injection H;intros ;subst;constructor 30.
   injection H;intros ;subst;constructor 29.
   injection H;intros ;subst;constructor 28.
   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.
   rewrite  <- or_ext_generated.or7_equiv in H|-.
   case H;clear H;intros H.
   injection H;intros ;subst;constructor 6.
   injection H;intros ;subst;constructor 5.
   injection H;intros ;subst;constructor 4.
   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_26 (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:Prop) :
  Prop := 
   | conj_26 :
    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->
     and_26 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
 .
 
 
 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_26 (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_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)) 
     (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 x14 x16, 
      t = (algebra.Alg.Term algebra.F.id_U51 (x12::x14::x16::nil)) ->
       exists x11,
         exists x13,
           exists x15,
             t' = (algebra.Alg.Term algebra.F.id_U51 (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_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)) 
     (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_U12 (x12::nil)) ->
       exists x11,
         t' = (algebra.Alg.Term algebra.F.id_U12 (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_U43 (x12::nil)) ->
       exists x11,
         t' = (algebra.Alg.Term algebra.F.id_U43 (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_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_U53 (x12::nil)) ->
       exists x11,
         t' = (algebra.Alg.Term algebra.F.id_U53 (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_and (x12::x14::nil)) ->
       exists x11,
         exists x13,
           t' = (algebra.Alg.Term algebra.F.id_and (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_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 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, 
      t = (algebra.Alg.Term algebra.F.id_U42 (x12::x14::nil)) ->
       exists x11,
         exists x13,
           t' = (algebra.Alg.Term algebra.F.id_U42 (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_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)) 
     (t = (algebra.Alg.Term algebra.F.id_tt nil) ->
      t' = (algebra.Alg.Term algebra.F.id_tt nil)) 
     (forall x12, 
      t = (algebra.Alg.Term algebra.F.id_U22 (x12::nil)) ->
       exists x11,
         t' = (algebra.Alg.Term algebra.F.id_U22 (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_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_U52 (x12::x14::nil)) ->
       exists x11,
         exists x13,
           t' = (algebra.Alg.Term algebra.F.id_U52 (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_nil nil) ->
      t' = (algebra.Alg.Term algebra.F.id_nil nil)) 
     (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_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, 
      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)) 
     (forall x12, 
      t = (algebra.Alg.Term algebra.F.id_U32 (x12::nil)) ->
       exists x11,
         t' = (algebra.Alg.Term algebra.F.id_U32 (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_U61 (x12::x14::nil)) ->
       exists x11,
         exists x13,
           t' = (algebra.Alg.Term algebra.F.id_U61 (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_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.
   
   induction H as [|y IH z z_to_y] using 
   closure_extension.refl_trans_clos_ind2.
   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 x12 H;exists x12;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 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 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 x12 H;exists x12;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 H;exists x12;exists x14;intuition;constructor 1.
   intros H;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 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.
   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_isNatKind H_id_U41 H_id_U21 H_id_U51 H_id_isNatIListKind H_id_0 
    H_id_U12 H_id_U43 H_id_U31 H_id_U53 H_id_and H_id_cons H_id_U11 H_id_U42 
    H_id_length H_id_tt H_id_U22 H_id_s H_id_U52 H_id_nil H_id_zeros 
    H_id_isNatList H_id_isNatIList H_id_U32 H_id_U61 H_id_isNat].
   constructor.
   
   clear H_id_U41 H_id_U21 H_id_U51 H_id_isNatIListKind H_id_0 H_id_U12 
   H_id_U43 H_id_U31 H_id_U53 H_id_and H_id_cons H_id_U11 H_id_U42 
   H_id_length H_id_tt H_id_U22 H_id_s H_id_U52 H_id_nil H_id_zeros 
   H_id_isNatList H_id_isNatIList H_id_U32 H_id_U61 H_id_isNat;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_isNatKind H_id_U21 H_id_U51 H_id_isNatIListKind H_id_0 
   H_id_U12 H_id_U43 H_id_U31 H_id_U53 H_id_and H_id_cons H_id_U11 H_id_U42 
   H_id_length H_id_tt H_id_U22 H_id_s H_id_U52 H_id_nil H_id_zeros 
   H_id_isNatList H_id_isNatIList H_id_U32 H_id_U61 H_id_isNat;
    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_isNatKind H_id_U41 H_id_U51 H_id_isNatIListKind H_id_0 
   H_id_U12 H_id_U43 H_id_U31 H_id_U53 H_id_and H_id_cons H_id_U11 H_id_U42 
   H_id_length H_id_tt H_id_U22 H_id_s H_id_U52 H_id_nil H_id_zeros 
   H_id_isNatList H_id_isNatIList H_id_U32 H_id_U61 H_id_isNat;
    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_isNatKind H_id_U41 H_id_U21 H_id_isNatIListKind H_id_0 
   H_id_U12 H_id_U43 H_id_U31 H_id_U53 H_id_and H_id_cons H_id_U11 H_id_U42 
   H_id_length H_id_tt H_id_U22 H_id_s H_id_U52 H_id_nil H_id_zeros 
   H_id_isNatList H_id_isNatIList H_id_U32 H_id_U61 H_id_isNat;
    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_U51 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_U51 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_U51 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_isNatKind H_id_U41 H_id_U21 H_id_U51 H_id_0 H_id_U12 H_id_U43 
   H_id_U31 H_id_U53 H_id_and H_id_cons H_id_U11 H_id_U42 H_id_length 
   H_id_tt H_id_U22 H_id_s H_id_U52 H_id_nil H_id_zeros H_id_isNatList 
   H_id_isNatIList H_id_U32 H_id_U61 H_id_isNat;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_isNatKind H_id_U41 H_id_U21 H_id_U51 H_id_isNatIListKind 
   H_id_U12 H_id_U43 H_id_U31 H_id_U53 H_id_and H_id_cons H_id_U11 H_id_U42 
   H_id_length H_id_tt H_id_U22 H_id_s H_id_U52 H_id_nil H_id_zeros 
   H_id_isNatList H_id_isNatIList H_id_U32 H_id_U61 H_id_isNat;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_isNatKind H_id_U41 H_id_U21 H_id_U51 H_id_isNatIListKind 
   H_id_0 H_id_U43 H_id_U31 H_id_U53 H_id_and H_id_cons H_id_U11 H_id_U42 
   H_id_length H_id_tt H_id_U22 H_id_s H_id_U52 H_id_nil H_id_zeros 
   H_id_isNatList H_id_isNatIList H_id_U32 H_id_U61 H_id_isNat;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_U12 y (refl_equal _)) as [x11];intros ;intuition;
       exists x11;intuition;eapply closure_extension.refl_trans_clos_R;
       eassumption
     end
   .
   
   clear H_id_isNatKind H_id_U41 H_id_U21 H_id_U51 H_id_isNatIListKind 
   H_id_0 H_id_U12 H_id_U31 H_id_U53 H_id_and H_id_cons H_id_U11 H_id_U42 
   H_id_length H_id_tt H_id_U22 H_id_s H_id_U52 H_id_nil H_id_zeros 
   H_id_isNatList H_id_isNatIList H_id_U32 H_id_U61 H_id_isNat;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_U43 y (refl_equal _)) as [x11];intros ;intuition;
       exists x11;intuition;eapply closure_extension.refl_trans_clos_R;
       eassumption
     end
   .
   
   clear H_id_isNatKind H_id_U41 H_id_U21 H_id_U51 H_id_isNatIListKind 
   H_id_0 H_id_U12 H_id_U43 H_id_U53 H_id_and H_id_cons H_id_U11 H_id_U42 
   H_id_length H_id_tt H_id_U22 H_id_s H_id_U52 H_id_nil H_id_zeros 
   H_id_isNatList H_id_isNatIList H_id_U32 H_id_U61 H_id_isNat;
    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_isNatKind H_id_U41 H_id_U21 H_id_U51 H_id_isNatIListKind 
   H_id_0 H_id_U12 H_id_U43 H_id_U31 H_id_and H_id_cons H_id_U11 H_id_U42 
   H_id_length H_id_tt H_id_U22 H_id_s H_id_U52 H_id_nil H_id_zeros 
   H_id_isNatList H_id_isNatIList H_id_U32 H_id_U61 H_id_isNat;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_U53 y (refl_equal _)) as [x11];intros ;intuition;
       exists x11;intuition;eapply closure_extension.refl_trans_clos_R;
       eassumption
     end
   .
   
   clear H_id_isNatKind H_id_U41 H_id_U21 H_id_U51 H_id_isNatIListKind 
   H_id_0 H_id_U12 H_id_U43 H_id_U31 H_id_U53 H_id_cons H_id_U11 H_id_U42 
   H_id_length H_id_tt H_id_U22 H_id_s H_id_U52 H_id_nil H_id_zeros 
   H_id_isNatList H_id_isNatIList H_id_U32 H_id_U61 H_id_isNat;
    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_and 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_and 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_isNatKind H_id_U41 H_id_U21 H_id_U51 H_id_isNatIListKind 
   H_id_0 H_id_U12 H_id_U43 H_id_U31 H_id_U53 H_id_and H_id_U11 H_id_U42 
   H_id_length H_id_tt H_id_U22 H_id_s H_id_U52 H_id_nil H_id_zeros 
   H_id_isNatList H_id_isNatIList H_id_U32 H_id_U61 H_id_isNat;
    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_isNatKind H_id_U41 H_id_U21 H_id_U51 H_id_isNatIListKind 
   H_id_0 H_id_U12 H_id_U43 H_id_U31 H_id_U53 H_id_and H_id_cons H_id_U42 
   H_id_length H_id_tt H_id_U22 H_id_s H_id_U52 H_id_nil H_id_zeros 
   H_id_isNatList H_id_isNatIList H_id_U32 H_id_U61 H_id_isNat;
    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_isNatKind H_id_U41 H_id_U21 H_id_U51 H_id_isNatIListKind 
   H_id_0 H_id_U12 H_id_U43 H_id_U31 H_id_U53 H_id_and H_id_cons H_id_U11 
   H_id_length H_id_tt H_id_U22 H_id_s H_id_U52 H_id_nil H_id_zeros 
   H_id_isNatList H_id_isNatIList H_id_U32 H_id_U61 H_id_isNat;
    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_U42 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_U42 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_isNatKind H_id_U41 H_id_U21 H_id_U51 H_id_isNatIListKind 
   H_id_0 H_id_U12 H_id_U43 H_id_U31 H_id_U53 H_id_and H_id_cons H_id_U11 
   H_id_U42 H_id_tt H_id_U22 H_id_s H_id_U52 H_id_nil H_id_zeros 
   H_id_isNatList H_id_isNatIList H_id_U32 H_id_U61 H_id_isNat;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_isNatKind H_id_U41 H_id_U21 H_id_U51 H_id_isNatIListKind 
   H_id_0 H_id_U12 H_id_U43 H_id_U31 H_id_U53 H_id_and H_id_cons H_id_U11 
   H_id_U42 H_id_length H_id_U22 H_id_s H_id_U52 H_id_nil H_id_zeros 
   H_id_isNatList H_id_isNatIList H_id_U32 H_id_U61 H_id_isNat;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_isNatKind H_id_U41 H_id_U21 H_id_U51 H_id_isNatIListKind 
   H_id_0 H_id_U12 H_id_U43 H_id_U31 H_id_U53 H_id_and H_id_cons H_id_U11 
   H_id_U42 H_id_length H_id_tt H_id_s H_id_U52 H_id_nil H_id_zeros 
   H_id_isNatList H_id_isNatIList H_id_U32 H_id_U61 H_id_isNat;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_U22 y (refl_equal _)) as [x11];intros ;intuition;
       exists x11;intuition;eapply closure_extension.refl_trans_clos_R;
       eassumption
     end
   .
   
   clear H_id_isNatKind H_id_U41 H_id_U21 H_id_U51 H_id_isNatIListKind 
   H_id_0 H_id_U12 H_id_U43 H_id_U31 H_id_U53 H_id_and H_id_cons H_id_U11 
   H_id_U42 H_id_length H_id_tt H_id_U22 H_id_U52 H_id_nil H_id_zeros 
   H_id_isNatList H_id_isNatIList H_id_U32 H_id_U61 H_id_isNat;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_isNatKind H_id_U41 H_id_U21 H_id_U51 H_id_isNatIListKind 
   H_id_0 H_id_U12 H_id_U43 H_id_U31 H_id_U53 H_id_and H_id_cons H_id_U11 
   H_id_U42 H_id_length H_id_tt H_id_U22 H_id_s H_id_nil H_id_zeros 
   H_id_isNatList H_id_isNatIList H_id_U32 H_id_U61 H_id_isNat;
    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_U52 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_U52 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_isNatKind H_id_U41 H_id_U21 H_id_U51 H_id_isNatIListKind 
   H_id_0 H_id_U12 H_id_U43 H_id_U31 H_id_U53 H_id_and H_id_cons H_id_U11 
   H_id_U42 H_id_length H_id_tt H_id_U22 H_id_s H_id_U52 H_id_zeros 
   H_id_isNatList H_id_isNatIList H_id_U32 H_id_U61 H_id_isNat;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_isNatKind H_id_U41 H_id_U21 H_id_U51 H_id_isNatIListKind 
   H_id_0 H_id_U12 H_id_U43 H_id_U31 H_id_U53 H_id_and H_id_cons H_id_U11 
   H_id_U42 H_id_length H_id_tt H_id_U22 H_id_s H_id_U52 H_id_nil 
   H_id_isNatList H_id_isNatIList H_id_U32 H_id_U61 H_id_isNat;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_isNatKind H_id_U41 H_id_U21 H_id_U51 H_id_isNatIListKind 
   H_id_0 H_id_U12 H_id_U43 H_id_U31 H_id_U53 H_id_and H_id_cons H_id_U11 
   H_id_U42 H_id_length H_id_tt H_id_U22 H_id_s H_id_U52 H_id_nil H_id_zeros 
   H_id_isNatIList H_id_U32 H_id_U61 H_id_isNat;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_isNatKind H_id_U41 H_id_U21 H_id_U51 H_id_isNatIListKind 
   H_id_0 H_id_U12 H_id_U43 H_id_U31 H_id_U53 H_id_and H_id_cons H_id_U11 
   H_id_U42 H_id_length H_id_tt H_id_U22 H_id_s H_id_U52 H_id_nil H_id_zeros 
   H_id_isNatList H_id_U32 H_id_U61 H_id_isNat;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
   .
   
   clear H_id_isNatKind H_id_U41 H_id_U21 H_id_U51 H_id_isNatIListKind 
   H_id_0 H_id_U12 H_id_U43 H_id_U31 H_id_U53 H_id_and H_id_cons H_id_U11 
   H_id_U42 H_id_length H_id_tt H_id_U22 H_id_s H_id_U52 H_id_nil H_id_zeros 
   H_id_isNatList H_id_isNatIList H_id_U61 H_id_isNat;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_U32 y (refl_equal _)) as [x11];intros ;intuition;
       exists x11;intuition;eapply closure_extension.refl_trans_clos_R;
       eassumption
     end
   .
   
   clear H_id_isNatKind H_id_U41 H_id_U21 H_id_U51 H_id_isNatIListKind 
   H_id_0 H_id_U12 H_id_U43 H_id_U31 H_id_U53 H_id_and H_id_cons H_id_U11 
   H_id_U42 H_id_length H_id_tt H_id_U22 H_id_s H_id_U52 H_id_nil H_id_zeros 
   H_id_isNatList H_id_isNatIList H_id_U32 H_id_isNat;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_U61 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_U61 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_isNatKind H_id_U41 H_id_U21 H_id_U51 H_id_isNatIListKind 
   H_id_0 H_id_U12 H_id_U43 H_id_U31 H_id_U53 H_id_and H_id_cons H_id_U11 
   H_id_U42 H_id_length H_id_tt H_id_U22 H_id_s H_id_U52 H_id_nil H_id_zeros 
   H_id_isNatList H_id_isNatIList H_id_U32 H_id_U61;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
   .
 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_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_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_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 x16, 
     t = (algebra.Alg.Term algebra.F.id_U51 (x12::x14::x16::nil)) ->
      exists x11,
        exists x13,
          exists x15,
            t' = (algebra.Alg.Term algebra.F.id_U51 (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_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_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_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, 
     t = (algebra.Alg.Term algebra.F.id_U12 (x12::nil)) ->
      exists x11,
        t' = (algebra.Alg.Term algebra.F.id_U12 (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_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, 
     t = (algebra.Alg.Term algebra.F.id_U43 (x12::nil)) ->
      exists x11,
        t' = (algebra.Alg.Term algebra.F.id_U43 (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_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_U53_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_U53 (x12::nil)) ->
      exists x11,
        t' = (algebra.Alg.Term algebra.F.id_U53 (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_and_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_and (x12::x14::nil)) ->
      exists x11,
        exists x13,
          t' = (algebra.Alg.Term algebra.F.id_and (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_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_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_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, 
     t = (algebra.Alg.Term algebra.F.id_U42 (x12::x14::nil)) ->
      exists x11,
        exists x13,
          t' = (algebra.Alg.Term algebra.F.id_U42 (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_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_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_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, 
     t = (algebra.Alg.Term algebra.F.id_U22 (x12::nil)) ->
      exists x11,
        t' = (algebra.Alg.Term algebra.F.id_U22 (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_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_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 x14, 
     t = (algebra.Alg.Term algebra.F.id_U52 (x12::x14::nil)) ->
      exists x11,
        exists x13,
          t' = (algebra.Alg.Term algebra.F.id_U52 (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_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_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_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_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.
 
 
 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, 
     t = (algebra.Alg.Term algebra.F.id_U32 (x12::nil)) ->
      exists x11,
        t' = (algebra.Alg.Term algebra.F.id_U32 (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_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 x14, 
     t = (algebra.Alg.Term algebra.F.id_U61 (x12::x14::nil)) ->
      exists x11,
        exists x13,
          t' = (algebra.Alg.Term algebra.F.id_U61 (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_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.
 
 
 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_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_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_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_U51 (?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_U51_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_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_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_U12 (?x11::nil)) |- _ =>
     let x11 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_U12_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_U43 (?x11::nil)) |- _ =>
     let x11 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_U43_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_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_U53 (?x11::nil)) |- _ =>
     let x11 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_U53_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_and (?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_and_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_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_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_U42 (?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_U42_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_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_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_U22 (?x11::nil)) |- _ =>
     let x11 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_U22_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_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_U52 (?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_U52_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_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_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_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_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 ))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U32 (?x11::nil)) |- _ =>
     let x11 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_U32_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_U61 (?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_U61_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_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 ))))
    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_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_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_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_U51 (?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_U51_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_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_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_U12 (?x11::nil)) |- _ =>
     let x11 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_U12_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_U43 (?x11::nil)) |- _ =>
     let x11 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_U43_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_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_U53 (?x11::nil)) |- _ =>
     let x11 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_U53_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_and (?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_and_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_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_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_U42 (?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_U42_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_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_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_U22 (?x11::nil)) |- _ =>
     let x11 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_U22_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_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_U52 (?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_U52_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_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_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_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_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 )))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_U32 (?x11::nil)) |- _ =>
     let x11 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_U32_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_U61 (?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_U61_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_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 )))))
    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_isNatKind : A ->A.
   
   Hypothesis P_id_a__isNatIList : A ->A.
   
   Hypothesis P_id_a__U21 : A ->A ->A.
   
   Hypothesis P_id_U41 : A ->A ->A ->A.
   
   Hypothesis P_id_a__and : A ->A ->A.
   
   Hypothesis P_id_a__U11 : A ->A ->A.
   
   Hypothesis P_id_U21 : A ->A ->A.
   
   Hypothesis P_id_a__U61 : A ->A ->A.
   
   Hypothesis P_id_a__U32 : A ->A.
   
   Hypothesis P_id_U51 : A ->A ->A ->A.
   
   Hypothesis P_id_isNatIListKind : A ->A.
   
   Hypothesis P_id_0 : A.
   
   Hypothesis P_id_U12 : A ->A.
   
   Hypothesis P_id_a__U52 : A ->A ->A.
   
   Hypothesis P_id_a__isNat : A ->A.
   
   Hypothesis P_id_U43 : A ->A.
   
   Hypothesis P_id_a__isNatIListKind : A ->A.
   
   Hypothesis P_id_a__U12 : A ->A.
   
   Hypothesis P_id_U31 : A ->A ->A.
   
   Hypothesis P_id_a__length : A ->A.
   
   Hypothesis P_id_a__U42 : A ->A ->A.
   
   Hypothesis P_id_U53 : A ->A.
   
   Hypothesis P_id_and : A ->A ->A.
   
   Hypothesis P_id_cons : A ->A ->A.
   
   Hypothesis P_id_U11 : A ->A ->A.
   
   Hypothesis P_id_a__U51 : A ->A ->A ->A.
   
   Hypothesis P_id_a__U22 : A ->A.
   
   Hypothesis P_id_U42 : A ->A ->A.
   
   Hypothesis P_id_length : A ->A.
   
   Hypothesis P_id_tt : A.
   
   Hypothesis P_id_U22 : A ->A.
   
   Hypothesis P_id_s : A ->A.
   
   Hypothesis P_id_a__U41 : A ->A ->A ->A.
   
   Hypothesis P_id_U52 : A ->A ->A.
   
   Hypothesis P_id_nil : A.
   
   Hypothesis P_id_zeros : A.
   
   Hypothesis P_id_isNatList : A ->A.
   
   Hypothesis P_id_a__U53 : A ->A.
   
   Hypothesis P_id_a__U31 : A ->A ->A.
   
   Hypothesis P_id_isNatIList : A ->A.
   
   Hypothesis P_id_a__isNatKind : A ->A.
   
   Hypothesis P_id_a__isNatList : A ->A.
   
   Hypothesis P_id_U32 : A ->A.
   
   Hypothesis P_id_mark : A ->A.
   
   Hypothesis P_id_a__U43 : A ->A.
   
   Hypothesis P_id_U61 : A ->A ->A.
   
   Hypothesis P_id_isNat : A ->A.
   
   Hypothesis P_id_isNatKind_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->P_id_isNatKind x12 <= P_id_isNatKind 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_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_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_a__and_monotonic :
    forall x12 x14 x13 x11, 
     (A0 <= x14)/\ (x14 <= x13) ->
      (A0 <= x12)/\ (x12 <= x11) ->P_id_a__and x12 x14 <= P_id_a__and x11 x13.
   
   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_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__U61_monotonic :
    forall x12 x14 x13 x11, 
     (A0 <= x14)/\ (x14 <= x13) ->
      (A0 <= x12)/\ (x12 <= x11) ->P_id_a__U61 x12 x14 <= P_id_a__U61 x11 x13.
   
   Hypothesis P_id_a__U32_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->P_id_a__U32 x12 <= P_id_a__U32 x11.
   
   Hypothesis P_id_U51_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (A0 <= x16)/\ (x16 <= x15) ->
      (A0 <= x14)/\ (x14 <= x13) ->
       (A0 <= x12)/\ (x12 <= x11) ->
        P_id_U51 x12 x14 x16 <= P_id_U51 x11 x13 x15.
   
   Hypothesis P_id_isNatIListKind_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->
      P_id_isNatIListKind x12 <= P_id_isNatIListKind x11.
   
   Hypothesis P_id_U12_monotonic :
    forall x12 x11, (A0 <= x12)/\ (x12 <= x11) ->P_id_U12 x12 <= P_id_U12 x11.
   
   Hypothesis P_id_a__U52_monotonic :
    forall x12 x14 x13 x11, 
     (A0 <= x14)/\ (x14 <= x13) ->
      (A0 <= x12)/\ (x12 <= x11) ->P_id_a__U52 x12 x14 <= P_id_a__U52 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_U43_monotonic :
    forall x12 x11, (A0 <= x12)/\ (x12 <= x11) ->P_id_U43 x12 <= P_id_U43 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__U12_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->P_id_a__U12 x12 <= P_id_a__U12 x11.
   
   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__length_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->P_id_a__length x12 <= P_id_a__length x11.
   
   Hypothesis P_id_a__U42_monotonic :
    forall x12 x14 x13 x11, 
     (A0 <= x14)/\ (x14 <= x13) ->
      (A0 <= x12)/\ (x12 <= x11) ->P_id_a__U42 x12 x14 <= P_id_a__U42 x11 x13.
   
   Hypothesis P_id_U53_monotonic :
    forall x12 x11, (A0 <= x12)/\ (x12 <= x11) ->P_id_U53 x12 <= P_id_U53 x11.
   
   Hypothesis P_id_and_monotonic :
    forall x12 x14 x13 x11, 
     (A0 <= x14)/\ (x14 <= x13) ->
      (A0 <= x12)/\ (x12 <= x11) ->P_id_and x12 x14 <= P_id_and x11 x13.
   
   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_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__U51_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (A0 <= x16)/\ (x16 <= x15) ->
      (A0 <= x14)/\ (x14 <= x13) ->
       (A0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U51 x12 x14 x16 <= P_id_a__U51 x11 x13 x15.
   
   Hypothesis P_id_a__U22_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->P_id_a__U22 x12 <= P_id_a__U22 x11.
   
   Hypothesis P_id_U42_monotonic :
    forall x12 x14 x13 x11, 
     (A0 <= x14)/\ (x14 <= x13) ->
      (A0 <= x12)/\ (x12 <= x11) ->P_id_U42 x12 x14 <= P_id_U42 x11 x13.
   
   Hypothesis P_id_length_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->P_id_length x12 <= P_id_length x11.
   
   Hypothesis P_id_U22_monotonic :
    forall x12 x11, (A0 <= x12)/\ (x12 <= x11) ->P_id_U22 x12 <= P_id_U22 x11.
   
   Hypothesis P_id_s_monotonic :
    forall x12 x11, (A0 <= x12)/\ (x12 <= x11) ->P_id_s x12 <= P_id_s 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_U52_monotonic :
    forall x12 x14 x13 x11, 
     (A0 <= x14)/\ (x14 <= x13) ->
      (A0 <= x12)/\ (x12 <= x11) ->P_id_U52 x12 x14 <= P_id_U52 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__U53_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->P_id_a__U53 x12 <= P_id_a__U53 x11.
   
   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_isNatIList_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->P_id_isNatIList x12 <= P_id_isNatIList 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_a__isNatList_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->
      P_id_a__isNatList x12 <= P_id_a__isNatList x11.
   
   Hypothesis P_id_U32_monotonic :
    forall x12 x11, (A0 <= x12)/\ (x12 <= x11) ->P_id_U32 x12 <= P_id_U32 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__U43_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->P_id_a__U43 x12 <= P_id_a__U43 x11.
   
   Hypothesis P_id_U61_monotonic :
    forall x12 x14 x13 x11, 
     (A0 <= x14)/\ (x14 <= x13) ->
      (A0 <= x12)/\ (x12 <= x11) ->P_id_U61 x12 x14 <= P_id_U61 x11 x13.
   
   Hypothesis P_id_isNat_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->P_id_isNat x12 <= P_id_isNat x11.
   
   Hypothesis P_id_a__zeros_bounded : A0 <= P_id_a__zeros .
   
   Hypothesis P_id_isNatKind_bounded :
    forall x11, (A0 <= x11) ->A0 <= P_id_isNatKind x11.
   
   Hypothesis P_id_a__isNatIList_bounded :
    forall x11, (A0 <= x11) ->A0 <= P_id_a__isNatIList x11.
   
   Hypothesis P_id_a__U21_bounded :
    forall x12 x11, (A0 <= x11) ->(A0 <= x12) ->A0 <= P_id_a__U21 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_a__and_bounded :
    forall x12 x11, (A0 <= x11) ->(A0 <= x12) ->A0 <= P_id_a__and x12 x11.
   
   Hypothesis P_id_a__U11_bounded :
    forall x12 x11, (A0 <= x11) ->(A0 <= x12) ->A0 <= P_id_a__U11 x12 x11.
   
   Hypothesis P_id_U21_bounded :
    forall x12 x11, (A0 <= x11) ->(A0 <= x12) ->A0 <= P_id_U21 x12 x11.
   
   Hypothesis P_id_a__U61_bounded :
    forall x12 x11, (A0 <= x11) ->(A0 <= x12) ->A0 <= P_id_a__U61 x12 x11.
   
   Hypothesis P_id_a__U32_bounded :
    forall x11, (A0 <= x11) ->A0 <= P_id_a__U32 x11.
   
   Hypothesis P_id_U51_bounded :
    forall x12 x13 x11, 
     (A0 <= x11) ->(A0 <= x12) ->(A0 <= x13) ->A0 <= P_id_U51 x13 x12 x11.
   
   Hypothesis P_id_isNatIListKind_bounded :
    forall x11, (A0 <= x11) ->A0 <= P_id_isNatIListKind x11.
   
   Hypothesis P_id_0_bounded : A0 <= P_id_0 .
   
   Hypothesis P_id_U12_bounded :
    forall x11, (A0 <= x11) ->A0 <= P_id_U12 x11.
   
   Hypothesis P_id_a__U52_bounded :
    forall x12 x11, (A0 <= x11) ->(A0 <= x12) ->A0 <= P_id_a__U52 x12 x11.
   
   Hypothesis P_id_a__isNat_bounded :
    forall x11, (A0 <= x11) ->A0 <= P_id_a__isNat x11.
   
   Hypothesis P_id_U43_bounded :
    forall x11, (A0 <= x11) ->A0 <= P_id_U43 x11.
   
   Hypothesis P_id_a__isNatIListKind_bounded :
    forall x11, (A0 <= x11) ->A0 <= P_id_a__isNatIListKind x11.
   
   Hypothesis P_id_a__U12_bounded :
    forall x11, (A0 <= x11) ->A0 <= P_id_a__U12 x11.
   
   Hypothesis P_id_U31_bounded :
    forall x12 x11, (A0 <= x11) ->(A0 <= x12) ->A0 <= P_id_U31 x12 x11.
   
   Hypothesis P_id_a__length_bounded :
    forall x11, (A0 <= x11) ->A0 <= P_id_a__length x11.
   
   Hypothesis P_id_a__U42_bounded :
    forall x12 x11, (A0 <= x11) ->(A0 <= x12) ->A0 <= P_id_a__U42 x12 x11.
   
   Hypothesis P_id_U53_bounded :
    forall x11, (A0 <= x11) ->A0 <= P_id_U53 x11.
   
   Hypothesis P_id_and_bounded :
    forall x12 x11, (A0 <= x11) ->(A0 <= x12) ->A0 <= P_id_and x12 x11.
   
   Hypothesis P_id_cons_bounded :
    forall x12 x11, (A0 <= x11) ->(A0 <= x12) ->A0 <= P_id_cons x12 x11.
   
   Hypothesis P_id_U11_bounded :
    forall x12 x11, (A0 <= x11) ->(A0 <= x12) ->A0 <= P_id_U11 x12 x11.
   
   Hypothesis P_id_a__U51_bounded :
    forall x12 x13 x11, 
     (A0 <= x11) ->(A0 <= x12) ->(A0 <= x13) ->A0 <= P_id_a__U51 x13 x12 x11.
   
   Hypothesis P_id_a__U22_bounded :
    forall x11, (A0 <= x11) ->A0 <= P_id_a__U22 x11.
   
   Hypothesis P_id_U42_bounded :
    forall x12 x11, (A0 <= x11) ->(A0 <= x12) ->A0 <= P_id_U42 x12 x11.
   
   Hypothesis P_id_length_bounded :
    forall x11, (A0 <= x11) ->A0 <= P_id_length x11.
   
   Hypothesis P_id_tt_bounded : A0 <= P_id_tt .
   
   Hypothesis P_id_U22_bounded :
    forall x11, (A0 <= x11) ->A0 <= P_id_U22 x11.
   
   Hypothesis P_id_s_bounded : forall x11, (A0 <= x11) ->A0 <= P_id_s 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_U52_bounded :
    forall x12 x11, (A0 <= x11) ->(A0 <= x12) ->A0 <= P_id_U52 x12 x11.
   
   Hypothesis P_id_nil_bounded : A0 <= P_id_nil .
   
   Hypothesis P_id_zeros_bounded : A0 <= P_id_zeros .
   
   Hypothesis P_id_isNatList_bounded :
    forall x11, (A0 <= x11) ->A0 <= P_id_isNatList x11.
   
   Hypothesis P_id_a__U53_bounded :
    forall x11, (A0 <= x11) ->A0 <= P_id_a__U53 x11.
   
   Hypothesis P_id_a__U31_bounded :
    forall x12 x11, (A0 <= x11) ->(A0 <= x12) ->A0 <= P_id_a__U31 x12 x11.
   
   Hypothesis P_id_isNatIList_bounded :
    forall x11, (A0 <= x11) ->A0 <= P_id_isNatIList x11.
   
   Hypothesis P_id_a__isNatKind_bounded :
    forall x11, (A0 <= x11) ->A0 <= P_id_a__isNatKind x11.
   
   Hypothesis P_id_a__isNatList_bounded :
    forall x11, (A0 <= x11) ->A0 <= P_id_a__isNatList x11.
   
   Hypothesis P_id_U32_bounded :
    forall x11, (A0 <= x11) ->A0 <= P_id_U32 x11.
   
   Hypothesis P_id_mark_bounded :
    forall x11, (A0 <= x11) ->A0 <= P_id_mark x11.
   
   Hypothesis P_id_a__U43_bounded :
    forall x11, (A0 <= x11) ->A0 <= P_id_a__U43 x11.
   
   Hypothesis P_id_U61_bounded :
    forall x12 x11, (A0 <= x11) ->(A0 <= x12) ->A0 <= P_id_U61 x12 x11.
   
   Hypothesis P_id_isNat_bounded :
    forall x11, (A0 <= x11) ->A0 <= P_id_isNat 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_isNatKind (x11::nil)) =>
        P_id_isNatKind (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__U21 (x12::x11::nil)) =>
        P_id_a__U21 (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_a__and (x12::x11::nil)) =>
        P_id_a__and (measure x12) (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_U21 (x12::x11::nil)) =>
        P_id_U21 (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U61 (x12::x11::nil)) =>
        P_id_a__U61 (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U32 (x11::nil)) =>
        P_id_a__U32 (measure x11)
       | (algebra.Alg.Term algebra.F.id_U51 (x13::x12::x11::nil)) =>
        P_id_U51 (measure x13) (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_isNatIListKind (x11::nil)) =>
        P_id_isNatIListKind (measure x11)
       | (algebra.Alg.Term algebra.F.id_0 nil) => P_id_0 
       | (algebra.Alg.Term algebra.F.id_U12 (x11::nil)) =>
        P_id_U12 (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U52 (x12::x11::nil)) =>
        P_id_a__U52 (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_U43 (x11::nil)) =>
        P_id_U43 (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__U12 (x11::nil)) =>
        P_id_a__U12 (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__length (x11::nil)) =>
        P_id_a__length (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U42 (x12::x11::nil)) =>
        P_id_a__U42 (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_U53 (x11::nil)) =>
        P_id_U53 (measure x11)
       | (algebra.Alg.Term algebra.F.id_and (x12::x11::nil)) =>
        P_id_and (measure x12) (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_U11 (x12::x11::nil)) =>
        P_id_U11 (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U51 (x13::x12::x11::nil)) =>
        P_id_a__U51 (measure x13) (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U22 (x11::nil)) =>
        P_id_a__U22 (measure x11)
       | (algebra.Alg.Term algebra.F.id_U42 (x12::x11::nil)) =>
        P_id_U42 (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_length (x11::nil)) =>
        P_id_length (measure x11)
       | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt 
       | (algebra.Alg.Term algebra.F.id_U22 (x11::nil)) =>
        P_id_U22 (measure x11)
       | (algebra.Alg.Term algebra.F.id_s (x11::nil)) => P_id_s (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_U52 (x12::x11::nil)) =>
        P_id_U52 (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil 
       | (algebra.Alg.Term algebra.F.id_zeros nil) => P_id_zeros 
       | (algebra.Alg.Term algebra.F.id_isNatList (x11::nil)) =>
        P_id_isNatList (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U53 (x11::nil)) =>
        P_id_a__U53 (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_isNatIList (x11::nil)) =>
        P_id_isNatIList (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__isNatList (x11::nil)) =>
        P_id_a__isNatList (measure x11)
       | (algebra.Alg.Term algebra.F.id_U32 (x11::nil)) =>
        P_id_U32 (measure x11)
       | (algebra.Alg.Term algebra.F.id_mark (x11::nil)) =>
        P_id_mark (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U43 (x11::nil)) =>
        P_id_a__U43 (measure x11)
       | (algebra.Alg.Term algebra.F.id_U61 (x12::x11::nil)) =>
        P_id_U61 (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_isNat (x11::nil)) =>
        P_id_isNat (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_isNatKind (x11::nil)) =>
                    P_id_isNatKind (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__U21 (x12::x11::nil)) =>
                    P_id_a__U21 (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_a__and (x12::x11::nil)) =>
                    P_id_a__and (measure x12) (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_U21 (x12::x11::nil)) =>
                    P_id_U21 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U61 (x12::x11::nil)) =>
                    P_id_a__U61 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U32 (x11::nil)) =>
                    P_id_a__U32 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U51 (x13::x12::x11::nil)) =>
                    P_id_U51 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_isNatIListKind 
                      (x11::nil)) =>
                    P_id_isNatIListKind (measure x11)
                   | (algebra.Alg.Term algebra.F.id_0 nil) => P_id_0 
                   | (algebra.Alg.Term algebra.F.id_U12 (x11::nil)) =>
                    P_id_U12 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U52 (x12::x11::nil)) =>
                    P_id_a__U52 (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_U43 (x11::nil)) =>
                    P_id_U43 (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__U12 (x11::nil)) =>
                    P_id_a__U12 (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__length (x11::nil)) =>
                    P_id_a__length (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U42 (x12::x11::nil)) =>
                    P_id_a__U42 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U53 (x11::nil)) =>
                    P_id_U53 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_and (x12::x11::nil)) =>
                    P_id_and (measure x12) (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_U11 (x12::x11::nil)) =>
                    P_id_U11 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U51 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U51 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U22 (x11::nil)) =>
                    P_id_a__U22 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U42 (x12::x11::nil)) =>
                    P_id_U42 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_length (x11::nil)) =>
                    P_id_length (measure x11)
                   | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt 
                   | (algebra.Alg.Term algebra.F.id_U22 (x11::nil)) =>
                    P_id_U22 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_s (x11::nil)) =>
                    P_id_s (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_U52 (x12::x11::nil)) =>
                    P_id_U52 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil 
                   | (algebra.Alg.Term algebra.F.id_zeros nil) => P_id_zeros 
                   | (algebra.Alg.Term algebra.F.id_isNatList (x11::nil)) =>
                    P_id_isNatList (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U53 (x11::nil)) =>
                    P_id_a__U53 (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_isNatIList (x11::nil)) =>
                    P_id_isNatIList (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__isNatList (x11::nil)) =>
                    P_id_a__isNatList (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U32 (x11::nil)) =>
                    P_id_U32 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_mark (x11::nil)) =>
                    P_id_mark (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U43 (x11::nil)) =>
                    P_id_a__U43 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U61 (x12::x11::nil)) =>
                    P_id_U61 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_isNat (x11::nil)) =>
                    P_id_isNat (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_isNatKind => P_id_isNatKind
       | algebra.F.id_a__isNatIList => P_id_a__isNatIList
       | algebra.F.id_a__U21 => P_id_a__U21
       | algebra.F.id_U41 => P_id_U41
       | algebra.F.id_a__and => P_id_a__and
       | algebra.F.id_a__U11 => P_id_a__U11
       | algebra.F.id_U21 => P_id_U21
       | algebra.F.id_a__U61 => P_id_a__U61
       | algebra.F.id_a__U32 => P_id_a__U32
       | algebra.F.id_U51 => P_id_U51
       | algebra.F.id_isNatIListKind => P_id_isNatIListKind
       | algebra.F.id_0 => P_id_0
       | algebra.F.id_U12 => P_id_U12
       | algebra.F.id_a__U52 => P_id_a__U52
       | algebra.F.id_a__isNat => P_id_a__isNat
       | algebra.F.id_U43 => P_id_U43
       | algebra.F.id_a__isNatIListKind => P_id_a__isNatIListKind
       | algebra.F.id_a__U12 => P_id_a__U12
       | algebra.F.id_U31 => P_id_U31
       | algebra.F.id_a__length => P_id_a__length
       | algebra.F.id_a__U42 => P_id_a__U42
       | algebra.F.id_U53 => P_id_U53
       | algebra.F.id_and => P_id_and
       | algebra.F.id_cons => P_id_cons
       | algebra.F.id_U11 => P_id_U11
       | algebra.F.id_a__U51 => P_id_a__U51
       | algebra.F.id_a__U22 => P_id_a__U22
       | algebra.F.id_U42 => P_id_U42
       | algebra.F.id_length => P_id_length
       | algebra.F.id_tt => P_id_tt
       | algebra.F.id_U22 => P_id_U22
       | algebra.F.id_s => P_id_s
       | algebra.F.id_a__U41 => P_id_a__U41
       | algebra.F.id_U52 => P_id_U52
       | algebra.F.id_nil => P_id_nil
       | algebra.F.id_zeros => P_id_zeros
       | algebra.F.id_isNatList => P_id_isNatList
       | algebra.F.id_a__U53 => P_id_a__U53
       | algebra.F.id_a__U31 => P_id_a__U31
       | algebra.F.id_isNatIList => P_id_isNatIList
       | algebra.F.id_a__isNatKind => P_id_a__isNatKind
       | algebra.F.id_a__isNatList => P_id_a__isNatList
       | algebra.F.id_U32 => P_id_U32
       | algebra.F.id_mark => P_id_mark
       | algebra.F.id_a__U43 => P_id_a__U43
       | algebra.F.id_U61 => P_id_U61
       | algebra.F.id_isNat => P_id_isNat
       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_isNatKind =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__isNatIList =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__U21 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_U41 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_a__and =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_a__U11 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_U21 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_a__U61 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_a__U32 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_U51 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_isNatIListKind =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_0 => match l with
                                    | nil => _
                                    | _::_ => _
                                    end
              | algebra.F.id_U12 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__U52 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_a__isNat =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_U43 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__isNatIListKind =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__U12 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_U31 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_a__length =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__U42 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_U53 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_and =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_cons =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_U11 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_a__U51 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_a__U22 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_U42 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_length =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_tt => match l with
                                     | nil => _
                                     | _::_ => _
                                     end
              | algebra.F.id_U22 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_s =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__U41 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_U52 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_nil => match l with
                                      | nil => _
                                      | _::_ => _
                                      end
              | algebra.F.id_zeros => match l with
                                        | nil => _
                                        | _::_ => _
                                        end
              | algebra.F.id_isNatList =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__U53 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__U31 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_isNatIList =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__isNatKind =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__isNatList =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_U32 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_mark =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__U43 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_U61 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_isNat =>
               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_isNatKind_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isNatIList_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U21_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U41_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__and_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U11_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U21_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U61_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U32_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U51_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_isNatIListKind_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_0_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U12_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U52_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isNat_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_a__U12_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U31_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__length_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U42_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U53_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_and_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_cons_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U11_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U51_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U22_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U42_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_length_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_tt_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U22_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_s_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U41_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U52_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_nil_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_zeros_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_isNatList_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U53_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U31_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_isNatIList_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isNatKind_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isNatList_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U32_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_mark_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U43_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U61_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_isNat_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_isNatKind_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isNatIList_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U21_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U41_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__and_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U11_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U21_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U61_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U32_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U51_monotonic;assumption.
     
     vm_compute in |-*;intros ;apply P_id_isNatIListKind_monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id_U12_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U52_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isNat_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_a__U12_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U31_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__length_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U42_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U53_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_and_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_cons_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U11_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U51_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U22_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U42_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_length_monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id_U22_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_s_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U41_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U52_monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id_isNatList_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U53_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U31_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_isNatIList_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isNatKind_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isNatList_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U32_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_mark_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U43_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U61_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_isNat_monotonic;assumption.
     intros f.
     case f.
     vm_compute in |-*;intros ;apply P_id_a__zeros_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_isNatKind_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isNatIList_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U21_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U41_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__and_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U11_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U21_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U61_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U32_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U51_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_isNatIListKind_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_0_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U12_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U52_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isNat_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_a__U12_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U31_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__length_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U42_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U53_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_and_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_cons_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U11_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U51_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U22_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U42_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_length_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_tt_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U22_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_s_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U41_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U52_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_nil_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_zeros_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_isNatList_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U53_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U31_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_isNatIList_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isNatKind_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isNatList_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U32_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_mark_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U43_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U61_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_isNat_bounded;assumption.
     intros .
     do 2 (rewrite  <- same_measure in |-*).
     apply rules_monotonic;assumption.
     assumption.
   Qed.
   
   Hypothesis P_id_A__LENGTH : A ->A.
   
   Hypothesis P_id_A__U11 : A ->A ->A.
   
   Hypothesis P_id_A__U41 : A ->A ->A ->A.
   
   Hypothesis P_id_A__ISNATKIND : A ->A.
   
   Hypothesis P_id_A__U22 : A ->A.
   
   Hypothesis P_id_A__U51 : A ->A ->A ->A.
   
   Hypothesis P_id_A__AND : A ->A ->A.
   
   Hypothesis P_id_A__ISNATLIST : A ->A.
   
   Hypothesis P_id_A__U43 : A ->A.
   
   Hypothesis P_id_A__U31 : A ->A ->A.
   
   Hypothesis P_id_A__U53 : A ->A.
   
   Hypothesis P_id_MARK : A ->A.
   
   Hypothesis P_id_A__U12 : A ->A.
   
   Hypothesis P_id_A__U42 : A ->A ->A.
   
   Hypothesis P_id_A__ZEROS : A.
   
   Hypothesis P_id_A__ISNAT : A ->A.
   
   Hypothesis P_id_A__U52 : A ->A ->A.
   
   Hypothesis P_id_A__ISNATILISTKIND : A ->A.
   
   Hypothesis P_id_A__U21 : A ->A ->A.
   
   Hypothesis P_id_A__ISNATILIST : A ->A.
   
   Hypothesis P_id_A__U32 : A ->A.
   
   Hypothesis P_id_A__U61 : A ->A ->A.
   
   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__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__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__ISNATKIND_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->
      P_id_A__ISNATKIND x12 <= P_id_A__ISNATKIND x11.
   
   Hypothesis P_id_A__U22_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->P_id_A__U22 x12 <= P_id_A__U22 x11.
   
   Hypothesis P_id_A__U51_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (A0 <= x16)/\ (x16 <= x15) ->
      (A0 <= x14)/\ (x14 <= x13) ->
       (A0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U51 x12 x14 x16 <= P_id_A__U51 x11 x13 x15.
   
   Hypothesis P_id_A__AND_monotonic :
    forall x12 x14 x13 x11, 
     (A0 <= x14)/\ (x14 <= x13) ->
      (A0 <= x12)/\ (x12 <= x11) ->P_id_A__AND x12 x14 <= P_id_A__AND 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__U43_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->P_id_A__U43 x12 <= P_id_A__U43 x11.
   
   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__U53_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->P_id_A__U53 x12 <= P_id_A__U53 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__U12_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->P_id_A__U12 x12 <= P_id_A__U12 x11.
   
   Hypothesis P_id_A__U42_monotonic :
    forall x12 x14 x13 x11, 
     (A0 <= x14)/\ (x14 <= x13) ->
      (A0 <= x12)/\ (x12 <= x11) ->P_id_A__U42 x12 x14 <= P_id_A__U42 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_A__U52_monotonic :
    forall x12 x14 x13 x11, 
     (A0 <= x14)/\ (x14 <= x13) ->
      (A0 <= x12)/\ (x12 <= x11) ->P_id_A__U52 x12 x14 <= P_id_A__U52 x11 x13.
   
   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__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__ISNATILIST_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->
      P_id_A__ISNATILIST x12 <= P_id_A__ISNATILIST x11.
   
   Hypothesis P_id_A__U32_monotonic :
    forall x12 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->P_id_A__U32 x12 <= P_id_A__U32 x11.
   
   Hypothesis P_id_A__U61_monotonic :
    forall x12 x14 x13 x11, 
     (A0 <= x14)/\ (x14 <= x13) ->
      (A0 <= x12)/\ (x12 <= x11) ->P_id_A__U61 x12 x14 <= P_id_A__U61 x11 x13.
   
   Definition marked_measure t := 
     match t with
       | (algebra.Alg.Term algebra.F.id_a__length (x11::nil)) =>
        P_id_A__LENGTH (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__U41 (x13::x12::x11::nil)) =>
        P_id_A__U41 (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__U22 (x11::nil)) =>
        P_id_A__U22 (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U51 (x13::x12::x11::nil)) =>
        P_id_A__U51 (measure x13) (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__and (x12::x11::nil)) =>
        P_id_A__AND (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__U43 (x11::nil)) =>
        P_id_A__U43 (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__U53 (x11::nil)) =>
        P_id_A__U53 (measure x11)
       | (algebra.Alg.Term algebra.F.id_mark (x11::nil)) =>
        P_id_MARK (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U12 (x11::nil)) =>
        P_id_A__U12 (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U42 (x12::x11::nil)) =>
        P_id_A__U42 (measure x12) (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__zeros nil) => P_id_A__ZEROS 
       | (algebra.Alg.Term algebra.F.id_a__isNat (x11::nil)) =>
        P_id_A__ISNAT (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U52 (x12::x11::nil)) =>
        P_id_A__U52 (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_a__U21 (x12::x11::nil)) =>
        P_id_A__U21 (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__U32 (x11::nil)) =>
        P_id_A__U32 (measure x11)
       | (algebra.Alg.Term algebra.F.id_a__U61 (x12::x11::nil)) =>
        P_id_A__U61 (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 x11;apply (P_id_A__U43 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__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 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__U53 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__U22 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__U51 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__U42 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__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__U12 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__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__U52 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__U32 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__U61 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 x12 x11;apply (P_id_A__AND 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__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 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 ;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_isNatKind =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__isNatIList =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__U21 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_U41 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_a__and =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_a__U11 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_U21 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_a__U61 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_a__U32 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_U51 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_isNatIListKind =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_0 => match l with
                                    | nil => _
                                    | _::_ => _
                                    end
              | algebra.F.id_U12 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__U52 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_a__isNat =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_U43 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__isNatIListKind =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__U12 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_U31 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_a__length =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__U42 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_U53 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_and =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_cons =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_U11 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_a__U51 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_a__U22 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_U42 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_length =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_tt => match l with
                                     | nil => _
                                     | _::_ => _
                                     end
              | algebra.F.id_U22 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_s =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__U41 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_U52 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_nil => match l with
                                      | nil => _
                                      | _::_ => _
                                      end
              | algebra.F.id_zeros => match l with
                                        | nil => _
                                        | _::_ => _
                                        end
              | algebra.F.id_isNatList =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__U53 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__U31 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_isNatIList =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__isNatKind =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__isNatList =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_U32 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_mark =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__U43 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_U61 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_isNat =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              end.
     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 .
     
     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 .
     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 .
     
     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 .
     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 .
     
     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 .
     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 .
     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 .
     
     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 .
   Qed.
   
   Lemma marked_measure_equation :
    forall t, 
     marked_measure t = match t with
                          | (algebra.Alg.Term algebra.F.id_a__length 
                             (x11::nil)) =>
                           P_id_A__LENGTH (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__U41 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U41 (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__U22 (x11::nil)) =>
                           P_id_A__U22 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U51 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U51 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__and (x12::
                             x11::nil)) =>
                           P_id_A__AND (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__U43 (x11::nil)) =>
                           P_id_A__U43 (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__U53 (x11::nil)) =>
                           P_id_A__U53 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_mark (x11::nil)) =>
                           P_id_MARK (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U12 (x11::nil)) =>
                           P_id_A__U12 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U42 (x12::
                             x11::nil)) =>
                           P_id_A__U42 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__zeros nil) =>
                           P_id_A__ZEROS 
                          | (algebra.Alg.Term algebra.F.id_a__isNat 
                             (x11::nil)) =>
                           P_id_A__ISNAT (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U52 (x12::
                             x11::nil)) =>
                           P_id_A__U52 (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_a__U21 (x12::
                             x11::nil)) =>
                           P_id_A__U21 (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__U32 (x11::nil)) =>
                           P_id_A__U32 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U61 (x12::
                             x11::nil)) =>
                           P_id_A__U61 (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_isNatKind_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isNatIList_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U21_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U41_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__and_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U11_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U21_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U61_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U32_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U51_monotonic;assumption.
     
     vm_compute in |-*;intros ;apply P_id_isNatIListKind_monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id_U12_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U52_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isNat_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_a__U12_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U31_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__length_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U42_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U53_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_and_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_cons_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U11_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U51_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U22_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U42_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_length_monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id_U22_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_s_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U41_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U52_monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id_isNatList_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U53_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U31_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_isNatIList_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isNatKind_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isNatList_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U32_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_mark_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U43_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U61_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_isNat_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_isNatKind_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isNatIList_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U21_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U41_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__and_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U11_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U21_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U61_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U32_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U51_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_isNatIListKind_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_0_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U12_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U52_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isNat_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_a__U12_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U31_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__length_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U42_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U53_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_and_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_cons_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U11_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U51_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U22_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U42_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_length_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_tt_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U22_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_s_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U41_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U52_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_nil_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_zeros_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_isNatList_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U53_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U31_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_isNatIList_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isNatKind_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isNatList_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U32_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_mark_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__U43_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U61_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_isNat_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__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_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__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__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__U53_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__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__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_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__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__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__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__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__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__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__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__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__AND_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__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 (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_isNatKind : Z ->Z.
   
   Hypothesis P_id_a__isNatIList : Z ->Z.
   
   Hypothesis P_id_a__U21 : Z ->Z ->Z.
   
   Hypothesis P_id_U41 : Z ->Z ->Z ->Z.
   
   Hypothesis P_id_a__and : Z ->Z ->Z.
   
   Hypothesis P_id_a__U11 : Z ->Z ->Z.
   
   Hypothesis P_id_U21 : Z ->Z ->Z.
   
   Hypothesis P_id_a__U61 : Z ->Z ->Z.
   
   Hypothesis P_id_a__U32 : Z ->Z.
   
   Hypothesis P_id_U51 : Z ->Z ->Z ->Z.
   
   Hypothesis P_id_isNatIListKind : Z ->Z.
   
   Hypothesis P_id_0 : Z.
   
   Hypothesis P_id_U12 : Z ->Z.
   
   Hypothesis P_id_a__U52 : Z ->Z ->Z.
   
   Hypothesis P_id_a__isNat : Z ->Z.
   
   Hypothesis P_id_U43 : Z ->Z.
   
   Hypothesis P_id_a__isNatIListKind : Z ->Z.
   
   Hypothesis P_id_a__U12 : Z ->Z.
   
   Hypothesis P_id_U31 : Z ->Z ->Z.
   
   Hypothesis P_id_a__length : Z ->Z.
   
   Hypothesis P_id_a__U42 : Z ->Z ->Z.
   
   Hypothesis P_id_U53 : Z ->Z.
   
   Hypothesis P_id_and : Z ->Z ->Z.
   
   Hypothesis P_id_cons : Z ->Z ->Z.
   
   Hypothesis P_id_U11 : Z ->Z ->Z.
   
   Hypothesis P_id_a__U51 : Z ->Z ->Z ->Z.
   
   Hypothesis P_id_a__U22 : Z ->Z.
   
   Hypothesis P_id_U42 : Z ->Z ->Z.
   
   Hypothesis P_id_length : Z ->Z.
   
   Hypothesis P_id_tt : Z.
   
   Hypothesis P_id_U22 : Z ->Z.
   
   Hypothesis P_id_s : Z ->Z.
   
   Hypothesis P_id_a__U41 : Z ->Z ->Z ->Z.
   
   Hypothesis P_id_U52 : Z ->Z ->Z.
   
   Hypothesis P_id_nil : Z.
   
   Hypothesis P_id_zeros : Z.
   
   Hypothesis P_id_isNatList : Z ->Z.
   
   Hypothesis P_id_a__U53 : Z ->Z.
   
   Hypothesis P_id_a__U31 : Z ->Z ->Z.
   
   Hypothesis P_id_isNatIList : Z ->Z.
   
   Hypothesis P_id_a__isNatKind : Z ->Z.
   
   Hypothesis P_id_a__isNatList : Z ->Z.
   
   Hypothesis P_id_U32 : Z ->Z.
   
   Hypothesis P_id_mark : Z ->Z.
   
   Hypothesis P_id_a__U43 : Z ->Z.
   
   Hypothesis P_id_U61 : Z ->Z ->Z.
   
   Hypothesis P_id_isNat : Z ->Z.
   
   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__isNatIList_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->
      P_id_a__isNatIList x12 <= P_id_a__isNatIList 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_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_a__and_monotonic :
    forall x12 x14 x13 x11, 
     (min_value <= x14)/\ (x14 <= x13) ->
      (min_value <= x12)/\ (x12 <= x11) ->
       P_id_a__and x12 x14 <= P_id_a__and x11 x13.
   
   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_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__U61_monotonic :
    forall x12 x14 x13 x11, 
     (min_value <= x14)/\ (x14 <= x13) ->
      (min_value <= x12)/\ (x12 <= x11) ->
       P_id_a__U61 x12 x14 <= P_id_a__U61 x11 x13.
   
   Hypothesis P_id_a__U32_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->P_id_a__U32 x12 <= P_id_a__U32 x11.
   
   Hypothesis P_id_U51_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_U51 x12 x14 x16 <= P_id_U51 x11 x13 x15.
   
   Hypothesis P_id_isNatIListKind_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->
      P_id_isNatIListKind x12 <= P_id_isNatIListKind x11.
   
   Hypothesis P_id_U12_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->P_id_U12 x12 <= P_id_U12 x11.
   
   Hypothesis P_id_a__U52_monotonic :
    forall x12 x14 x13 x11, 
     (min_value <= x14)/\ (x14 <= x13) ->
      (min_value <= x12)/\ (x12 <= x11) ->
       P_id_a__U52 x12 x14 <= P_id_a__U52 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_U43_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->P_id_U43 x12 <= P_id_U43 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__U12_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->P_id_a__U12 x12 <= P_id_a__U12 x11.
   
   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__length_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->
      P_id_a__length x12 <= P_id_a__length x11.
   
   Hypothesis P_id_a__U42_monotonic :
    forall x12 x14 x13 x11, 
     (min_value <= x14)/\ (x14 <= x13) ->
      (min_value <= x12)/\ (x12 <= x11) ->
       P_id_a__U42 x12 x14 <= P_id_a__U42 x11 x13.
   
   Hypothesis P_id_U53_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->P_id_U53 x12 <= P_id_U53 x11.
   
   Hypothesis P_id_and_monotonic :
    forall x12 x14 x13 x11, 
     (min_value <= x14)/\ (x14 <= x13) ->
      (min_value <= x12)/\ (x12 <= x11) ->
       P_id_and x12 x14 <= P_id_and x11 x13.
   
   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_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__U51_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__U51 x12 x14 x16 <= P_id_a__U51 x11 x13 x15.
   
   Hypothesis P_id_a__U22_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->P_id_a__U22 x12 <= P_id_a__U22 x11.
   
   Hypothesis P_id_U42_monotonic :
    forall x12 x14 x13 x11, 
     (min_value <= x14)/\ (x14 <= x13) ->
      (min_value <= x12)/\ (x12 <= x11) ->
       P_id_U42 x12 x14 <= P_id_U42 x11 x13.
   
   Hypothesis P_id_length_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->P_id_length x12 <= P_id_length x11.
   
   Hypothesis P_id_U22_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->P_id_U22 x12 <= P_id_U22 x11.
   
   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__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_U52_monotonic :
    forall x12 x14 x13 x11, 
     (min_value <= x14)/\ (x14 <= x13) ->
      (min_value <= x12)/\ (x12 <= x11) ->
       P_id_U52 x12 x14 <= P_id_U52 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__U53_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->P_id_a__U53 x12 <= P_id_a__U53 x11.
   
   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_isNatIList_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->
      P_id_isNatIList x12 <= P_id_isNatIList 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_a__isNatList_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->
      P_id_a__isNatList x12 <= P_id_a__isNatList x11.
   
   Hypothesis P_id_U32_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->P_id_U32 x12 <= P_id_U32 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__U43_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->P_id_a__U43 x12 <= P_id_a__U43 x11.
   
   Hypothesis P_id_U61_monotonic :
    forall x12 x14 x13 x11, 
     (min_value <= x14)/\ (x14 <= x13) ->
      (min_value <= x12)/\ (x12 <= x11) ->
       P_id_U61 x12 x14 <= P_id_U61 x11 x13.
   
   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__zeros_bounded : min_value <= P_id_a__zeros .
   
   Hypothesis P_id_isNatKind_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_isNatKind x11.
   
   Hypothesis P_id_a__isNatIList_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_a__isNatIList 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_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_a__and_bounded :
    forall x12 x11, 
     (min_value <= x11) ->
      (min_value <= x12) ->min_value <= P_id_a__and x12 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_U21_bounded :
    forall x12 x11, 
     (min_value <= x11) ->(min_value <= x12) ->min_value <= P_id_U21 x12 x11.
   
   Hypothesis P_id_a__U61_bounded :
    forall x12 x11, 
     (min_value <= x11) ->
      (min_value <= x12) ->min_value <= P_id_a__U61 x12 x11.
   
   Hypothesis P_id_a__U32_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_a__U32 x11.
   
   Hypothesis P_id_U51_bounded :
    forall x12 x13 x11, 
     (min_value <= x11) ->
      (min_value <= x12) ->
       (min_value <= x13) ->min_value <= P_id_U51 x13 x12 x11.
   
   Hypothesis P_id_isNatIListKind_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_isNatIListKind x11.
   
   Hypothesis P_id_0_bounded : min_value <= P_id_0 .
   
   Hypothesis P_id_U12_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_U12 x11.
   
   Hypothesis P_id_a__U52_bounded :
    forall x12 x11, 
     (min_value <= x11) ->
      (min_value <= x12) ->min_value <= P_id_a__U52 x12 x11.
   
   Hypothesis P_id_a__isNat_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_a__isNat x11.
   
   Hypothesis P_id_U43_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_U43 x11.
   
   Hypothesis P_id_a__isNatIListKind_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_a__isNatIListKind x11.
   
   Hypothesis P_id_a__U12_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_a__U12 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__length_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_a__length x11.
   
   Hypothesis P_id_a__U42_bounded :
    forall x12 x11, 
     (min_value <= x11) ->
      (min_value <= x12) ->min_value <= P_id_a__U42 x12 x11.
   
   Hypothesis P_id_U53_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_U53 x11.
   
   Hypothesis P_id_and_bounded :
    forall x12 x11, 
     (min_value <= x11) ->(min_value <= x12) ->min_value <= P_id_and x12 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_U11_bounded :
    forall x12 x11, 
     (min_value <= x11) ->(min_value <= x12) ->min_value <= P_id_U11 x12 x11.
   
   Hypothesis P_id_a__U51_bounded :
    forall x12 x13 x11, 
     (min_value <= x11) ->
      (min_value <= x12) ->
       (min_value <= x13) ->min_value <= P_id_a__U51 x13 x12 x11.
   
   Hypothesis P_id_a__U22_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_a__U22 x11.
   
   Hypothesis P_id_U42_bounded :
    forall x12 x11, 
     (min_value <= x11) ->(min_value <= x12) ->min_value <= P_id_U42 x12 x11.
   
   Hypothesis P_id_length_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_length x11.
   
   Hypothesis P_id_tt_bounded : min_value <= P_id_tt .
   
   Hypothesis P_id_U22_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_U22 x11.
   
   Hypothesis P_id_s_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_s 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_U52_bounded :
    forall x12 x11, 
     (min_value <= x11) ->(min_value <= x12) ->min_value <= P_id_U52 x12 x11.
   
   Hypothesis P_id_nil_bounded : min_value <= P_id_nil .
   
   Hypothesis P_id_zeros_bounded : min_value <= P_id_zeros .
   
   Hypothesis P_id_isNatList_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_isNatList x11.
   
   Hypothesis P_id_a__U53_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_a__U53 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_isNatIList_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_isNatIList x11.
   
   Hypothesis P_id_a__isNatKind_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_a__isNatKind x11.
   
   Hypothesis P_id_a__isNatList_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_a__isNatList x11.
   
   Hypothesis P_id_U32_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_U32 x11.
   
   Hypothesis P_id_mark_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_mark x11.
   
   Hypothesis P_id_a__U43_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_a__U43 x11.
   
   Hypothesis P_id_U61_bounded :
    forall x12 x11, 
     (min_value <= x11) ->(min_value <= x12) ->min_value <= P_id_U61 x12 x11.
   
   Hypothesis P_id_isNat_bounded :
    forall x11, (min_value <= x11) ->min_value <= P_id_isNat x11.
   
   Definition measure  := 
     Interp.measure min_value P_id_a__zeros P_id_isNatKind 
      P_id_a__isNatIList P_id_a__U21 P_id_U41 P_id_a__and P_id_a__U11 
      P_id_U21 P_id_a__U61 P_id_a__U32 P_id_U51 P_id_isNatIListKind P_id_0 
      P_id_U12 P_id_a__U52 P_id_a__isNat P_id_U43 P_id_a__isNatIListKind 
      P_id_a__U12 P_id_U31 P_id_a__length P_id_a__U42 P_id_U53 P_id_and 
      P_id_cons P_id_U11 P_id_a__U51 P_id_a__U22 P_id_U42 P_id_length 
      P_id_tt P_id_U22 P_id_s P_id_a__U41 P_id_U52 P_id_nil P_id_zeros 
      P_id_isNatList P_id_a__U53 P_id_a__U31 P_id_isNatIList 
      P_id_a__isNatKind P_id_a__isNatList P_id_U32 P_id_mark P_id_a__U43 
      P_id_U61 P_id_isNat.
   
   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_isNatKind (x11::nil)) =>
                    P_id_isNatKind (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__U21 (x12::x11::nil)) =>
                    P_id_a__U21 (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_a__and (x12::x11::nil)) =>
                    P_id_a__and (measure x12) (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_U21 (x12::x11::nil)) =>
                    P_id_U21 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U61 (x12::x11::nil)) =>
                    P_id_a__U61 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U32 (x11::nil)) =>
                    P_id_a__U32 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U51 (x13::x12::x11::nil)) =>
                    P_id_U51 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_isNatIListKind 
                      (x11::nil)) =>
                    P_id_isNatIListKind (measure x11)
                   | (algebra.Alg.Term algebra.F.id_0 nil) => P_id_0 
                   | (algebra.Alg.Term algebra.F.id_U12 (x11::nil)) =>
                    P_id_U12 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U52 (x12::x11::nil)) =>
                    P_id_a__U52 (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_U43 (x11::nil)) =>
                    P_id_U43 (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__U12 (x11::nil)) =>
                    P_id_a__U12 (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__length (x11::nil)) =>
                    P_id_a__length (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U42 (x12::x11::nil)) =>
                    P_id_a__U42 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U53 (x11::nil)) =>
                    P_id_U53 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_and (x12::x11::nil)) =>
                    P_id_and (measure x12) (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_U11 (x12::x11::nil)) =>
                    P_id_U11 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U51 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U51 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U22 (x11::nil)) =>
                    P_id_a__U22 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U42 (x12::x11::nil)) =>
                    P_id_U42 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_length (x11::nil)) =>
                    P_id_length (measure x11)
                   | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt 
                   | (algebra.Alg.Term algebra.F.id_U22 (x11::nil)) =>
                    P_id_U22 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_s (x11::nil)) =>
                    P_id_s (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_U52 (x12::x11::nil)) =>
                    P_id_U52 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil 
                   | (algebra.Alg.Term algebra.F.id_zeros nil) => P_id_zeros 
                   | (algebra.Alg.Term algebra.F.id_isNatList (x11::nil)) =>
                    P_id_isNatList (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U53 (x11::nil)) =>
                    P_id_a__U53 (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_isNatIList (x11::nil)) =>
                    P_id_isNatIList (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__isNatList (x11::nil)) =>
                    P_id_a__isNatList (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U32 (x11::nil)) =>
                    P_id_U32 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_mark (x11::nil)) =>
                    P_id_mark (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U43 (x11::nil)) =>
                    P_id_a__U43 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U61 (x12::x11::nil)) =>
                    P_id_U61 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_isNat (x11::nil)) =>
                    P_id_isNat (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_isNatKind_monotonic;assumption.
     intros ;apply P_id_a__isNatIList_monotonic;assumption.
     intros ;apply P_id_a__U21_monotonic;assumption.
     intros ;apply P_id_U41_monotonic;assumption.
     intros ;apply P_id_a__and_monotonic;assumption.
     intros ;apply P_id_a__U11_monotonic;assumption.
     intros ;apply P_id_U21_monotonic;assumption.
     intros ;apply P_id_a__U61_monotonic;assumption.
     intros ;apply P_id_a__U32_monotonic;assumption.
     intros ;apply P_id_U51_monotonic;assumption.
     intros ;apply P_id_isNatIListKind_monotonic;assumption.
     intros ;apply P_id_U12_monotonic;assumption.
     intros ;apply P_id_a__U52_monotonic;assumption.
     intros ;apply P_id_a__isNat_monotonic;assumption.
     intros ;apply P_id_U43_monotonic;assumption.
     intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
     intros ;apply P_id_a__U12_monotonic;assumption.
     intros ;apply P_id_U31_monotonic;assumption.
     intros ;apply P_id_a__length_monotonic;assumption.
     intros ;apply P_id_a__U42_monotonic;assumption.
     intros ;apply P_id_U53_monotonic;assumption.
     intros ;apply P_id_and_monotonic;assumption.
     intros ;apply P_id_cons_monotonic;assumption.
     intros ;apply P_id_U11_monotonic;assumption.
     intros ;apply P_id_a__U51_monotonic;assumption.
     intros ;apply P_id_a__U22_monotonic;assumption.
     intros ;apply P_id_U42_monotonic;assumption.
     intros ;apply P_id_length_monotonic;assumption.
     intros ;apply P_id_U22_monotonic;assumption.
     intros ;apply P_id_s_monotonic;assumption.
     intros ;apply P_id_a__U41_monotonic;assumption.
     intros ;apply P_id_U52_monotonic;assumption.
     intros ;apply P_id_isNatList_monotonic;assumption.
     intros ;apply P_id_a__U53_monotonic;assumption.
     intros ;apply P_id_a__U31_monotonic;assumption.
     intros ;apply P_id_isNatIList_monotonic;assumption.
     intros ;apply P_id_a__isNatKind_monotonic;assumption.
     intros ;apply P_id_a__isNatList_monotonic;assumption.
     intros ;apply P_id_U32_monotonic;assumption.
     intros ;apply P_id_mark_monotonic;assumption.
     intros ;apply P_id_a__U43_monotonic;assumption.
     intros ;apply P_id_U61_monotonic;assumption.
     intros ;apply P_id_isNat_monotonic;assumption.
     intros ;apply P_id_a__zeros_bounded;assumption.
     intros ;apply P_id_isNatKind_bounded;assumption.
     intros ;apply P_id_a__isNatIList_bounded;assumption.
     intros ;apply P_id_a__U21_bounded;assumption.
     intros ;apply P_id_U41_bounded;assumption.
     intros ;apply P_id_a__and_bounded;assumption.
     intros ;apply P_id_a__U11_bounded;assumption.
     intros ;apply P_id_U21_bounded;assumption.
     intros ;apply P_id_a__U61_bounded;assumption.
     intros ;apply P_id_a__U32_bounded;assumption.
     intros ;apply P_id_U51_bounded;assumption.
     intros ;apply P_id_isNatIListKind_bounded;assumption.
     intros ;apply P_id_0_bounded;assumption.
     intros ;apply P_id_U12_bounded;assumption.
     intros ;apply P_id_a__U52_bounded;assumption.
     intros ;apply P_id_a__isNat_bounded;assumption.
     intros ;apply P_id_U43_bounded;assumption.
     intros ;apply P_id_a__isNatIListKind_bounded;assumption.
     intros ;apply P_id_a__U12_bounded;assumption.
     intros ;apply P_id_U31_bounded;assumption.
     intros ;apply P_id_a__length_bounded;assumption.
     intros ;apply P_id_a__U42_bounded;assumption.
     intros ;apply P_id_U53_bounded;assumption.
     intros ;apply P_id_and_bounded;assumption.
     intros ;apply P_id_cons_bounded;assumption.
     intros ;apply P_id_U11_bounded;assumption.
     intros ;apply P_id_a__U51_bounded;assumption.
     intros ;apply P_id_a__U22_bounded;assumption.
     intros ;apply P_id_U42_bounded;assumption.
     intros ;apply P_id_length_bounded;assumption.
     intros ;apply P_id_tt_bounded;assumption.
     intros ;apply P_id_U22_bounded;assumption.
     intros ;apply P_id_s_bounded;assumption.
     intros ;apply P_id_a__U41_bounded;assumption.
     intros ;apply P_id_U52_bounded;assumption.
     intros ;apply P_id_nil_bounded;assumption.
     intros ;apply P_id_zeros_bounded;assumption.
     intros ;apply P_id_isNatList_bounded;assumption.
     intros ;apply P_id_a__U53_bounded;assumption.
     intros ;apply P_id_a__U31_bounded;assumption.
     intros ;apply P_id_isNatIList_bounded;assumption.
     intros ;apply P_id_a__isNatKind_bounded;assumption.
     intros ;apply P_id_a__isNatList_bounded;assumption.
     intros ;apply P_id_U32_bounded;assumption.
     intros ;apply P_id_mark_bounded;assumption.
     intros ;apply P_id_a__U43_bounded;assumption.
     intros ;apply P_id_U61_bounded;assumption.
     intros ;apply P_id_isNat_bounded;assumption.
     apply rules_monotonic.
   Qed.
   
   Hypothesis P_id_A__LENGTH : Z ->Z.
   
   Hypothesis P_id_A__U11 : Z ->Z ->Z.
   
   Hypothesis P_id_A__U41 : Z ->Z ->Z ->Z.
   
   Hypothesis P_id_A__ISNATKIND : Z ->Z.
   
   Hypothesis P_id_A__U22 : Z ->Z.
   
   Hypothesis P_id_A__U51 : Z ->Z ->Z ->Z.
   
   Hypothesis P_id_A__AND : Z ->Z ->Z.
   
   Hypothesis P_id_A__ISNATLIST : Z ->Z.
   
   Hypothesis P_id_A__U43 : Z ->Z.
   
   Hypothesis P_id_A__U31 : Z ->Z ->Z.
   
   Hypothesis P_id_A__U53 : Z ->Z.
   
   Hypothesis P_id_MARK : Z ->Z.
   
   Hypothesis P_id_A__U12 : Z ->Z.
   
   Hypothesis P_id_A__U42 : Z ->Z ->Z.
   
   Hypothesis P_id_A__ZEROS : Z.
   
   Hypothesis P_id_A__ISNAT : Z ->Z.
   
   Hypothesis P_id_A__U52 : Z ->Z ->Z.
   
   Hypothesis P_id_A__ISNATILISTKIND : Z ->Z.
   
   Hypothesis P_id_A__U21 : Z ->Z ->Z.
   
   Hypothesis P_id_A__ISNATILIST : Z ->Z.
   
   Hypothesis P_id_A__U32 : Z ->Z.
   
   Hypothesis P_id_A__U61 : Z ->Z ->Z.
   
   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__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__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__ISNATKIND_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->
      P_id_A__ISNATKIND x12 <= P_id_A__ISNATKIND x11.
   
   Hypothesis P_id_A__U22_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->P_id_A__U22 x12 <= P_id_A__U22 x11.
   
   Hypothesis P_id_A__U51_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__U51 x12 x14 x16 <= P_id_A__U51 x11 x13 x15.
   
   Hypothesis P_id_A__AND_monotonic :
    forall x12 x14 x13 x11, 
     (min_value <= x14)/\ (x14 <= x13) ->
      (min_value <= x12)/\ (x12 <= x11) ->
       P_id_A__AND x12 x14 <= P_id_A__AND 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__U43_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->P_id_A__U43 x12 <= P_id_A__U43 x11.
   
   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__U53_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->P_id_A__U53 x12 <= P_id_A__U53 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__U12_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->P_id_A__U12 x12 <= P_id_A__U12 x11.
   
   Hypothesis P_id_A__U42_monotonic :
    forall x12 x14 x13 x11, 
     (min_value <= x14)/\ (x14 <= x13) ->
      (min_value <= x12)/\ (x12 <= x11) ->
       P_id_A__U42 x12 x14 <= P_id_A__U42 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_A__U52_monotonic :
    forall x12 x14 x13 x11, 
     (min_value <= x14)/\ (x14 <= x13) ->
      (min_value <= x12)/\ (x12 <= x11) ->
       P_id_A__U52 x12 x14 <= P_id_A__U52 x11 x13.
   
   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__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__ISNATILIST_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->
      P_id_A__ISNATILIST x12 <= P_id_A__ISNATILIST x11.
   
   Hypothesis P_id_A__U32_monotonic :
    forall x12 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->P_id_A__U32 x12 <= P_id_A__U32 x11.
   
   Hypothesis P_id_A__U61_monotonic :
    forall x12 x14 x13 x11, 
     (min_value <= x14)/\ (x14 <= x13) ->
      (min_value <= x12)/\ (x12 <= x11) ->
       P_id_A__U61 x12 x14 <= P_id_A__U61 x11 x13.
   
   Definition marked_measure  := 
     Interp.marked_measure min_value P_id_a__zeros P_id_isNatKind 
      P_id_a__isNatIList P_id_a__U21 P_id_U41 P_id_a__and P_id_a__U11 
      P_id_U21 P_id_a__U61 P_id_a__U32 P_id_U51 P_id_isNatIListKind P_id_0 
      P_id_U12 P_id_a__U52 P_id_a__isNat P_id_U43 P_id_a__isNatIListKind 
      P_id_a__U12 P_id_U31 P_id_a__length P_id_a__U42 P_id_U53 P_id_and 
      P_id_cons P_id_U11 P_id_a__U51 P_id_a__U22 P_id_U42 P_id_length 
      P_id_tt P_id_U22 P_id_s P_id_a__U41 P_id_U52 P_id_nil P_id_zeros 
      P_id_isNatList P_id_a__U53 P_id_a__U31 P_id_isNatIList 
      P_id_a__isNatKind P_id_a__isNatList P_id_U32 P_id_mark P_id_a__U43 
      P_id_U61 P_id_isNat P_id_A__LENGTH P_id_A__U11 P_id_A__U41 
      P_id_A__ISNATKIND P_id_A__U22 P_id_A__U51 P_id_A__AND 
      P_id_A__ISNATLIST P_id_A__U43 P_id_A__U31 P_id_A__U53 P_id_MARK 
      P_id_A__U12 P_id_A__U42 P_id_A__ZEROS P_id_A__ISNAT P_id_A__U52 
      P_id_A__ISNATILISTKIND P_id_A__U21 P_id_A__ISNATILIST P_id_A__U32 
      P_id_A__U61.
   
   Lemma marked_measure_equation :
    forall t, 
     marked_measure t = match t with
                          | (algebra.Alg.Term algebra.F.id_a__length 
                             (x11::nil)) =>
                           P_id_A__LENGTH (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__U41 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U41 (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__U22 (x11::nil)) =>
                           P_id_A__U22 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U51 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U51 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__and (x12::
                             x11::nil)) =>
                           P_id_A__AND (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__U43 (x11::nil)) =>
                           P_id_A__U43 (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__U53 (x11::nil)) =>
                           P_id_A__U53 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_mark (x11::nil)) =>
                           P_id_MARK (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U12 (x11::nil)) =>
                           P_id_A__U12 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U42 (x12::
                             x11::nil)) =>
                           P_id_A__U42 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__zeros nil) =>
                           P_id_A__ZEROS 
                          | (algebra.Alg.Term algebra.F.id_a__isNat 
                             (x11::nil)) =>
                           P_id_A__ISNAT (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U52 (x12::
                             x11::nil)) =>
                           P_id_A__U52 (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_a__U21 (x12::
                             x11::nil)) =>
                           P_id_A__U21 (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__U32 (x11::nil)) =>
                           P_id_A__U32 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U61 (x12::
                             x11::nil)) =>
                           P_id_A__U61 (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_isNatKind_monotonic;assumption.
     intros ;apply P_id_a__isNatIList_monotonic;assumption.
     intros ;apply P_id_a__U21_monotonic;assumption.
     intros ;apply P_id_U41_monotonic;assumption.
     intros ;apply P_id_a__and_monotonic;assumption.
     intros ;apply P_id_a__U11_monotonic;assumption.
     intros ;apply P_id_U21_monotonic;assumption.
     intros ;apply P_id_a__U61_monotonic;assumption.
     intros ;apply P_id_a__U32_monotonic;assumption.
     intros ;apply P_id_U51_monotonic;assumption.
     intros ;apply P_id_isNatIListKind_monotonic;assumption.
     intros ;apply P_id_U12_monotonic;assumption.
     intros ;apply P_id_a__U52_monotonic;assumption.
     intros ;apply P_id_a__isNat_monotonic;assumption.
     intros ;apply P_id_U43_monotonic;assumption.
     intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
     intros ;apply P_id_a__U12_monotonic;assumption.
     intros ;apply P_id_U31_monotonic;assumption.
     intros ;apply P_id_a__length_monotonic;assumption.
     intros ;apply P_id_a__U42_monotonic;assumption.
     intros ;apply P_id_U53_monotonic;assumption.
     intros ;apply P_id_and_monotonic;assumption.
     intros ;apply P_id_cons_monotonic;assumption.
     intros ;apply P_id_U11_monotonic;assumption.
     intros ;apply P_id_a__U51_monotonic;assumption.
     intros ;apply P_id_a__U22_monotonic;assumption.
     intros ;apply P_id_U42_monotonic;assumption.
     intros ;apply P_id_length_monotonic;assumption.
     intros ;apply P_id_U22_monotonic;assumption.
     intros ;apply P_id_s_monotonic;assumption.
     intros ;apply P_id_a__U41_monotonic;assumption.
     intros ;apply P_id_U52_monotonic;assumption.
     intros ;apply P_id_isNatList_monotonic;assumption.
     intros ;apply P_id_a__U53_monotonic;assumption.
     intros ;apply P_id_a__U31_monotonic;assumption.
     intros ;apply P_id_isNatIList_monotonic;assumption.
     intros ;apply P_id_a__isNatKind_monotonic;assumption.
     intros ;apply P_id_a__isNatList_monotonic;assumption.
     intros ;apply P_id_U32_monotonic;assumption.
     intros ;apply P_id_mark_monotonic;assumption.
     intros ;apply P_id_a__U43_monotonic;assumption.
     intros ;apply P_id_U61_monotonic;assumption.
     intros ;apply P_id_isNat_monotonic;assumption.
     intros ;apply P_id_a__zeros_bounded;assumption.
     intros ;apply P_id_isNatKind_bounded;assumption.
     intros ;apply P_id_a__isNatIList_bounded;assumption.
     intros ;apply P_id_a__U21_bounded;assumption.
     intros ;apply P_id_U41_bounded;assumption.
     intros ;apply P_id_a__and_bounded;assumption.
     intros ;apply P_id_a__U11_bounded;assumption.
     intros ;apply P_id_U21_bounded;assumption.
     intros ;apply P_id_a__U61_bounded;assumption.
     intros ;apply P_id_a__U32_bounded;assumption.
     intros ;apply P_id_U51_bounded;assumption.
     intros ;apply P_id_isNatIListKind_bounded;assumption.
     intros ;apply P_id_0_bounded;assumption.
     intros ;apply P_id_U12_bounded;assumption.
     intros ;apply P_id_a__U52_bounded;assumption.
     intros ;apply P_id_a__isNat_bounded;assumption.
     intros ;apply P_id_U43_bounded;assumption.
     intros ;apply P_id_a__isNatIListKind_bounded;assumption.
     intros ;apply P_id_a__U12_bounded;assumption.
     intros ;apply P_id_U31_bounded;assumption.
     intros ;apply P_id_a__length_bounded;assumption.
     intros ;apply P_id_a__U42_bounded;assumption.
     intros ;apply P_id_U53_bounded;assumption.
     intros ;apply P_id_and_bounded;assumption.
     intros ;apply P_id_cons_bounded;assumption.
     intros ;apply P_id_U11_bounded;assumption.
     intros ;apply P_id_a__U51_bounded;assumption.
     intros ;apply P_id_a__U22_bounded;assumption.
     intros ;apply P_id_U42_bounded;assumption.
     intros ;apply P_id_length_bounded;assumption.
     intros ;apply P_id_tt_bounded;assumption.
     intros ;apply P_id_U22_bounded;assumption.
     intros ;apply P_id_s_bounded;assumption.
     intros ;apply P_id_a__U41_bounded;assumption.
     intros ;apply P_id_U52_bounded;assumption.
     intros ;apply P_id_nil_bounded;assumption.
     intros ;apply P_id_zeros_bounded;assumption.
     intros ;apply P_id_isNatList_bounded;assumption.
     intros ;apply P_id_a__U53_bounded;assumption.
     intros ;apply P_id_a__U31_bounded;assumption.
     intros ;apply P_id_isNatIList_bounded;assumption.
     intros ;apply P_id_a__isNatKind_bounded;assumption.
     intros ;apply P_id_a__isNatList_bounded;assumption.
     intros ;apply P_id_U32_bounded;assumption.
     intros ;apply P_id_mark_bounded;assumption.
     intros ;apply P_id_a__U43_bounded;assumption.
     intros ;apply P_id_U61_bounded;assumption.
     intros ;apply P_id_isNat_bounded;assumption.
     apply rules_monotonic.
     intros ;apply P_id_A__LENGTH_monotonic;assumption.
     intros ;apply P_id_A__U11_monotonic;assumption.
     intros ;apply P_id_A__U41_monotonic;assumption.
     intros ;apply P_id_A__ISNATKIND_monotonic;assumption.
     intros ;apply P_id_A__U22_monotonic;assumption.
     intros ;apply P_id_A__U51_monotonic;assumption.
     intros ;apply P_id_A__AND_monotonic;assumption.
     intros ;apply P_id_A__ISNATLIST_monotonic;assumption.
     intros ;apply P_id_A__U43_monotonic;assumption.
     intros ;apply P_id_A__U31_monotonic;assumption.
     intros ;apply P_id_A__U53_monotonic;assumption.
     intros ;apply P_id_MARK_monotonic;assumption.
     intros ;apply P_id_A__U12_monotonic;assumption.
     intros ;apply P_id_A__U42_monotonic;assumption.
     intros ;apply P_id_A__ISNAT_monotonic;assumption.
     intros ;apply P_id_A__U52_monotonic;assumption.
     intros ;apply P_id_A__ISNATILISTKIND_monotonic;assumption.
     intros ;apply P_id_A__U21_monotonic;assumption.
     intros ;apply P_id_A__ISNATILIST_monotonic;assumption.
     intros ;apply P_id_A__U32_monotonic;assumption.
     intros ;apply P_id_A__U61_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__isNatList(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__isNatList (x1::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_a__U11 (x12::x11::nil))
    (* <a__U11(tt,V1_),a__isNatList(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__isNatList (x1::nil)) 
        (algebra.Alg.Term algebra.F.id_a__U11 (x12::x11::nil))
    (* <a__U21(tt,V1_),a__U22(a__isNat(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__U22 ((algebra.Alg.Term 
                   algebra.F.id_a__isNat (x1::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_a__U21 (x12::x11::nil))
    (* <a__U21(tt,V1_),a__isNat(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__isNat (x1::nil)) 
        (algebra.Alg.Term algebra.F.id_a__U21 (x12::x11::nil))
    (* <a__U31(tt,V_),a__U32(a__isNatList(V_))> *)
   | DP_R_xml_0_4 :
    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__isNatList (x2::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_a__U31 (x12::x11::nil))
    (* <a__U31(tt,V_),a__isNatList(V_)> *)
   | DP_R_xml_0_5 :
    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__U31 (x12::x11::nil))
    (* <a__U41(tt,V1_,V2_),a__U42(a__isNat(V1_),V2_)> *)
   | DP_R_xml_0_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 (algebra.Alg.Term algebra.F.id_a__U42 ((algebra.Alg.Term 
                    algebra.F.id_a__isNat (x1::nil))::x3::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U41 (x13::x12::x11::nil))
    (* <a__U41(tt,V1_,V2_),a__isNat(V1_)> *)
   | DP_R_xml_0_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 (algebra.Alg.Term algebra.F.id_a__isNat (x1::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U41 (x13::x12::x11::nil))
    (* <a__U42(tt,V2_),a__U43(a__isNatIList(V2_))> *)
   | DP_R_xml_0_8 :
    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__U43 ((algebra.Alg.Term 
                   algebra.F.id_a__isNatIList (x3::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_a__U42 (x12::x11::nil))
    (* <a__U42(tt,V2_),a__isNatIList(V2_)> *)
   | DP_R_xml_0_9 :
    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__U42 (x12::x11::nil))
    (* <a__U51(tt,V1_,V2_),a__U52(a__isNat(V1_),V2_)> *)
   | DP_R_xml_0_10 :
    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__U52 ((algebra.Alg.Term 
                    algebra.F.id_a__isNat (x1::nil))::x3::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U51 (x13::x12::x11::nil))
    (* <a__U51(tt,V1_,V2_),a__isNat(V1_)> *)
   | DP_R_xml_0_11 :
    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__U51 (x13::x12::x11::nil))
    (* <a__U52(tt,V2_),a__U53(a__isNatList(V2_))> *)
   | DP_R_xml_0_12 :
    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__U53 ((algebra.Alg.Term 
                   algebra.F.id_a__isNatList (x3::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_a__U52 (x12::x11::nil))
    (* <a__U52(tt,V2_),a__isNatList(V2_)> *)
   | DP_R_xml_0_13 :
    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__U52 (x12::x11::nil))
    (* <a__U61(tt,L_),a__length(mark(L_))> *)
   | DP_R_xml_0_14 :
    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__U61 (x12::x11::nil))
    (* <a__U61(tt,L_),mark(L_)> *)
   | DP_R_xml_0_15 :
    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__U61 (x12::x11::nil))
    (* <a__and(tt,X_),mark(X_)> *)
   | DP_R_xml_0_16 :
    forall x12 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_tt nil) 
       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_mark (x5::nil)) 
        (algebra.Alg.Term algebra.F.id_a__and (x12::x11::nil))
   
    (* <a__isNat(length(V1_)),a__U11(a__isNatIListKind(V1_),V1_)> *)
   | DP_R_xml_0_17 :
    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_18 :
    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_19 :
    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_20 :
    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_21 :
    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_22 :
    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__and(a__isNatKind(V1_),isNatIListKind(V2_)),V1_,V2_)> *)
   | DP_R_xml_0_23 :
    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__and ((algebra.Alg.Term 
                  algebra.F.id_a__isNatKind (x1::nil))::(algebra.Alg.Term 
                  algebra.F.id_isNatIListKind (x3::nil))::nil))::x1::
                  x3::nil)) 
       (algebra.Alg.Term algebra.F.id_a__isNatIList (x11::nil))
   
    (* <a__isNatIList(cons(V1_,V2_)),a__and(a__isNatKind(V1_),isNatIListKind(V2_))> *)
   | DP_R_xml_0_24 :
    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__and ((algebra.Alg.Term 
                  algebra.F.id_a__isNatKind (x1::nil))::(algebra.Alg.Term 
                  algebra.F.id_isNatIListKind (x3::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_a__isNatIList (x11::nil))
    (* <a__isNatIList(cons(V1_,V2_)),a__isNatKind(V1_)> *)
   | DP_R_xml_0_25 :
    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__and(a__isNatKind(V1_),isNatIListKind(V2_))> *)
   | DP_R_xml_0_26 :
    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__and ((algebra.Alg.Term 
                  algebra.F.id_a__isNatKind (x1::nil))::(algebra.Alg.Term 
                  algebra.F.id_isNatIListKind (x3::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_a__isNatIListKind (x11::nil))
   
    (* <a__isNatIListKind(cons(V1_,V2_)),a__isNatKind(V1_)> *)
   | DP_R_xml_0_27 :
    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__isNatIListKind(V1_)> *)
   | DP_R_xml_0_28 :
    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__isNatKind(V1_)> *)
   | DP_R_xml_0_29 :
    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__U51(a__and(a__isNatKind(V1_),isNatIListKind(V2_)),V1_,V2_)> *)
   | DP_R_xml_0_30 :
    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__and ((algebra.Alg.Term 
                  algebra.F.id_a__isNatKind (x1::nil))::(algebra.Alg.Term 
                  algebra.F.id_isNatIListKind (x3::nil))::nil))::x1::
                  x3::nil)) 
       (algebra.Alg.Term algebra.F.id_a__isNatList (x11::nil))
   
    (* <a__isNatList(cons(V1_,V2_)),a__and(a__isNatKind(V1_),isNatIListKind(V2_))> *)
   | DP_R_xml_0_31 :
    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__and ((algebra.Alg.Term 
                  algebra.F.id_a__isNatKind (x1::nil))::(algebra.Alg.Term 
                  algebra.F.id_isNatIListKind (x3::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_a__isNatList (x11::nil))
    (* <a__isNatList(cons(V1_,V2_)),a__isNatKind(V1_)> *)
   | DP_R_xml_0_32 :
    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__U61(a__and(a__and(a__isNatList(L_),isNatIListKind(L_)),
        and(isNat(N_),isNatKind(N_))),L_)> *)
   | DP_R_xml_0_33 :
    forall x4 x6 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 (x6::x4::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U61 ((algebra.Alg.Term 
                  algebra.F.id_a__and ((algebra.Alg.Term algebra.F.id_a__and 
                  ((algebra.Alg.Term algebra.F.id_a__isNatList (x4::nil))::
                  (algebra.Alg.Term algebra.F.id_isNatIListKind 
                  (x4::nil))::nil))::(algebra.Alg.Term algebra.F.id_and 
                  ((algebra.Alg.Term algebra.F.id_isNat (x6::nil))::
                  (algebra.Alg.Term algebra.F.id_isNatKind 
                  (x6::nil))::nil))::nil))::x4::nil)) 
       (algebra.Alg.Term algebra.F.id_a__length (x11::nil))
   
    (* <a__length(cons(N_,L_)),a__and(a__and(a__isNatList(L_),isNatIListKind(L_)),
 and(isNat(N_),isNatKind(N_)))> *)
   | DP_R_xml_0_34 :
    forall x4 x6 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 (x6::x4::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__and ((algebra.Alg.Term 
                  algebra.F.id_a__and ((algebra.Alg.Term 
                  algebra.F.id_a__isNatList (x4::nil))::(algebra.Alg.Term 
                  algebra.F.id_isNatIListKind (x4::nil))::nil))::
                  (algebra.Alg.Term algebra.F.id_and ((algebra.Alg.Term 
                  algebra.F.id_isNat (x6::nil))::(algebra.Alg.Term 
                  algebra.F.id_isNatKind (x6::nil))::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_a__length (x11::nil))
   
    (* <a__length(cons(N_,L_)),a__and(a__isNatList(L_),isNatIListKind(L_))> *)
   | DP_R_xml_0_35 :
    forall x4 x6 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 (x6::x4::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__and ((algebra.Alg.Term 
                  algebra.F.id_a__isNatList (x4::nil))::(algebra.Alg.Term 
                  algebra.F.id_isNatIListKind (x4::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_a__length (x11::nil))
    (* <a__length(cons(N_,L_)),a__isNatList(L_)> *)
   | DP_R_xml_0_36 :
    forall x4 x6 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 (x6::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_37 :
    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_38 :
    forall x8 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 (x7::x8::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U11 ((algebra.Alg.Term 
                  algebra.F.id_mark (x7::nil))::x8::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U11(X1_,X2_)),mark(X1_)> *)
   | DP_R_xml_0_39 :
    forall x8 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 (x7::x8::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x7::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U12(X_)),a__U12(mark(X_))> *)
   | DP_R_xml_0_40 :
    forall 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_U12 (x5::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U12 ((algebra.Alg.Term 
                  algebra.F.id_mark (x5::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U12(X_)),mark(X_)> *)
   | DP_R_xml_0_41 :
    forall 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_U12 (x5::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(isNatList(X_)),a__isNatList(X_)> *)
   | DP_R_xml_0_42 :
    forall 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_isNatList (x5::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isNatList (x5::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U21(X1_,X2_)),a__U21(mark(X1_),X2_)> *)
   | DP_R_xml_0_43 :
    forall x8 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 (x7::x8::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U21 ((algebra.Alg.Term 
                  algebra.F.id_mark (x7::nil))::x8::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U21(X1_,X2_)),mark(X1_)> *)
   | DP_R_xml_0_44 :
    forall x8 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 (x7::x8::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x7::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U22(X_)),a__U22(mark(X_))> *)
   | DP_R_xml_0_45 :
    forall 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_U22 (x5::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U22 ((algebra.Alg.Term 
                  algebra.F.id_mark (x5::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U22(X_)),mark(X_)> *)
   | DP_R_xml_0_46 :
    forall 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_U22 (x5::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(isNat(X_)),a__isNat(X_)> *)
   | DP_R_xml_0_47 :
    forall 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_isNat (x5::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isNat (x5::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U31(X1_,X2_)),a__U31(mark(X1_),X2_)> *)
   | DP_R_xml_0_48 :
    forall x8 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 (x7::x8::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U31 ((algebra.Alg.Term 
                  algebra.F.id_mark (x7::nil))::x8::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U31(X1_,X2_)),mark(X1_)> *)
   | DP_R_xml_0_49 :
    forall x8 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 (x7::x8::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x7::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U32(X_)),a__U32(mark(X_))> *)
   | DP_R_xml_0_50 :
    forall 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_U32 (x5::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U32 ((algebra.Alg.Term 
                  algebra.F.id_mark (x5::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U32(X_)),mark(X_)> *)
   | DP_R_xml_0_51 :
    forall 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_U32 (x5::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x5::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_52 :
    forall x8 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 (x7::x8::x9::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U41 ((algebra.Alg.Term 
                  algebra.F.id_mark (x7::nil))::x8::x9::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U41(X1_,X2_,X3_)),mark(X1_)> *)
   | DP_R_xml_0_53 :
    forall x8 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 (x7::x8::x9::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x7::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U42(X1_,X2_)),a__U42(mark(X1_),X2_)> *)
   | DP_R_xml_0_54 :
    forall x8 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 (x7::x8::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U42 ((algebra.Alg.Term 
                  algebra.F.id_mark (x7::nil))::x8::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U42(X1_,X2_)),mark(X1_)> *)
   | DP_R_xml_0_55 :
    forall x8 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 (x7::x8::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x7::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U43(X_)),a__U43(mark(X_))> *)
   | DP_R_xml_0_56 :
    forall 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_U43 (x5::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U43 ((algebra.Alg.Term 
                  algebra.F.id_mark (x5::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U43(X_)),mark(X_)> *)
   | DP_R_xml_0_57 :
    forall 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_U43 (x5::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(isNatIList(X_)),a__isNatIList(X_)> *)
   | DP_R_xml_0_58 :
    forall 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_isNatIList (x5::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isNatIList (x5::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U51(X1_,X2_,X3_)),a__U51(mark(X1_),X2_,X3_)> *)
   | DP_R_xml_0_59 :
    forall x8 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_U51 (x7::x8::x9::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U51 ((algebra.Alg.Term 
                  algebra.F.id_mark (x7::nil))::x8::x9::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U51(X1_,X2_,X3_)),mark(X1_)> *)
   | DP_R_xml_0_60 :
    forall x8 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_U51 (x7::x8::x9::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x7::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U52(X1_,X2_)),a__U52(mark(X1_),X2_)> *)
   | DP_R_xml_0_61 :
    forall x8 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_U52 (x7::x8::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U52 ((algebra.Alg.Term 
                  algebra.F.id_mark (x7::nil))::x8::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U52(X1_,X2_)),mark(X1_)> *)
   | DP_R_xml_0_62 :
    forall x8 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_U52 (x7::x8::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x7::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U53(X_)),a__U53(mark(X_))> *)
   | DP_R_xml_0_63 :
    forall 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_U53 (x5::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U53 ((algebra.Alg.Term 
                  algebra.F.id_mark (x5::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U53(X_)),mark(X_)> *)
   | DP_R_xml_0_64 :
    forall 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_U53 (x5::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U61(X1_,X2_)),a__U61(mark(X1_),X2_)> *)
   | DP_R_xml_0_65 :
    forall x8 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_U61 (x7::x8::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U61 ((algebra.Alg.Term 
                  algebra.F.id_mark (x7::nil))::x8::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(U61(X1_,X2_)),mark(X1_)> *)
   | DP_R_xml_0_66 :
    forall x8 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_U61 (x7::x8::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x7::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(length(X_)),a__length(mark(X_))> *)
   | DP_R_xml_0_67 :
    forall 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_length (x5::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__length ((algebra.Alg.Term 
                  algebra.F.id_mark (x5::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(length(X_)),mark(X_)> *)
   | DP_R_xml_0_68 :
    forall 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_length (x5::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(and(X1_,X2_)),a__and(mark(X1_),X2_)> *)
   | DP_R_xml_0_69 :
    forall x8 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_and (x7::x8::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__and ((algebra.Alg.Term 
                  algebra.F.id_mark (x7::nil))::x8::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(and(X1_,X2_)),mark(X1_)> *)
   | DP_R_xml_0_70 :
    forall x8 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_and (x7::x8::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x7::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(isNatIListKind(X_)),a__isNatIListKind(X_)> *)
   | DP_R_xml_0_71 :
    forall 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_isNatIListKind (x5::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isNatIListKind (x5::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(isNatKind(X_)),a__isNatKind(X_)> *)
   | DP_R_xml_0_72 :
    forall 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_isNatKind (x5::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isNatKind (x5::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(cons(X1_,X2_)),mark(X1_)> *)
   | DP_R_xml_0_73 :
    forall x8 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 (x7::x8::nil)) x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x7::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    (* <mark(s(X_)),mark(X_)> *)
   | DP_R_xml_0_74 :
    forall 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_s (x5::nil)) 
       x11) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x5::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(U53(X_)),a__U53(mark(X_))> *)
    | DP_R_xml_0_non_scc_1_0 :
     forall 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_U53 (x5::nil)) x11) ->
       DP_R_xml_0_non_scc_1 (algebra.Alg.Term algebra.F.id_a__U53 
                             ((algebra.Alg.Term algebra.F.id_mark 
                             (x5::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(U43(X_)),a__U43(mark(X_))> *)
    | DP_R_xml_0_non_scc_2_0 :
     forall 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_U43 (x5::nil)) x11) ->
       DP_R_xml_0_non_scc_2 (algebra.Alg.Term algebra.F.id_a__U43 
                             ((algebra.Alg.Term algebra.F.id_mark 
                             (x5::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(U32(X_)),a__U32(mark(X_))> *)
    | DP_R_xml_0_non_scc_3_0 :
     forall 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_U32 (x5::nil)) x11) ->
       DP_R_xml_0_non_scc_3 (algebra.Alg.Term algebra.F.id_a__U32 
                             ((algebra.Alg.Term algebra.F.id_mark 
                             (x5::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(U22(X_)),a__U22(mark(X_))> *)
    | DP_R_xml_0_non_scc_4_0 :
     forall 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_U22 (x5::nil)) x11) ->
       DP_R_xml_0_non_scc_4 (algebra.Alg.Term algebra.F.id_a__U22 
                             ((algebra.Alg.Term algebra.F.id_mark 
                             (x5::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(U12(X_)),a__U12(mark(X_))> *)
    | DP_R_xml_0_non_scc_5_0 :
     forall 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_U12 (x5::nil)) x11) ->
       DP_R_xml_0_non_scc_5 (algebra.Alg.Term algebra.F.id_a__U12 
                             ((algebra.Alg.Term algebra.F.id_mark 
                             (x5::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(zeros),a__zeros> *)
    | DP_R_xml_0_non_scc_6_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_6 (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_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 := 
     (* <a__U52(tt,V2_),a__U53(a__isNatList(V2_))> *)
    | DP_R_xml_0_non_scc_7_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_7 (algebra.Alg.Term algebra.F.id_a__U53 
                              ((algebra.Alg.Term algebra.F.id_a__isNatList 
                              (x3::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U52 (x12::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 := 
     (* <a__U42(tt,V2_),a__U43(a__isNatIList(V2_))> *)
    | DP_R_xml_0_non_scc_8_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_8 (algebra.Alg.Term algebra.F.id_a__U43 
                              ((algebra.Alg.Term algebra.F.id_a__isNatIList 
                              (x3::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U42 (x12::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 := 
     (* <a__U31(tt,V_),a__U32(a__isNatList(V_))> *)
    | DP_R_xml_0_non_scc_9_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_9 (algebra.Alg.Term algebra.F.id_a__U32 
                              ((algebra.Alg.Term algebra.F.id_a__isNatList 
                              (x2::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U31 (x12::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__U21(tt,V1_),a__U22(a__isNat(V1_))> *)
    | DP_R_xml_0_non_scc_10_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_10 (algebra.Alg.Term algebra.F.id_a__U22 
                               ((algebra.Alg.Term algebra.F.id_a__isNat 
                               (x1::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U21 (x12::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__U11(tt,V1_),a__U12(a__isNatList(V1_))> *)
    | DP_R_xml_0_non_scc_11_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_11 (algebra.Alg.Term algebra.F.id_a__U12 
                               ((algebra.Alg.Term algebra.F.id_a__isNatList 
                               (x1::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U11 (x12::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_scc_12  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <a__U11(tt,V1_),a__isNatList(V1_)> *)
    | DP_R_xml_0_scc_12_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_12 (algebra.Alg.Term algebra.F.id_a__isNatList 
                           (x1::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U11 (x12::x11::nil))
    
     (* <a__isNatList(cons(V1_,V2_)),a__U51(a__and(a__isNatKind(V1_),isNatIListKind(V2_)),V1_,V2_)> *)
    | DP_R_xml_0_scc_12_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_12 (algebra.Alg.Term algebra.F.id_a__U51 
                          ((algebra.Alg.Term algebra.F.id_a__and 
                          ((algebra.Alg.Term algebra.F.id_a__isNatKind 
                          (x1::nil))::(algebra.Alg.Term 
                          algebra.F.id_isNatIListKind (x3::nil))::nil))::x1::
                          x3::nil)) 
        (algebra.Alg.Term algebra.F.id_a__isNatList (x11::nil))
    
     (* <a__U51(tt,V1_,V2_),a__U52(a__isNat(V1_),V2_)> *)
    | DP_R_xml_0_scc_12_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_12 (algebra.Alg.Term algebra.F.id_a__U52 
                            ((algebra.Alg.Term algebra.F.id_a__isNat 
                            (x1::nil))::x3::nil)) 
          (algebra.Alg.Term algebra.F.id_a__U51 (x13::x12::x11::nil))
     (* <a__U52(tt,V2_),a__isNatList(V2_)> *)
    | DP_R_xml_0_scc_12_3 :
     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_12 (algebra.Alg.Term algebra.F.id_a__isNatList 
                           (x3::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U52 (x12::x11::nil))
    
     (* <a__isNatList(cons(V1_,V2_)),a__and(a__isNatKind(V1_),isNatIListKind(V2_))> *)
    | DP_R_xml_0_scc_12_4 :
     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_12 (algebra.Alg.Term algebra.F.id_a__and 
                          ((algebra.Alg.Term algebra.F.id_a__isNatKind 
                          (x1::nil))::(algebra.Alg.Term 
                          algebra.F.id_isNatIListKind (x3::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_a__isNatList (x11::nil))
     (* <a__and(tt,X_),mark(X_)> *)
    | DP_R_xml_0_scc_12_5 :
     forall x12 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_tt nil) 
        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_12 (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
         (algebra.Alg.Term algebra.F.id_a__and (x12::x11::nil))
     (* <mark(U11(X1_,X2_)),a__U11(mark(X1_),X2_)> *)
    | DP_R_xml_0_scc_12_6 :
     forall x8 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 (x7::x8::nil)) x11) ->
       DP_R_xml_0_scc_12 (algebra.Alg.Term algebra.F.id_a__U11 
                          ((algebra.Alg.Term algebra.F.id_mark (x7::nil))::
                          x8::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <mark(U11(X1_,X2_)),mark(X1_)> *)
    | DP_R_xml_0_scc_12_7 :
     forall x8 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 (x7::x8::nil)) x11) ->
       DP_R_xml_0_scc_12 (algebra.Alg.Term algebra.F.id_mark (x7::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <mark(U12(X_)),mark(X_)> *)
    | DP_R_xml_0_scc_12_8 :
     forall 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_U12 (x5::nil)) x11) ->
       DP_R_xml_0_scc_12 (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <mark(isNatList(X_)),a__isNatList(X_)> *)
    | DP_R_xml_0_scc_12_9 :
     forall 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_isNatList (x5::nil)) x11) ->
       DP_R_xml_0_scc_12 (algebra.Alg.Term algebra.F.id_a__isNatList 
                          (x5::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    
     (* <a__isNatList(cons(V1_,V2_)),a__isNatKind(V1_)> *)
    | DP_R_xml_0_scc_12_10 :
     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_12 (algebra.Alg.Term algebra.F.id_a__isNatKind 
                          (x1::nil)) 
        (algebra.Alg.Term algebra.F.id_a__isNatList (x11::nil))
    
     (* <a__isNatKind(length(V1_)),a__isNatIListKind(V1_)> *)
    | DP_R_xml_0_scc_12_11 :
     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_12 (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
                          (x1::nil)) 
        (algebra.Alg.Term algebra.F.id_a__isNatKind (x11::nil))
    
     (* <a__isNatIListKind(cons(V1_,V2_)),a__and(a__isNatKind(V1_),isNatIListKind(V2_))> *)
    | DP_R_xml_0_scc_12_12 :
     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_12 (algebra.Alg.Term algebra.F.id_a__and 
                          ((algebra.Alg.Term algebra.F.id_a__isNatKind 
                          (x1::nil))::(algebra.Alg.Term 
                          algebra.F.id_isNatIListKind (x3::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_a__isNatIListKind (x11::nil))
    
     (* <a__isNatIListKind(cons(V1_,V2_)),a__isNatKind(V1_)> *)
    | DP_R_xml_0_scc_12_13 :
     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_12 (algebra.Alg.Term algebra.F.id_a__isNatKind 
                          (x1::nil)) 
        (algebra.Alg.Term algebra.F.id_a__isNatIListKind (x11::nil))
     (* <a__isNatKind(s(V1_)),a__isNatKind(V1_)> *)
    | DP_R_xml_0_scc_12_14 :
     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_12 (algebra.Alg.Term algebra.F.id_a__isNatKind 
                          (x1::nil)) 
        (algebra.Alg.Term algebra.F.id_a__isNatKind (x11::nil))
     (* <mark(U21(X1_,X2_)),a__U21(mark(X1_),X2_)> *)
    | DP_R_xml_0_scc_12_15 :
     forall x8 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 (x7::x8::nil)) x11) ->
       DP_R_xml_0_scc_12 (algebra.Alg.Term algebra.F.id_a__U21 
                          ((algebra.Alg.Term algebra.F.id_mark (x7::nil))::
                          x8::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <a__U21(tt,V1_),a__isNat(V1_)> *)
    | DP_R_xml_0_scc_12_16 :
     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_12 (algebra.Alg.Term algebra.F.id_a__isNat (x1::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U21 (x12::x11::nil))
    
     (* <a__isNat(length(V1_)),a__U11(a__isNatIListKind(V1_),V1_)> *)
    | DP_R_xml_0_scc_12_17 :
     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_12 (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_scc_12_18 :
     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_12 (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_scc_12_19 :
     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_12 (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_scc_12_20 :
     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_12 (algebra.Alg.Term algebra.F.id_a__isNatKind 
                          (x1::nil)) 
        (algebra.Alg.Term algebra.F.id_a__isNat (x11::nil))
     (* <mark(U21(X1_,X2_)),mark(X1_)> *)
    | DP_R_xml_0_scc_12_21 :
     forall x8 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 (x7::x8::nil)) x11) ->
       DP_R_xml_0_scc_12 (algebra.Alg.Term algebra.F.id_mark (x7::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <mark(U22(X_)),mark(X_)> *)
    | DP_R_xml_0_scc_12_22 :
     forall 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_U22 (x5::nil)) x11) ->
       DP_R_xml_0_scc_12 (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <mark(isNat(X_)),a__isNat(X_)> *)
    | DP_R_xml_0_scc_12_23 :
     forall 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_isNat (x5::nil)) x11) ->
       DP_R_xml_0_scc_12 (algebra.Alg.Term algebra.F.id_a__isNat (x5::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <mark(U31(X1_,X2_)),a__U31(mark(X1_),X2_)> *)
    | DP_R_xml_0_scc_12_24 :
     forall x8 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 (x7::x8::nil)) x11) ->
       DP_R_xml_0_scc_12 (algebra.Alg.Term algebra.F.id_a__U31 
                          ((algebra.Alg.Term algebra.F.id_mark (x7::nil))::
                          x8::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <a__U31(tt,V_),a__isNatList(V_)> *)
    | DP_R_xml_0_scc_12_25 :
     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_scc_12 (algebra.Alg.Term algebra.F.id_a__isNatList 
                           (x2::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U31 (x12::x11::nil))
     (* <mark(U31(X1_,X2_)),mark(X1_)> *)
    | DP_R_xml_0_scc_12_26 :
     forall x8 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 (x7::x8::nil)) x11) ->
       DP_R_xml_0_scc_12 (algebra.Alg.Term algebra.F.id_mark (x7::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <mark(U32(X_)),mark(X_)> *)
    | DP_R_xml_0_scc_12_27 :
     forall 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_U32 (x5::nil)) x11) ->
       DP_R_xml_0_scc_12 (algebra.Alg.Term algebra.F.id_mark (x5::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_scc_12_28 :
     forall x8 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 (x7::x8::x9::nil)) x11) ->
       DP_R_xml_0_scc_12 (algebra.Alg.Term algebra.F.id_a__U41 
                          ((algebra.Alg.Term algebra.F.id_mark (x7::nil))::
                          x8::x9::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    
     (* <a__U41(tt,V1_,V2_),a__U42(a__isNat(V1_),V2_)> *)
    | DP_R_xml_0_scc_12_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_scc_12 (algebra.Alg.Term algebra.F.id_a__U42 
                            ((algebra.Alg.Term algebra.F.id_a__isNat 
                            (x1::nil))::x3::nil)) 
          (algebra.Alg.Term algebra.F.id_a__U41 (x13::x12::x11::nil))
     (* <a__U42(tt,V2_),a__isNatIList(V2_)> *)
    | DP_R_xml_0_scc_12_30 :
     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_12 (algebra.Alg.Term algebra.F.id_a__isNatIList 
                           (x3::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U42 (x12::x11::nil))
    
     (* <a__isNatIList(V_),a__U31(a__isNatIListKind(V_),V_)> *)
    | DP_R_xml_0_scc_12_31 :
     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_scc_12 (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_scc_12_32 :
     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_scc_12 (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__and(a__isNatKind(V1_),isNatIListKind(V2_)),V1_,V2_)> *)
    | DP_R_xml_0_scc_12_33 :
     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_12 (algebra.Alg.Term algebra.F.id_a__U41 
                          ((algebra.Alg.Term algebra.F.id_a__and 
                          ((algebra.Alg.Term algebra.F.id_a__isNatKind 
                          (x1::nil))::(algebra.Alg.Term 
                          algebra.F.id_isNatIListKind (x3::nil))::nil))::x1::
                          x3::nil)) 
        (algebra.Alg.Term algebra.F.id_a__isNatIList (x11::nil))
     (* <a__U41(tt,V1_,V2_),a__isNat(V1_)> *)
    | DP_R_xml_0_scc_12_34 :
     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_12 (algebra.Alg.Term algebra.F.id_a__isNat (x1::nil)) 
          (algebra.Alg.Term algebra.F.id_a__U41 (x13::x12::x11::nil))
    
     (* <a__isNatIList(cons(V1_,V2_)),a__and(a__isNatKind(V1_),isNatIListKind(V2_))> *)
    | DP_R_xml_0_scc_12_35 :
     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_12 (algebra.Alg.Term algebra.F.id_a__and 
                          ((algebra.Alg.Term algebra.F.id_a__isNatKind 
                          (x1::nil))::(algebra.Alg.Term 
                          algebra.F.id_isNatIListKind (x3::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_a__isNatIList (x11::nil))
    
     (* <a__isNatIList(cons(V1_,V2_)),a__isNatKind(V1_)> *)
    | DP_R_xml_0_scc_12_36 :
     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_12 (algebra.Alg.Term algebra.F.id_a__isNatKind 
                          (x1::nil)) 
        (algebra.Alg.Term algebra.F.id_a__isNatIList (x11::nil))
     (* <mark(U41(X1_,X2_,X3_)),mark(X1_)> *)
    | DP_R_xml_0_scc_12_37 :
     forall x8 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 (x7::x8::x9::nil)) x11) ->
       DP_R_xml_0_scc_12 (algebra.Alg.Term algebra.F.id_mark (x7::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <mark(U42(X1_,X2_)),a__U42(mark(X1_),X2_)> *)
    | DP_R_xml_0_scc_12_38 :
     forall x8 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 (x7::x8::nil)) x11) ->
       DP_R_xml_0_scc_12 (algebra.Alg.Term algebra.F.id_a__U42 
                          ((algebra.Alg.Term algebra.F.id_mark (x7::nil))::
                          x8::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <mark(U42(X1_,X2_)),mark(X1_)> *)
    | DP_R_xml_0_scc_12_39 :
     forall x8 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 (x7::x8::nil)) x11) ->
       DP_R_xml_0_scc_12 (algebra.Alg.Term algebra.F.id_mark (x7::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <mark(U43(X_)),mark(X_)> *)
    | DP_R_xml_0_scc_12_40 :
     forall 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_U43 (x5::nil)) x11) ->
       DP_R_xml_0_scc_12 (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <mark(isNatIList(X_)),a__isNatIList(X_)> *)
    | DP_R_xml_0_scc_12_41 :
     forall 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_isNatIList (x5::nil)) x11) ->
       DP_R_xml_0_scc_12 (algebra.Alg.Term algebra.F.id_a__isNatIList 
                          (x5::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    
     (* <mark(U51(X1_,X2_,X3_)),a__U51(mark(X1_),X2_,X3_)> *)
    | DP_R_xml_0_scc_12_42 :
     forall x8 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_U51 (x7::x8::x9::nil)) x11) ->
       DP_R_xml_0_scc_12 (algebra.Alg.Term algebra.F.id_a__U51 
                          ((algebra.Alg.Term algebra.F.id_mark (x7::nil))::
                          x8::x9::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <a__U51(tt,V1_,V2_),a__isNat(V1_)> *)
    | DP_R_xml_0_scc_12_43 :
     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_12 (algebra.Alg.Term algebra.F.id_a__isNat (x1::nil)) 
          (algebra.Alg.Term algebra.F.id_a__U51 (x13::x12::x11::nil))
     (* <mark(U51(X1_,X2_,X3_)),mark(X1_)> *)
    | DP_R_xml_0_scc_12_44 :
     forall x8 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_U51 (x7::x8::x9::nil)) x11) ->
       DP_R_xml_0_scc_12 (algebra.Alg.Term algebra.F.id_mark (x7::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <mark(U52(X1_,X2_)),a__U52(mark(X1_),X2_)> *)
    | DP_R_xml_0_scc_12_45 :
     forall x8 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_U52 (x7::x8::nil)) x11) ->
       DP_R_xml_0_scc_12 (algebra.Alg.Term algebra.F.id_a__U52 
                          ((algebra.Alg.Term algebra.F.id_mark (x7::nil))::
                          x8::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <mark(U52(X1_,X2_)),mark(X1_)> *)
    | DP_R_xml_0_scc_12_46 :
     forall x8 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_U52 (x7::x8::nil)) x11) ->
       DP_R_xml_0_scc_12 (algebra.Alg.Term algebra.F.id_mark (x7::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <mark(U53(X_)),mark(X_)> *)
    | DP_R_xml_0_scc_12_47 :
     forall 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_U53 (x5::nil)) x11) ->
       DP_R_xml_0_scc_12 (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <mark(U61(X1_,X2_)),a__U61(mark(X1_),X2_)> *)
    | DP_R_xml_0_scc_12_48 :
     forall x8 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_U61 (x7::x8::nil)) x11) ->
       DP_R_xml_0_scc_12 (algebra.Alg.Term algebra.F.id_a__U61 
                          ((algebra.Alg.Term algebra.F.id_mark (x7::nil))::
                          x8::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <a__U61(tt,L_),a__length(mark(L_))> *)
    | DP_R_xml_0_scc_12_49 :
     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_12 (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__U61 (x12::x11::nil))
    
     (* <a__length(cons(N_,L_)),a__U61(a__and(a__and(a__isNatList(L_),isNatIListKind(L_)),
        and(isNat(N_),isNatKind(N_))),L_)> *)
    | DP_R_xml_0_scc_12_50 :
     forall x4 x6 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 (x6::x4::nil)) x11) ->
       DP_R_xml_0_scc_12 (algebra.Alg.Term algebra.F.id_a__U61 
                          ((algebra.Alg.Term algebra.F.id_a__and 
                          ((algebra.Alg.Term algebra.F.id_a__and 
                          ((algebra.Alg.Term algebra.F.id_a__isNatList 
                          (x4::nil))::(algebra.Alg.Term 
                          algebra.F.id_isNatIListKind (x4::nil))::nil))::
                          (algebra.Alg.Term algebra.F.id_and 
                          ((algebra.Alg.Term algebra.F.id_isNat (x6::nil))::
                          (algebra.Alg.Term algebra.F.id_isNatKind 
                          (x6::nil))::nil))::nil))::x4::nil)) 
        (algebra.Alg.Term algebra.F.id_a__length (x11::nil))
     (* <a__U61(tt,L_),mark(L_)> *)
    | DP_R_xml_0_scc_12_51 :
     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_12 (algebra.Alg.Term algebra.F.id_mark (x4::nil)) 
         (algebra.Alg.Term algebra.F.id_a__U61 (x12::x11::nil))
     (* <mark(U61(X1_,X2_)),mark(X1_)> *)
    | DP_R_xml_0_scc_12_52 :
     forall x8 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_U61 (x7::x8::nil)) x11) ->
       DP_R_xml_0_scc_12 (algebra.Alg.Term algebra.F.id_mark (x7::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <mark(length(X_)),a__length(mark(X_))> *)
    | DP_R_xml_0_scc_12_53 :
     forall 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_length (x5::nil)) x11) ->
       DP_R_xml_0_scc_12 (algebra.Alg.Term algebra.F.id_a__length 
                          ((algebra.Alg.Term algebra.F.id_mark 
                          (x5::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    
     (* <a__length(cons(N_,L_)),a__and(a__and(a__isNatList(L_),isNatIListKind(L_)),
 and(isNat(N_),isNatKind(N_)))> *)
    | DP_R_xml_0_scc_12_54 :
     forall x4 x6 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 (x6::x4::nil)) x11) ->
       DP_R_xml_0_scc_12 (algebra.Alg.Term algebra.F.id_a__and 
                          ((algebra.Alg.Term algebra.F.id_a__and 
                          ((algebra.Alg.Term algebra.F.id_a__isNatList 
                          (x4::nil))::(algebra.Alg.Term 
                          algebra.F.id_isNatIListKind (x4::nil))::nil))::
                          (algebra.Alg.Term algebra.F.id_and 
                          ((algebra.Alg.Term algebra.F.id_isNat (x6::nil))::
                          (algebra.Alg.Term algebra.F.id_isNatKind 
                          (x6::nil))::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_a__length (x11::nil))
    
     (* <a__length(cons(N_,L_)),a__and(a__isNatList(L_),isNatIListKind(L_))> *)
    | DP_R_xml_0_scc_12_55 :
     forall x4 x6 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 (x6::x4::nil)) x11) ->
       DP_R_xml_0_scc_12 (algebra.Alg.Term algebra.F.id_a__and 
                          ((algebra.Alg.Term algebra.F.id_a__isNatList 
                          (x4::nil))::(algebra.Alg.Term 
                          algebra.F.id_isNatIListKind (x4::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_a__length (x11::nil))
     (* <a__length(cons(N_,L_)),a__isNatList(L_)> *)
    | DP_R_xml_0_scc_12_56 :
     forall x4 x6 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 (x6::x4::nil)) x11) ->
       DP_R_xml_0_scc_12 (algebra.Alg.Term algebra.F.id_a__isNatList 
                          (x4::nil)) 
        (algebra.Alg.Term algebra.F.id_a__length (x11::nil))
     (* <mark(length(X_)),mark(X_)> *)
    | DP_R_xml_0_scc_12_57 :
     forall 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_length (x5::nil)) x11) ->
       DP_R_xml_0_scc_12 (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <mark(and(X1_,X2_)),a__and(mark(X1_),X2_)> *)
    | DP_R_xml_0_scc_12_58 :
     forall x8 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_and (x7::x8::nil)) x11) ->
       DP_R_xml_0_scc_12 (algebra.Alg.Term algebra.F.id_a__and 
                          ((algebra.Alg.Term algebra.F.id_mark (x7::nil))::
                          x8::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <mark(and(X1_,X2_)),mark(X1_)> *)
    | DP_R_xml_0_scc_12_59 :
     forall x8 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_and (x7::x8::nil)) x11) ->
       DP_R_xml_0_scc_12 (algebra.Alg.Term algebra.F.id_mark (x7::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    
     (* <mark(isNatIListKind(X_)),a__isNatIListKind(X_)> *)
    | DP_R_xml_0_scc_12_60 :
     forall 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_isNatIListKind (x5::nil)) x11) ->
       DP_R_xml_0_scc_12 (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
                          (x5::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <mark(isNatKind(X_)),a__isNatKind(X_)> *)
    | DP_R_xml_0_scc_12_61 :
     forall 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_isNatKind (x5::nil)) x11) ->
       DP_R_xml_0_scc_12 (algebra.Alg.Term algebra.F.id_a__isNatKind 
                          (x5::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <mark(cons(X1_,X2_)),mark(X1_)> *)
    | DP_R_xml_0_scc_12_62 :
     forall x8 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 (x7::x8::nil)) x11) ->
       DP_R_xml_0_scc_12 (algebra.Alg.Term algebra.F.id_mark (x7::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     (* <mark(s(X_)),mark(X_)> *)
    | DP_R_xml_0_scc_12_63 :
     forall 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_s (x5::nil)) 
        x11) ->
       DP_R_xml_0_scc_12 (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
  .
  
  
  Module WF_DP_R_xml_0_scc_12.
   Inductive DP_R_xml_0_scc_12_large  :
    algebra.Alg.term ->algebra.Alg.term ->Prop := 
      (* <a__U11(tt,V1_),a__isNatList(V1_)> *)
     | DP_R_xml_0_scc_12_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_12_large (algebra.Alg.Term algebra.F.id_a__isNatList 
                                  (x1::nil)) 
          (algebra.Alg.Term algebra.F.id_a__U11 (x12::x11::nil))
     
      (* <a__isNatList(cons(V1_,V2_)),a__U51(a__and(a__isNatKind(V1_),isNatIListKind(V2_)),V1_,V2_)> *)
     | DP_R_xml_0_scc_12_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_12_large (algebra.Alg.Term algebra.F.id_a__U51 
                                 ((algebra.Alg.Term algebra.F.id_a__and 
                                 ((algebra.Alg.Term 
                                 algebra.F.id_a__isNatKind (x1::nil))::
                                 (algebra.Alg.Term 
                                 algebra.F.id_isNatIListKind 
                                 (x3::nil))::nil))::x1::x3::nil)) 
         (algebra.Alg.Term algebra.F.id_a__isNatList (x11::nil))
     
      (* <a__U51(tt,V1_,V2_),a__U52(a__isNat(V1_),V2_)> *)
     | DP_R_xml_0_scc_12_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_12_large (algebra.Alg.Term algebra.F.id_a__U52 
                                   ((algebra.Alg.Term algebra.F.id_a__isNat 
                                   (x1::nil))::x3::nil)) 
           (algebra.Alg.Term algebra.F.id_a__U51 (x13::x12::x11::nil))
      (* <a__U52(tt,V2_),a__isNatList(V2_)> *)
     | DP_R_xml_0_scc_12_large_3 :
      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_12_large (algebra.Alg.Term algebra.F.id_a__isNatList 
                                  (x3::nil)) 
          (algebra.Alg.Term algebra.F.id_a__U52 (x12::x11::nil))
     
      (* <a__isNatList(cons(V1_,V2_)),a__and(a__isNatKind(V1_),isNatIListKind(V2_))> *)
     | DP_R_xml_0_scc_12_large_4 :
      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_12_large (algebra.Alg.Term algebra.F.id_a__and 
                                 ((algebra.Alg.Term 
                                 algebra.F.id_a__isNatKind (x1::nil))::
                                 (algebra.Alg.Term 
                                 algebra.F.id_isNatIListKind 
                                 (x3::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_a__isNatList (x11::nil))
      (* <a__and(tt,X_),mark(X_)> *)
     | DP_R_xml_0_scc_12_large_5 :
      forall x12 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_tt nil) 
         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_12_large (algebra.Alg.Term algebra.F.id_mark 
                                  (x5::nil)) 
          (algebra.Alg.Term algebra.F.id_a__and (x12::x11::nil))
     
      (* <mark(U11(X1_,X2_)),a__U11(mark(X1_),X2_)> *)
     | DP_R_xml_0_scc_12_large_6 :
      forall x8 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 (x7::x8::nil)) x11) ->
        DP_R_xml_0_scc_12_large (algebra.Alg.Term algebra.F.id_a__U11 
                                 ((algebra.Alg.Term algebra.F.id_mark 
                                 (x7::nil))::x8::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      (* <mark(U11(X1_,X2_)),mark(X1_)> *)
     | DP_R_xml_0_scc_12_large_7 :
      forall x8 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 (x7::x8::nil)) x11) ->
        DP_R_xml_0_scc_12_large (algebra.Alg.Term algebra.F.id_mark 
                                 (x7::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      (* <mark(U12(X_)),mark(X_)> *)
     | DP_R_xml_0_scc_12_large_8 :
      forall 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_U12 (x5::nil)) x11) ->
        DP_R_xml_0_scc_12_large (algebra.Alg.Term algebra.F.id_mark 
                                 (x5::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     
      (* <mark(isNatList(X_)),a__isNatList(X_)> *)
     | DP_R_xml_0_scc_12_large_9 :
      forall 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_isNatList (x5::nil)) x11) ->
        DP_R_xml_0_scc_12_large (algebra.Alg.Term algebra.F.id_a__isNatList 
                                 (x5::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     
      (* <a__isNatList(cons(V1_,V2_)),a__isNatKind(V1_)> *)
     | DP_R_xml_0_scc_12_large_10 :
      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_12_large (algebra.Alg.Term algebra.F.id_a__isNatKind 
                                 (x1::nil)) 
         (algebra.Alg.Term algebra.F.id_a__isNatList (x11::nil))
     
      (* <a__isNatKind(length(V1_)),a__isNatIListKind(V1_)> *)
     | DP_R_xml_0_scc_12_large_11 :
      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_12_large (algebra.Alg.Term 
                                 algebra.F.id_a__isNatIListKind (x1::nil)) 
         (algebra.Alg.Term algebra.F.id_a__isNatKind (x11::nil))
     
      (* <a__isNatIListKind(cons(V1_,V2_)),a__and(a__isNatKind(V1_),isNatIListKind(V2_))> *)
     | DP_R_xml_0_scc_12_large_12 :
      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_12_large (algebra.Alg.Term algebra.F.id_a__and 
                                 ((algebra.Alg.Term 
                                 algebra.F.id_a__isNatKind (x1::nil))::
                                 (algebra.Alg.Term 
                                 algebra.F.id_isNatIListKind 
                                 (x3::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_a__isNatIListKind (x11::nil))
     
      (* <a__isNatIListKind(cons(V1_,V2_)),a__isNatKind(V1_)> *)
     | DP_R_xml_0_scc_12_large_13 :
      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_12_large (algebra.Alg.Term algebra.F.id_a__isNatKind 
                                 (x1::nil)) 
         (algebra.Alg.Term algebra.F.id_a__isNatIListKind (x11::nil))
     
      (* <a__isNatKind(s(V1_)),a__isNatKind(V1_)> *)
     | DP_R_xml_0_scc_12_large_14 :
      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_12_large (algebra.Alg.Term algebra.F.id_a__isNatKind 
                                 (x1::nil)) 
         (algebra.Alg.Term algebra.F.id_a__isNatKind (x11::nil))
     
      (* <mark(U21(X1_,X2_)),a__U21(mark(X1_),X2_)> *)
     | DP_R_xml_0_scc_12_large_15 :
      forall x8 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 (x7::x8::nil)) x11) ->
        DP_R_xml_0_scc_12_large (algebra.Alg.Term algebra.F.id_a__U21 
                                 ((algebra.Alg.Term algebra.F.id_mark 
                                 (x7::nil))::x8::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      (* <a__U21(tt,V1_),a__isNat(V1_)> *)
     | DP_R_xml_0_scc_12_large_16 :
      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_12_large (algebra.Alg.Term algebra.F.id_a__isNat 
                                  (x1::nil)) 
          (algebra.Alg.Term algebra.F.id_a__U21 (x12::x11::nil))
     
      (* <a__isNat(length(V1_)),a__U11(a__isNatIListKind(V1_),V1_)> *)
     | DP_R_xml_0_scc_12_large_17 :
      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_12_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__isNat(length(V1_)),a__isNatIListKind(V1_)> *)
     | DP_R_xml_0_scc_12_large_18 :
      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_12_large (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_scc_12_large_19 :
      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_12_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__isNat(s(V1_)),a__isNatKind(V1_)> *)
     | DP_R_xml_0_scc_12_large_20 :
      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_12_large (algebra.Alg.Term algebra.F.id_a__isNatKind 
                                 (x1::nil)) 
         (algebra.Alg.Term algebra.F.id_a__isNat (x11::nil))
      (* <mark(U21(X1_,X2_)),mark(X1_)> *)
     | DP_R_xml_0_scc_12_large_21 :
      forall x8 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 (x7::x8::nil)) x11) ->
        DP_R_xml_0_scc_12_large (algebra.Alg.Term algebra.F.id_mark 
                                 (x7::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      (* <mark(U22(X_)),mark(X_)> *)
     | DP_R_xml_0_scc_12_large_22 :
      forall 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_U22 (x5::nil)) x11) ->
        DP_R_xml_0_scc_12_large (algebra.Alg.Term algebra.F.id_mark 
                                 (x5::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      (* <mark(isNat(X_)),a__isNat(X_)> *)
     | DP_R_xml_0_scc_12_large_23 :
      forall 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_isNat (x5::nil)) x11) ->
        DP_R_xml_0_scc_12_large (algebra.Alg.Term algebra.F.id_a__isNat 
                                 (x5::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     
      (* <mark(U31(X1_,X2_)),a__U31(mark(X1_),X2_)> *)
     | DP_R_xml_0_scc_12_large_24 :
      forall x8 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 (x7::x8::nil)) x11) ->
        DP_R_xml_0_scc_12_large (algebra.Alg.Term algebra.F.id_a__U31 
                                 ((algebra.Alg.Term algebra.F.id_mark 
                                 (x7::nil))::x8::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      (* <a__U31(tt,V_),a__isNatList(V_)> *)
     | DP_R_xml_0_scc_12_large_25 :
      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_scc_12_large (algebra.Alg.Term algebra.F.id_a__isNatList 
                                  (x2::nil)) 
          (algebra.Alg.Term algebra.F.id_a__U31 (x12::x11::nil))
      (* <mark(U31(X1_,X2_)),mark(X1_)> *)
     | DP_R_xml_0_scc_12_large_26 :
      forall x8 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 (x7::x8::nil)) x11) ->
        DP_R_xml_0_scc_12_large (algebra.Alg.Term algebra.F.id_mark 
                                 (x7::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      (* <mark(U32(X_)),mark(X_)> *)
     | DP_R_xml_0_scc_12_large_27 :
      forall 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_U32 (x5::nil)) x11) ->
        DP_R_xml_0_scc_12_large (algebra.Alg.Term algebra.F.id_mark 
                                 (x5::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_scc_12_large_28 :
      forall x8 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 (x7::x8::x9::nil)) x11) ->
        DP_R_xml_0_scc_12_large (algebra.Alg.Term algebra.F.id_a__U41 
                                 ((algebra.Alg.Term algebra.F.id_mark 
                                 (x7::nil))::x8::x9::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     
      (* <a__U41(tt,V1_,V2_),a__U42(a__isNat(V1_),V2_)> *)
     | DP_R_xml_0_scc_12_large_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_scc_12_large (algebra.Alg.Term algebra.F.id_a__U42 
                                   ((algebra.Alg.Term algebra.F.id_a__isNat 
                                   (x1::nil))::x3::nil)) 
           (algebra.Alg.Term algebra.F.id_a__U41 (x13::x12::x11::nil))
      (* <a__U42(tt,V2_),a__isNatIList(V2_)> *)
     | DP_R_xml_0_scc_12_large_30 :
      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_12_large (algebra.Alg.Term 
                                  algebra.F.id_a__isNatIList (x3::nil)) 
          (algebra.Alg.Term algebra.F.id_a__U42 (x12::x11::nil))
     
      (* <a__isNatIList(V_),a__U31(a__isNatIListKind(V_),V_)> *)
     | DP_R_xml_0_scc_12_large_31 :
      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_scc_12_large (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_scc_12_large_32 :
      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_scc_12_large (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__and(a__isNatKind(V1_),isNatIListKind(V2_)),V1_,V2_)> *)
     | DP_R_xml_0_scc_12_large_33 :
      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_12_large (algebra.Alg.Term algebra.F.id_a__U41 
                                 ((algebra.Alg.Term algebra.F.id_a__and 
                                 ((algebra.Alg.Term 
                                 algebra.F.id_a__isNatKind (x1::nil))::
                                 (algebra.Alg.Term 
                                 algebra.F.id_isNatIListKind 
                                 (x3::nil))::nil))::x1::x3::nil)) 
         (algebra.Alg.Term algebra.F.id_a__isNatIList (x11::nil))
      (* <a__U41(tt,V1_,V2_),a__isNat(V1_)> *)
     | DP_R_xml_0_scc_12_large_34 :
      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_12_large (algebra.Alg.Term algebra.F.id_a__isNat 
                                   (x1::nil)) 
           (algebra.Alg.Term algebra.F.id_a__U41 (x13::x12::x11::nil))
     
      (* <a__isNatIList(cons(V1_,V2_)),a__and(a__isNatKind(V1_),isNatIListKind(V2_))> *)
     | DP_R_xml_0_scc_12_large_35 :
      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_12_large (algebra.Alg.Term algebra.F.id_a__and 
                                 ((algebra.Alg.Term 
                                 algebra.F.id_a__isNatKind (x1::nil))::
                                 (algebra.Alg.Term 
                                 algebra.F.id_isNatIListKind 
                                 (x3::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_a__isNatIList (x11::nil))
     
      (* <a__isNatIList(cons(V1_,V2_)),a__isNatKind(V1_)> *)
     | DP_R_xml_0_scc_12_large_36 :
      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_12_large (algebra.Alg.Term algebra.F.id_a__isNatKind 
                                 (x1::nil)) 
         (algebra.Alg.Term algebra.F.id_a__isNatIList (x11::nil))
      (* <mark(U41(X1_,X2_,X3_)),mark(X1_)> *)
     | DP_R_xml_0_scc_12_large_37 :
      forall x8 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 (x7::x8::x9::nil)) x11) ->
        DP_R_xml_0_scc_12_large (algebra.Alg.Term algebra.F.id_mark 
                                 (x7::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     
      (* <mark(U42(X1_,X2_)),a__U42(mark(X1_),X2_)> *)
     | DP_R_xml_0_scc_12_large_38 :
      forall x8 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 (x7::x8::nil)) x11) ->
        DP_R_xml_0_scc_12_large (algebra.Alg.Term algebra.F.id_a__U42 
                                 ((algebra.Alg.Term algebra.F.id_mark 
                                 (x7::nil))::x8::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      (* <mark(U42(X1_,X2_)),mark(X1_)> *)
     | DP_R_xml_0_scc_12_large_39 :
      forall x8 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 (x7::x8::nil)) x11) ->
        DP_R_xml_0_scc_12_large (algebra.Alg.Term algebra.F.id_mark 
                                 (x7::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      (* <mark(U43(X_)),mark(X_)> *)
     | DP_R_xml_0_scc_12_large_40 :
      forall 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_U43 (x5::nil)) x11) ->
        DP_R_xml_0_scc_12_large (algebra.Alg.Term algebra.F.id_mark 
                                 (x5::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     
      (* <mark(isNatIList(X_)),a__isNatIList(X_)> *)
     | DP_R_xml_0_scc_12_large_41 :
      forall 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_isNatIList (x5::nil)) x11) ->
        DP_R_xml_0_scc_12_large (algebra.Alg.Term algebra.F.id_a__isNatIList 
                                 (x5::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     
      (* <mark(U51(X1_,X2_,X3_)),a__U51(mark(X1_),X2_,X3_)> *)
     | DP_R_xml_0_scc_12_large_42 :
      forall x8 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_U51 (x7::x8::x9::nil)) x11) ->
        DP_R_xml_0_scc_12_large (algebra.Alg.Term algebra.F.id_a__U51 
                                 ((algebra.Alg.Term algebra.F.id_mark 
                                 (x7::nil))::x8::x9::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      (* <a__U51(tt,V1_,V2_),a__isNat(V1_)> *)
     | DP_R_xml_0_scc_12_large_43 :
      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_12_large (algebra.Alg.Term algebra.F.id_a__isNat 
                                   (x1::nil)) 
           (algebra.Alg.Term algebra.F.id_a__U51 (x13::x12::x11::nil))
      (* <mark(U51(X1_,X2_,X3_)),mark(X1_)> *)
     | DP_R_xml_0_scc_12_large_44 :
      forall x8 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_U51 (x7::x8::x9::nil)) x11) ->
        DP_R_xml_0_scc_12_large (algebra.Alg.Term algebra.F.id_mark 
                                 (x7::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     
      (* <mark(U52(X1_,X2_)),a__U52(mark(X1_),X2_)> *)
     | DP_R_xml_0_scc_12_large_45 :
      forall x8 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_U52 (x7::x8::nil)) x11) ->
        DP_R_xml_0_scc_12_large (algebra.Alg.Term algebra.F.id_a__U52 
                                 ((algebra.Alg.Term algebra.F.id_mark 
                                 (x7::nil))::x8::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      (* <mark(U52(X1_,X2_)),mark(X1_)> *)
     | DP_R_xml_0_scc_12_large_46 :
      forall x8 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_U52 (x7::x8::nil)) x11) ->
        DP_R_xml_0_scc_12_large (algebra.Alg.Term algebra.F.id_mark 
                                 (x7::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      (* <mark(U53(X_)),mark(X_)> *)
     | DP_R_xml_0_scc_12_large_47 :
      forall 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_U53 (x5::nil)) x11) ->
        DP_R_xml_0_scc_12_large (algebra.Alg.Term algebra.F.id_mark 
                                 (x5::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      (* <a__U61(tt,L_),a__length(mark(L_))> *)
     | DP_R_xml_0_scc_12_large_48 :
      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_12_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__U61 (x12::x11::nil))
     
      (* <a__length(cons(N_,L_)),a__U61(a__and(a__and(a__isNatList(L_),isNatIListKind(L_)),
        and(isNat(N_),isNatKind(N_))),L_)> *)
     | DP_R_xml_0_scc_12_large_49 :
      forall x4 x6 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 (x6::x4::nil)) x11) ->
        DP_R_xml_0_scc_12_large (algebra.Alg.Term algebra.F.id_a__U61 
                                 ((algebra.Alg.Term algebra.F.id_a__and 
                                 ((algebra.Alg.Term algebra.F.id_a__and 
                                 ((algebra.Alg.Term 
                                 algebra.F.id_a__isNatList (x4::nil))::
                                 (algebra.Alg.Term 
                                 algebra.F.id_isNatIListKind 
                                 (x4::nil))::nil))::(algebra.Alg.Term 
                                 algebra.F.id_and ((algebra.Alg.Term 
                                 algebra.F.id_isNat (x6::nil))::
                                 (algebra.Alg.Term algebra.F.id_isNatKind 
                                 (x6::nil))::nil))::nil))::x4::nil)) 
         (algebra.Alg.Term algebra.F.id_a__length (x11::nil))
     
      (* <mark(and(X1_,X2_)),a__and(mark(X1_),X2_)> *)
     | DP_R_xml_0_scc_12_large_50 :
      forall x8 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_and (x7::x8::nil)) x11) ->
        DP_R_xml_0_scc_12_large (algebra.Alg.Term algebra.F.id_a__and 
                                 ((algebra.Alg.Term algebra.F.id_mark 
                                 (x7::nil))::x8::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      (* <mark(and(X1_,X2_)),mark(X1_)> *)
     | DP_R_xml_0_scc_12_large_51 :
      forall x8 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_and (x7::x8::nil)) x11) ->
        DP_R_xml_0_scc_12_large (algebra.Alg.Term algebra.F.id_mark 
                                 (x7::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     
      (* <mark(isNatIListKind(X_)),a__isNatIListKind(X_)> *)
     | DP_R_xml_0_scc_12_large_52 :
      forall 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_isNatIListKind (x5::nil)) x11) ->
        DP_R_xml_0_scc_12_large (algebra.Alg.Term 
                                 algebra.F.id_a__isNatIListKind (x5::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     
      (* <mark(isNatKind(X_)),a__isNatKind(X_)> *)
     | DP_R_xml_0_scc_12_large_53 :
      forall 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_isNatKind (x5::nil)) x11) ->
        DP_R_xml_0_scc_12_large (algebra.Alg.Term algebra.F.id_a__isNatKind 
                                 (x5::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      (* <mark(cons(X1_,X2_)),mark(X1_)> *)
     | DP_R_xml_0_scc_12_large_54 :
      forall x8 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 (x7::x8::nil)) x11) ->
        DP_R_xml_0_scc_12_large (algebra.Alg.Term algebra.F.id_mark 
                                 (x7::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      (* <mark(s(X_)),mark(X_)> *)
     | DP_R_xml_0_scc_12_large_55 :
      forall 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_s (x5::nil)) x11) ->
        DP_R_xml_0_scc_12_large (algebra.Alg.Term algebra.F.id_mark 
                                 (x5::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
   .
   
   
   Inductive DP_R_xml_0_scc_12_strict  :
    algebra.Alg.term ->algebra.Alg.term ->Prop := 
      (* <mark(U61(X1_,X2_)),a__U61(mark(X1_),X2_)> *)
     | DP_R_xml_0_scc_12_strict_0 :
      forall x8 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_U61 (x7::x8::nil)) x11) ->
        DP_R_xml_0_scc_12_strict (algebra.Alg.Term algebra.F.id_a__U61 
                                  ((algebra.Alg.Term algebra.F.id_mark 
                                  (x7::nil))::x8::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      (* <a__U61(tt,L_),mark(L_)> *)
     | DP_R_xml_0_scc_12_strict_1 :
      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_12_strict (algebra.Alg.Term algebra.F.id_mark 
                                   (x4::nil)) 
          (algebra.Alg.Term algebra.F.id_a__U61 (x12::x11::nil))
      (* <mark(U61(X1_,X2_)),mark(X1_)> *)
     | DP_R_xml_0_scc_12_strict_2 :
      forall x8 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_U61 (x7::x8::nil)) x11) ->
        DP_R_xml_0_scc_12_strict (algebra.Alg.Term algebra.F.id_mark 
                                  (x7::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     
      (* <mark(length(X_)),a__length(mark(X_))> *)
     | DP_R_xml_0_scc_12_strict_3 :
      forall 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_length (x5::nil)) x11) ->
        DP_R_xml_0_scc_12_strict (algebra.Alg.Term algebra.F.id_a__length 
                                  ((algebra.Alg.Term algebra.F.id_mark 
                                  (x5::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     
      (* <a__length(cons(N_,L_)),a__and(a__and(a__isNatList(L_),isNatIListKind(L_)),
 and(isNat(N_),isNatKind(N_)))> *)
     | DP_R_xml_0_scc_12_strict_4 :
      forall x4 x6 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 (x6::x4::nil)) x11) ->
        DP_R_xml_0_scc_12_strict (algebra.Alg.Term algebra.F.id_a__and 
                                  ((algebra.Alg.Term algebra.F.id_a__and 
                                  ((algebra.Alg.Term 
                                  algebra.F.id_a__isNatList (x4::nil))::
                                  (algebra.Alg.Term 
                                  algebra.F.id_isNatIListKind 
                                  (x4::nil))::nil))::(algebra.Alg.Term 
                                  algebra.F.id_and ((algebra.Alg.Term 
                                  algebra.F.id_isNat (x6::nil))::
                                  (algebra.Alg.Term algebra.F.id_isNatKind 
                                  (x6::nil))::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_a__length (x11::nil))
     
      (* <a__length(cons(N_,L_)),a__and(a__isNatList(L_),isNatIListKind(L_))> *)
     | DP_R_xml_0_scc_12_strict_5 :
      forall x4 x6 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 (x6::x4::nil)) x11) ->
        DP_R_xml_0_scc_12_strict (algebra.Alg.Term algebra.F.id_a__and 
                                  ((algebra.Alg.Term 
                                  algebra.F.id_a__isNatList (x4::nil))::
                                  (algebra.Alg.Term 
                                  algebra.F.id_isNatIListKind 
                                  (x4::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_a__length (x11::nil))
     
      (* <a__length(cons(N_,L_)),a__isNatList(L_)> *)
     | DP_R_xml_0_scc_12_strict_6 :
      forall x4 x6 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 (x6::x4::nil)) x11) ->
        DP_R_xml_0_scc_12_strict (algebra.Alg.Term algebra.F.id_a__isNatList 
                                  (x4::nil)) 
         (algebra.Alg.Term algebra.F.id_a__length (x11::nil))
      (* <mark(length(X_)),mark(X_)> *)
     | DP_R_xml_0_scc_12_strict_7 :
      forall 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_length (x5::nil)) x11) ->
        DP_R_xml_0_scc_12_strict (algebra.Alg.Term algebra.F.id_mark 
                                  (x5::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
   .
   
   
   Module WF_DP_R_xml_0_scc_12_large.
    Inductive DP_R_xml_0_scc_12_large_scc_1  :
     algebra.Alg.term ->algebra.Alg.term ->Prop := 
       (* <a__length(cons(N_,L_)),a__U61(a__and(a__and(a__isNatList(L_),isNatIListKind(L_)),
        and(isNat(N_),isNatKind(N_))),L_)> *)
      | DP_R_xml_0_scc_12_large_scc_1_0 :
       forall x4 x6 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 (x6::x4::nil)) x11) ->
         DP_R_xml_0_scc_12_large_scc_1 (algebra.Alg.Term algebra.F.id_a__U61 
                                        ((algebra.Alg.Term 
                                        algebra.F.id_a__and 
                                        ((algebra.Alg.Term 
                                        algebra.F.id_a__and 
                                        ((algebra.Alg.Term 
                                        algebra.F.id_a__isNatList 
                                        (x4::nil))::(algebra.Alg.Term 
                                        algebra.F.id_isNatIListKind 
                                        (x4::nil))::nil))::(algebra.Alg.Term 
                                        algebra.F.id_and ((algebra.Alg.Term 
                                        algebra.F.id_isNat (x6::nil))::
                                        (algebra.Alg.Term 
                                        algebra.F.id_isNatKind 
                                        (x6::nil))::nil))::nil))::x4::nil)) 
          (algebra.Alg.Term algebra.F.id_a__length (x11::nil))
      
       (* <a__U61(tt,L_),a__length(mark(L_))> *)
      | DP_R_xml_0_scc_12_large_scc_1_1 :
       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_12_large_scc_1 (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__U61 (x12::x11::nil))
    .
    
    
    Module WF_DP_R_xml_0_scc_12_large_scc_1.
     Inductive DP_R_xml_0_scc_12_large_scc_1_large  :
      algebra.Alg.term ->algebra.Alg.term ->Prop := 
        (* <a__length(cons(N_,L_)),a__U61(a__and(a__and(a__isNatList(L_),isNatIListKind(L_)),
        and(isNat(N_),isNatKind(N_))),L_)> *)
       | DP_R_xml_0_scc_12_large_scc_1_large_0 :
        forall x4 x6 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 (x6::x4::nil)) x11) ->
          DP_R_xml_0_scc_12_large_scc_1_large (algebra.Alg.Term 
                                               algebra.F.id_a__U61 
                                               ((algebra.Alg.Term 
                                               algebra.F.id_a__and 
                                               ((algebra.Alg.Term 
                                               algebra.F.id_a__and 
                                               ((algebra.Alg.Term 
                                               algebra.F.id_a__isNatList 
                                               (x4::nil))::(algebra.Alg.Term 
                                               algebra.F.id_isNatIListKind 
                                               (x4::nil))::nil))::
                                               (algebra.Alg.Term 
                                               algebra.F.id_and 
                                               ((algebra.Alg.Term 
                                               algebra.F.id_isNat 
                                               (x6::nil))::(algebra.Alg.Term 
                                               algebra.F.id_isNatKind 
                                               (x6::nil))::nil))::nil))::
                                               x4::nil)) 
           (algebra.Alg.Term algebra.F.id_a__length (x11::nil))
     .
     
     
     Inductive DP_R_xml_0_scc_12_large_scc_1_strict  :
      algebra.Alg.term ->algebra.Alg.term ->Prop := 
        (* <a__U61(tt,L_),a__length(mark(L_))> *)
       | DP_R_xml_0_scc_12_large_scc_1_strict_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_12_large_scc_1_strict (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__U61 (x12::x11::nil))
     .
     
     
     Module WF_DP_R_xml_0_scc_12_large_scc_1_large.
      Inductive DP_R_xml_0_scc_12_large_scc_1_large_non_scc_1  :
       algebra.Alg.term ->algebra.Alg.term ->Prop := 
         (* <a__length(cons(N_,L_)),a__U61(a__and(a__and(a__isNatList(L_),isNatIListKind(L_)),
        and(isNat(N_),isNatKind(N_))),L_)> *)
        | DP_R_xml_0_scc_12_large_scc_1_large_non_scc_1_0 :
         forall x4 x6 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 (x6::x4::nil)) x11) ->
           DP_R_xml_0_scc_12_large_scc_1_large_non_scc_1 (algebra.Alg.Term 
                                                          algebra.F.id_a__U61 
                                                          ((algebra.Alg.Term 
                                                          algebra.F.id_a__and 
                                                          ((algebra.Alg.Term 
                                                          algebra.F.id_a__and 
                                                          ((algebra.Alg.Term 
                                                          algebra.F.id_a__isNatList 
                                                          (x4::nil))::
                                                          (algebra.Alg.Term 
                                                          algebra.F.id_isNatIListKind 
                                                          (x4::nil))::nil))::
                                                          (algebra.Alg.Term 
                                                          algebra.F.id_and 
                                                          ((algebra.Alg.Term 
                                                          algebra.F.id_isNat 
                                                          (x6::nil))::
                                                          (algebra.Alg.Term 
                                                          algebra.F.id_isNatKind 
                                                          (x6::nil))::nil))::nil))::
                                                          x4::nil)) 
            (algebra.Alg.Term algebra.F.id_a__length (x11::nil))
      .
      
      
      Lemma acc_DP_R_xml_0_scc_12_large_scc_1_large_non_scc_1 :
       forall x y, 
        (DP_R_xml_0_scc_12_large_scc_1_large_non_scc_1 x y) ->
         Acc WF_DP_R_xml_0_scc_12_large_scc_1.DP_R_xml_0_scc_12_large_scc_1_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_12_large_scc_1.DP_R_xml_0_scc_12_large_scc_1_large
        .
      Proof.
        constructor;intros _y _h;inversion _h;clear _h;subst;
         (eapply acc_DP_R_xml_0_scc_12_large_scc_1_large_non_scc_1;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
         ((eapply acc_DP_R_xml_0_scc_12_large_scc_1_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_12_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_isNatKind (x11:Z) := 0.
     
     Definition P_id_a__isNatIList (x11:Z) := 2 + 1* x11.
     
     Definition P_id_a__U21 (x11:Z) (x12:Z) := 1.
     
     Definition P_id_U41 (x11:Z) (x12:Z) (x13:Z) := 1 + 1* x11 + 1* x13.
     
     Definition P_id_a__and (x11:Z) (x12:Z) := 1* x11 + 2* x12.
     
     Definition P_id_a__U11 (x11:Z) (x12:Z) := 1* x11.
     
     Definition P_id_U21 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_a__U61 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_a__U32 (x11:Z) := 1* x11.
     
     Definition P_id_U51 (x11:Z) (x12:Z) (x13:Z) := 3* x13.
     
     Definition P_id_isNatIListKind (x11:Z) := 0.
     
     Definition P_id_0  := 0.
     
     Definition P_id_U12 (x11:Z) := 0.
     
     Definition P_id_a__U52 (x11:Z) (x12:Z) := 1* x12.
     
     Definition P_id_a__isNat (x11:Z) := 1.
     
     Definition P_id_U43 (x11:Z) := 1* x11.
     
     Definition P_id_a__isNatIListKind (x11:Z) := 1.
     
     Definition P_id_a__U12 (x11:Z) := 1.
     
     Definition P_id_U31 (x11:Z) (x12:Z) := 1* x12.
     
     Definition P_id_a__length (x11:Z) := 0.
     
     Definition P_id_a__U42 (x11:Z) (x12:Z) := 1 + 1* x11 + 1* x12.
     
     Definition P_id_U53 (x11:Z) := 1* x11.
     
     Definition P_id_and (x11:Z) (x12:Z) := 1* x11 + 2* x12.
     
     Definition P_id_cons (x11:Z) (x12:Z) := 3* x12.
     
     Definition P_id_U11 (x11:Z) (x12:Z) := 1* x11.
     
     Definition P_id_a__U51 (x11:Z) (x12:Z) (x13:Z) := 3* x13.
     
     Definition P_id_a__U22 (x11:Z) := 1* x11.
     
     Definition P_id_U42 (x11:Z) (x12:Z) := 1 + 1* x11 + 1* x12.
     
     Definition P_id_length (x11:Z) := 0.
     
     Definition P_id_tt  := 1.
     
     Definition P_id_U22 (x11:Z) := 1* x11.
     
     Definition P_id_s (x11:Z) := 0.
     
     Definition P_id_a__U41 (x11:Z) (x12:Z) (x13:Z) := 1 + 1* x11 + 1* x13.
     
     Definition P_id_U52 (x11:Z) (x12:Z) := 1* x12.
     
     Definition P_id_nil  := 1.
     
     Definition P_id_zeros  := 0.
     
     Definition P_id_isNatList (x11:Z) := 1* x11.
     
     Definition P_id_a__U53 (x11:Z) := 1* x11.
     
     Definition P_id_a__U31 (x11:Z) (x12:Z) := 1* x12.
     
     Definition P_id_isNatIList (x11:Z) := 1 + 1* x11.
     
     Definition P_id_a__isNatKind (x11:Z) := 1.
     
     Definition P_id_a__isNatList (x11:Z) := 1* x11.
     
     Definition P_id_U32 (x11:Z) := 1* x11.
     
     Definition P_id_mark (x11:Z) := 1 + 1* x11.
     
     Definition P_id_a__U43 (x11:Z) := 1* x11.
     
     Definition P_id_U61 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_isNat (x11:Z) := 0.
     
     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__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__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_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_a__and_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__and x12 x14 <= P_id_a__and x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       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_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__U61_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U61 x12 x14 <= P_id_a__U61 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       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 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U32 x12 <= P_id_a__U32 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U51_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U51 x12 x14 x16 <= P_id_U51 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       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_U12_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U12 x12 <= P_id_U12 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_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 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U52 x12 x14 <= P_id_a__U52 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_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_U43_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U43 x12 <= P_id_U43 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_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__U12_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U12 x12 <= P_id_a__U12 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       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__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__U42_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U42 x12 x14 <= P_id_a__U42 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U53_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U53 x12 <= P_id_U53 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_and_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_and x12 x14 <= P_id_and x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       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_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__U51_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U51 x12 x14 x16 <= P_id_a__U51 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_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 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U22 x12 <= P_id_a__U22 x11.
     Proof.
       intros x12 x11.
       intros [H_1 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 x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U42 x12 x14 <= P_id_U42 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       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_U22_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U22 x12 <= P_id_U22 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       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__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_U52_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U52 x12 x14 <= P_id_U52 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 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__U53_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U53 x12 <= P_id_a__U53 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       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_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__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__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_U32_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U32 x12 <= P_id_U32 x11.
     Proof.
       intros x12 x11.
       intros [H_1 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__U43_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U43 x12 <= P_id_a__U43 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       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 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U61 x12 x14 <= P_id_U61 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       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__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_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__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_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_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_a__and_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__and x12 x11.
     Proof.
       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_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__U61_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U61 x12 x11.
     Proof.
       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 x11, (0 <= x11) ->0 <= P_id_a__U32 x11.
     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 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U51 x13 x12 x11.
     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_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_U12_bounded : forall x11, (0 <= x11) ->0 <= P_id_U12 x11.
     Proof.
       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 x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U52 x12 x11.
     Proof.
       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_U43_bounded : forall x11, (0 <= x11) ->0 <= P_id_U43 x11.
     Proof.
       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_a__U12_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U12 x11.
     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__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__U42_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U42 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U53_bounded : forall x11, (0 <= x11) ->0 <= P_id_U53 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_and_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_and x12 x11.
     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_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__U51_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U51 x13 x12 x11.
     Proof.
       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 x11, (0 <= x11) ->0 <= P_id_a__U22 x11.
     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 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U42 x12 x11.
     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_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_U22_bounded : forall x11, (0 <= x11) ->0 <= P_id_U22 x11.
     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__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_U52_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U52 x12 x11.
     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_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_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__U53_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U53 x11.
     Proof.
       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_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.
     
     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_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_U32_bounded : forall x11, (0 <= x11) ->0 <= P_id_U32 x11.
     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__U43_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U43 x11.
     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 x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U61 x12 x11.
     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.
     
     Definition measure  := 
       InterpZ.measure 0 P_id_a__zeros P_id_isNatKind P_id_a__isNatIList 
        P_id_a__U21 P_id_U41 P_id_a__and P_id_a__U11 P_id_U21 P_id_a__U61 
        P_id_a__U32 P_id_U51 P_id_isNatIListKind P_id_0 P_id_U12 P_id_a__U52 
        P_id_a__isNat P_id_U43 P_id_a__isNatIListKind P_id_a__U12 P_id_U31 
        P_id_a__length P_id_a__U42 P_id_U53 P_id_and P_id_cons P_id_U11 
        P_id_a__U51 P_id_a__U22 P_id_U42 P_id_length P_id_tt P_id_U22 
        P_id_s P_id_a__U41 P_id_U52 P_id_nil P_id_zeros P_id_isNatList 
        P_id_a__U53 P_id_a__U31 P_id_isNatIList P_id_a__isNatKind 
        P_id_a__isNatList P_id_U32 P_id_mark P_id_a__U43 P_id_U61 P_id_isNat.
     
     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_isNatKind (x11::nil)) =>
                      P_id_isNatKind (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__U21 (x12::x11::nil)) =>
                      P_id_a__U21 (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_a__and (x12::x11::nil)) =>
                      P_id_a__and (measure x12) (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_U21 (x12::x11::nil)) =>
                      P_id_U21 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U61 (x12::x11::nil)) =>
                      P_id_a__U61 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U32 (x11::nil)) =>
                      P_id_a__U32 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U51 (x13::x12::
                        x11::nil)) =>
                      P_id_U51 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_isNatIListKind 
                        (x11::nil)) =>
                      P_id_isNatIListKind (measure x11)
                     | (algebra.Alg.Term algebra.F.id_0 nil) => P_id_0 
                     | (algebra.Alg.Term algebra.F.id_U12 (x11::nil)) =>
                      P_id_U12 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U52 (x12::x11::nil)) =>
                      P_id_a__U52 (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_U43 (x11::nil)) =>
                      P_id_U43 (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__U12 (x11::nil)) =>
                      P_id_a__U12 (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__length (x11::nil)) =>
                      P_id_a__length (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U42 (x12::x11::nil)) =>
                      P_id_a__U42 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U53 (x11::nil)) =>
                      P_id_U53 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_and (x12::x11::nil)) =>
                      P_id_and (measure x12) (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_U11 (x12::x11::nil)) =>
                      P_id_U11 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U51 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U51 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U22 (x11::nil)) =>
                      P_id_a__U22 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U42 (x12::x11::nil)) =>
                      P_id_U42 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_length (x11::nil)) =>
                      P_id_length (measure x11)
                     | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt 
                     | (algebra.Alg.Term algebra.F.id_U22 (x11::nil)) =>
                      P_id_U22 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_s (x11::nil)) =>
                      P_id_s (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_U52 (x12::x11::nil)) =>
                      P_id_U52 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil 
                     | (algebra.Alg.Term algebra.F.id_zeros nil) =>
                      P_id_zeros 
                     | (algebra.Alg.Term algebra.F.id_isNatList (x11::nil)) =>
                      P_id_isNatList (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U53 (x11::nil)) =>
                      P_id_a__U53 (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_isNatIList (x11::nil)) =>
                      P_id_isNatIList (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__isNatList 
                        (x11::nil)) =>
                      P_id_a__isNatList (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U32 (x11::nil)) =>
                      P_id_U32 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_mark (x11::nil)) =>
                      P_id_mark (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U43 (x11::nil)) =>
                      P_id_a__U43 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U61 (x12::x11::nil)) =>
                      P_id_U61 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_isNat (x11::nil)) =>
                      P_id_isNat (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_isNatKind_monotonic;assumption.
       intros ;apply P_id_a__isNatIList_monotonic;assumption.
       intros ;apply P_id_a__U21_monotonic;assumption.
       intros ;apply P_id_U41_monotonic;assumption.
       intros ;apply P_id_a__and_monotonic;assumption.
       intros ;apply P_id_a__U11_monotonic;assumption.
       intros ;apply P_id_U21_monotonic;assumption.
       intros ;apply P_id_a__U61_monotonic;assumption.
       intros ;apply P_id_a__U32_monotonic;assumption.
       intros ;apply P_id_U51_monotonic;assumption.
       intros ;apply P_id_isNatIListKind_monotonic;assumption.
       intros ;apply P_id_U12_monotonic;assumption.
       intros ;apply P_id_a__U52_monotonic;assumption.
       intros ;apply P_id_a__isNat_monotonic;assumption.
       intros ;apply P_id_U43_monotonic;assumption.
       intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
       intros ;apply P_id_a__U12_monotonic;assumption.
       intros ;apply P_id_U31_monotonic;assumption.
       intros ;apply P_id_a__length_monotonic;assumption.
       intros ;apply P_id_a__U42_monotonic;assumption.
       intros ;apply P_id_U53_monotonic;assumption.
       intros ;apply P_id_and_monotonic;assumption.
       intros ;apply P_id_cons_monotonic;assumption.
       intros ;apply P_id_U11_monotonic;assumption.
       intros ;apply P_id_a__U51_monotonic;assumption.
       intros ;apply P_id_a__U22_monotonic;assumption.
       intros ;apply P_id_U42_monotonic;assumption.
       intros ;apply P_id_length_monotonic;assumption.
       intros ;apply P_id_U22_monotonic;assumption.
       intros ;apply P_id_s_monotonic;assumption.
       intros ;apply P_id_a__U41_monotonic;assumption.
       intros ;apply P_id_U52_monotonic;assumption.
       intros ;apply P_id_isNatList_monotonic;assumption.
       intros ;apply P_id_a__U53_monotonic;assumption.
       intros ;apply P_id_a__U31_monotonic;assumption.
       intros ;apply P_id_isNatIList_monotonic;assumption.
       intros ;apply P_id_a__isNatKind_monotonic;assumption.
       intros ;apply P_id_a__isNatList_monotonic;assumption.
       intros ;apply P_id_U32_monotonic;assumption.
       intros ;apply P_id_mark_monotonic;assumption.
       intros ;apply P_id_a__U43_monotonic;assumption.
       intros ;apply P_id_U61_monotonic;assumption.
       intros ;apply P_id_isNat_monotonic;assumption.
       intros ;apply P_id_a__zeros_bounded;assumption.
       intros ;apply P_id_isNatKind_bounded;assumption.
       intros ;apply P_id_a__isNatIList_bounded;assumption.
       intros ;apply P_id_a__U21_bounded;assumption.
       intros ;apply P_id_U41_bounded;assumption.
       intros ;apply P_id_a__and_bounded;assumption.
       intros ;apply P_id_a__U11_bounded;assumption.
       intros ;apply P_id_U21_bounded;assumption.
       intros ;apply P_id_a__U61_bounded;assumption.
       intros ;apply P_id_a__U32_bounded;assumption.
       intros ;apply P_id_U51_bounded;assumption.
       intros ;apply P_id_isNatIListKind_bounded;assumption.
       intros ;apply P_id_0_bounded;assumption.
       intros ;apply P_id_U12_bounded;assumption.
       intros ;apply P_id_a__U52_bounded;assumption.
       intros ;apply P_id_a__isNat_bounded;assumption.
       intros ;apply P_id_U43_bounded;assumption.
       intros ;apply P_id_a__isNatIListKind_bounded;assumption.
       intros ;apply P_id_a__U12_bounded;assumption.
       intros ;apply P_id_U31_bounded;assumption.
       intros ;apply P_id_a__length_bounded;assumption.
       intros ;apply P_id_a__U42_bounded;assumption.
       intros ;apply P_id_U53_bounded;assumption.
       intros ;apply P_id_and_bounded;assumption.
       intros ;apply P_id_cons_bounded;assumption.
       intros ;apply P_id_U11_bounded;assumption.
       intros ;apply P_id_a__U51_bounded;assumption.
       intros ;apply P_id_a__U22_bounded;assumption.
       intros ;apply P_id_U42_bounded;assumption.
       intros ;apply P_id_length_bounded;assumption.
       intros ;apply P_id_tt_bounded;assumption.
       intros ;apply P_id_U22_bounded;assumption.
       intros ;apply P_id_s_bounded;assumption.
       intros ;apply P_id_a__U41_bounded;assumption.
       intros ;apply P_id_U52_bounded;assumption.
       intros ;apply P_id_nil_bounded;assumption.
       intros ;apply P_id_zeros_bounded;assumption.
       intros ;apply P_id_isNatList_bounded;assumption.
       intros ;apply P_id_a__U53_bounded;assumption.
       intros ;apply P_id_a__U31_bounded;assumption.
       intros ;apply P_id_isNatIList_bounded;assumption.
       intros ;apply P_id_a__isNatKind_bounded;assumption.
       intros ;apply P_id_a__isNatList_bounded;assumption.
       intros ;apply P_id_U32_bounded;assumption.
       intros ;apply P_id_mark_bounded;assumption.
       intros ;apply P_id_a__U43_bounded;assumption.
       intros ;apply P_id_U61_bounded;assumption.
       intros ;apply P_id_isNat_bounded;assumption.
       apply rules_monotonic.
     Qed.
     
     Definition P_id_A__LENGTH (x11:Z) := 1* x11.
     
     Definition P_id_A__U11 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__U41 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__ISNATKIND (x11:Z) := 0.
     
     Definition P_id_A__U22 (x11:Z) := 0.
     
     Definition P_id_A__U51 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__AND (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__ISNATLIST (x11:Z) := 0.
     
     Definition P_id_A__U43 (x11:Z) := 0.
     
     Definition P_id_A__U31 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__U53 (x11:Z) := 0.
     
     Definition P_id_MARK (x11:Z) := 0.
     
     Definition P_id_A__U12 (x11:Z) := 0.
     
     Definition P_id_A__U42 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__ZEROS  := 0.
     
     Definition P_id_A__ISNAT (x11:Z) := 0.
     
     Definition P_id_A__U52 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__ISNATILISTKIND (x11:Z) := 0.
     
     Definition P_id_A__U21 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__ISNATILIST (x11:Z) := 0.
     
     Definition P_id_A__U32 (x11:Z) := 0.
     
     Definition P_id_A__U61 (x11:Z) (x12:Z) := 2* x11 + 1* x12.
     
     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__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__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__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__U22_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U22 x12 <= P_id_A__U22 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_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 x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U51 x12 x14 x16 <= P_id_A__U51 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__AND_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__AND x12 x14 <= P_id_A__AND x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_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__U43_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U43 x12 <= P_id_A__U43 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       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__U53_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U53 x12 <= P_id_A__U53 x11.
     Proof.
       intros x12 x11.
       intros [H_1 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__U12_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U12 x12 <= P_id_A__U12 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       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 x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U42 x12 x14 <= P_id_A__U42 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_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_A__U52_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U52 x12 x14 <= P_id_A__U52 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       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__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__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__U32_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U32 x12 <= P_id_A__U32 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_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 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U61 x12 x14 <= P_id_A__U61 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       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_isNatKind 
        P_id_a__isNatIList P_id_a__U21 P_id_U41 P_id_a__and P_id_a__U11 
        P_id_U21 P_id_a__U61 P_id_a__U32 P_id_U51 P_id_isNatIListKind 
        P_id_0 P_id_U12 P_id_a__U52 P_id_a__isNat P_id_U43 
        P_id_a__isNatIListKind P_id_a__U12 P_id_U31 P_id_a__length 
        P_id_a__U42 P_id_U53 P_id_and P_id_cons P_id_U11 P_id_a__U51 
        P_id_a__U22 P_id_U42 P_id_length P_id_tt P_id_U22 P_id_s P_id_a__U41 
        P_id_U52 P_id_nil P_id_zeros P_id_isNatList P_id_a__U53 P_id_a__U31 
        P_id_isNatIList P_id_a__isNatKind P_id_a__isNatList P_id_U32 
        P_id_mark P_id_a__U43 P_id_U61 P_id_isNat P_id_A__LENGTH P_id_A__U11 
        P_id_A__U41 P_id_A__ISNATKIND P_id_A__U22 P_id_A__U51 P_id_A__AND 
        P_id_A__ISNATLIST P_id_A__U43 P_id_A__U31 P_id_A__U53 P_id_MARK 
        P_id_A__U12 P_id_A__U42 P_id_A__ZEROS P_id_A__ISNAT P_id_A__U52 
        P_id_A__ISNATILISTKIND P_id_A__U21 P_id_A__ISNATILIST P_id_A__U32 
        P_id_A__U61.
     
     Lemma marked_measure_equation :
      forall t, 
       marked_measure t = match t with
                            | (algebra.Alg.Term algebra.F.id_a__length 
                               (x11::nil)) =>
                             P_id_A__LENGTH (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__U41 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U41 (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__U22 
                               (x11::nil)) =>
                             P_id_A__U22 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U51 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U51 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__and (x12::
                               x11::nil)) =>
                             P_id_A__AND (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__U43 
                               (x11::nil)) =>
                             P_id_A__U43 (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__U53 
                               (x11::nil)) =>
                             P_id_A__U53 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_mark (x11::nil)) =>
                             P_id_MARK (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U12 
                               (x11::nil)) =>
                             P_id_A__U12 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U42 (x12::
                               x11::nil)) =>
                             P_id_A__U42 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__zeros nil) =>
                             P_id_A__ZEROS 
                            | (algebra.Alg.Term algebra.F.id_a__isNat 
                               (x11::nil)) =>
                             P_id_A__ISNAT (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U52 (x12::
                               x11::nil)) =>
                             P_id_A__U52 (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_a__U21 (x12::
                               x11::nil)) =>
                             P_id_A__U21 (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__U32 
                               (x11::nil)) =>
                             P_id_A__U32 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U61 (x12::
                               x11::nil)) =>
                             P_id_A__U61 (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_isNatKind_monotonic;assumption.
       intros ;apply P_id_a__isNatIList_monotonic;assumption.
       intros ;apply P_id_a__U21_monotonic;assumption.
       intros ;apply P_id_U41_monotonic;assumption.
       intros ;apply P_id_a__and_monotonic;assumption.
       intros ;apply P_id_a__U11_monotonic;assumption.
       intros ;apply P_id_U21_monotonic;assumption.
       intros ;apply P_id_a__U61_monotonic;assumption.
       intros ;apply P_id_a__U32_monotonic;assumption.
       intros ;apply P_id_U51_monotonic;assumption.
       intros ;apply P_id_isNatIListKind_monotonic;assumption.
       intros ;apply P_id_U12_monotonic;assumption.
       intros ;apply P_id_a__U52_monotonic;assumption.
       intros ;apply P_id_a__isNat_monotonic;assumption.
       intros ;apply P_id_U43_monotonic;assumption.
       intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
       intros ;apply P_id_a__U12_monotonic;assumption.
       intros ;apply P_id_U31_monotonic;assumption.
       intros ;apply P_id_a__length_monotonic;assumption.
       intros ;apply P_id_a__U42_monotonic;assumption.
       intros ;apply P_id_U53_monotonic;assumption.
       intros ;apply P_id_and_monotonic;assumption.
       intros ;apply P_id_cons_monotonic;assumption.
       intros ;apply P_id_U11_monotonic;assumption.
       intros ;apply P_id_a__U51_monotonic;assumption.
       intros ;apply P_id_a__U22_monotonic;assumption.
       intros ;apply P_id_U42_monotonic;assumption.
       intros ;apply P_id_length_monotonic;assumption.
       intros ;apply P_id_U22_monotonic;assumption.
       intros ;apply P_id_s_monotonic;assumption.
       intros ;apply P_id_a__U41_monotonic;assumption.
       intros ;apply P_id_U52_monotonic;assumption.
       intros ;apply P_id_isNatList_monotonic;assumption.
       intros ;apply P_id_a__U53_monotonic;assumption.
       intros ;apply P_id_a__U31_monotonic;assumption.
       intros ;apply P_id_isNatIList_monotonic;assumption.
       intros ;apply P_id_a__isNatKind_monotonic;assumption.
       intros ;apply P_id_a__isNatList_monotonic;assumption.
       intros ;apply P_id_U32_monotonic;assumption.
       intros ;apply P_id_mark_monotonic;assumption.
       intros ;apply P_id_a__U43_monotonic;assumption.
       intros ;apply P_id_U61_monotonic;assumption.
       intros ;apply P_id_isNat_monotonic;assumption.
       intros ;apply P_id_a__zeros_bounded;assumption.
       intros ;apply P_id_isNatKind_bounded;assumption.
       intros ;apply P_id_a__isNatIList_bounded;assumption.
       intros ;apply P_id_a__U21_bounded;assumption.
       intros ;apply P_id_U41_bounded;assumption.
       intros ;apply P_id_a__and_bounded;assumption.
       intros ;apply P_id_a__U11_bounded;assumption.
       intros ;apply P_id_U21_bounded;assumption.
       intros ;apply P_id_a__U61_bounded;assumption.
       intros ;apply P_id_a__U32_bounded;assumption.
       intros ;apply P_id_U51_bounded;assumption.
       intros ;apply P_id_isNatIListKind_bounded;assumption.
       intros ;apply P_id_0_bounded;assumption.
       intros ;apply P_id_U12_bounded;assumption.
       intros ;apply P_id_a__U52_bounded;assumption.
       intros ;apply P_id_a__isNat_bounded;assumption.
       intros ;apply P_id_U43_bounded;assumption.
       intros ;apply P_id_a__isNatIListKind_bounded;assumption.
       intros ;apply P_id_a__U12_bounded;assumption.
       intros ;apply P_id_U31_bounded;assumption.
       intros ;apply P_id_a__length_bounded;assumption.
       intros ;apply P_id_a__U42_bounded;assumption.
       intros ;apply P_id_U53_bounded;assumption.
       intros ;apply P_id_and_bounded;assumption.
       intros ;apply P_id_cons_bounded;assumption.
       intros ;apply P_id_U11_bounded;assumption.
       intros ;apply P_id_a__U51_bounded;assumption.
       intros ;apply P_id_a__U22_bounded;assumption.
       intros ;apply P_id_U42_bounded;assumption.
       intros ;apply P_id_length_bounded;assumption.
       intros ;apply P_id_tt_bounded;assumption.
       intros ;apply P_id_U22_bounded;assumption.
       intros ;apply P_id_s_bounded;assumption.
       intros ;apply P_id_a__U41_bounded;assumption.
       intros ;apply P_id_U52_bounded;assumption.
       intros ;apply P_id_nil_bounded;assumption.
       intros ;apply P_id_zeros_bounded;assumption.
       intros ;apply P_id_isNatList_bounded;assumption.
       intros ;apply P_id_a__U53_bounded;assumption.
       intros ;apply P_id_a__U31_bounded;assumption.
       intros ;apply P_id_isNatIList_bounded;assumption.
       intros ;apply P_id_a__isNatKind_bounded;assumption.
       intros ;apply P_id_a__isNatList_bounded;assumption.
       intros ;apply P_id_U32_bounded;assumption.
       intros ;apply P_id_mark_bounded;assumption.
       intros ;apply P_id_a__U43_bounded;assumption.
       intros ;apply P_id_U61_bounded;assumption.
       intros ;apply P_id_isNat_bounded;assumption.
       apply rules_monotonic.
       intros ;apply P_id_A__LENGTH_monotonic;assumption.
       intros ;apply P_id_A__U11_monotonic;assumption.
       intros ;apply P_id_A__U41_monotonic;assumption.
       intros ;apply P_id_A__ISNATKIND_monotonic;assumption.
       intros ;apply P_id_A__U22_monotonic;assumption.
       intros ;apply P_id_A__U51_monotonic;assumption.
       intros ;apply P_id_A__AND_monotonic;assumption.
       intros ;apply P_id_A__ISNATLIST_monotonic;assumption.
       intros ;apply P_id_A__U43_monotonic;assumption.
       intros ;apply P_id_A__U31_monotonic;assumption.
       intros ;apply P_id_A__U53_monotonic;assumption.
       intros ;apply P_id_MARK_monotonic;assumption.
       intros ;apply P_id_A__U12_monotonic;assumption.
       intros ;apply P_id_A__U42_monotonic;assumption.
       intros ;apply P_id_A__ISNAT_monotonic;assumption.
       intros ;apply P_id_A__U52_monotonic;assumption.
       intros ;apply P_id_A__ISNATILISTKIND_monotonic;assumption.
       intros ;apply P_id_A__U21_monotonic;assumption.
       intros ;apply P_id_A__ISNATILIST_monotonic;assumption.
       intros ;apply P_id_A__U32_monotonic;assumption.
       intros ;apply P_id_A__U61_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_12_large_scc_1_strict_in_lt :
      Relation_Definitions.inclusion _ DP_R_xml_0_scc_12_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_12_large_scc_1_large_in_le :
      Relation_Definitions.inclusion _ DP_R_xml_0_scc_12_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_12_large_scc_1_large  := 
       WF_DP_R_xml_0_scc_12_large_scc_1_large.wf.
     
     
     Lemma wf :
      well_founded WF_DP_R_xml_0_scc_12_large.DP_R_xml_0_scc_12_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_12_large_scc_1_large).
       clear x.
       intros x _ IHx IHx'.
       constructor.
       intros y H.
       
       destruct H;
        (apply IHx';apply DP_R_xml_0_scc_12_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_12_large_scc_1_large_in_le;
           econstructor eassumption])).
       apply wf_DP_R_xml_0_scc_12_large_scc_1_large.
     Qed.
    End WF_DP_R_xml_0_scc_12_large_scc_1.
    
    Definition wf_DP_R_xml_0_scc_12_large_scc_1  := 
      WF_DP_R_xml_0_scc_12_large_scc_1.wf.
    
    
    Lemma acc_DP_R_xml_0_scc_12_large_scc_1 :
     forall x y, 
      (DP_R_xml_0_scc_12_large_scc_1 x y) ->
       Acc WF_DP_R_xml_0_scc_12.DP_R_xml_0_scc_12_large x.
    Proof.
      intros x.
      pattern x.
      apply (@Acc_ind _ DP_R_xml_0_scc_12_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_12_large_scc_1.
    Qed.
    
    
    Inductive DP_R_xml_0_scc_12_large_scc_2  :
     algebra.Alg.term ->algebra.Alg.term ->Prop := 
       (* <a__isNatList(cons(V1_,V2_)),a__U51(a__and(a__isNatKind(V1_),isNatIListKind(V2_)),V1_,V2_)> *)
      | DP_R_xml_0_scc_12_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_12_large_scc_2 (algebra.Alg.Term algebra.F.id_a__U51 
                                        ((algebra.Alg.Term 
                                        algebra.F.id_a__and 
                                        ((algebra.Alg.Term 
                                        algebra.F.id_a__isNatKind 
                                        (x1::nil))::(algebra.Alg.Term 
                                        algebra.F.id_isNatIListKind 
                                        (x3::nil))::nil))::x1::x3::nil)) 
          (algebra.Alg.Term algebra.F.id_a__isNatList (x11::nil))
      
       (* <a__U51(tt,V1_,V2_),a__U52(a__isNat(V1_),V2_)> *)
      | DP_R_xml_0_scc_12_large_scc_2_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_12_large_scc_2 (algebra.Alg.Term 
                                          algebra.F.id_a__U52 
                                          ((algebra.Alg.Term 
                                          algebra.F.id_a__isNat (x1::nil))::
                                          x3::nil)) 
            (algebra.Alg.Term algebra.F.id_a__U51 (x13::x12::x11::nil))
      
       (* <a__U52(tt,V2_),a__isNatList(V2_)> *)
      | DP_R_xml_0_scc_12_large_scc_2_2 :
       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_12_large_scc_2 (algebra.Alg.Term 
                                         algebra.F.id_a__isNatList (x3::nil)) 
           (algebra.Alg.Term algebra.F.id_a__U52 (x12::x11::nil))
      
       (* <a__isNatList(cons(V1_,V2_)),a__and(a__isNatKind(V1_),isNatIListKind(V2_))> *)
      | DP_R_xml_0_scc_12_large_scc_2_3 :
       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_12_large_scc_2 (algebra.Alg.Term algebra.F.id_a__and 
                                        ((algebra.Alg.Term 
                                        algebra.F.id_a__isNatKind 
                                        (x1::nil))::(algebra.Alg.Term 
                                        algebra.F.id_isNatIListKind 
                                        (x3::nil))::nil)) 
          (algebra.Alg.Term algebra.F.id_a__isNatList (x11::nil))
       (* <a__and(tt,X_),mark(X_)> *)
      | DP_R_xml_0_scc_12_large_scc_2_4 :
       forall x12 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_tt nil) 
          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_12_large_scc_2 (algebra.Alg.Term algebra.F.id_mark 
                                         (x5::nil)) 
           (algebra.Alg.Term algebra.F.id_a__and (x12::x11::nil))
      
       (* <mark(U11(X1_,X2_)),a__U11(mark(X1_),X2_)> *)
      | DP_R_xml_0_scc_12_large_scc_2_5 :
       forall x8 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 (x7::x8::nil)) x11) ->
         DP_R_xml_0_scc_12_large_scc_2 (algebra.Alg.Term algebra.F.id_a__U11 
                                        ((algebra.Alg.Term algebra.F.id_mark 
                                        (x7::nil))::x8::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      
       (* <a__U11(tt,V1_),a__isNatList(V1_)> *)
      | DP_R_xml_0_scc_12_large_scc_2_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_scc_12_large_scc_2 (algebra.Alg.Term 
                                         algebra.F.id_a__isNatList (x1::nil)) 
           (algebra.Alg.Term algebra.F.id_a__U11 (x12::x11::nil))
      
       (* <a__isNatList(cons(V1_,V2_)),a__isNatKind(V1_)> *)
      | DP_R_xml_0_scc_12_large_scc_2_7 :
       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_12_large_scc_2 (algebra.Alg.Term 
                                        algebra.F.id_a__isNatKind (x1::nil)) 
          (algebra.Alg.Term algebra.F.id_a__isNatList (x11::nil))
      
       (* <a__isNatKind(length(V1_)),a__isNatIListKind(V1_)> *)
      | DP_R_xml_0_scc_12_large_scc_2_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_12_large_scc_2 (algebra.Alg.Term 
                                        algebra.F.id_a__isNatIListKind 
                                        (x1::nil)) 
          (algebra.Alg.Term algebra.F.id_a__isNatKind (x11::nil))
      
       (* <a__isNatIListKind(cons(V1_,V2_)),a__and(a__isNatKind(V1_),isNatIListKind(V2_))> *)
      | DP_R_xml_0_scc_12_large_scc_2_9 :
       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_12_large_scc_2 (algebra.Alg.Term algebra.F.id_a__and 
                                        ((algebra.Alg.Term 
                                        algebra.F.id_a__isNatKind 
                                        (x1::nil))::(algebra.Alg.Term 
                                        algebra.F.id_isNatIListKind 
                                        (x3::nil))::nil)) 
          (algebra.Alg.Term algebra.F.id_a__isNatIListKind (x11::nil))
      
       (* <a__isNatIListKind(cons(V1_,V2_)),a__isNatKind(V1_)> *)
      | DP_R_xml_0_scc_12_large_scc_2_10 :
       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_12_large_scc_2 (algebra.Alg.Term 
                                        algebra.F.id_a__isNatKind (x1::nil)) 
          (algebra.Alg.Term algebra.F.id_a__isNatIListKind (x11::nil))
      
       (* <a__isNatKind(s(V1_)),a__isNatKind(V1_)> *)
      | DP_R_xml_0_scc_12_large_scc_2_11 :
       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_12_large_scc_2 (algebra.Alg.Term 
                                        algebra.F.id_a__isNatKind (x1::nil)) 
          (algebra.Alg.Term algebra.F.id_a__isNatKind (x11::nil))
      
       (* <mark(U11(X1_,X2_)),mark(X1_)> *)
      | DP_R_xml_0_scc_12_large_scc_2_12 :
       forall x8 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 (x7::x8::nil)) x11) ->
         DP_R_xml_0_scc_12_large_scc_2 (algebra.Alg.Term algebra.F.id_mark 
                                        (x7::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       (* <mark(U12(X_)),mark(X_)> *)
      | DP_R_xml_0_scc_12_large_scc_2_13 :
       forall 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_U12 (x5::nil)) x11) ->
         DP_R_xml_0_scc_12_large_scc_2 (algebra.Alg.Term algebra.F.id_mark 
                                        (x5::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      
       (* <mark(isNatList(X_)),a__isNatList(X_)> *)
      | DP_R_xml_0_scc_12_large_scc_2_14 :
       forall 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_isNatList (x5::nil)) x11) ->
         DP_R_xml_0_scc_12_large_scc_2 (algebra.Alg.Term 
                                        algebra.F.id_a__isNatList (x5::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      
       (* <mark(U21(X1_,X2_)),a__U21(mark(X1_),X2_)> *)
      | DP_R_xml_0_scc_12_large_scc_2_15 :
       forall x8 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 (x7::x8::nil)) x11) ->
         DP_R_xml_0_scc_12_large_scc_2 (algebra.Alg.Term algebra.F.id_a__U21 
                                        ((algebra.Alg.Term algebra.F.id_mark 
                                        (x7::nil))::x8::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      
       (* <a__U21(tt,V1_),a__isNat(V1_)> *)
      | DP_R_xml_0_scc_12_large_scc_2_16 :
       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_12_large_scc_2 (algebra.Alg.Term 
                                         algebra.F.id_a__isNat (x1::nil)) 
           (algebra.Alg.Term algebra.F.id_a__U21 (x12::x11::nil))
      
       (* <a__isNat(length(V1_)),a__U11(a__isNatIListKind(V1_),V1_)> *)
      | DP_R_xml_0_scc_12_large_scc_2_17 :
       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_12_large_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))
      
       (* <a__isNat(length(V1_)),a__isNatIListKind(V1_)> *)
      | DP_R_xml_0_scc_12_large_scc_2_18 :
       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_12_large_scc_2 (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_scc_12_large_scc_2_19 :
       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_12_large_scc_2 (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_scc_12_large_scc_2_20 :
       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_12_large_scc_2 (algebra.Alg.Term 
                                        algebra.F.id_a__isNatKind (x1::nil)) 
          (algebra.Alg.Term algebra.F.id_a__isNat (x11::nil))
      
       (* <mark(U21(X1_,X2_)),mark(X1_)> *)
      | DP_R_xml_0_scc_12_large_scc_2_21 :
       forall x8 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 (x7::x8::nil)) x11) ->
         DP_R_xml_0_scc_12_large_scc_2 (algebra.Alg.Term algebra.F.id_mark 
                                        (x7::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       (* <mark(U22(X_)),mark(X_)> *)
      | DP_R_xml_0_scc_12_large_scc_2_22 :
       forall 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_U22 (x5::nil)) x11) ->
         DP_R_xml_0_scc_12_large_scc_2 (algebra.Alg.Term algebra.F.id_mark 
                                        (x5::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      
       (* <mark(isNat(X_)),a__isNat(X_)> *)
      | DP_R_xml_0_scc_12_large_scc_2_23 :
       forall 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_isNat (x5::nil)) x11) ->
         DP_R_xml_0_scc_12_large_scc_2 (algebra.Alg.Term 
                                        algebra.F.id_a__isNat (x5::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      
       (* <mark(U31(X1_,X2_)),a__U31(mark(X1_),X2_)> *)
      | DP_R_xml_0_scc_12_large_scc_2_24 :
       forall x8 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 (x7::x8::nil)) x11) ->
         DP_R_xml_0_scc_12_large_scc_2 (algebra.Alg.Term algebra.F.id_a__U31 
                                        ((algebra.Alg.Term algebra.F.id_mark 
                                        (x7::nil))::x8::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      
       (* <a__U31(tt,V_),a__isNatList(V_)> *)
      | DP_R_xml_0_scc_12_large_scc_2_25 :
       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_scc_12_large_scc_2 (algebra.Alg.Term 
                                         algebra.F.id_a__isNatList (x2::nil)) 
           (algebra.Alg.Term algebra.F.id_a__U31 (x12::x11::nil))
      
       (* <mark(U31(X1_,X2_)),mark(X1_)> *)
      | DP_R_xml_0_scc_12_large_scc_2_26 :
       forall x8 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 (x7::x8::nil)) x11) ->
         DP_R_xml_0_scc_12_large_scc_2 (algebra.Alg.Term algebra.F.id_mark 
                                        (x7::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       (* <mark(U32(X_)),mark(X_)> *)
      | DP_R_xml_0_scc_12_large_scc_2_27 :
       forall 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_U32 (x5::nil)) x11) ->
         DP_R_xml_0_scc_12_large_scc_2 (algebra.Alg.Term algebra.F.id_mark 
                                        (x5::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_scc_12_large_scc_2_28 :
       forall x8 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 (x7::x8::x9::nil)) x11) ->
         DP_R_xml_0_scc_12_large_scc_2 (algebra.Alg.Term algebra.F.id_a__U41 
                                        ((algebra.Alg.Term algebra.F.id_mark 
                                        (x7::nil))::x8::x9::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      
       (* <a__U41(tt,V1_,V2_),a__U42(a__isNat(V1_),V2_)> *)
      | DP_R_xml_0_scc_12_large_scc_2_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_scc_12_large_scc_2 (algebra.Alg.Term 
                                          algebra.F.id_a__U42 
                                          ((algebra.Alg.Term 
                                          algebra.F.id_a__isNat (x1::nil))::
                                          x3::nil)) 
            (algebra.Alg.Term algebra.F.id_a__U41 (x13::x12::x11::nil))
      
       (* <a__U42(tt,V2_),a__isNatIList(V2_)> *)
      | DP_R_xml_0_scc_12_large_scc_2_30 :
       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_12_large_scc_2 (algebra.Alg.Term 
                                         algebra.F.id_a__isNatIList 
                                         (x3::nil)) 
           (algebra.Alg.Term algebra.F.id_a__U42 (x12::x11::nil))
      
       (* <a__isNatIList(V_),a__U31(a__isNatIListKind(V_),V_)> *)
      | DP_R_xml_0_scc_12_large_scc_2_31 :
       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_scc_12_large_scc_2 (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_scc_12_large_scc_2_32 :
       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_scc_12_large_scc_2 (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__and(a__isNatKind(V1_),isNatIListKind(V2_)),V1_,V2_)> *)
      | DP_R_xml_0_scc_12_large_scc_2_33 :
       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_12_large_scc_2 (algebra.Alg.Term algebra.F.id_a__U41 
                                        ((algebra.Alg.Term 
                                        algebra.F.id_a__and 
                                        ((algebra.Alg.Term 
                                        algebra.F.id_a__isNatKind 
                                        (x1::nil))::(algebra.Alg.Term 
                                        algebra.F.id_isNatIListKind 
                                        (x3::nil))::nil))::x1::x3::nil)) 
          (algebra.Alg.Term algebra.F.id_a__isNatIList (x11::nil))
      
       (* <a__U41(tt,V1_,V2_),a__isNat(V1_)> *)
      | DP_R_xml_0_scc_12_large_scc_2_34 :
       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_12_large_scc_2 (algebra.Alg.Term 
                                          algebra.F.id_a__isNat (x1::nil)) 
            (algebra.Alg.Term algebra.F.id_a__U41 (x13::x12::x11::nil))
      
       (* <a__isNatIList(cons(V1_,V2_)),a__and(a__isNatKind(V1_),isNatIListKind(V2_))> *)
      | DP_R_xml_0_scc_12_large_scc_2_35 :
       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_12_large_scc_2 (algebra.Alg.Term algebra.F.id_a__and 
                                        ((algebra.Alg.Term 
                                        algebra.F.id_a__isNatKind 
                                        (x1::nil))::(algebra.Alg.Term 
                                        algebra.F.id_isNatIListKind 
                                        (x3::nil))::nil)) 
          (algebra.Alg.Term algebra.F.id_a__isNatIList (x11::nil))
      
       (* <a__isNatIList(cons(V1_,V2_)),a__isNatKind(V1_)> *)
      | DP_R_xml_0_scc_12_large_scc_2_36 :
       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_12_large_scc_2 (algebra.Alg.Term 
                                        algebra.F.id_a__isNatKind (x1::nil)) 
          (algebra.Alg.Term algebra.F.id_a__isNatIList (x11::nil))
      
       (* <mark(U41(X1_,X2_,X3_)),mark(X1_)> *)
      | DP_R_xml_0_scc_12_large_scc_2_37 :
       forall x8 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 (x7::x8::x9::nil)) x11) ->
         DP_R_xml_0_scc_12_large_scc_2 (algebra.Alg.Term algebra.F.id_mark 
                                        (x7::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      
       (* <mark(U42(X1_,X2_)),a__U42(mark(X1_),X2_)> *)
      | DP_R_xml_0_scc_12_large_scc_2_38 :
       forall x8 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 (x7::x8::nil)) x11) ->
         DP_R_xml_0_scc_12_large_scc_2 (algebra.Alg.Term algebra.F.id_a__U42 
                                        ((algebra.Alg.Term algebra.F.id_mark 
                                        (x7::nil))::x8::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      
       (* <mark(U42(X1_,X2_)),mark(X1_)> *)
      | DP_R_xml_0_scc_12_large_scc_2_39 :
       forall x8 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 (x7::x8::nil)) x11) ->
         DP_R_xml_0_scc_12_large_scc_2 (algebra.Alg.Term algebra.F.id_mark 
                                        (x7::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       (* <mark(U43(X_)),mark(X_)> *)
      | DP_R_xml_0_scc_12_large_scc_2_40 :
       forall 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_U43 (x5::nil)) x11) ->
         DP_R_xml_0_scc_12_large_scc_2 (algebra.Alg.Term algebra.F.id_mark 
                                        (x5::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      
       (* <mark(isNatIList(X_)),a__isNatIList(X_)> *)
      | DP_R_xml_0_scc_12_large_scc_2_41 :
       forall 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_isNatIList (x5::nil)) x11) ->
         DP_R_xml_0_scc_12_large_scc_2 (algebra.Alg.Term 
                                        algebra.F.id_a__isNatIList (x5::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      
       (* <mark(U51(X1_,X2_,X3_)),a__U51(mark(X1_),X2_,X3_)> *)
      | DP_R_xml_0_scc_12_large_scc_2_42 :
       forall x8 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_U51 (x7::x8::x9::nil)) x11) ->
         DP_R_xml_0_scc_12_large_scc_2 (algebra.Alg.Term algebra.F.id_a__U51 
                                        ((algebra.Alg.Term algebra.F.id_mark 
                                        (x7::nil))::x8::x9::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      
       (* <a__U51(tt,V1_,V2_),a__isNat(V1_)> *)
      | DP_R_xml_0_scc_12_large_scc_2_43 :
       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_12_large_scc_2 (algebra.Alg.Term 
                                          algebra.F.id_a__isNat (x1::nil)) 
            (algebra.Alg.Term algebra.F.id_a__U51 (x13::x12::x11::nil))
      
       (* <mark(U51(X1_,X2_,X3_)),mark(X1_)> *)
      | DP_R_xml_0_scc_12_large_scc_2_44 :
       forall x8 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_U51 (x7::x8::x9::nil)) x11) ->
         DP_R_xml_0_scc_12_large_scc_2 (algebra.Alg.Term algebra.F.id_mark 
                                        (x7::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      
       (* <mark(U52(X1_,X2_)),a__U52(mark(X1_),X2_)> *)
      | DP_R_xml_0_scc_12_large_scc_2_45 :
       forall x8 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_U52 (x7::x8::nil)) x11) ->
         DP_R_xml_0_scc_12_large_scc_2 (algebra.Alg.Term algebra.F.id_a__U52 
                                        ((algebra.Alg.Term algebra.F.id_mark 
                                        (x7::nil))::x8::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      
       (* <mark(U52(X1_,X2_)),mark(X1_)> *)
      | DP_R_xml_0_scc_12_large_scc_2_46 :
       forall x8 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_U52 (x7::x8::nil)) x11) ->
         DP_R_xml_0_scc_12_large_scc_2 (algebra.Alg.Term algebra.F.id_mark 
                                        (x7::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       (* <mark(U53(X_)),mark(X_)> *)
      | DP_R_xml_0_scc_12_large_scc_2_47 :
       forall 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_U53 (x5::nil)) x11) ->
         DP_R_xml_0_scc_12_large_scc_2 (algebra.Alg.Term algebra.F.id_mark 
                                        (x5::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      
       (* <mark(and(X1_,X2_)),a__and(mark(X1_),X2_)> *)
      | DP_R_xml_0_scc_12_large_scc_2_48 :
       forall x8 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_and (x7::x8::nil)) x11) ->
         DP_R_xml_0_scc_12_large_scc_2 (algebra.Alg.Term algebra.F.id_a__and 
                                        ((algebra.Alg.Term algebra.F.id_mark 
                                        (x7::nil))::x8::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      
       (* <mark(and(X1_,X2_)),mark(X1_)> *)
      | DP_R_xml_0_scc_12_large_scc_2_49 :
       forall x8 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_and (x7::x8::nil)) x11) ->
         DP_R_xml_0_scc_12_large_scc_2 (algebra.Alg.Term algebra.F.id_mark 
                                        (x7::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      
       (* <mark(isNatIListKind(X_)),a__isNatIListKind(X_)> *)
      | DP_R_xml_0_scc_12_large_scc_2_50 :
       forall 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_isNatIListKind (x5::nil)) x11) ->
         DP_R_xml_0_scc_12_large_scc_2 (algebra.Alg.Term 
                                        algebra.F.id_a__isNatIListKind 
                                        (x5::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      
       (* <mark(isNatKind(X_)),a__isNatKind(X_)> *)
      | DP_R_xml_0_scc_12_large_scc_2_51 :
       forall 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_isNatKind (x5::nil)) x11) ->
         DP_R_xml_0_scc_12_large_scc_2 (algebra.Alg.Term 
                                        algebra.F.id_a__isNatKind (x5::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      
       (* <mark(cons(X1_,X2_)),mark(X1_)> *)
      | DP_R_xml_0_scc_12_large_scc_2_52 :
       forall x8 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 (x7::x8::nil)) x11) ->
         DP_R_xml_0_scc_12_large_scc_2 (algebra.Alg.Term algebra.F.id_mark 
                                        (x7::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       (* <mark(s(X_)),mark(X_)> *)
      | DP_R_xml_0_scc_12_large_scc_2_53 :
       forall 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_s (x5::nil)) x11) ->
         DP_R_xml_0_scc_12_large_scc_2 (algebra.Alg.Term algebra.F.id_mark 
                                        (x5::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
    .
    
    
    Module WF_DP_R_xml_0_scc_12_large_scc_2.
     Inductive DP_R_xml_0_scc_12_large_scc_2_large  :
      algebra.Alg.term ->algebra.Alg.term ->Prop := 
        (* <a__isNatList(cons(V1_,V2_)),a__U51(a__and(a__isNatKind(V1_),isNatIListKind(V2_)),V1_,V2_)> *)
       | DP_R_xml_0_scc_12_large_scc_2_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_12_large_scc_2_large (algebra.Alg.Term 
                                               algebra.F.id_a__U51 
                                               ((algebra.Alg.Term 
                                               algebra.F.id_a__and 
                                               ((algebra.Alg.Term 
                                               algebra.F.id_a__isNatKind 
                                               (x1::nil))::(algebra.Alg.Term 
                                               algebra.F.id_isNatIListKind 
                                               (x3::nil))::nil))::x1::
                                               x3::nil)) 
           (algebra.Alg.Term algebra.F.id_a__isNatList (x11::nil))
       
        (* <a__U51(tt,V1_,V2_),a__U52(a__isNat(V1_),V2_)> *)
       | DP_R_xml_0_scc_12_large_scc_2_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_12_large_scc_2_large (algebra.Alg.Term 
                                                 algebra.F.id_a__U52 
                                                 ((algebra.Alg.Term 
                                                 algebra.F.id_a__isNat 
                                                 (x1::nil))::x3::nil)) 
             (algebra.Alg.Term algebra.F.id_a__U51 (x13::x12::x11::nil))
       
        (* <a__U52(tt,V2_),a__isNatList(V2_)> *)
       | DP_R_xml_0_scc_12_large_scc_2_large_2 :
        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_12_large_scc_2_large (algebra.Alg.Term 
                                                algebra.F.id_a__isNatList 
                                                (x3::nil)) 
            (algebra.Alg.Term algebra.F.id_a__U52 (x12::x11::nil))
       
        (* <a__isNatList(cons(V1_,V2_)),a__and(a__isNatKind(V1_),isNatIListKind(V2_))> *)
       | DP_R_xml_0_scc_12_large_scc_2_large_3 :
        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_12_large_scc_2_large (algebra.Alg.Term 
                                               algebra.F.id_a__and 
                                               ((algebra.Alg.Term 
                                               algebra.F.id_a__isNatKind 
                                               (x1::nil))::(algebra.Alg.Term 
                                               algebra.F.id_isNatIListKind 
                                               (x3::nil))::nil)) 
           (algebra.Alg.Term algebra.F.id_a__isNatList (x11::nil))
       
        (* <a__and(tt,X_),mark(X_)> *)
       | DP_R_xml_0_scc_12_large_scc_2_large_4 :
        forall x12 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_tt nil) 
           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_12_large_scc_2_large (algebra.Alg.Term 
                                                algebra.F.id_mark (x5::nil)) 
            (algebra.Alg.Term algebra.F.id_a__and (x12::x11::nil))
       
        (* <mark(U11(X1_,X2_)),a__U11(mark(X1_),X2_)> *)
       | DP_R_xml_0_scc_12_large_scc_2_large_5 :
        forall x8 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 (x7::x8::nil)) x11) ->
          DP_R_xml_0_scc_12_large_scc_2_large (algebra.Alg.Term 
                                               algebra.F.id_a__U11 
                                               ((algebra.Alg.Term 
                                               algebra.F.id_mark (x7::nil))::
                                               x8::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       
        (* <a__U11(tt,V1_),a__isNatList(V1_)> *)
       | DP_R_xml_0_scc_12_large_scc_2_large_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_scc_12_large_scc_2_large (algebra.Alg.Term 
                                                algebra.F.id_a__isNatList 
                                                (x1::nil)) 
            (algebra.Alg.Term algebra.F.id_a__U11 (x12::x11::nil))
       
        (* <a__isNatList(cons(V1_,V2_)),a__isNatKind(V1_)> *)
       | DP_R_xml_0_scc_12_large_scc_2_large_7 :
        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_12_large_scc_2_large (algebra.Alg.Term 
                                               algebra.F.id_a__isNatKind 
                                               (x1::nil)) 
           (algebra.Alg.Term algebra.F.id_a__isNatList (x11::nil))
       
        (* <a__isNatKind(length(V1_)),a__isNatIListKind(V1_)> *)
       | DP_R_xml_0_scc_12_large_scc_2_large_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_12_large_scc_2_large (algebra.Alg.Term 
                                               algebra.F.id_a__isNatIListKind 
                                               (x1::nil)) 
           (algebra.Alg.Term algebra.F.id_a__isNatKind (x11::nil))
       
        (* <a__isNatIListKind(cons(V1_,V2_)),a__and(a__isNatKind(V1_),isNatIListKind(V2_))> *)
       | DP_R_xml_0_scc_12_large_scc_2_large_9 :
        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_12_large_scc_2_large (algebra.Alg.Term 
                                               algebra.F.id_a__and 
                                               ((algebra.Alg.Term 
                                               algebra.F.id_a__isNatKind 
                                               (x1::nil))::(algebra.Alg.Term 
                                               algebra.F.id_isNatIListKind 
                                               (x3::nil))::nil)) 
           (algebra.Alg.Term algebra.F.id_a__isNatIListKind (x11::nil))
       
        (* <a__isNatIListKind(cons(V1_,V2_)),a__isNatKind(V1_)> *)
       | DP_R_xml_0_scc_12_large_scc_2_large_10 :
        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_12_large_scc_2_large (algebra.Alg.Term 
                                               algebra.F.id_a__isNatKind 
                                               (x1::nil)) 
           (algebra.Alg.Term algebra.F.id_a__isNatIListKind (x11::nil))
       
        (* <a__isNatKind(s(V1_)),a__isNatKind(V1_)> *)
       | DP_R_xml_0_scc_12_large_scc_2_large_11 :
        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_12_large_scc_2_large (algebra.Alg.Term 
                                               algebra.F.id_a__isNatKind 
                                               (x1::nil)) 
           (algebra.Alg.Term algebra.F.id_a__isNatKind (x11::nil))
       
        (* <mark(U11(X1_,X2_)),mark(X1_)> *)
       | DP_R_xml_0_scc_12_large_scc_2_large_12 :
        forall x8 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 (x7::x8::nil)) x11) ->
          DP_R_xml_0_scc_12_large_scc_2_large (algebra.Alg.Term 
                                               algebra.F.id_mark (x7::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       
        (* <mark(U12(X_)),mark(X_)> *)
       | DP_R_xml_0_scc_12_large_scc_2_large_13 :
        forall 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_U12 (x5::nil)) x11) ->
          DP_R_xml_0_scc_12_large_scc_2_large (algebra.Alg.Term 
                                               algebra.F.id_mark (x5::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       
        (* <mark(isNatList(X_)),a__isNatList(X_)> *)
       | DP_R_xml_0_scc_12_large_scc_2_large_14 :
        forall 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_isNatList (x5::nil)) x11) ->
          DP_R_xml_0_scc_12_large_scc_2_large (algebra.Alg.Term 
                                               algebra.F.id_a__isNatList 
                                               (x5::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       
        (* <mark(U21(X1_,X2_)),a__U21(mark(X1_),X2_)> *)
       | DP_R_xml_0_scc_12_large_scc_2_large_15 :
        forall x8 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 (x7::x8::nil)) x11) ->
          DP_R_xml_0_scc_12_large_scc_2_large (algebra.Alg.Term 
                                               algebra.F.id_a__U21 
                                               ((algebra.Alg.Term 
                                               algebra.F.id_mark (x7::nil))::
                                               x8::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       
        (* <a__U21(tt,V1_),a__isNat(V1_)> *)
       | DP_R_xml_0_scc_12_large_scc_2_large_16 :
        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_12_large_scc_2_large (algebra.Alg.Term 
                                                algebra.F.id_a__isNat 
                                                (x1::nil)) 
            (algebra.Alg.Term algebra.F.id_a__U21 (x12::x11::nil))
       
        (* <a__isNat(length(V1_)),a__U11(a__isNatIListKind(V1_),V1_)> *)
       | DP_R_xml_0_scc_12_large_scc_2_large_17 :
        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_12_large_scc_2_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__isNat(length(V1_)),a__isNatIListKind(V1_)> *)
       | DP_R_xml_0_scc_12_large_scc_2_large_18 :
        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_12_large_scc_2_large (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_scc_12_large_scc_2_large_19 :
        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_12_large_scc_2_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__isNat(s(V1_)),a__isNatKind(V1_)> *)
       | DP_R_xml_0_scc_12_large_scc_2_large_20 :
        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_12_large_scc_2_large (algebra.Alg.Term 
                                               algebra.F.id_a__isNatKind 
                                               (x1::nil)) 
           (algebra.Alg.Term algebra.F.id_a__isNat (x11::nil))
       
        (* <mark(U21(X1_,X2_)),mark(X1_)> *)
       | DP_R_xml_0_scc_12_large_scc_2_large_21 :
        forall x8 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 (x7::x8::nil)) x11) ->
          DP_R_xml_0_scc_12_large_scc_2_large (algebra.Alg.Term 
                                               algebra.F.id_mark (x7::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       
        (* <mark(U22(X_)),mark(X_)> *)
       | DP_R_xml_0_scc_12_large_scc_2_large_22 :
        forall 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_U22 (x5::nil)) x11) ->
          DP_R_xml_0_scc_12_large_scc_2_large (algebra.Alg.Term 
                                               algebra.F.id_mark (x5::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       
        (* <mark(isNat(X_)),a__isNat(X_)> *)
       | DP_R_xml_0_scc_12_large_scc_2_large_23 :
        forall 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_isNat (x5::nil)) x11) ->
          DP_R_xml_0_scc_12_large_scc_2_large (algebra.Alg.Term 
                                               algebra.F.id_a__isNat 
                                               (x5::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       
        (* <mark(U31(X1_,X2_)),a__U31(mark(X1_),X2_)> *)
       | DP_R_xml_0_scc_12_large_scc_2_large_24 :
        forall x8 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 (x7::x8::nil)) x11) ->
          DP_R_xml_0_scc_12_large_scc_2_large (algebra.Alg.Term 
                                               algebra.F.id_a__U31 
                                               ((algebra.Alg.Term 
                                               algebra.F.id_mark (x7::nil))::
                                               x8::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       
        (* <a__U31(tt,V_),a__isNatList(V_)> *)
       | DP_R_xml_0_scc_12_large_scc_2_large_25 :
        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_scc_12_large_scc_2_large (algebra.Alg.Term 
                                                algebra.F.id_a__isNatList 
                                                (x2::nil)) 
            (algebra.Alg.Term algebra.F.id_a__U31 (x12::x11::nil))
       
        (* <mark(U31(X1_,X2_)),mark(X1_)> *)
       | DP_R_xml_0_scc_12_large_scc_2_large_26 :
        forall x8 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 (x7::x8::nil)) x11) ->
          DP_R_xml_0_scc_12_large_scc_2_large (algebra.Alg.Term 
                                               algebra.F.id_mark (x7::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       
        (* <mark(U32(X_)),mark(X_)> *)
       | DP_R_xml_0_scc_12_large_scc_2_large_27 :
        forall 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_U32 (x5::nil)) x11) ->
          DP_R_xml_0_scc_12_large_scc_2_large (algebra.Alg.Term 
                                               algebra.F.id_mark (x5::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_scc_12_large_scc_2_large_28 :
        forall x8 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 (x7::x8::x9::nil)) x11) ->
          DP_R_xml_0_scc_12_large_scc_2_large (algebra.Alg.Term 
                                               algebra.F.id_a__U41 
                                               ((algebra.Alg.Term 
                                               algebra.F.id_mark (x7::nil))::
                                               x8::x9::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       
        (* <a__U41(tt,V1_,V2_),a__U42(a__isNat(V1_),V2_)> *)
       | DP_R_xml_0_scc_12_large_scc_2_large_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_scc_12_large_scc_2_large (algebra.Alg.Term 
                                                 algebra.F.id_a__U42 
                                                 ((algebra.Alg.Term 
                                                 algebra.F.id_a__isNat 
                                                 (x1::nil))::x3::nil)) 
             (algebra.Alg.Term algebra.F.id_a__U41 (x13::x12::x11::nil))
       
        (* <a__U42(tt,V2_),a__isNatIList(V2_)> *)
       | DP_R_xml_0_scc_12_large_scc_2_large_30 :
        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_12_large_scc_2_large (algebra.Alg.Term 
                                                algebra.F.id_a__isNatIList 
                                                (x3::nil)) 
            (algebra.Alg.Term algebra.F.id_a__U42 (x12::x11::nil))
       
        (* <a__isNatIList(V_),a__U31(a__isNatIListKind(V_),V_)> *)
       | DP_R_xml_0_scc_12_large_scc_2_large_31 :
        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_scc_12_large_scc_2_large (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_scc_12_large_scc_2_large_32 :
        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_scc_12_large_scc_2_large (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__and(a__isNatKind(V1_),isNatIListKind(V2_)),V1_,V2_)> *)
       | DP_R_xml_0_scc_12_large_scc_2_large_33 :
        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_12_large_scc_2_large (algebra.Alg.Term 
                                               algebra.F.id_a__U41 
                                               ((algebra.Alg.Term 
                                               algebra.F.id_a__and 
                                               ((algebra.Alg.Term 
                                               algebra.F.id_a__isNatKind 
                                               (x1::nil))::(algebra.Alg.Term 
                                               algebra.F.id_isNatIListKind 
                                               (x3::nil))::nil))::x1::
                                               x3::nil)) 
           (algebra.Alg.Term algebra.F.id_a__isNatIList (x11::nil))
       
        (* <a__U41(tt,V1_,V2_),a__isNat(V1_)> *)
       | DP_R_xml_0_scc_12_large_scc_2_large_34 :
        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_12_large_scc_2_large (algebra.Alg.Term 
                                                 algebra.F.id_a__isNat 
                                                 (x1::nil)) 
             (algebra.Alg.Term algebra.F.id_a__U41 (x13::x12::x11::nil))
       
        (* <a__isNatIList(cons(V1_,V2_)),a__and(a__isNatKind(V1_),isNatIListKind(V2_))> *)
       | DP_R_xml_0_scc_12_large_scc_2_large_35 :
        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_12_large_scc_2_large (algebra.Alg.Term 
                                               algebra.F.id_a__and 
                                               ((algebra.Alg.Term 
                                               algebra.F.id_a__isNatKind 
                                               (x1::nil))::(algebra.Alg.Term 
                                               algebra.F.id_isNatIListKind 
                                               (x3::nil))::nil)) 
           (algebra.Alg.Term algebra.F.id_a__isNatIList (x11::nil))
       
        (* <a__isNatIList(cons(V1_,V2_)),a__isNatKind(V1_)> *)
       | DP_R_xml_0_scc_12_large_scc_2_large_36 :
        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_12_large_scc_2_large (algebra.Alg.Term 
                                               algebra.F.id_a__isNatKind 
                                               (x1::nil)) 
           (algebra.Alg.Term algebra.F.id_a__isNatIList (x11::nil))
       
        (* <mark(U41(X1_,X2_,X3_)),mark(X1_)> *)
       | DP_R_xml_0_scc_12_large_scc_2_large_37 :
        forall x8 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 (x7::x8::x9::nil)) x11) ->
          DP_R_xml_0_scc_12_large_scc_2_large (algebra.Alg.Term 
                                               algebra.F.id_mark (x7::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       
        (* <mark(U42(X1_,X2_)),a__U42(mark(X1_),X2_)> *)
       | DP_R_xml_0_scc_12_large_scc_2_large_38 :
        forall x8 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 (x7::x8::nil)) x11) ->
          DP_R_xml_0_scc_12_large_scc_2_large (algebra.Alg.Term 
                                               algebra.F.id_a__U42 
                                               ((algebra.Alg.Term 
                                               algebra.F.id_mark (x7::nil))::
                                               x8::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       
        (* <mark(U42(X1_,X2_)),mark(X1_)> *)
       | DP_R_xml_0_scc_12_large_scc_2_large_39 :
        forall x8 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 (x7::x8::nil)) x11) ->
          DP_R_xml_0_scc_12_large_scc_2_large (algebra.Alg.Term 
                                               algebra.F.id_mark (x7::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       
        (* <mark(U43(X_)),mark(X_)> *)
       | DP_R_xml_0_scc_12_large_scc_2_large_40 :
        forall 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_U43 (x5::nil)) x11) ->
          DP_R_xml_0_scc_12_large_scc_2_large (algebra.Alg.Term 
                                               algebra.F.id_mark (x5::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       
        (* <mark(isNatIList(X_)),a__isNatIList(X_)> *)
       | DP_R_xml_0_scc_12_large_scc_2_large_41 :
        forall 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_isNatIList (x5::nil)) x11) ->
          DP_R_xml_0_scc_12_large_scc_2_large (algebra.Alg.Term 
                                               algebra.F.id_a__isNatIList 
                                               (x5::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       
        (* <mark(U51(X1_,X2_,X3_)),a__U51(mark(X1_),X2_,X3_)> *)
       | DP_R_xml_0_scc_12_large_scc_2_large_42 :
        forall x8 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_U51 (x7::x8::x9::nil)) x11) ->
          DP_R_xml_0_scc_12_large_scc_2_large (algebra.Alg.Term 
                                               algebra.F.id_a__U51 
                                               ((algebra.Alg.Term 
                                               algebra.F.id_mark (x7::nil))::
                                               x8::x9::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       
        (* <a__U51(tt,V1_,V2_),a__isNat(V1_)> *)
       | DP_R_xml_0_scc_12_large_scc_2_large_43 :
        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_12_large_scc_2_large (algebra.Alg.Term 
                                                 algebra.F.id_a__isNat 
                                                 (x1::nil)) 
             (algebra.Alg.Term algebra.F.id_a__U51 (x13::x12::x11::nil))
       
        (* <mark(U51(X1_,X2_,X3_)),mark(X1_)> *)
       | DP_R_xml_0_scc_12_large_scc_2_large_44 :
        forall x8 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_U51 (x7::x8::x9::nil)) x11) ->
          DP_R_xml_0_scc_12_large_scc_2_large (algebra.Alg.Term 
                                               algebra.F.id_mark (x7::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       
        (* <mark(U52(X1_,X2_)),a__U52(mark(X1_),X2_)> *)
       | DP_R_xml_0_scc_12_large_scc_2_large_45 :
        forall x8 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_U52 (x7::x8::nil)) x11) ->
          DP_R_xml_0_scc_12_large_scc_2_large (algebra.Alg.Term 
                                               algebra.F.id_a__U52 
                                               ((algebra.Alg.Term 
                                               algebra.F.id_mark (x7::nil))::
                                               x8::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       
        (* <mark(U52(X1_,X2_)),mark(X1_)> *)
       | DP_R_xml_0_scc_12_large_scc_2_large_46 :
        forall x8 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_U52 (x7::x8::nil)) x11) ->
          DP_R_xml_0_scc_12_large_scc_2_large (algebra.Alg.Term 
                                               algebra.F.id_mark (x7::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       
        (* <mark(U53(X_)),mark(X_)> *)
       | DP_R_xml_0_scc_12_large_scc_2_large_47 :
        forall 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_U53 (x5::nil)) x11) ->
          DP_R_xml_0_scc_12_large_scc_2_large (algebra.Alg.Term 
                                               algebra.F.id_mark (x5::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       
        (* <mark(and(X1_,X2_)),a__and(mark(X1_),X2_)> *)
       | DP_R_xml_0_scc_12_large_scc_2_large_48 :
        forall x8 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_and (x7::x8::nil)) x11) ->
          DP_R_xml_0_scc_12_large_scc_2_large (algebra.Alg.Term 
                                               algebra.F.id_a__and 
                                               ((algebra.Alg.Term 
                                               algebra.F.id_mark (x7::nil))::
                                               x8::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       
        (* <mark(and(X1_,X2_)),mark(X1_)> *)
       | DP_R_xml_0_scc_12_large_scc_2_large_49 :
        forall x8 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_and (x7::x8::nil)) x11) ->
          DP_R_xml_0_scc_12_large_scc_2_large (algebra.Alg.Term 
                                               algebra.F.id_mark (x7::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       
        (* <mark(isNatIListKind(X_)),a__isNatIListKind(X_)> *)
       | DP_R_xml_0_scc_12_large_scc_2_large_50 :
        forall 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_isNatIListKind (x5::nil)) x11) ->
          DP_R_xml_0_scc_12_large_scc_2_large (algebra.Alg.Term 
                                               algebra.F.id_a__isNatIListKind 
                                               (x5::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       
        (* <mark(isNatKind(X_)),a__isNatKind(X_)> *)
       | DP_R_xml_0_scc_12_large_scc_2_large_51 :
        forall 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_isNatKind (x5::nil)) x11) ->
          DP_R_xml_0_scc_12_large_scc_2_large (algebra.Alg.Term 
                                               algebra.F.id_a__isNatKind 
                                               (x5::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       
        (* <mark(s(X_)),mark(X_)> *)
       | DP_R_xml_0_scc_12_large_scc_2_large_52 :
        forall 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_s (x5::nil)) x11) ->
          DP_R_xml_0_scc_12_large_scc_2_large (algebra.Alg.Term 
                                               algebra.F.id_mark (x5::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     .
     
     
     Inductive DP_R_xml_0_scc_12_large_scc_2_strict  :
      algebra.Alg.term ->algebra.Alg.term ->Prop := 
        (* <mark(cons(X1_,X2_)),mark(X1_)> *)
       | DP_R_xml_0_scc_12_large_scc_2_strict_0 :
        forall x8 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 (x7::x8::nil)) x11) ->
          DP_R_xml_0_scc_12_large_scc_2_strict (algebra.Alg.Term 
                                                algebra.F.id_mark (x7::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x11::nil))
     .
     
     
     Module WF_DP_R_xml_0_scc_12_large_scc_2_large.
      Inductive DP_R_xml_0_scc_12_large_scc_2_large_large  :
       algebra.Alg.term ->algebra.Alg.term ->Prop := 
         (* <a__isNatList(cons(V1_,V2_)),a__U51(a__and(a__isNatKind(V1_),isNatIListKind(V2_)),V1_,V2_)> *)
        | DP_R_xml_0_scc_12_large_scc_2_large_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_12_large_scc_2_large_large (algebra.Alg.Term 
                                                      algebra.F.id_a__U51 
                                                      ((algebra.Alg.Term 
                                                      algebra.F.id_a__and 
                                                      ((algebra.Alg.Term 
                                                      algebra.F.id_a__isNatKind 
                                                      (x1::nil))::
                                                      (algebra.Alg.Term 
                                                      algebra.F.id_isNatIListKind 
                                                      (x3::nil))::nil))::x1::
                                                      x3::nil)) 
            (algebra.Alg.Term algebra.F.id_a__isNatList (x11::nil))
        
         (* <a__U51(tt,V1_,V2_),a__U52(a__isNat(V1_),V2_)> *)
        | DP_R_xml_0_scc_12_large_scc_2_large_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_12_large_scc_2_large_large (algebra.Alg.Term 
                                                        algebra.F.id_a__U52 
                                                        ((algebra.Alg.Term 
                                                        algebra.F.id_a__isNat 
                                                        (x1::nil))::x3::nil)) 
              (algebra.Alg.Term algebra.F.id_a__U51 (x13::x12::x11::nil))
        
         (* <a__U52(tt,V2_),a__isNatList(V2_)> *)
        | DP_R_xml_0_scc_12_large_scc_2_large_large_2 :
         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_12_large_scc_2_large_large (algebra.Alg.Term 
                                                       algebra.F.id_a__isNatList 
                                                       (x3::nil)) 
             (algebra.Alg.Term algebra.F.id_a__U52 (x12::x11::nil))
        
         (* <a__isNatList(cons(V1_,V2_)),a__and(a__isNatKind(V1_),isNatIListKind(V2_))> *)
        | DP_R_xml_0_scc_12_large_scc_2_large_large_3 :
         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_12_large_scc_2_large_large (algebra.Alg.Term 
                                                      algebra.F.id_a__and 
                                                      ((algebra.Alg.Term 
                                                      algebra.F.id_a__isNatKind 
                                                      (x1::nil))::
                                                      (algebra.Alg.Term 
                                                      algebra.F.id_isNatIListKind 
                                                      (x3::nil))::nil)) 
            (algebra.Alg.Term algebra.F.id_a__isNatList (x11::nil))
        
         (* <a__and(tt,X_),mark(X_)> *)
        | DP_R_xml_0_scc_12_large_scc_2_large_large_4 :
         forall x12 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_tt nil) 
            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_12_large_scc_2_large_large (algebra.Alg.Term 
                                                       algebra.F.id_mark 
                                                       (x5::nil)) 
             (algebra.Alg.Term algebra.F.id_a__and (x12::x11::nil))
        
         (* <mark(U11(X1_,X2_)),a__U11(mark(X1_),X2_)> *)
        | DP_R_xml_0_scc_12_large_scc_2_large_large_5 :
         forall x8 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 (x7::x8::nil)) x11) ->
           DP_R_xml_0_scc_12_large_scc_2_large_large (algebra.Alg.Term 
                                                      algebra.F.id_a__U11 
                                                      ((algebra.Alg.Term 
                                                      algebra.F.id_mark 
                                                      (x7::nil))::x8::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x11::nil))
        
         (* <a__U11(tt,V1_),a__isNatList(V1_)> *)
        | DP_R_xml_0_scc_12_large_scc_2_large_large_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_scc_12_large_scc_2_large_large (algebra.Alg.Term 
                                                       algebra.F.id_a__isNatList 
                                                       (x1::nil)) 
             (algebra.Alg.Term algebra.F.id_a__U11 (x12::x11::nil))
        
         (* <a__isNatList(cons(V1_,V2_)),a__isNatKind(V1_)> *)
        | DP_R_xml_0_scc_12_large_scc_2_large_large_7 :
         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_12_large_scc_2_large_large (algebra.Alg.Term 
                                                      algebra.F.id_a__isNatKind 
                                                      (x1::nil)) 
            (algebra.Alg.Term algebra.F.id_a__isNatList (x11::nil))
        
         (* <a__isNatKind(length(V1_)),a__isNatIListKind(V1_)> *)
        | DP_R_xml_0_scc_12_large_scc_2_large_large_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_12_large_scc_2_large_large (algebra.Alg.Term 
                                                      algebra.F.id_a__isNatIListKind 
                                                      (x1::nil)) 
            (algebra.Alg.Term algebra.F.id_a__isNatKind (x11::nil))
        
         (* <a__isNatIListKind(cons(V1_,V2_)),a__and(a__isNatKind(V1_),isNatIListKind(V2_))> *)
        | DP_R_xml_0_scc_12_large_scc_2_large_large_9 :
         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_12_large_scc_2_large_large (algebra.Alg.Term 
                                                      algebra.F.id_a__and 
                                                      ((algebra.Alg.Term 
                                                      algebra.F.id_a__isNatKind 
                                                      (x1::nil))::
                                                      (algebra.Alg.Term 
                                                      algebra.F.id_isNatIListKind 
                                                      (x3::nil))::nil)) 
            (algebra.Alg.Term algebra.F.id_a__isNatIListKind (x11::nil))
        
         (* <a__isNatIListKind(cons(V1_,V2_)),a__isNatKind(V1_)> *)
        | DP_R_xml_0_scc_12_large_scc_2_large_large_10 :
         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_12_large_scc_2_large_large (algebra.Alg.Term 
                                                      algebra.F.id_a__isNatKind 
                                                      (x1::nil)) 
            (algebra.Alg.Term algebra.F.id_a__isNatIListKind (x11::nil))
        
         (* <a__isNatKind(s(V1_)),a__isNatKind(V1_)> *)
        | DP_R_xml_0_scc_12_large_scc_2_large_large_11 :
         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_12_large_scc_2_large_large (algebra.Alg.Term 
                                                      algebra.F.id_a__isNatKind 
                                                      (x1::nil)) 
            (algebra.Alg.Term algebra.F.id_a__isNatKind (x11::nil))
        
         (* <mark(U11(X1_,X2_)),mark(X1_)> *)
        | DP_R_xml_0_scc_12_large_scc_2_large_large_12 :
         forall x8 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 (x7::x8::nil)) x11) ->
           DP_R_xml_0_scc_12_large_scc_2_large_large (algebra.Alg.Term 
                                                      algebra.F.id_mark 
                                                      (x7::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x11::nil))
        
         (* <mark(U12(X_)),mark(X_)> *)
        | DP_R_xml_0_scc_12_large_scc_2_large_large_13 :
         forall 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_U12 (x5::nil)) x11) ->
           DP_R_xml_0_scc_12_large_scc_2_large_large (algebra.Alg.Term 
                                                      algebra.F.id_mark 
                                                      (x5::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x11::nil))
        
         (* <mark(isNatList(X_)),a__isNatList(X_)> *)
        | DP_R_xml_0_scc_12_large_scc_2_large_large_14 :
         forall 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_isNatList (x5::nil)) x11) ->
           DP_R_xml_0_scc_12_large_scc_2_large_large (algebra.Alg.Term 
                                                      algebra.F.id_a__isNatList 
                                                      (x5::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x11::nil))
        
         (* <mark(U21(X1_,X2_)),a__U21(mark(X1_),X2_)> *)
        | DP_R_xml_0_scc_12_large_scc_2_large_large_15 :
         forall x8 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 (x7::x8::nil)) x11) ->
           DP_R_xml_0_scc_12_large_scc_2_large_large (algebra.Alg.Term 
                                                      algebra.F.id_a__U21 
                                                      ((algebra.Alg.Term 
                                                      algebra.F.id_mark 
                                                      (x7::nil))::x8::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x11::nil))
        
         (* <a__U21(tt,V1_),a__isNat(V1_)> *)
        | DP_R_xml_0_scc_12_large_scc_2_large_large_16 :
         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_12_large_scc_2_large_large (algebra.Alg.Term 
                                                       algebra.F.id_a__isNat 
                                                       (x1::nil)) 
             (algebra.Alg.Term algebra.F.id_a__U21 (x12::x11::nil))
        
         (* <a__isNat(length(V1_)),a__U11(a__isNatIListKind(V1_),V1_)> *)
        | DP_R_xml_0_scc_12_large_scc_2_large_large_17 :
         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_12_large_scc_2_large_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__isNat(length(V1_)),a__isNatIListKind(V1_)> *)
        | DP_R_xml_0_scc_12_large_scc_2_large_large_18 :
         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_12_large_scc_2_large_large (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_scc_12_large_scc_2_large_large_19 :
         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_12_large_scc_2_large_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__isNat(s(V1_)),a__isNatKind(V1_)> *)
        | DP_R_xml_0_scc_12_large_scc_2_large_large_20 :
         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_12_large_scc_2_large_large (algebra.Alg.Term 
                                                      algebra.F.id_a__isNatKind 
                                                      (x1::nil)) 
            (algebra.Alg.Term algebra.F.id_a__isNat (x11::nil))
        
         (* <mark(U21(X1_,X2_)),mark(X1_)> *)
        | DP_R_xml_0_scc_12_large_scc_2_large_large_21 :
         forall x8 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 (x7::x8::nil)) x11) ->
           DP_R_xml_0_scc_12_large_scc_2_large_large (algebra.Alg.Term 
                                                      algebra.F.id_mark 
                                                      (x7::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x11::nil))
        
         (* <mark(U22(X_)),mark(X_)> *)
        | DP_R_xml_0_scc_12_large_scc_2_large_large_22 :
         forall 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_U22 (x5::nil)) x11) ->
           DP_R_xml_0_scc_12_large_scc_2_large_large (algebra.Alg.Term 
                                                      algebra.F.id_mark 
                                                      (x5::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x11::nil))
        
         (* <mark(isNat(X_)),a__isNat(X_)> *)
        | DP_R_xml_0_scc_12_large_scc_2_large_large_23 :
         forall 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_isNat (x5::nil)) x11) ->
           DP_R_xml_0_scc_12_large_scc_2_large_large (algebra.Alg.Term 
                                                      algebra.F.id_a__isNat 
                                                      (x5::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x11::nil))
        
         (* <a__U31(tt,V_),a__isNatList(V_)> *)
        | DP_R_xml_0_scc_12_large_scc_2_large_large_24 :
         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_scc_12_large_scc_2_large_large (algebra.Alg.Term 
                                                       algebra.F.id_a__isNatList 
                                                       (x2::nil)) 
             (algebra.Alg.Term algebra.F.id_a__U31 (x12::x11::nil))
        
         (* <a__U41(tt,V1_,V2_),a__U42(a__isNat(V1_),V2_)> *)
        | DP_R_xml_0_scc_12_large_scc_2_large_large_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_scc_12_large_scc_2_large_large (algebra.Alg.Term 
                                                        algebra.F.id_a__U42 
                                                        ((algebra.Alg.Term 
                                                        algebra.F.id_a__isNat 
                                                        (x1::nil))::x3::nil)) 
              (algebra.Alg.Term algebra.F.id_a__U41 (x13::x12::x11::nil))
        
         (* <a__U42(tt,V2_),a__isNatIList(V2_)> *)
        | DP_R_xml_0_scc_12_large_scc_2_large_large_26 :
         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_12_large_scc_2_large_large (algebra.Alg.Term 
                                                       algebra.F.id_a__isNatIList 
                                                       (x3::nil)) 
             (algebra.Alg.Term algebra.F.id_a__U42 (x12::x11::nil))
        
         (* <a__isNatIList(V_),a__U31(a__isNatIListKind(V_),V_)> *)
        | DP_R_xml_0_scc_12_large_scc_2_large_large_27 :
         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_scc_12_large_scc_2_large_large (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_scc_12_large_scc_2_large_large_28 :
         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_scc_12_large_scc_2_large_large (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__and(a__isNatKind(V1_),isNatIListKind(V2_)),V1_,V2_)> *)
        | DP_R_xml_0_scc_12_large_scc_2_large_large_29 :
         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_12_large_scc_2_large_large (algebra.Alg.Term 
                                                      algebra.F.id_a__U41 
                                                      ((algebra.Alg.Term 
                                                      algebra.F.id_a__and 
                                                      ((algebra.Alg.Term 
                                                      algebra.F.id_a__isNatKind 
                                                      (x1::nil))::
                                                      (algebra.Alg.Term 
                                                      algebra.F.id_isNatIListKind 
                                                      (x3::nil))::nil))::x1::
                                                      x3::nil)) 
            (algebra.Alg.Term algebra.F.id_a__isNatIList (x11::nil))
        
         (* <a__U41(tt,V1_,V2_),a__isNat(V1_)> *)
        | DP_R_xml_0_scc_12_large_scc_2_large_large_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_scc_12_large_scc_2_large_large (algebra.Alg.Term 
                                                        algebra.F.id_a__isNat 
                                                        (x1::nil)) 
              (algebra.Alg.Term algebra.F.id_a__U41 (x13::x12::x11::nil))
        
         (* <a__isNatIList(cons(V1_,V2_)),a__and(a__isNatKind(V1_),isNatIListKind(V2_))> *)
        | DP_R_xml_0_scc_12_large_scc_2_large_large_31 :
         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_12_large_scc_2_large_large (algebra.Alg.Term 
                                                      algebra.F.id_a__and 
                                                      ((algebra.Alg.Term 
                                                      algebra.F.id_a__isNatKind 
                                                      (x1::nil))::
                                                      (algebra.Alg.Term 
                                                      algebra.F.id_isNatIListKind 
                                                      (x3::nil))::nil)) 
            (algebra.Alg.Term algebra.F.id_a__isNatIList (x11::nil))
        
         (* <a__isNatIList(cons(V1_,V2_)),a__isNatKind(V1_)> *)
        | DP_R_xml_0_scc_12_large_scc_2_large_large_32 :
         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_12_large_scc_2_large_large (algebra.Alg.Term 
                                                      algebra.F.id_a__isNatKind 
                                                      (x1::nil)) 
            (algebra.Alg.Term algebra.F.id_a__isNatIList (x11::nil))
        
         (* <mark(U43(X_)),mark(X_)> *)
        | DP_R_xml_0_scc_12_large_scc_2_large_large_33 :
         forall 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_U43 (x5::nil)) x11) ->
           DP_R_xml_0_scc_12_large_scc_2_large_large (algebra.Alg.Term 
                                                      algebra.F.id_mark 
                                                      (x5::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x11::nil))
        
         (* <mark(U51(X1_,X2_,X3_)),a__U51(mark(X1_),X2_,X3_)> *)
        | DP_R_xml_0_scc_12_large_scc_2_large_large_34 :
         forall x8 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_U51 (x7::x8::x9::nil)) x11) ->
           DP_R_xml_0_scc_12_large_scc_2_large_large (algebra.Alg.Term 
                                                      algebra.F.id_a__U51 
                                                      ((algebra.Alg.Term 
                                                      algebra.F.id_mark 
                                                      (x7::nil))::x8::
                                                      x9::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x11::nil))
        
         (* <a__U51(tt,V1_,V2_),a__isNat(V1_)> *)
        | DP_R_xml_0_scc_12_large_scc_2_large_large_35 :
         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_12_large_scc_2_large_large (algebra.Alg.Term 
                                                        algebra.F.id_a__isNat 
                                                        (x1::nil)) 
              (algebra.Alg.Term algebra.F.id_a__U51 (x13::x12::x11::nil))
        
         (* <mark(U51(X1_,X2_,X3_)),mark(X1_)> *)
        | DP_R_xml_0_scc_12_large_scc_2_large_large_36 :
         forall x8 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_U51 (x7::x8::x9::nil)) x11) ->
           DP_R_xml_0_scc_12_large_scc_2_large_large (algebra.Alg.Term 
                                                      algebra.F.id_mark 
                                                      (x7::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x11::nil))
        
         (* <mark(U52(X1_,X2_)),a__U52(mark(X1_),X2_)> *)
        | DP_R_xml_0_scc_12_large_scc_2_large_large_37 :
         forall x8 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_U52 (x7::x8::nil)) x11) ->
           DP_R_xml_0_scc_12_large_scc_2_large_large (algebra.Alg.Term 
                                                      algebra.F.id_a__U52 
                                                      ((algebra.Alg.Term 
                                                      algebra.F.id_mark 
                                                      (x7::nil))::x8::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x11::nil))
        
         (* <mark(U52(X1_,X2_)),mark(X1_)> *)
        | DP_R_xml_0_scc_12_large_scc_2_large_large_38 :
         forall x8 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_U52 (x7::x8::nil)) x11) ->
           DP_R_xml_0_scc_12_large_scc_2_large_large (algebra.Alg.Term 
                                                      algebra.F.id_mark 
                                                      (x7::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x11::nil))
        
         (* <mark(U53(X_)),mark(X_)> *)
        | DP_R_xml_0_scc_12_large_scc_2_large_large_39 :
         forall 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_U53 (x5::nil)) x11) ->
           DP_R_xml_0_scc_12_large_scc_2_large_large (algebra.Alg.Term 
                                                      algebra.F.id_mark 
                                                      (x5::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x11::nil))
        
         (* <mark(and(X1_,X2_)),a__and(mark(X1_),X2_)> *)
        | DP_R_xml_0_scc_12_large_scc_2_large_large_40 :
         forall x8 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_and (x7::x8::nil)) x11) ->
           DP_R_xml_0_scc_12_large_scc_2_large_large (algebra.Alg.Term 
                                                      algebra.F.id_a__and 
                                                      ((algebra.Alg.Term 
                                                      algebra.F.id_mark 
                                                      (x7::nil))::x8::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x11::nil))
        
         (* <mark(and(X1_,X2_)),mark(X1_)> *)
        | DP_R_xml_0_scc_12_large_scc_2_large_large_41 :
         forall x8 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_and (x7::x8::nil)) x11) ->
           DP_R_xml_0_scc_12_large_scc_2_large_large (algebra.Alg.Term 
                                                      algebra.F.id_mark 
                                                      (x7::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x11::nil))
        
         (* <mark(isNatIListKind(X_)),a__isNatIListKind(X_)> *)
        | DP_R_xml_0_scc_12_large_scc_2_large_large_42 :
         forall 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_isNatIListKind (x5::nil)) 
            x11) ->
           DP_R_xml_0_scc_12_large_scc_2_large_large (algebra.Alg.Term 
                                                      algebra.F.id_a__isNatIListKind 
                                                      (x5::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x11::nil))
        
         (* <mark(isNatKind(X_)),a__isNatKind(X_)> *)
        | DP_R_xml_0_scc_12_large_scc_2_large_large_43 :
         forall 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_isNatKind (x5::nil)) x11) ->
           DP_R_xml_0_scc_12_large_scc_2_large_large (algebra.Alg.Term 
                                                      algebra.F.id_a__isNatKind 
                                                      (x5::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x11::nil))
        
         (* <mark(s(X_)),mark(X_)> *)
        | DP_R_xml_0_scc_12_large_scc_2_large_large_44 :
         forall 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_s (x5::nil)) x11) ->
           DP_R_xml_0_scc_12_large_scc_2_large_large (algebra.Alg.Term 
                                                      algebra.F.id_mark 
                                                      (x5::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      .
      
      
      Inductive DP_R_xml_0_scc_12_large_scc_2_large_strict  :
       algebra.Alg.term ->algebra.Alg.term ->Prop := 
         (* <mark(U31(X1_,X2_)),a__U31(mark(X1_),X2_)> *)
        | DP_R_xml_0_scc_12_large_scc_2_large_strict_0 :
         forall x8 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 (x7::x8::nil)) x11) ->
           DP_R_xml_0_scc_12_large_scc_2_large_strict (algebra.Alg.Term 
                                                       algebra.F.id_a__U31 
                                                       ((algebra.Alg.Term 
                                                       algebra.F.id_mark 
                                                       (x7::nil))::x8::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x11::nil))
        
         (* <mark(U31(X1_,X2_)),mark(X1_)> *)
        | DP_R_xml_0_scc_12_large_scc_2_large_strict_1 :
         forall x8 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 (x7::x8::nil)) x11) ->
           DP_R_xml_0_scc_12_large_scc_2_large_strict (algebra.Alg.Term 
                                                       algebra.F.id_mark 
                                                       (x7::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x11::nil))
        
         (* <mark(U32(X_)),mark(X_)> *)
        | DP_R_xml_0_scc_12_large_scc_2_large_strict_2 :
         forall 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_U32 (x5::nil)) x11) ->
           DP_R_xml_0_scc_12_large_scc_2_large_strict (algebra.Alg.Term 
                                                       algebra.F.id_mark 
                                                       (x5::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_scc_12_large_scc_2_large_strict_3 :
         forall x8 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 (x7::x8::x9::nil)) x11) ->
           DP_R_xml_0_scc_12_large_scc_2_large_strict (algebra.Alg.Term 
                                                       algebra.F.id_a__U41 
                                                       ((algebra.Alg.Term 
                                                       algebra.F.id_mark 
                                                       (x7::nil))::x8::
                                                       x9::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x11::nil))
        
         (* <mark(U41(X1_,X2_,X3_)),mark(X1_)> *)
        | DP_R_xml_0_scc_12_large_scc_2_large_strict_4 :
         forall x8 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 (x7::x8::x9::nil)) x11) ->
           DP_R_xml_0_scc_12_large_scc_2_large_strict (algebra.Alg.Term 
                                                       algebra.F.id_mark 
                                                       (x7::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x11::nil))
        
         (* <mark(U42(X1_,X2_)),a__U42(mark(X1_),X2_)> *)
        | DP_R_xml_0_scc_12_large_scc_2_large_strict_5 :
         forall x8 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 (x7::x8::nil)) x11) ->
           DP_R_xml_0_scc_12_large_scc_2_large_strict (algebra.Alg.Term 
                                                       algebra.F.id_a__U42 
                                                       ((algebra.Alg.Term 
                                                       algebra.F.id_mark 
                                                       (x7::nil))::x8::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x11::nil))
        
         (* <mark(U42(X1_,X2_)),mark(X1_)> *)
        | DP_R_xml_0_scc_12_large_scc_2_large_strict_6 :
         forall x8 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 (x7::x8::nil)) x11) ->
           DP_R_xml_0_scc_12_large_scc_2_large_strict (algebra.Alg.Term 
                                                       algebra.F.id_mark 
                                                       (x7::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x11::nil))
        
         (* <mark(isNatIList(X_)),a__isNatIList(X_)> *)
        | DP_R_xml_0_scc_12_large_scc_2_large_strict_7 :
         forall 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_isNatIList (x5::nil)) x11) ->
           DP_R_xml_0_scc_12_large_scc_2_large_strict (algebra.Alg.Term 
                                                       algebra.F.id_a__isNatIList 
                                                       (x5::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x11::nil))
      .
      
      
      Module WF_DP_R_xml_0_scc_12_large_scc_2_large_large.
       Inductive DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1  :
        algebra.Alg.term ->algebra.Alg.term ->Prop := 
          (* <a__U51(tt,V1_,V2_),a__U52(a__isNat(V1_),V2_)> *)
         | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_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_12_large_scc_2_large_large_scc_1 (algebra.Alg.Term 
                                                               algebra.F.id_a__U52 
                                                               ((algebra.Alg.Term 
                                                               algebra.F.id_a__isNat 
                                                               (x1::nil))::
                                                               x3::nil)) 
               (algebra.Alg.Term algebra.F.id_a__U51 (x13::x12::x11::nil))
         
          (* <a__U52(tt,V2_),a__isNatList(V2_)> *)
         | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_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_12_large_scc_2_large_large_scc_1 (algebra.Alg.Term 
                                                              algebra.F.id_a__isNatList 
                                                              (x3::nil)) 
              (algebra.Alg.Term algebra.F.id_a__U52 (x12::x11::nil))
         
          (* <a__isNatList(cons(V1_,V2_)),a__U51(a__and(a__isNatKind(V1_),isNatIListKind(V2_)),V1_,V2_)> *)
         | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_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_12_large_scc_2_large_large_scc_1 (algebra.Alg.Term 
                                                             algebra.F.id_a__U51 
                                                             ((algebra.Alg.Term 
                                                             algebra.F.id_a__and 
                                                             ((algebra.Alg.Term 
                                                             algebra.F.id_a__isNatKind 
                                                             (x1::nil))::
                                                             (algebra.Alg.Term 
                                                             algebra.F.id_isNatIListKind 
                                                             (x3::nil))::nil))::
                                                             x1::x3::nil)) 
             (algebra.Alg.Term algebra.F.id_a__isNatList (x11::nil))
         
          (* <a__U51(tt,V1_,V2_),a__isNat(V1_)> *)
         | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_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_12_large_scc_2_large_large_scc_1 (algebra.Alg.Term 
                                                               algebra.F.id_a__isNat 
                                                               (x1::nil)) 
               (algebra.Alg.Term algebra.F.id_a__U51 (x13::x12::x11::nil))
         
          (* <a__isNat(length(V1_)),a__U11(a__isNatIListKind(V1_),V1_)> *)
         | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_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_length (x1::nil)) x11) ->
            DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1 (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__isNatList(V1_)> *)
         | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_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_scc_12_large_scc_2_large_large_scc_1 (algebra.Alg.Term 
                                                              algebra.F.id_a__isNatList 
                                                              (x1::nil)) 
              (algebra.Alg.Term algebra.F.id_a__U11 (x12::x11::nil))
         
          (* <a__isNatList(cons(V1_,V2_)),a__and(a__isNatKind(V1_),isNatIListKind(V2_))> *)
         | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_6 :
          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_12_large_scc_2_large_large_scc_1 (algebra.Alg.Term 
                                                             algebra.F.id_a__and 
                                                             ((algebra.Alg.Term 
                                                             algebra.F.id_a__isNatKind 
                                                             (x1::nil))::
                                                             (algebra.Alg.Term 
                                                             algebra.F.id_isNatIListKind 
                                                             (x3::nil))::nil)) 
             (algebra.Alg.Term algebra.F.id_a__isNatList (x11::nil))
         
          (* <a__and(tt,X_),mark(X_)> *)
         | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_7 :
          forall x12 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_tt nil) 
             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_12_large_scc_2_large_large_scc_1 (algebra.Alg.Term 
                                                              algebra.F.id_mark 
                                                              (x5::nil)) 
              (algebra.Alg.Term algebra.F.id_a__and (x12::x11::nil))
         
          (* <mark(U11(X1_,X2_)),a__U11(mark(X1_),X2_)> *)
         | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_8 :
          forall x8 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 (x7::x8::nil)) x11) ->
            DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1 (algebra.Alg.Term 
                                                             algebra.F.id_a__U11 
                                                             ((algebra.Alg.Term 
                                                             algebra.F.id_mark 
                                                             (x7::nil))::
                                                             x8::nil)) 
             (algebra.Alg.Term algebra.F.id_mark (x11::nil))
         
          (* <mark(U11(X1_,X2_)),mark(X1_)> *)
         | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_9 :
          forall x8 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 (x7::x8::nil)) x11) ->
            DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1 (algebra.Alg.Term 
                                                             algebra.F.id_mark 
                                                             (x7::nil)) 
             (algebra.Alg.Term algebra.F.id_mark (x11::nil))
         
          (* <mark(U12(X_)),mark(X_)> *)
         | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_10 :
          forall 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_U12 (x5::nil)) x11) ->
            DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1 (algebra.Alg.Term 
                                                             algebra.F.id_mark 
                                                             (x5::nil)) 
             (algebra.Alg.Term algebra.F.id_mark (x11::nil))
         
          (* <mark(isNatList(X_)),a__isNatList(X_)> *)
         | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_11 :
          forall 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_isNatList (x5::nil)) x11) ->
            DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1 (algebra.Alg.Term 
                                                             algebra.F.id_a__isNatList 
                                                             (x5::nil)) 
             (algebra.Alg.Term algebra.F.id_mark (x11::nil))
         
          (* <a__isNatList(cons(V1_,V2_)),a__isNatKind(V1_)> *)
         | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_12 :
          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_12_large_scc_2_large_large_scc_1 (algebra.Alg.Term 
                                                             algebra.F.id_a__isNatKind 
                                                             (x1::nil)) 
             (algebra.Alg.Term algebra.F.id_a__isNatList (x11::nil))
         
          (* <a__isNatKind(length(V1_)),a__isNatIListKind(V1_)> *)
         | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_13 :
          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_12_large_scc_2_large_large_scc_1 (algebra.Alg.Term 
                                                             algebra.F.id_a__isNatIListKind 
                                                             (x1::nil)) 
             (algebra.Alg.Term algebra.F.id_a__isNatKind (x11::nil))
         
          (* <a__isNatIListKind(cons(V1_,V2_)),a__and(a__isNatKind(V1_),isNatIListKind(V2_))> *)
         | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_14 :
          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_12_large_scc_2_large_large_scc_1 (algebra.Alg.Term 
                                                             algebra.F.id_a__and 
                                                             ((algebra.Alg.Term 
                                                             algebra.F.id_a__isNatKind 
                                                             (x1::nil))::
                                                             (algebra.Alg.Term 
                                                             algebra.F.id_isNatIListKind 
                                                             (x3::nil))::nil)) 
             (algebra.Alg.Term algebra.F.id_a__isNatIListKind (x11::nil))
         
          (* <a__isNatIListKind(cons(V1_,V2_)),a__isNatKind(V1_)> *)
         | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_15 :
          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_12_large_scc_2_large_large_scc_1 (algebra.Alg.Term 
                                                             algebra.F.id_a__isNatKind 
                                                             (x1::nil)) 
             (algebra.Alg.Term algebra.F.id_a__isNatIListKind (x11::nil))
         
          (* <a__isNatKind(s(V1_)),a__isNatKind(V1_)> *)
         | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_16 :
          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_12_large_scc_2_large_large_scc_1 (algebra.Alg.Term 
                                                             algebra.F.id_a__isNatKind 
                                                             (x1::nil)) 
             (algebra.Alg.Term algebra.F.id_a__isNatKind (x11::nil))
         
          (* <mark(U21(X1_,X2_)),a__U21(mark(X1_),X2_)> *)
         | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_17 :
          forall x8 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 (x7::x8::nil)) x11) ->
            DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1 (algebra.Alg.Term 
                                                             algebra.F.id_a__U21 
                                                             ((algebra.Alg.Term 
                                                             algebra.F.id_mark 
                                                             (x7::nil))::
                                                             x8::nil)) 
             (algebra.Alg.Term algebra.F.id_mark (x11::nil))
         
          (* <a__U21(tt,V1_),a__isNat(V1_)> *)
         | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_18 :
          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_12_large_scc_2_large_large_scc_1 (algebra.Alg.Term 
                                                              algebra.F.id_a__isNat 
                                                              (x1::nil)) 
              (algebra.Alg.Term algebra.F.id_a__U21 (x12::x11::nil))
         
          (* <a__isNat(length(V1_)),a__isNatIListKind(V1_)> *)
         | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_19 :
          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_12_large_scc_2_large_large_scc_1 (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_scc_12_large_scc_2_large_large_scc_1_20 :
          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_12_large_scc_2_large_large_scc_1 (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_scc_12_large_scc_2_large_large_scc_1_21 :
          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_12_large_scc_2_large_large_scc_1 (algebra.Alg.Term 
                                                             algebra.F.id_a__isNatKind 
                                                             (x1::nil)) 
             (algebra.Alg.Term algebra.F.id_a__isNat (x11::nil))
         
          (* <mark(U21(X1_,X2_)),mark(X1_)> *)
         | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_22 :
          forall x8 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 (x7::x8::nil)) x11) ->
            DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1 (algebra.Alg.Term 
                                                             algebra.F.id_mark 
                                                             (x7::nil)) 
             (algebra.Alg.Term algebra.F.id_mark (x11::nil))
         
          (* <mark(U22(X_)),mark(X_)> *)
         | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_23 :
          forall 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_U22 (x5::nil)) x11) ->
            DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1 (algebra.Alg.Term 
                                                             algebra.F.id_mark 
                                                             (x5::nil)) 
             (algebra.Alg.Term algebra.F.id_mark (x11::nil))
         
          (* <mark(isNat(X_)),a__isNat(X_)> *)
         | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_24 :
          forall 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_isNat (x5::nil)) x11) ->
            DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1 (algebra.Alg.Term 
                                                             algebra.F.id_a__isNat 
                                                             (x5::nil)) 
             (algebra.Alg.Term algebra.F.id_mark (x11::nil))
         
          (* <mark(U43(X_)),mark(X_)> *)
         | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_25 :
          forall 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_U43 (x5::nil)) x11) ->
            DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1 (algebra.Alg.Term 
                                                             algebra.F.id_mark 
                                                             (x5::nil)) 
             (algebra.Alg.Term algebra.F.id_mark (x11::nil))
         
          (* <mark(U51(X1_,X2_,X3_)),a__U51(mark(X1_),X2_,X3_)> *)
         | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_26 :
          forall x8 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_U51 (x7::x8::x9::nil)) x11) ->
            DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1 (algebra.Alg.Term 
                                                             algebra.F.id_a__U51 
                                                             ((algebra.Alg.Term 
                                                             algebra.F.id_mark 
                                                             (x7::nil))::x8::
                                                             x9::nil)) 
             (algebra.Alg.Term algebra.F.id_mark (x11::nil))
         
          (* <mark(U51(X1_,X2_,X3_)),mark(X1_)> *)
         | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_27 :
          forall x8 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_U51 (x7::x8::x9::nil)) x11) ->
            DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1 (algebra.Alg.Term 
                                                             algebra.F.id_mark 
                                                             (x7::nil)) 
             (algebra.Alg.Term algebra.F.id_mark (x11::nil))
         
          (* <mark(U52(X1_,X2_)),a__U52(mark(X1_),X2_)> *)
         | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_28 :
          forall x8 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_U52 (x7::x8::nil)) x11) ->
            DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1 (algebra.Alg.Term 
                                                             algebra.F.id_a__U52 
                                                             ((algebra.Alg.Term 
                                                             algebra.F.id_mark 
                                                             (x7::nil))::
                                                             x8::nil)) 
             (algebra.Alg.Term algebra.F.id_mark (x11::nil))
         
          (* <mark(U52(X1_,X2_)),mark(X1_)> *)
         | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_29 :
          forall x8 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_U52 (x7::x8::nil)) x11) ->
            DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1 (algebra.Alg.Term 
                                                             algebra.F.id_mark 
                                                             (x7::nil)) 
             (algebra.Alg.Term algebra.F.id_mark (x11::nil))
         
          (* <mark(U53(X_)),mark(X_)> *)
         | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_30 :
          forall 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_U53 (x5::nil)) x11) ->
            DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1 (algebra.Alg.Term 
                                                             algebra.F.id_mark 
                                                             (x5::nil)) 
             (algebra.Alg.Term algebra.F.id_mark (x11::nil))
         
          (* <mark(and(X1_,X2_)),a__and(mark(X1_),X2_)> *)
         | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_31 :
          forall x8 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_and (x7::x8::nil)) x11) ->
            DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1 (algebra.Alg.Term 
                                                             algebra.F.id_a__and 
                                                             ((algebra.Alg.Term 
                                                             algebra.F.id_mark 
                                                             (x7::nil))::
                                                             x8::nil)) 
             (algebra.Alg.Term algebra.F.id_mark (x11::nil))
         
          (* <mark(and(X1_,X2_)),mark(X1_)> *)
         | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_32 :
          forall x8 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_and (x7::x8::nil)) x11) ->
            DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1 (algebra.Alg.Term 
                                                             algebra.F.id_mark 
                                                             (x7::nil)) 
             (algebra.Alg.Term algebra.F.id_mark (x11::nil))
         
          (* <mark(isNatIListKind(X_)),a__isNatIListKind(X_)> *)
         | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_33 :
          forall 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_isNatIListKind (x5::nil)) 
             x11) ->
            DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1 (algebra.Alg.Term 
                                                             algebra.F.id_a__isNatIListKind 
                                                             (x5::nil)) 
             (algebra.Alg.Term algebra.F.id_mark (x11::nil))
         
          (* <mark(isNatKind(X_)),a__isNatKind(X_)> *)
         | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_34 :
          forall 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_isNatKind (x5::nil)) x11) ->
            DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1 (algebra.Alg.Term 
                                                             algebra.F.id_a__isNatKind 
                                                             (x5::nil)) 
             (algebra.Alg.Term algebra.F.id_mark (x11::nil))
         
          (* <mark(s(X_)),mark(X_)> *)
         | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_35 :
          forall 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_s (x5::nil)) x11) ->
            DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1 (algebra.Alg.Term 
                                                             algebra.F.id_mark 
                                                             (x5::nil)) 
             (algebra.Alg.Term algebra.F.id_mark (x11::nil))
       .
       
       
       Module WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1.
        Inductive DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large  :
         algebra.Alg.term ->algebra.Alg.term ->Prop := 
           (* <a__U51(tt,V1_,V2_),a__U52(a__isNat(V1_),V2_)> *)
          | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_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_12_large_scc_2_large_large_scc_1_large 
                (algebra.Alg.Term algebra.F.id_a__U52 ((algebra.Alg.Term 
                 algebra.F.id_a__isNat (x1::nil))::x3::nil)) 
                (algebra.Alg.Term algebra.F.id_a__U51 (x13::x12::x11::nil))
          
           (* <a__U52(tt,V2_),a__isNatList(V2_)> *)
          | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_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_12_large_scc_2_large_large_scc_1_large (algebra.Alg.Term 
                                                                    algebra.F.id_a__isNatList 
                                                                    (x3::nil)) 
               (algebra.Alg.Term algebra.F.id_a__U52 (x12::x11::nil))
          
           (* <a__isNatList(cons(V1_,V2_)),a__U51(a__and(a__isNatKind(V1_),isNatIListKind(V2_)),V1_,V2_)> *)
          | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_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_12_large_scc_2_large_large_scc_1_large (algebra.Alg.Term 
                                                                    algebra.F.id_a__U51 
                                                                    ((
                                                                    algebra.Alg.Term 
                                                                    algebra.F.id_a__and 
                                                                    ((
                                                                    algebra.Alg.Term 
                                                                    algebra.F.id_a__isNatKind 
                                                                    (x1::nil))::
                                                                    
                                                                    (
                                                                    algebra.Alg.Term 
                                                                    algebra.F.id_isNatIListKind 
                                                                    (x3::nil))::nil))::
                                                                    x1::
                                                                    x3::nil)) 
              (algebra.Alg.Term algebra.F.id_a__isNatList (x11::nil))
          
           (* <a__U51(tt,V1_,V2_),a__isNat(V1_)> *)
          | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_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_12_large_scc_2_large_large_scc_1_large 
                (algebra.Alg.Term algebra.F.id_a__isNat (x1::nil)) 
                (algebra.Alg.Term algebra.F.id_a__U51 (x13::x12::x11::nil))
          
           (* <a__isNat(length(V1_)),a__U11(a__isNatIListKind(V1_),V1_)> *)
          | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_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_length (x1::nil)) x11) ->
             DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_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__isNatList(V1_)> *)
          | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_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_scc_12_large_scc_2_large_large_scc_1_large (algebra.Alg.Term 
                                                                    algebra.F.id_a__isNatList 
                                                                    (x1::nil)) 
               (algebra.Alg.Term algebra.F.id_a__U11 (x12::x11::nil))
          
           (* <a__isNatList(cons(V1_,V2_)),a__and(a__isNatKind(V1_),isNatIListKind(V2_))> *)
          | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_6 :
           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_12_large_scc_2_large_large_scc_1_large (algebra.Alg.Term 
                                                                    algebra.F.id_a__and 
                                                                    ((
                                                                    algebra.Alg.Term 
                                                                    algebra.F.id_a__isNatKind 
                                                                    (x1::nil))::
                                                                    
                                                                    (
                                                                    algebra.Alg.Term 
                                                                    algebra.F.id_isNatIListKind 
                                                                    (x3::nil))::nil))
                                                                     
              (algebra.Alg.Term algebra.F.id_a__isNatList (x11::nil))
          
           (* <a__and(tt,X_),mark(X_)> *)
          | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_7 :
           forall x12 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_tt nil) 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_12_large_scc_2_large_large_scc_1_large (algebra.Alg.Term 
                                                                    algebra.F.id_mark 
                                                                    (x5::nil)) 
               (algebra.Alg.Term algebra.F.id_a__and (x12::x11::nil))
          
           (* <mark(U12(X_)),mark(X_)> *)
          | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_8 :
           forall 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_U12 (x5::nil)) x11) ->
             DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large (algebra.Alg.Term 
                                                                    algebra.F.id_mark 
                                                                    (x5::nil)) 
              (algebra.Alg.Term algebra.F.id_mark (x11::nil))
          
           (* <mark(isNatList(X_)),a__isNatList(X_)> *)
          | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_9 :
           forall 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_isNatList (x5::nil)) x11) ->
             DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large (algebra.Alg.Term 
                                                                    algebra.F.id_a__isNatList 
                                                                    (x5::nil)) 
              (algebra.Alg.Term algebra.F.id_mark (x11::nil))
          
           (* <a__isNatList(cons(V1_,V2_)),a__isNatKind(V1_)> *)
          | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_10 :
           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_12_large_scc_2_large_large_scc_1_large (algebra.Alg.Term 
                                                                    algebra.F.id_a__isNatKind 
                                                                    (x1::nil)) 
              (algebra.Alg.Term algebra.F.id_a__isNatList (x11::nil))
          
           (* <a__isNatKind(length(V1_)),a__isNatIListKind(V1_)> *)
          | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_11 :
           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_12_large_scc_2_large_large_scc_1_large (algebra.Alg.Term 
                                                                    algebra.F.id_a__isNatIListKind 
                                                                    (x1::nil)) 
              (algebra.Alg.Term algebra.F.id_a__isNatKind (x11::nil))
          
           (* <a__isNatIListKind(cons(V1_,V2_)),a__and(a__isNatKind(V1_),isNatIListKind(V2_))> *)
          | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_12 :
           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_12_large_scc_2_large_large_scc_1_large (algebra.Alg.Term 
                                                                    algebra.F.id_a__and 
                                                                    ((
                                                                    algebra.Alg.Term 
                                                                    algebra.F.id_a__isNatKind 
                                                                    (x1::nil))::
                                                                    
                                                                    (
                                                                    algebra.Alg.Term 
                                                                    algebra.F.id_isNatIListKind 
                                                                    (x3::nil))::nil))
                                                                     
              (algebra.Alg.Term algebra.F.id_a__isNatIListKind (x11::nil))
          
           (* <a__isNatIListKind(cons(V1_,V2_)),a__isNatKind(V1_)> *)
          | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_13 :
           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_12_large_scc_2_large_large_scc_1_large (algebra.Alg.Term 
                                                                    algebra.F.id_a__isNatKind 
                                                                    (x1::nil)) 
              (algebra.Alg.Term algebra.F.id_a__isNatIListKind (x11::nil))
          
           (* <a__isNatKind(s(V1_)),a__isNatKind(V1_)> *)
          | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_14 :
           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_12_large_scc_2_large_large_scc_1_large (algebra.Alg.Term 
                                                                    algebra.F.id_a__isNatKind 
                                                                    (x1::nil)) 
              (algebra.Alg.Term algebra.F.id_a__isNatKind (x11::nil))
          
           (* <mark(U21(X1_,X2_)),a__U21(mark(X1_),X2_)> *)
          | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_15 :
           forall x8 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 (x7::x8::nil)) x11) ->
             DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large (algebra.Alg.Term 
                                                                    algebra.F.id_a__U21 
                                                                    ((
                                                                    algebra.Alg.Term 
                                                                    algebra.F.id_mark 
                                                                    (x7::nil))::
                                                                    x8::nil)) 
              (algebra.Alg.Term algebra.F.id_mark (x11::nil))
          
           (* <a__U21(tt,V1_),a__isNat(V1_)> *)
          | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_16 :
           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_12_large_scc_2_large_large_scc_1_large (algebra.Alg.Term 
                                                                    algebra.F.id_a__isNat 
                                                                    (x1::nil)) 
               (algebra.Alg.Term algebra.F.id_a__U21 (x12::x11::nil))
          
           (* <a__isNat(length(V1_)),a__isNatIListKind(V1_)> *)
          | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_17 :
           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_12_large_scc_2_large_large_scc_1_large (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_scc_12_large_scc_2_large_large_scc_1_large_18 :
           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_12_large_scc_2_large_large_scc_1_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__isNat(s(V1_)),a__isNatKind(V1_)> *)
          | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_19 :
           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_12_large_scc_2_large_large_scc_1_large (algebra.Alg.Term 
                                                                    algebra.F.id_a__isNatKind 
                                                                    (x1::nil)) 
              (algebra.Alg.Term algebra.F.id_a__isNat (x11::nil))
          
           (* <mark(U21(X1_,X2_)),mark(X1_)> *)
          | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_20 :
           forall x8 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 (x7::x8::nil)) x11) ->
             DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large (algebra.Alg.Term 
                                                                    algebra.F.id_mark 
                                                                    (x7::nil)) 
              (algebra.Alg.Term algebra.F.id_mark (x11::nil))
          
           (* <mark(U22(X_)),mark(X_)> *)
          | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_21 :
           forall 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_U22 (x5::nil)) x11) ->
             DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large (algebra.Alg.Term 
                                                                    algebra.F.id_mark 
                                                                    (x5::nil)) 
              (algebra.Alg.Term algebra.F.id_mark (x11::nil))
          
           (* <mark(isNat(X_)),a__isNat(X_)> *)
          | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_22 :
           forall 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_isNat (x5::nil)) x11) ->
             DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large (algebra.Alg.Term 
                                                                    algebra.F.id_a__isNat 
                                                                    (x5::nil)) 
              (algebra.Alg.Term algebra.F.id_mark (x11::nil))
          
           (* <mark(U43(X_)),mark(X_)> *)
          | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_23 :
           forall 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_U43 (x5::nil)) x11) ->
             DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large (algebra.Alg.Term 
                                                                    algebra.F.id_mark 
                                                                    (x5::nil)) 
              (algebra.Alg.Term algebra.F.id_mark (x11::nil))
          
           (* <mark(U51(X1_,X2_,X3_)),a__U51(mark(X1_),X2_,X3_)> *)
          | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_24 :
           forall x8 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_U51 (x7::x8::x9::nil)) x11) ->
             DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large (algebra.Alg.Term 
                                                                    algebra.F.id_a__U51 
                                                                    ((
                                                                    algebra.Alg.Term 
                                                                    algebra.F.id_mark 
                                                                    (x7::nil))::
                                                                    x8::
                                                                    x9::nil)) 
              (algebra.Alg.Term algebra.F.id_mark (x11::nil))
          
           (* <mark(U51(X1_,X2_,X3_)),mark(X1_)> *)
          | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_25 :
           forall x8 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_U51 (x7::x8::x9::nil)) x11) ->
             DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large (algebra.Alg.Term 
                                                                    algebra.F.id_mark 
                                                                    (x7::nil)) 
              (algebra.Alg.Term algebra.F.id_mark (x11::nil))
          
           (* <mark(U52(X1_,X2_)),a__U52(mark(X1_),X2_)> *)
          | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_26 :
           forall x8 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_U52 (x7::x8::nil)) x11) ->
             DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large (algebra.Alg.Term 
                                                                    algebra.F.id_a__U52 
                                                                    ((
                                                                    algebra.Alg.Term 
                                                                    algebra.F.id_mark 
                                                                    (x7::nil))::
                                                                    x8::nil)) 
              (algebra.Alg.Term algebra.F.id_mark (x11::nil))
          
           (* <mark(U52(X1_,X2_)),mark(X1_)> *)
          | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_27 :
           forall x8 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_U52 (x7::x8::nil)) x11) ->
             DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large (algebra.Alg.Term 
                                                                    algebra.F.id_mark 
                                                                    (x7::nil)) 
              (algebra.Alg.Term algebra.F.id_mark (x11::nil))
          
           (* <mark(U53(X_)),mark(X_)> *)
          | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_28 :
           forall 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_U53 (x5::nil)) x11) ->
             DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large (algebra.Alg.Term 
                                                                    algebra.F.id_mark 
                                                                    (x5::nil)) 
              (algebra.Alg.Term algebra.F.id_mark (x11::nil))
          
           (* <mark(and(X1_,X2_)),a__and(mark(X1_),X2_)> *)
          | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_29 :
           forall x8 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_and (x7::x8::nil)) x11) ->
             DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large (algebra.Alg.Term 
                                                                    algebra.F.id_a__and 
                                                                    ((
                                                                    algebra.Alg.Term 
                                                                    algebra.F.id_mark 
                                                                    (x7::nil))::
                                                                    x8::nil)) 
              (algebra.Alg.Term algebra.F.id_mark (x11::nil))
          
           (* <mark(and(X1_,X2_)),mark(X1_)> *)
          | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_30 :
           forall x8 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_and (x7::x8::nil)) x11) ->
             DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large (algebra.Alg.Term 
                                                                    algebra.F.id_mark 
                                                                    (x7::nil)) 
              (algebra.Alg.Term algebra.F.id_mark (x11::nil))
          
           (* <mark(isNatIListKind(X_)),a__isNatIListKind(X_)> *)
          | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_31 :
           forall 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_isNatIListKind (x5::nil)) 
              x11) ->
             DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large (algebra.Alg.Term 
                                                                    algebra.F.id_a__isNatIListKind 
                                                                    (x5::nil)) 
              (algebra.Alg.Term algebra.F.id_mark (x11::nil))
          
           (* <mark(isNatKind(X_)),a__isNatKind(X_)> *)
          | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_32 :
           forall 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_isNatKind (x5::nil)) x11) ->
             DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large (algebra.Alg.Term 
                                                                    algebra.F.id_a__isNatKind 
                                                                    (x5::nil)) 
              (algebra.Alg.Term algebra.F.id_mark (x11::nil))
          
           (* <mark(s(X_)),mark(X_)> *)
          | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_33 :
           forall 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_s (x5::nil)) x11) ->
             DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large (algebra.Alg.Term 
                                                                    algebra.F.id_mark 
                                                                    (x5::nil)) 
              (algebra.Alg.Term algebra.F.id_mark (x11::nil))
        .
        
        
        Inductive DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_strict  :
         algebra.Alg.term ->algebra.Alg.term ->Prop := 
           (* <mark(U11(X1_,X2_)),a__U11(mark(X1_),X2_)> *)
          | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_strict_0 :
           forall x8 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 (x7::x8::nil)) x11) ->
             DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_strict (algebra.Alg.Term 
                                                                    algebra.F.id_a__U11 
                                                                    ((
                                                                    algebra.Alg.Term 
                                                                    algebra.F.id_mark 
                                                                    (x7::nil))::
                                                                    x8::nil)) 
              (algebra.Alg.Term algebra.F.id_mark (x11::nil))
          
           (* <mark(U11(X1_,X2_)),mark(X1_)> *)
          | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_strict_1 :
           forall x8 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 (x7::x8::nil)) x11) ->
             DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_strict (algebra.Alg.Term 
                                                                    algebra.F.id_mark 
                                                                    (x7::nil)) 
              (algebra.Alg.Term algebra.F.id_mark (x11::nil))
        .
        
        
        Module WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large.
         Inductive DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large  :
          algebra.Alg.term ->algebra.Alg.term ->Prop := 
            (* <a__U51(tt,V1_,V2_),a__U52(a__isNat(V1_),V2_)> *)
           | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_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_12_large_scc_2_large_large_scc_1_large_large 
                 (algebra.Alg.Term algebra.F.id_a__U52 ((algebra.Alg.Term 
                  algebra.F.id_a__isNat (x1::nil))::x3::nil)) 
                 (algebra.Alg.Term algebra.F.id_a__U51 (x13::x12::x11::nil))
           
            (* <a__U52(tt,V2_),a__isNatList(V2_)> *)
           | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_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_12_large_scc_2_large_large_scc_1_large_large 
                (algebra.Alg.Term algebra.F.id_a__isNatList (x3::nil)) 
                (algebra.Alg.Term algebra.F.id_a__U52 (x12::x11::nil))
           
            (* <a__isNatList(cons(V1_,V2_)),a__U51(a__and(a__isNatKind(V1_),isNatIListKind(V2_)),V1_,V2_)> *)
           | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_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_12_large_scc_2_large_large_scc_1_large_large 
               (algebra.Alg.Term algebra.F.id_a__U51 ((algebra.Alg.Term 
                algebra.F.id_a__and ((algebra.Alg.Term 
                algebra.F.id_a__isNatKind (x1::nil))::(algebra.Alg.Term 
                algebra.F.id_isNatIListKind (x3::nil))::nil))::x1::x3::nil)) 
               (algebra.Alg.Term algebra.F.id_a__isNatList (x11::nil))
           
            (* <a__U51(tt,V1_,V2_),a__isNat(V1_)> *)
           | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_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_12_large_scc_2_large_large_scc_1_large_large 
                 (algebra.Alg.Term algebra.F.id_a__isNat (x1::nil)) 
                 (algebra.Alg.Term algebra.F.id_a__U51 (x13::x12::x11::nil))
           
            (* <a__isNat(length(V1_)),a__U11(a__isNatIListKind(V1_),V1_)> *)
           | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_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_length (x1::nil)) x11) ->
              DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_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__isNatList(V1_)> *)
           | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_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_scc_12_large_scc_2_large_large_scc_1_large_large 
                (algebra.Alg.Term algebra.F.id_a__isNatList (x1::nil)) 
                (algebra.Alg.Term algebra.F.id_a__U11 (x12::x11::nil))
           
            (* <a__isNatList(cons(V1_,V2_)),a__and(a__isNatKind(V1_),isNatIListKind(V2_))> *)
           | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_6 :
            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_12_large_scc_2_large_large_scc_1_large_large 
               (algebra.Alg.Term algebra.F.id_a__and ((algebra.Alg.Term 
                algebra.F.id_a__isNatKind (x1::nil))::(algebra.Alg.Term 
                algebra.F.id_isNatIListKind (x3::nil))::nil)) 
               (algebra.Alg.Term algebra.F.id_a__isNatList (x11::nil))
           
            (* <a__and(tt,X_),mark(X_)> *)
           | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_7 :
            forall x12 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_tt nil) 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_12_large_scc_2_large_large_scc_1_large_large 
                (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
                (algebra.Alg.Term algebra.F.id_a__and (x12::x11::nil))
           
            (* <mark(U12(X_)),mark(X_)> *)
           | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_8 :
            forall 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_U12 (x5::nil)) x11) ->
              DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large 
               (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
               (algebra.Alg.Term algebra.F.id_mark (x11::nil))
           
            (* <mark(isNatList(X_)),a__isNatList(X_)> *)
           | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_9 :
            forall 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_isNatList (x5::nil)) x11) ->
              DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large 
               (algebra.Alg.Term algebra.F.id_a__isNatList (x5::nil)) 
               (algebra.Alg.Term algebra.F.id_mark (x11::nil))
           
            (* <a__isNatList(cons(V1_,V2_)),a__isNatKind(V1_)> *)
           | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_10 :
            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_12_large_scc_2_large_large_scc_1_large_large 
               (algebra.Alg.Term algebra.F.id_a__isNatKind (x1::nil)) 
               (algebra.Alg.Term algebra.F.id_a__isNatList (x11::nil))
           
            (* <a__isNatKind(length(V1_)),a__isNatIListKind(V1_)> *)
           | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_11 :
            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_12_large_scc_2_large_large_scc_1_large_large 
               (algebra.Alg.Term algebra.F.id_a__isNatIListKind (x1::nil)) 
               (algebra.Alg.Term algebra.F.id_a__isNatKind (x11::nil))
           
            (* <a__isNatIListKind(cons(V1_,V2_)),a__and(a__isNatKind(V1_),isNatIListKind(V2_))> *)
           | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_12 :
            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_12_large_scc_2_large_large_scc_1_large_large 
               (algebra.Alg.Term algebra.F.id_a__and ((algebra.Alg.Term 
                algebra.F.id_a__isNatKind (x1::nil))::(algebra.Alg.Term 
                algebra.F.id_isNatIListKind (x3::nil))::nil)) 
               (algebra.Alg.Term algebra.F.id_a__isNatIListKind (x11::nil))
           
            (* <a__isNatIListKind(cons(V1_,V2_)),a__isNatKind(V1_)> *)
           | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_13 :
            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_12_large_scc_2_large_large_scc_1_large_large 
               (algebra.Alg.Term algebra.F.id_a__isNatKind (x1::nil)) 
               (algebra.Alg.Term algebra.F.id_a__isNatIListKind (x11::nil))
           
            (* <a__isNatKind(s(V1_)),a__isNatKind(V1_)> *)
           | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_14 :
            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_12_large_scc_2_large_large_scc_1_large_large 
               (algebra.Alg.Term algebra.F.id_a__isNatKind (x1::nil)) 
               (algebra.Alg.Term algebra.F.id_a__isNatKind (x11::nil))
           
            (* <mark(U21(X1_,X2_)),a__U21(mark(X1_),X2_)> *)
           | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_15 :
            forall x8 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 (x7::x8::nil)) x11) ->
              DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large 
               (algebra.Alg.Term algebra.F.id_a__U21 ((algebra.Alg.Term 
                algebra.F.id_mark (x7::nil))::x8::nil)) 
               (algebra.Alg.Term algebra.F.id_mark (x11::nil))
           
            (* <a__U21(tt,V1_),a__isNat(V1_)> *)
           | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_16 :
            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_12_large_scc_2_large_large_scc_1_large_large 
                (algebra.Alg.Term algebra.F.id_a__isNat (x1::nil)) 
                (algebra.Alg.Term algebra.F.id_a__U21 (x12::x11::nil))
           
            (* <a__isNat(length(V1_)),a__isNatIListKind(V1_)> *)
           | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_17 :
            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_12_large_scc_2_large_large_scc_1_large_large 
               (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_scc_12_large_scc_2_large_large_scc_1_large_large_18 :
            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_12_large_scc_2_large_large_scc_1_large_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__isNat(s(V1_)),a__isNatKind(V1_)> *)
           | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_19 :
            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_12_large_scc_2_large_large_scc_1_large_large 
               (algebra.Alg.Term algebra.F.id_a__isNatKind (x1::nil)) 
               (algebra.Alg.Term algebra.F.id_a__isNat (x11::nil))
           
            (* <mark(U21(X1_,X2_)),mark(X1_)> *)
           | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_20 :
            forall x8 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 (x7::x8::nil)) x11) ->
              DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large 
               (algebra.Alg.Term algebra.F.id_mark (x7::nil)) 
               (algebra.Alg.Term algebra.F.id_mark (x11::nil))
           
            (* <mark(U22(X_)),mark(X_)> *)
           | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_21 :
            forall 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_U22 (x5::nil)) x11) ->
              DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large 
               (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
               (algebra.Alg.Term algebra.F.id_mark (x11::nil))
           
            (* <mark(isNat(X_)),a__isNat(X_)> *)
           | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_22 :
            forall 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_isNat (x5::nil)) x11) ->
              DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large 
               (algebra.Alg.Term algebra.F.id_a__isNat (x5::nil)) 
               (algebra.Alg.Term algebra.F.id_mark (x11::nil))
           
            (* <mark(U51(X1_,X2_,X3_)),a__U51(mark(X1_),X2_,X3_)> *)
           | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_23 :
            forall x8 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_U51 (x7::x8::x9::nil)) 
               x11) ->
              DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large 
               (algebra.Alg.Term algebra.F.id_a__U51 ((algebra.Alg.Term 
                algebra.F.id_mark (x7::nil))::x8::x9::nil)) 
               (algebra.Alg.Term algebra.F.id_mark (x11::nil))
           
            (* <mark(U51(X1_,X2_,X3_)),mark(X1_)> *)
           | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_24 :
            forall x8 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_U51 (x7::x8::x9::nil)) 
               x11) ->
              DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large 
               (algebra.Alg.Term algebra.F.id_mark (x7::nil)) 
               (algebra.Alg.Term algebra.F.id_mark (x11::nil))
           
            (* <mark(U52(X1_,X2_)),a__U52(mark(X1_),X2_)> *)
           | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_25 :
            forall x8 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_U52 (x7::x8::nil)) x11) ->
              DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large 
               (algebra.Alg.Term algebra.F.id_a__U52 ((algebra.Alg.Term 
                algebra.F.id_mark (x7::nil))::x8::nil)) 
               (algebra.Alg.Term algebra.F.id_mark (x11::nil))
           
            (* <mark(U52(X1_,X2_)),mark(X1_)> *)
           | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_26 :
            forall x8 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_U52 (x7::x8::nil)) x11) ->
              DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large 
               (algebra.Alg.Term algebra.F.id_mark (x7::nil)) 
               (algebra.Alg.Term algebra.F.id_mark (x11::nil))
           
            (* <mark(U53(X_)),mark(X_)> *)
           | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_27 :
            forall 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_U53 (x5::nil)) x11) ->
              DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large 
               (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
               (algebra.Alg.Term algebra.F.id_mark (x11::nil))
           
            (* <mark(and(X1_,X2_)),a__and(mark(X1_),X2_)> *)
           | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_28 :
            forall x8 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_and (x7::x8::nil)) x11) ->
              DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large 
               (algebra.Alg.Term algebra.F.id_a__and ((algebra.Alg.Term 
                algebra.F.id_mark (x7::nil))::x8::nil)) 
               (algebra.Alg.Term algebra.F.id_mark (x11::nil))
           
            (* <mark(and(X1_,X2_)),mark(X1_)> *)
           | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_29 :
            forall x8 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_and (x7::x8::nil)) x11) ->
              DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large 
               (algebra.Alg.Term algebra.F.id_mark (x7::nil)) 
               (algebra.Alg.Term algebra.F.id_mark (x11::nil))
           
            (* <mark(isNatIListKind(X_)),a__isNatIListKind(X_)> *)
           | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_30 :
            forall 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_isNatIListKind (x5::nil)) 
               x11) ->
              DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large 
               (algebra.Alg.Term algebra.F.id_a__isNatIListKind (x5::nil)) 
               (algebra.Alg.Term algebra.F.id_mark (x11::nil))
           
            (* <mark(isNatKind(X_)),a__isNatKind(X_)> *)
           | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_31 :
            forall 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_isNatKind (x5::nil)) x11) ->
              DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large 
               (algebra.Alg.Term algebra.F.id_a__isNatKind (x5::nil)) 
               (algebra.Alg.Term algebra.F.id_mark (x11::nil))
           
            (* <mark(s(X_)),mark(X_)> *)
           | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_32 :
            forall 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_s (x5::nil)) x11) ->
              DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large 
               (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
               (algebra.Alg.Term algebra.F.id_mark (x11::nil))
         .
         
         
         Inductive DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_strict  :
          algebra.Alg.term ->algebra.Alg.term ->Prop := 
            (* <mark(U43(X_)),mark(X_)> *)
           | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_strict_0 :
            forall 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_U43 (x5::nil)) x11) ->
              DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_strict 
               (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
               (algebra.Alg.Term algebra.F.id_mark (x11::nil))
         .
         
         
         Module WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large.
          Inductive DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large  :
           algebra.Alg.term ->algebra.Alg.term ->Prop := 
             (* <a__U51(tt,V1_,V2_),a__U52(a__isNat(V1_),V2_)> *)
            | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_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_12_large_scc_2_large_large_scc_1_large_large_large 
                  (algebra.Alg.Term algebra.F.id_a__U52 ((algebra.Alg.Term 
                   algebra.F.id_a__isNat (x1::nil))::x3::nil)) 
                  (algebra.Alg.Term algebra.F.id_a__U51 (x13::x12::x11::nil))
            
             (* <a__U52(tt,V2_),a__isNatList(V2_)> *)
            | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_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_12_large_scc_2_large_large_scc_1_large_large_large 
                 (algebra.Alg.Term algebra.F.id_a__isNatList (x3::nil)) 
                 (algebra.Alg.Term algebra.F.id_a__U52 (x12::x11::nil))
            
             (* <a__isNatList(cons(V1_,V2_)),a__U51(a__and(a__isNatKind(V1_),isNatIListKind(V2_)),V1_,V2_)> *)
            | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_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_12_large_scc_2_large_large_scc_1_large_large_large 
                (algebra.Alg.Term algebra.F.id_a__U51 ((algebra.Alg.Term 
                 algebra.F.id_a__and ((algebra.Alg.Term 
                 algebra.F.id_a__isNatKind (x1::nil))::(algebra.Alg.Term 
                 algebra.F.id_isNatIListKind (x3::nil))::nil))::x1::x3::nil)) 
                (algebra.Alg.Term algebra.F.id_a__isNatList (x11::nil))
            
             (* <a__isNat(length(V1_)),a__U11(a__isNatIListKind(V1_),V1_)> *)
            | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_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_12_large_scc_2_large_large_scc_1_large_large_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__isNatList(V1_)> *)
            | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_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_scc_12_large_scc_2_large_large_scc_1_large_large_large 
                 (algebra.Alg.Term algebra.F.id_a__isNatList (x1::nil)) 
                 (algebra.Alg.Term algebra.F.id_a__U11 (x12::x11::nil))
            
             (* <a__and(tt,X_),mark(X_)> *)
            | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_5 :
             forall x12 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_tt nil) 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_12_large_scc_2_large_large_scc_1_large_large_large 
                 (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
                 (algebra.Alg.Term algebra.F.id_a__and (x12::x11::nil))
            
             (* <mark(U12(X_)),mark(X_)> *)
            | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_6 :
             forall 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_U12 (x5::nil)) x11) ->
               DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large 
                (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
                (algebra.Alg.Term algebra.F.id_mark (x11::nil))
            
             (* <mark(isNatList(X_)),a__isNatList(X_)> *)
            | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_7 :
             forall 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_isNatList (x5::nil)) x11) ->
               DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large 
                (algebra.Alg.Term algebra.F.id_a__isNatList (x5::nil)) 
                (algebra.Alg.Term algebra.F.id_mark (x11::nil))
            
             (* <a__isNatKind(length(V1_)),a__isNatIListKind(V1_)> *)
            | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_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_12_large_scc_2_large_large_scc_1_large_large_large 
                (algebra.Alg.Term algebra.F.id_a__isNatIListKind (x1::nil)) 
                (algebra.Alg.Term algebra.F.id_a__isNatKind (x11::nil))
            
             (* <a__isNatIListKind(cons(V1_,V2_)),a__and(a__isNatKind(V1_),isNatIListKind(V2_))> *)
            | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_9 :
             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_12_large_scc_2_large_large_scc_1_large_large_large 
                (algebra.Alg.Term algebra.F.id_a__and ((algebra.Alg.Term 
                 algebra.F.id_a__isNatKind (x1::nil))::(algebra.Alg.Term 
                 algebra.F.id_isNatIListKind (x3::nil))::nil)) 
                (algebra.Alg.Term algebra.F.id_a__isNatIListKind (x11::nil))
            
             (* <a__isNatIListKind(cons(V1_,V2_)),a__isNatKind(V1_)> *)
            | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_10 :
             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_12_large_scc_2_large_large_scc_1_large_large_large 
                (algebra.Alg.Term algebra.F.id_a__isNatKind (x1::nil)) 
                (algebra.Alg.Term algebra.F.id_a__isNatIListKind (x11::nil))
            
             (* <a__isNatKind(s(V1_)),a__isNatKind(V1_)> *)
            | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_11 :
             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_12_large_scc_2_large_large_scc_1_large_large_large 
                (algebra.Alg.Term algebra.F.id_a__isNatKind (x1::nil)) 
                (algebra.Alg.Term algebra.F.id_a__isNatKind (x11::nil))
            
             (* <mark(U21(X1_,X2_)),a__U21(mark(X1_),X2_)> *)
            | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_12 :
             forall x8 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 (x7::x8::nil)) x11) ->
               DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large 
                (algebra.Alg.Term algebra.F.id_a__U21 ((algebra.Alg.Term 
                 algebra.F.id_mark (x7::nil))::x8::nil)) 
                (algebra.Alg.Term algebra.F.id_mark (x11::nil))
            
             (* <a__U21(tt,V1_),a__isNat(V1_)> *)
            | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_13 :
             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_12_large_scc_2_large_large_scc_1_large_large_large 
                 (algebra.Alg.Term algebra.F.id_a__isNat (x1::nil)) 
                 (algebra.Alg.Term algebra.F.id_a__U21 (x12::x11::nil))
            
             (* <a__isNat(s(V1_)),a__U21(a__isNatKind(V1_),V1_)> *)
            | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_14 :
             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_12_large_scc_2_large_large_scc_1_large_large_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__isNat(s(V1_)),a__isNatKind(V1_)> *)
            | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_15 :
             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_12_large_scc_2_large_large_scc_1_large_large_large 
                (algebra.Alg.Term algebra.F.id_a__isNatKind (x1::nil)) 
                (algebra.Alg.Term algebra.F.id_a__isNat (x11::nil))
            
             (* <mark(U21(X1_,X2_)),mark(X1_)> *)
            | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_16 :
             forall x8 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 (x7::x8::nil)) x11) ->
               DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large 
                (algebra.Alg.Term algebra.F.id_mark (x7::nil)) 
                (algebra.Alg.Term algebra.F.id_mark (x11::nil))
            
             (* <mark(U22(X_)),mark(X_)> *)
            | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_17 :
             forall 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_U22 (x5::nil)) x11) ->
               DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large 
                (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
                (algebra.Alg.Term algebra.F.id_mark (x11::nil))
            
             (* <mark(isNat(X_)),a__isNat(X_)> *)
            | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_18 :
             forall 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_isNat (x5::nil)) x11) ->
               DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large 
                (algebra.Alg.Term algebra.F.id_a__isNat (x5::nil)) 
                (algebra.Alg.Term algebra.F.id_mark (x11::nil))
            
             (* <mark(U51(X1_,X2_,X3_)),a__U51(mark(X1_),X2_,X3_)> *)
            | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_19 :
             forall x8 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_U51 (x7::x8::x9::nil)) 
                x11) ->
               DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large 
                (algebra.Alg.Term algebra.F.id_a__U51 ((algebra.Alg.Term 
                 algebra.F.id_mark (x7::nil))::x8::x9::nil)) 
                (algebra.Alg.Term algebra.F.id_mark (x11::nil))
            
             (* <mark(U52(X1_,X2_)),a__U52(mark(X1_),X2_)> *)
            | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_20 :
             forall x8 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_U52 (x7::x8::nil)) x11) ->
               DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large 
                (algebra.Alg.Term algebra.F.id_a__U52 ((algebra.Alg.Term 
                 algebra.F.id_mark (x7::nil))::x8::nil)) 
                (algebra.Alg.Term algebra.F.id_mark (x11::nil))
            
             (* <mark(U53(X_)),mark(X_)> *)
            | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_21 :
             forall 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_U53 (x5::nil)) x11) ->
               DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large 
                (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
                (algebra.Alg.Term algebra.F.id_mark (x11::nil))
            
             (* <mark(and(X1_,X2_)),a__and(mark(X1_),X2_)> *)
            | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_22 :
             forall x8 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_and (x7::x8::nil)) x11) ->
               DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large 
                (algebra.Alg.Term algebra.F.id_a__and ((algebra.Alg.Term 
                 algebra.F.id_mark (x7::nil))::x8::nil)) 
                (algebra.Alg.Term algebra.F.id_mark (x11::nil))
            
             (* <mark(and(X1_,X2_)),mark(X1_)> *)
            | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_23 :
             forall x8 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_and (x7::x8::nil)) x11) ->
               DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large 
                (algebra.Alg.Term algebra.F.id_mark (x7::nil)) 
                (algebra.Alg.Term algebra.F.id_mark (x11::nil))
            
             (* <mark(isNatIListKind(X_)),a__isNatIListKind(X_)> *)
            | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_24 :
             forall 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_isNatIListKind (x5::nil)) 
                x11) ->
               DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large 
                (algebra.Alg.Term algebra.F.id_a__isNatIListKind (x5::nil)) 
                (algebra.Alg.Term algebra.F.id_mark (x11::nil))
            
             (* <mark(isNatKind(X_)),a__isNatKind(X_)> *)
            | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_25 :
             forall 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_isNatKind (x5::nil)) x11) ->
               DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large 
                (algebra.Alg.Term algebra.F.id_a__isNatKind (x5::nil)) 
                (algebra.Alg.Term algebra.F.id_mark (x11::nil))
            
             (* <mark(s(X_)),mark(X_)> *)
            | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_26 :
             forall 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_s (x5::nil)) x11) ->
               DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large 
                (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
                (algebra.Alg.Term algebra.F.id_mark (x11::nil))
          .
          
          
          Inductive DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_strict  :
           algebra.Alg.term ->algebra.Alg.term ->Prop := 
             (* <a__U51(tt,V1_,V2_),a__isNat(V1_)> *)
            | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_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_12_large_scc_2_large_large_scc_1_large_large_strict 
                  (algebra.Alg.Term algebra.F.id_a__isNat (x1::nil)) 
                  (algebra.Alg.Term algebra.F.id_a__U51 (x13::x12::x11::nil))
            
             (* <a__isNatList(cons(V1_,V2_)),a__and(a__isNatKind(V1_),isNatIListKind(V2_))> *)
            | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_strict_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_12_large_scc_2_large_large_scc_1_large_large_strict 
                (algebra.Alg.Term algebra.F.id_a__and ((algebra.Alg.Term 
                 algebra.F.id_a__isNatKind (x1::nil))::(algebra.Alg.Term 
                 algebra.F.id_isNatIListKind (x3::nil))::nil)) 
                (algebra.Alg.Term algebra.F.id_a__isNatList (x11::nil))
            
             (* <a__isNatList(cons(V1_,V2_)),a__isNatKind(V1_)> *)
            | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_strict_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_12_large_scc_2_large_large_scc_1_large_large_strict 
                (algebra.Alg.Term algebra.F.id_a__isNatKind (x1::nil)) 
                (algebra.Alg.Term algebra.F.id_a__isNatList (x11::nil))
            
             (* <a__isNat(length(V1_)),a__isNatIListKind(V1_)> *)
            | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_strict_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_12_large_scc_2_large_large_scc_1_large_large_strict 
                (algebra.Alg.Term algebra.F.id_a__isNatIListKind (x1::nil)) 
                (algebra.Alg.Term algebra.F.id_a__isNat (x11::nil))
            
             (* <mark(U51(X1_,X2_,X3_)),mark(X1_)> *)
            | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_strict_4 :
             forall x8 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_U51 (x7::x8::x9::nil)) 
                x11) ->
               DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_strict 
                (algebra.Alg.Term algebra.F.id_mark (x7::nil)) 
                (algebra.Alg.Term algebra.F.id_mark (x11::nil))
            
             (* <mark(U52(X1_,X2_)),mark(X1_)> *)
            | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_strict_5 :
             forall x8 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_U52 (x7::x8::nil)) x11) ->
               DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_strict 
                (algebra.Alg.Term algebra.F.id_mark (x7::nil)) 
                (algebra.Alg.Term algebra.F.id_mark (x11::nil))
          .
          
          
          Module WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large.
           Inductive DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_1  :
            algebra.Alg.term ->algebra.Alg.term ->Prop := 
              (* <a__U52(tt,V2_),a__isNatList(V2_)> *)
             | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_1 
                  (algebra.Alg.Term algebra.F.id_a__isNatList (x3::nil)) 
                  (algebra.Alg.Term algebra.F.id_a__U52 (x12::x11::nil))
             
              (* <a__isNatList(cons(V1_,V2_)),a__U51(a__and(a__isNatKind(V1_),isNatIListKind(V2_)),V1_,V2_)> *)
             | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_1_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_1 
                 (algebra.Alg.Term algebra.F.id_a__U51 ((algebra.Alg.Term 
                  algebra.F.id_a__and ((algebra.Alg.Term 
                  algebra.F.id_a__isNatKind (x1::nil))::(algebra.Alg.Term 
                  algebra.F.id_isNatIListKind (x3::nil))::nil))::x1::
                  x3::nil)) 
                 (algebra.Alg.Term algebra.F.id_a__isNatList (x11::nil))
             
              (* <a__U51(tt,V1_,V2_),a__U52(a__isNat(V1_),V2_)> *)
             | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_1_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_1 
                   (algebra.Alg.Term algebra.F.id_a__U52 ((algebra.Alg.Term 
                    algebra.F.id_a__isNat (x1::nil))::x3::nil)) 
                   (algebra.Alg.Term algebra.F.id_a__U51 (x13::x12::
                    x11::nil))
           .
           
           
           Module WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_1.
            Inductive DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_1_large  :
             algebra.Alg.term ->algebra.Alg.term ->Prop := 
               (* <a__isNatList(cons(V1_,V2_)),a__U51(a__and(a__isNatKind(V1_),isNatIListKind(V2_)),V1_,V2_)> *)
              | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_1_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_1_large 
                  (algebra.Alg.Term algebra.F.id_a__U51 ((algebra.Alg.Term 
                   algebra.F.id_a__and ((algebra.Alg.Term 
                   algebra.F.id_a__isNatKind (x1::nil))::(algebra.Alg.Term 
                   algebra.F.id_isNatIListKind (x3::nil))::nil))::x1::
                   x3::nil)) 
                  (algebra.Alg.Term algebra.F.id_a__isNatList (x11::nil))
              
               (* <a__U51(tt,V1_,V2_),a__U52(a__isNat(V1_),V2_)> *)
              | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_1_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_1_large 
                    (algebra.Alg.Term algebra.F.id_a__U52 ((algebra.Alg.Term 
                     algebra.F.id_a__isNat (x1::nil))::x3::nil)) 
                    (algebra.Alg.Term algebra.F.id_a__U51 (x13::x12::
                     x11::nil))
            .
            
            
            Inductive DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_1_strict  :
             algebra.Alg.term ->algebra.Alg.term ->Prop := 
               (* <a__U52(tt,V2_),a__isNatList(V2_)> *)
              | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_1_strict_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_1_strict 
                   (algebra.Alg.Term algebra.F.id_a__isNatList (x3::nil)) 
                   (algebra.Alg.Term algebra.F.id_a__U52 (x12::x11::nil))
            .
            
            
            Module WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_1_large.
             Inductive DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_1_large_non_scc_1  :
              algebra.Alg.term ->algebra.Alg.term ->Prop := 
                (* <a__U51(tt,V1_,V2_),a__U52(a__isNat(V1_),V2_)> *)
               | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_1_large_non_scc_1_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_1_large_non_scc_1 
                     (algebra.Alg.Term algebra.F.id_a__U52 
                      ((algebra.Alg.Term algebra.F.id_a__isNat (x1::nil))::
                      x3::nil)) 
                     (algebra.Alg.Term algebra.F.id_a__U51 (x13::x12::
                      x11::nil))
             .
             
             
             Lemma acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_1_large_non_scc_1 :
              forall x y, 
               (DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_1_large_non_scc_1 
                 x y) ->
                Acc WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_1.DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_1_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_1_large_non_scc_2  :
              algebra.Alg.term ->algebra.Alg.term ->Prop := 
                (* <a__isNatList(cons(V1_,V2_)),a__U51(a__and(a__isNatKind(V1_),isNatIListKind(V2_)),V1_,V2_)> *)
               | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_1_large_non_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_1_large_non_scc_2 
                   (algebra.Alg.Term algebra.F.id_a__U51 ((algebra.Alg.Term 
                    algebra.F.id_a__and ((algebra.Alg.Term 
                    algebra.F.id_a__isNatKind (x1::nil))::(algebra.Alg.Term 
                    algebra.F.id_isNatIListKind (x3::nil))::nil))::x1::
                    x3::nil)) 
                   (algebra.Alg.Term algebra.F.id_a__isNatList (x11::nil))
             .
             
             
             Lemma acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_1_large_non_scc_2 :
              forall x y, 
               (DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_1_large_non_scc_2 
                 x y) ->
                Acc WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_1.DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_1_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_1_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.
             
             
             Lemma wf :
              well_founded WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_1.DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_1_large
               .
             Proof.
               constructor;intros _y _h;inversion _h;clear _h;subst;
                (eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_1_large_non_scc_2;
                  econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
                ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_1_large_non_scc_1;
                   econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
                 ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_1_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_12_large_scc_2_large_large_scc_1_large_large_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_isNatKind (x11:Z) := 0.
            
            Definition P_id_a__isNatIList (x11:Z) := 3 + 2* x11.
            
            Definition P_id_a__U21 (x11:Z) (x12:Z) := 1* x11.
            
            Definition P_id_U41 (x11:Z) (x12:Z) (x13:Z) := 3 + 2* x13.
            
            Definition P_id_a__and (x11:Z) (x12:Z) := 1 + 2* x12.
            
            Definition P_id_a__U11 (x11:Z) (x12:Z) := 1* x11.
            
            Definition P_id_U21 (x11:Z) (x12:Z) := 1* x11.
            
            Definition P_id_a__U61 (x11:Z) (x12:Z) := 1* x11.
            
            Definition P_id_a__U32 (x11:Z) := 1.
            
            Definition P_id_U51 (x11:Z) (x12:Z) (x13:Z) := 1.
            
            Definition P_id_isNatIListKind (x11:Z) := 0.
            
            Definition P_id_0  := 1.
            
            Definition P_id_U12 (x11:Z) := 0.
            
            Definition P_id_a__U52 (x11:Z) (x12:Z) := 1.
            
            Definition P_id_a__isNat (x11:Z) := 1* x11.
            
            Definition P_id_U43 (x11:Z) := 1.
            
            Definition P_id_a__isNatIListKind (x11:Z) := 1.
            
            Definition P_id_a__U12 (x11:Z) := 1.
            
            Definition P_id_U31 (x11:Z) (x12:Z) := 0.
            
            Definition P_id_a__length (x11:Z) := 1.
            
            Definition P_id_a__U42 (x11:Z) (x12:Z) := 1.
            
            Definition P_id_U53 (x11:Z) := 0.
            
            Definition P_id_and (x11:Z) (x12:Z) := 2* x12.
            
            Definition P_id_cons (x11:Z) (x12:Z) := 1* x11 + 1* x12.
            
            Definition P_id_U11 (x11:Z) (x12:Z) := 1* x11.
            
            Definition P_id_a__U51 (x11:Z) (x12:Z) (x13:Z) := 1.
            
            Definition P_id_a__U22 (x11:Z) := 1.
            
            Definition P_id_U42 (x11:Z) (x12:Z) := 0.
            
            Definition P_id_length (x11:Z) := 1.
            
            Definition P_id_tt  := 1.
            
            Definition P_id_U22 (x11:Z) := 0.
            
            Definition P_id_s (x11:Z) := 1.
            
            Definition P_id_a__U41 (x11:Z) (x12:Z) (x13:Z) := 3 + 2* x13.
            
            Definition P_id_U52 (x11:Z) (x12:Z) := 0.
            
            Definition P_id_nil  := 0.
            
            Definition P_id_zeros  := 0.
            
            Definition P_id_isNatList (x11:Z) := 1.
            
            Definition P_id_a__U53 (x11:Z) := 1.
            
            Definition P_id_a__U31 (x11:Z) (x12:Z) := 1.
            
            Definition P_id_isNatIList (x11:Z) := 3 + 2* x11.
            
            Definition P_id_a__isNatKind (x11:Z) := 1.
            
            Definition P_id_a__isNatList (x11:Z) := 1.
            
            Definition P_id_U32 (x11:Z) := 1.
            
            Definition P_id_mark (x11:Z) := 1 + 1* x11.
            
            Definition P_id_a__U43 (x11:Z) := 1.
            
            Definition P_id_U61 (x11:Z) (x12:Z) := 1* x11.
            
            Definition P_id_isNat (x11:Z) := 1* x11.
            
            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__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__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_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_a__and_monotonic :
             forall x12 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_a__and x12 x14 <= P_id_a__and x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              
              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_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__U61_monotonic :
             forall x12 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_a__U61 x12 x14 <= P_id_a__U61 x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              
              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 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_a__U32 x12 <= P_id_a__U32 x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U51_monotonic :
             forall x16 x12 x14 x13 x11 x15, 
              (0 <= x16)/\ (x16 <= x15) ->
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_U51 x12 x14 x16 <= P_id_U51 x11 x13 x15.
            Proof.
              intros x16 x15 x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              intros [H_5 H_4].
              
              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_U12_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_U12 x12 <= P_id_U12 x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_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 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_a__U52 x12 x14 <= P_id_a__U52 x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_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_U43_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_U43 x12 <= P_id_U43 x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_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__U12_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_a__U12 x12 <= P_id_a__U12 x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              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__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__U42_monotonic :
             forall x12 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_a__U42 x12 x14 <= P_id_a__U42 x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U53_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_U53 x12 <= P_id_U53 x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_and_monotonic :
             forall x12 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_and x12 x14 <= P_id_and x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              
              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_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__U51_monotonic :
             forall x16 x12 x14 x13 x11 x15, 
              (0 <= x16)/\ (x16 <= x15) ->
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_a__U51 x12 x14 x16 <= P_id_a__U51 x11 x13 x15.
            Proof.
              intros x16 x15 x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              intros [H_5 H_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 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_a__U22 x12 <= P_id_a__U22 x11.
            Proof.
              intros x12 x11.
              intros [H_1 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 x12 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_U42 x12 x14 <= P_id_U42 x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              
              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_U22_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_U22 x12 <= P_id_U22 x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              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__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_U52_monotonic :
             forall x12 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_U52 x12 x14 <= P_id_U52 x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 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__U53_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_a__U53 x12 <= P_id_a__U53 x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              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_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__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__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_U32_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_U32 x12 <= P_id_U32 x11.
            Proof.
              intros x12 x11.
              intros [H_1 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__U43_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_a__U43 x12 <= P_id_a__U43 x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              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 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_U61 x12 x14 <= P_id_U61 x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              
              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__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_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__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_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_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_a__and_bounded :
             forall x12 x11, 
              (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__and x12 x11.
            Proof.
              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_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__U61_bounded :
             forall x12 x11, 
              (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U61 x12 x11.
            Proof.
              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 x11, (0 <= x11) ->0 <= P_id_a__U32 x11.
            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 x13 x11, 
              (0 <= x11) ->
               (0 <= x12) ->(0 <= x13) ->0 <= P_id_U51 x13 x12 x11.
            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_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_U12_bounded :
             forall x11, (0 <= x11) ->0 <= P_id_U12 x11.
            Proof.
              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 x12 x11, 
              (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U52 x12 x11.
            Proof.
              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_U43_bounded :
             forall x11, (0 <= x11) ->0 <= P_id_U43 x11.
            Proof.
              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_a__U12_bounded :
             forall x11, (0 <= x11) ->0 <= P_id_a__U12 x11.
            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__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__U42_bounded :
             forall x12 x11, 
              (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U42 x12 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U53_bounded :
             forall x11, (0 <= x11) ->0 <= P_id_U53 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_and_bounded :
             forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_and x12 x11.
            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_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__U51_bounded :
             forall x12 x13 x11, 
              (0 <= x11) ->
               (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U51 x13 x12 x11.
            Proof.
              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 x11, (0 <= x11) ->0 <= P_id_a__U22 x11.
            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 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U42 x12 x11.
            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_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_U22_bounded :
             forall x11, (0 <= x11) ->0 <= P_id_U22 x11.
            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__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_U52_bounded :
             forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U52 x12 x11.
            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_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_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__U53_bounded :
             forall x11, (0 <= x11) ->0 <= P_id_a__U53 x11.
            Proof.
              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_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.
            
            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_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_U32_bounded :
             forall x11, (0 <= x11) ->0 <= P_id_U32 x11.
            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__U43_bounded :
             forall x11, (0 <= x11) ->0 <= P_id_a__U43 x11.
            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 x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U61 x12 x11.
            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.
            
            Definition measure  := 
              InterpZ.measure 0 P_id_a__zeros P_id_isNatKind 
               P_id_a__isNatIList P_id_a__U21 P_id_U41 P_id_a__and 
               P_id_a__U11 P_id_U21 P_id_a__U61 P_id_a__U32 P_id_U51 
               P_id_isNatIListKind P_id_0 P_id_U12 P_id_a__U52 P_id_a__isNat 
               P_id_U43 P_id_a__isNatIListKind P_id_a__U12 P_id_U31 
               P_id_a__length P_id_a__U42 P_id_U53 P_id_and P_id_cons 
               P_id_U11 P_id_a__U51 P_id_a__U22 P_id_U42 P_id_length 
               P_id_tt P_id_U22 P_id_s P_id_a__U41 P_id_U52 P_id_nil 
               P_id_zeros P_id_isNatList P_id_a__U53 P_id_a__U31 
               P_id_isNatIList P_id_a__isNatKind P_id_a__isNatList P_id_U32 
               P_id_mark P_id_a__U43 P_id_U61 P_id_isNat.
            
            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_isNatKind 
                               (x11::nil)) =>
                             P_id_isNatKind (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__U21 (x12::
                               x11::nil)) =>
                             P_id_a__U21 (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_a__and (x12::
                               x11::nil)) =>
                             P_id_a__and (measure x12) (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_U21 (x12::
                               x11::nil)) =>
                             P_id_U21 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U61 (x12::
                               x11::nil)) =>
                             P_id_a__U61 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U32 
                               (x11::nil)) =>
                             P_id_a__U32 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_U51 (x13::x12::
                               x11::nil)) =>
                             P_id_U51 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_isNatIListKind 
                               (x11::nil)) =>
                             P_id_isNatIListKind (measure x11)
                            | (algebra.Alg.Term algebra.F.id_0 nil) =>
                             P_id_0 
                            | (algebra.Alg.Term algebra.F.id_U12 (x11::nil)) =>
                             P_id_U12 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U52 (x12::
                               x11::nil)) =>
                             P_id_a__U52 (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_U43 (x11::nil)) =>
                             P_id_U43 (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__U12 
                               (x11::nil)) =>
                             P_id_a__U12 (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__length 
                               (x11::nil)) =>
                             P_id_a__length (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U42 (x12::
                               x11::nil)) =>
                             P_id_a__U42 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_U53 (x11::nil)) =>
                             P_id_U53 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_and (x12::
                               x11::nil)) =>
                             P_id_and (measure x12) (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_U11 (x12::
                               x11::nil)) =>
                             P_id_U11 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U51 (x13::
                               x12::x11::nil)) =>
                             P_id_a__U51 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U22 
                               (x11::nil)) =>
                             P_id_a__U22 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_U42 (x12::
                               x11::nil)) =>
                             P_id_U42 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_length 
                               (x11::nil)) =>
                             P_id_length (measure x11)
                            | (algebra.Alg.Term algebra.F.id_tt nil) =>
                             P_id_tt 
                            | (algebra.Alg.Term algebra.F.id_U22 (x11::nil)) =>
                             P_id_U22 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_s (x11::nil)) =>
                             P_id_s (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_U52 (x12::
                               x11::nil)) =>
                             P_id_U52 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_nil nil) =>
                             P_id_nil 
                            | (algebra.Alg.Term algebra.F.id_zeros nil) =>
                             P_id_zeros 
                            | (algebra.Alg.Term algebra.F.id_isNatList 
                               (x11::nil)) =>
                             P_id_isNatList (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U53 
                               (x11::nil)) =>
                             P_id_a__U53 (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_isNatIList 
                               (x11::nil)) =>
                             P_id_isNatIList (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__isNatList 
                               (x11::nil)) =>
                             P_id_a__isNatList (measure x11)
                            | (algebra.Alg.Term algebra.F.id_U32 (x11::nil)) =>
                             P_id_U32 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_mark (x11::nil)) =>
                             P_id_mark (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U43 
                               (x11::nil)) =>
                             P_id_a__U43 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_U61 (x12::
                               x11::nil)) =>
                             P_id_U61 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_isNat 
                               (x11::nil)) =>
                             P_id_isNat (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_isNatKind_monotonic;assumption.
              intros ;apply P_id_a__isNatIList_monotonic;assumption.
              intros ;apply P_id_a__U21_monotonic;assumption.
              intros ;apply P_id_U41_monotonic;assumption.
              intros ;apply P_id_a__and_monotonic;assumption.
              intros ;apply P_id_a__U11_monotonic;assumption.
              intros ;apply P_id_U21_monotonic;assumption.
              intros ;apply P_id_a__U61_monotonic;assumption.
              intros ;apply P_id_a__U32_monotonic;assumption.
              intros ;apply P_id_U51_monotonic;assumption.
              intros ;apply P_id_isNatIListKind_monotonic;assumption.
              intros ;apply P_id_U12_monotonic;assumption.
              intros ;apply P_id_a__U52_monotonic;assumption.
              intros ;apply P_id_a__isNat_monotonic;assumption.
              intros ;apply P_id_U43_monotonic;assumption.
              intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
              intros ;apply P_id_a__U12_monotonic;assumption.
              intros ;apply P_id_U31_monotonic;assumption.
              intros ;apply P_id_a__length_monotonic;assumption.
              intros ;apply P_id_a__U42_monotonic;assumption.
              intros ;apply P_id_U53_monotonic;assumption.
              intros ;apply P_id_and_monotonic;assumption.
              intros ;apply P_id_cons_monotonic;assumption.
              intros ;apply P_id_U11_monotonic;assumption.
              intros ;apply P_id_a__U51_monotonic;assumption.
              intros ;apply P_id_a__U22_monotonic;assumption.
              intros ;apply P_id_U42_monotonic;assumption.
              intros ;apply P_id_length_monotonic;assumption.
              intros ;apply P_id_U22_monotonic;assumption.
              intros ;apply P_id_s_monotonic;assumption.
              intros ;apply P_id_a__U41_monotonic;assumption.
              intros ;apply P_id_U52_monotonic;assumption.
              intros ;apply P_id_isNatList_monotonic;assumption.
              intros ;apply P_id_a__U53_monotonic;assumption.
              intros ;apply P_id_a__U31_monotonic;assumption.
              intros ;apply P_id_isNatIList_monotonic;assumption.
              intros ;apply P_id_a__isNatKind_monotonic;assumption.
              intros ;apply P_id_a__isNatList_monotonic;assumption.
              intros ;apply P_id_U32_monotonic;assumption.
              intros ;apply P_id_mark_monotonic;assumption.
              intros ;apply P_id_a__U43_monotonic;assumption.
              intros ;apply P_id_U61_monotonic;assumption.
              intros ;apply P_id_isNat_monotonic;assumption.
              intros ;apply P_id_a__zeros_bounded;assumption.
              intros ;apply P_id_isNatKind_bounded;assumption.
              intros ;apply P_id_a__isNatIList_bounded;assumption.
              intros ;apply P_id_a__U21_bounded;assumption.
              intros ;apply P_id_U41_bounded;assumption.
              intros ;apply P_id_a__and_bounded;assumption.
              intros ;apply P_id_a__U11_bounded;assumption.
              intros ;apply P_id_U21_bounded;assumption.
              intros ;apply P_id_a__U61_bounded;assumption.
              intros ;apply P_id_a__U32_bounded;assumption.
              intros ;apply P_id_U51_bounded;assumption.
              intros ;apply P_id_isNatIListKind_bounded;assumption.
              intros ;apply P_id_0_bounded;assumption.
              intros ;apply P_id_U12_bounded;assumption.
              intros ;apply P_id_a__U52_bounded;assumption.
              intros ;apply P_id_a__isNat_bounded;assumption.
              intros ;apply P_id_U43_bounded;assumption.
              intros ;apply P_id_a__isNatIListKind_bounded;assumption.
              intros ;apply P_id_a__U12_bounded;assumption.
              intros ;apply P_id_U31_bounded;assumption.
              intros ;apply P_id_a__length_bounded;assumption.
              intros ;apply P_id_a__U42_bounded;assumption.
              intros ;apply P_id_U53_bounded;assumption.
              intros ;apply P_id_and_bounded;assumption.
              intros ;apply P_id_cons_bounded;assumption.
              intros ;apply P_id_U11_bounded;assumption.
              intros ;apply P_id_a__U51_bounded;assumption.
              intros ;apply P_id_a__U22_bounded;assumption.
              intros ;apply P_id_U42_bounded;assumption.
              intros ;apply P_id_length_bounded;assumption.
              intros ;apply P_id_tt_bounded;assumption.
              intros ;apply P_id_U22_bounded;assumption.
              intros ;apply P_id_s_bounded;assumption.
              intros ;apply P_id_a__U41_bounded;assumption.
              intros ;apply P_id_U52_bounded;assumption.
              intros ;apply P_id_nil_bounded;assumption.
              intros ;apply P_id_zeros_bounded;assumption.
              intros ;apply P_id_isNatList_bounded;assumption.
              intros ;apply P_id_a__U53_bounded;assumption.
              intros ;apply P_id_a__U31_bounded;assumption.
              intros ;apply P_id_isNatIList_bounded;assumption.
              intros ;apply P_id_a__isNatKind_bounded;assumption.
              intros ;apply P_id_a__isNatList_bounded;assumption.
              intros ;apply P_id_U32_bounded;assumption.
              intros ;apply P_id_mark_bounded;assumption.
              intros ;apply P_id_a__U43_bounded;assumption.
              intros ;apply P_id_U61_bounded;assumption.
              intros ;apply P_id_isNat_bounded;assumption.
              apply rules_monotonic.
            Qed.
            
            Definition P_id_A__LENGTH (x11:Z) := 0.
            
            Definition P_id_A__U11 (x11:Z) (x12:Z) := 0.
            
            Definition P_id_A__U41 (x11:Z) (x12:Z) (x13:Z) := 0.
            
            Definition P_id_A__ISNATKIND (x11:Z) := 0.
            
            Definition P_id_A__U22 (x11:Z) := 0.
            
            Definition P_id_A__U51 (x11:Z) (x12:Z) (x13:Z) := 1* x12 + 1* x13
              .
            
            Definition P_id_A__AND (x11:Z) (x12:Z) := 0.
            
            Definition P_id_A__ISNATLIST (x11:Z) := 1* x11.
            
            Definition P_id_A__U43 (x11:Z) := 0.
            
            Definition P_id_A__U31 (x11:Z) (x12:Z) := 0.
            
            Definition P_id_A__U53 (x11:Z) := 0.
            
            Definition P_id_MARK (x11:Z) := 0.
            
            Definition P_id_A__U12 (x11:Z) := 0.
            
            Definition P_id_A__U42 (x11:Z) (x12:Z) := 0.
            
            Definition P_id_A__ZEROS  := 0.
            
            Definition P_id_A__ISNAT (x11:Z) := 0.
            
            Definition P_id_A__U52 (x11:Z) (x12:Z) := 1* x11 + 1* x12.
            
            Definition P_id_A__ISNATILISTKIND (x11:Z) := 0.
            
            Definition P_id_A__U21 (x11:Z) (x12:Z) := 0.
            
            Definition P_id_A__ISNATILIST (x11:Z) := 0.
            
            Definition P_id_A__U32 (x11:Z) := 0.
            
            Definition P_id_A__U61 (x11:Z) (x12:Z) := 0.
            
            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__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__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__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__U22_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_A__U22 x12 <= P_id_A__U22 x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_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 x16 x12 x14 x13 x11 x15, 
              (0 <= x16)/\ (x16 <= x15) ->
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_A__U51 x12 x14 x16 <= P_id_A__U51 x11 x13 x15.
            Proof.
              intros x16 x15 x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              intros [H_5 H_4].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_A__AND_monotonic :
             forall x12 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_A__AND x12 x14 <= P_id_A__AND x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_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__U43_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_A__U43 x12 <= P_id_A__U43 x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              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__U53_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_A__U53 x12 <= P_id_A__U53 x11.
            Proof.
              intros x12 x11.
              intros [H_1 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__U12_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_A__U12 x12 <= P_id_A__U12 x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              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 x12 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_A__U42 x12 x14 <= P_id_A__U42 x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_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_A__U52_monotonic :
             forall x12 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_A__U52 x12 x14 <= P_id_A__U52 x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              
              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__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__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__U32_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_A__U32 x12 <= P_id_A__U32 x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_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 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_A__U61 x12 x14 <= P_id_A__U61 x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              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_isNatKind 
               P_id_a__isNatIList P_id_a__U21 P_id_U41 P_id_a__and 
               P_id_a__U11 P_id_U21 P_id_a__U61 P_id_a__U32 P_id_U51 
               P_id_isNatIListKind P_id_0 P_id_U12 P_id_a__U52 P_id_a__isNat 
               P_id_U43 P_id_a__isNatIListKind P_id_a__U12 P_id_U31 
               P_id_a__length P_id_a__U42 P_id_U53 P_id_and P_id_cons 
               P_id_U11 P_id_a__U51 P_id_a__U22 P_id_U42 P_id_length 
               P_id_tt P_id_U22 P_id_s P_id_a__U41 P_id_U52 P_id_nil 
               P_id_zeros P_id_isNatList P_id_a__U53 P_id_a__U31 
               P_id_isNatIList P_id_a__isNatKind P_id_a__isNatList P_id_U32 
               P_id_mark P_id_a__U43 P_id_U61 P_id_isNat P_id_A__LENGTH 
               P_id_A__U11 P_id_A__U41 P_id_A__ISNATKIND P_id_A__U22 
               P_id_A__U51 P_id_A__AND P_id_A__ISNATLIST P_id_A__U43 
               P_id_A__U31 P_id_A__U53 P_id_MARK P_id_A__U12 P_id_A__U42 
               P_id_A__ZEROS P_id_A__ISNAT P_id_A__U52 
               P_id_A__ISNATILISTKIND P_id_A__U21 P_id_A__ISNATILIST 
               P_id_A__U32 P_id_A__U61.
            
            Lemma marked_measure_equation :
             forall t, 
              marked_measure t = match t with
                                   | (algebra.Alg.Term 
                                      algebra.F.id_a__length (x11::nil)) =>
                                    P_id_A__LENGTH (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__U41 
                                      (x13::x12::x11::nil)) =>
                                    P_id_A__U41 (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__U22 
                                      (x11::nil)) =>
                                    P_id_A__U22 (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U51 
                                      (x13::x12::x11::nil)) =>
                                    P_id_A__U51 (measure x13) (measure x12) 
                                     (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__and 
                                      (x12::x11::nil)) =>
                                    P_id_A__AND (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__U43 
                                      (x11::nil)) =>
                                    P_id_A__U43 (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__U53 
                                      (x11::nil)) =>
                                    P_id_A__U53 (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_mark 
                                      (x11::nil)) =>
                                    P_id_MARK (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U12 
                                      (x11::nil)) =>
                                    P_id_A__U12 (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U42 
                                      (x12::x11::nil)) =>
                                    P_id_A__U42 (measure x12) (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__zeros 
                                      nil) =>
                                    P_id_A__ZEROS 
                                   | (algebra.Alg.Term algebra.F.id_a__isNat 
                                      (x11::nil)) =>
                                    P_id_A__ISNAT (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U52 
                                      (x12::x11::nil)) =>
                                    P_id_A__U52 (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_a__U21 
                                      (x12::x11::nil)) =>
                                    P_id_A__U21 (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__U32 
                                      (x11::nil)) =>
                                    P_id_A__U32 (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U61 
                                      (x12::x11::nil)) =>
                                    P_id_A__U61 (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_isNatKind_monotonic;assumption.
              intros ;apply P_id_a__isNatIList_monotonic;assumption.
              intros ;apply P_id_a__U21_monotonic;assumption.
              intros ;apply P_id_U41_monotonic;assumption.
              intros ;apply P_id_a__and_monotonic;assumption.
              intros ;apply P_id_a__U11_monotonic;assumption.
              intros ;apply P_id_U21_monotonic;assumption.
              intros ;apply P_id_a__U61_monotonic;assumption.
              intros ;apply P_id_a__U32_monotonic;assumption.
              intros ;apply P_id_U51_monotonic;assumption.
              intros ;apply P_id_isNatIListKind_monotonic;assumption.
              intros ;apply P_id_U12_monotonic;assumption.
              intros ;apply P_id_a__U52_monotonic;assumption.
              intros ;apply P_id_a__isNat_monotonic;assumption.
              intros ;apply P_id_U43_monotonic;assumption.
              intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
              intros ;apply P_id_a__U12_monotonic;assumption.
              intros ;apply P_id_U31_monotonic;assumption.
              intros ;apply P_id_a__length_monotonic;assumption.
              intros ;apply P_id_a__U42_monotonic;assumption.
              intros ;apply P_id_U53_monotonic;assumption.
              intros ;apply P_id_and_monotonic;assumption.
              intros ;apply P_id_cons_monotonic;assumption.
              intros ;apply P_id_U11_monotonic;assumption.
              intros ;apply P_id_a__U51_monotonic;assumption.
              intros ;apply P_id_a__U22_monotonic;assumption.
              intros ;apply P_id_U42_monotonic;assumption.
              intros ;apply P_id_length_monotonic;assumption.
              intros ;apply P_id_U22_monotonic;assumption.
              intros ;apply P_id_s_monotonic;assumption.
              intros ;apply P_id_a__U41_monotonic;assumption.
              intros ;apply P_id_U52_monotonic;assumption.
              intros ;apply P_id_isNatList_monotonic;assumption.
              intros ;apply P_id_a__U53_monotonic;assumption.
              intros ;apply P_id_a__U31_monotonic;assumption.
              intros ;apply P_id_isNatIList_monotonic;assumption.
              intros ;apply P_id_a__isNatKind_monotonic;assumption.
              intros ;apply P_id_a__isNatList_monotonic;assumption.
              intros ;apply P_id_U32_monotonic;assumption.
              intros ;apply P_id_mark_monotonic;assumption.
              intros ;apply P_id_a__U43_monotonic;assumption.
              intros ;apply P_id_U61_monotonic;assumption.
              intros ;apply P_id_isNat_monotonic;assumption.
              intros ;apply P_id_a__zeros_bounded;assumption.
              intros ;apply P_id_isNatKind_bounded;assumption.
              intros ;apply P_id_a__isNatIList_bounded;assumption.
              intros ;apply P_id_a__U21_bounded;assumption.
              intros ;apply P_id_U41_bounded;assumption.
              intros ;apply P_id_a__and_bounded;assumption.
              intros ;apply P_id_a__U11_bounded;assumption.
              intros ;apply P_id_U21_bounded;assumption.
              intros ;apply P_id_a__U61_bounded;assumption.
              intros ;apply P_id_a__U32_bounded;assumption.
              intros ;apply P_id_U51_bounded;assumption.
              intros ;apply P_id_isNatIListKind_bounded;assumption.
              intros ;apply P_id_0_bounded;assumption.
              intros ;apply P_id_U12_bounded;assumption.
              intros ;apply P_id_a__U52_bounded;assumption.
              intros ;apply P_id_a__isNat_bounded;assumption.
              intros ;apply P_id_U43_bounded;assumption.
              intros ;apply P_id_a__isNatIListKind_bounded;assumption.
              intros ;apply P_id_a__U12_bounded;assumption.
              intros ;apply P_id_U31_bounded;assumption.
              intros ;apply P_id_a__length_bounded;assumption.
              intros ;apply P_id_a__U42_bounded;assumption.
              intros ;apply P_id_U53_bounded;assumption.
              intros ;apply P_id_and_bounded;assumption.
              intros ;apply P_id_cons_bounded;assumption.
              intros ;apply P_id_U11_bounded;assumption.
              intros ;apply P_id_a__U51_bounded;assumption.
              intros ;apply P_id_a__U22_bounded;assumption.
              intros ;apply P_id_U42_bounded;assumption.
              intros ;apply P_id_length_bounded;assumption.
              intros ;apply P_id_tt_bounded;assumption.
              intros ;apply P_id_U22_bounded;assumption.
              intros ;apply P_id_s_bounded;assumption.
              intros ;apply P_id_a__U41_bounded;assumption.
              intros ;apply P_id_U52_bounded;assumption.
              intros ;apply P_id_nil_bounded;assumption.
              intros ;apply P_id_zeros_bounded;assumption.
              intros ;apply P_id_isNatList_bounded;assumption.
              intros ;apply P_id_a__U53_bounded;assumption.
              intros ;apply P_id_a__U31_bounded;assumption.
              intros ;apply P_id_isNatIList_bounded;assumption.
              intros ;apply P_id_a__isNatKind_bounded;assumption.
              intros ;apply P_id_a__isNatList_bounded;assumption.
              intros ;apply P_id_U32_bounded;assumption.
              intros ;apply P_id_mark_bounded;assumption.
              intros ;apply P_id_a__U43_bounded;assumption.
              intros ;apply P_id_U61_bounded;assumption.
              intros ;apply P_id_isNat_bounded;assumption.
              apply rules_monotonic.
              intros ;apply P_id_A__LENGTH_monotonic;assumption.
              intros ;apply P_id_A__U11_monotonic;assumption.
              intros ;apply P_id_A__U41_monotonic;assumption.
              intros ;apply P_id_A__ISNATKIND_monotonic;assumption.
              intros ;apply P_id_A__U22_monotonic;assumption.
              intros ;apply P_id_A__U51_monotonic;assumption.
              intros ;apply P_id_A__AND_monotonic;assumption.
              intros ;apply P_id_A__ISNATLIST_monotonic;assumption.
              intros ;apply P_id_A__U43_monotonic;assumption.
              intros ;apply P_id_A__U31_monotonic;assumption.
              intros ;apply P_id_A__U53_monotonic;assumption.
              intros ;apply P_id_MARK_monotonic;assumption.
              intros ;apply P_id_A__U12_monotonic;assumption.
              intros ;apply P_id_A__U42_monotonic;assumption.
              intros ;apply P_id_A__ISNAT_monotonic;assumption.
              intros ;apply P_id_A__U52_monotonic;assumption.
              intros ;apply P_id_A__ISNATILISTKIND_monotonic;assumption.
              intros ;apply P_id_A__U21_monotonic;assumption.
              intros ;apply P_id_A__ISNATILIST_monotonic;assumption.
              intros ;apply P_id_A__U32_monotonic;assumption.
              intros ;apply P_id_A__U61_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_1_strict_in_lt :
             Relation_Definitions.inclusion _ 
              DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_1_large_in_le :
             Relation_Definitions.inclusion _ 
              DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_1_large  := 
              WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_1_large.wf
              .
            
            
            Lemma wf :
             well_founded WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large.DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_1_large)
                     .
              clear x.
              intros x _ IHx IHx'.
              constructor.
              intros y H.
              
              destruct H;
               (apply IHx';
                 apply DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_1_large_in_le;
                  econstructor eassumption])).
              
              apply wf_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_1_large.
            Qed.
           End WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_1.
           
           Definition wf_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_1  := 
             WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_1.wf
             .
           
           
           Lemma acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_1 :
            forall x y, 
             (DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_1 
               x y) ->
              Acc WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large.DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large 
               x.
           Proof.
             intros x.
             pattern x.
             
             apply (@Acc_ind _ 
                     DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_1.
           Qed.
           
           
           Inductive DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_non_scc_2  :
            algebra.Alg.term ->algebra.Alg.term ->Prop := 
              (* <mark(U51(X1_,X2_,X3_)),a__U51(mark(X1_),X2_,X3_)> *)
             | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_non_scc_2_0 :
              forall x8 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_U51 (x7::x8::x9::nil)) 
                 x11) ->
                DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_non_scc_2 
                 (algebra.Alg.Term algebra.F.id_a__U51 ((algebra.Alg.Term 
                  algebra.F.id_mark (x7::nil))::x8::x9::nil)) 
                 (algebra.Alg.Term algebra.F.id_mark (x11::nil))
           .
           
           
           Lemma acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_non_scc_2 :
            forall x y, 
             (DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_non_scc_2 
               x y) ->
              Acc WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large.DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_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_12_large_scc_2_large_large_scc_1_large_large_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_12_large_scc_2_large_large_scc_1_large_large_large_non_scc_3  :
            algebra.Alg.term ->algebra.Alg.term ->Prop := 
              (* <mark(isNatList(X_)),a__isNatList(X_)> *)
             | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_non_scc_3_0 :
              forall 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_isNatList (x5::nil)) 
                 x11) ->
                DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_non_scc_3 
                 (algebra.Alg.Term algebra.F.id_a__isNatList (x5::nil)) 
                 (algebra.Alg.Term algebra.F.id_mark (x11::nil))
           .
           
           
           Lemma acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_non_scc_3 :
            forall x y, 
             (DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_non_scc_3 
               x y) ->
              Acc WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large.DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_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_12_large_scc_2_large_large_scc_1_large_large_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_12_large_scc_2_large_large_scc_1_large_large_large_non_scc_4  :
            algebra.Alg.term ->algebra.Alg.term ->Prop := 
              (* <a__U11(tt,V1_),a__isNatList(V1_)> *)
             | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_non_scc_4_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_12_large_scc_2_large_large_scc_1_large_large_large_non_scc_4 
                  (algebra.Alg.Term algebra.F.id_a__isNatList (x1::nil)) 
                  (algebra.Alg.Term algebra.F.id_a__U11 (x12::x11::nil))
           .
           
           
           Lemma acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_non_scc_4 :
            forall x y, 
             (DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_non_scc_4 
               x y) ->
              Acc WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large.DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_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_12_large_scc_2_large_large_scc_1_large_large_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_12_large_scc_2_large_large_scc_1_large_large_large_non_scc_5  :
            algebra.Alg.term ->algebra.Alg.term ->Prop := 
              (* <a__isNat(length(V1_)),a__U11(a__isNatIListKind(V1_),V1_)> *)
             | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_non_scc_5_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_12_large_scc_2_large_large_scc_1_large_large_large_non_scc_5 
                 (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_12_large_scc_2_large_large_scc_1_large_large_large_non_scc_5 :
            forall x y, 
             (DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_non_scc_5 
               x y) ->
              Acc WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large.DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_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_12_large_scc_2_large_large_scc_1_large_large_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.
           
           
           Inductive DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_non_scc_6  :
            algebra.Alg.term ->algebra.Alg.term ->Prop := 
              (* <mark(U52(X1_,X2_)),a__U52(mark(X1_),X2_)> *)
             | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_non_scc_6_0 :
              forall x8 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_U52 (x7::x8::nil)) x11) ->
                DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_non_scc_6 
                 (algebra.Alg.Term algebra.F.id_a__U52 ((algebra.Alg.Term 
                  algebra.F.id_mark (x7::nil))::x8::nil)) 
                 (algebra.Alg.Term algebra.F.id_mark (x11::nil))
           .
           
           
           Lemma acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_non_scc_6 :
            forall x y, 
             (DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_non_scc_6 
               x y) ->
              Acc WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large.DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_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_12_large_scc_2_large_large_scc_1_large_large_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7  :
            algebra.Alg.term ->algebra.Alg.term ->Prop := 
              (* <mark(U12(X_)),mark(X_)> *)
             | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_0 :
              forall 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_U12 (x5::nil)) x11) ->
                DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7 
                 (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
                 (algebra.Alg.Term algebra.F.id_mark (x11::nil))
             
              (* <mark(U21(X1_,X2_)),a__U21(mark(X1_),X2_)> *)
             | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_1 :
              forall x8 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 (x7::x8::nil)) x11) ->
                DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7 
                 (algebra.Alg.Term algebra.F.id_a__U21 ((algebra.Alg.Term 
                  algebra.F.id_mark (x7::nil))::x8::nil)) 
                 (algebra.Alg.Term algebra.F.id_mark (x11::nil))
             
              (* <a__U21(tt,V1_),a__isNat(V1_)> *)
             | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7 
                  (algebra.Alg.Term algebra.F.id_a__isNat (x1::nil)) 
                  (algebra.Alg.Term algebra.F.id_a__U21 (x12::x11::nil))
             
              (* <a__isNat(s(V1_)),a__U21(a__isNatKind(V1_),V1_)> *)
             | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_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_s (x1::nil)) x11) ->
                DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7 
                 (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_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7 
                 (algebra.Alg.Term algebra.F.id_a__isNatKind (x1::nil)) 
                 (algebra.Alg.Term algebra.F.id_a__isNat (x11::nil))
             
              (* <a__isNatKind(length(V1_)),a__isNatIListKind(V1_)> *)
             | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_5 :
              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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7 
                 (algebra.Alg.Term algebra.F.id_a__isNatIListKind (x1::nil)) 
                 (algebra.Alg.Term algebra.F.id_a__isNatKind (x11::nil))
             
              (* <a__isNatIListKind(cons(V1_,V2_)),a__and(a__isNatKind(V1_),isNatIListKind(V2_))> *)
             | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_6 :
              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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7 
                 (algebra.Alg.Term algebra.F.id_a__and ((algebra.Alg.Term 
                  algebra.F.id_a__isNatKind (x1::nil))::(algebra.Alg.Term 
                  algebra.F.id_isNatIListKind (x3::nil))::nil)) 
                 (algebra.Alg.Term algebra.F.id_a__isNatIListKind (x11::nil))
             
              (* <a__and(tt,X_),mark(X_)> *)
             | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_7 :
              forall x12 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_tt nil) 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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7 
                  (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
                  (algebra.Alg.Term algebra.F.id_a__and (x12::x11::nil))
             
              (* <mark(U21(X1_,X2_)),mark(X1_)> *)
             | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_8 :
              forall x8 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 (x7::x8::nil)) x11) ->
                DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7 
                 (algebra.Alg.Term algebra.F.id_mark (x7::nil)) 
                 (algebra.Alg.Term algebra.F.id_mark (x11::nil))
             
              (* <mark(U22(X_)),mark(X_)> *)
             | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_9 :
              forall 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_U22 (x5::nil)) x11) ->
                DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7 
                 (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
                 (algebra.Alg.Term algebra.F.id_mark (x11::nil))
             
              (* <mark(isNat(X_)),a__isNat(X_)> *)
             | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_10 :
              forall 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_isNat (x5::nil)) x11) ->
                DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7 
                 (algebra.Alg.Term algebra.F.id_a__isNat (x5::nil)) 
                 (algebra.Alg.Term algebra.F.id_mark (x11::nil))
             
              (* <mark(U53(X_)),mark(X_)> *)
             | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_11 :
              forall 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_U53 (x5::nil)) x11) ->
                DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7 
                 (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
                 (algebra.Alg.Term algebra.F.id_mark (x11::nil))
             
              (* <mark(and(X1_,X2_)),a__and(mark(X1_),X2_)> *)
             | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_12 :
              forall x8 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_and (x7::x8::nil)) x11) ->
                DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7 
                 (algebra.Alg.Term algebra.F.id_a__and ((algebra.Alg.Term 
                  algebra.F.id_mark (x7::nil))::x8::nil)) 
                 (algebra.Alg.Term algebra.F.id_mark (x11::nil))
             
              (* <mark(and(X1_,X2_)),mark(X1_)> *)
             | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_13 :
              forall x8 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_and (x7::x8::nil)) x11) ->
                DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7 
                 (algebra.Alg.Term algebra.F.id_mark (x7::nil)) 
                 (algebra.Alg.Term algebra.F.id_mark (x11::nil))
             
              (* <mark(isNatIListKind(X_)),a__isNatIListKind(X_)> *)
             | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_14 :
              forall 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_isNatIListKind (x5::nil)) 
                 x11) ->
                DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7 
                 (algebra.Alg.Term algebra.F.id_a__isNatIListKind (x5::nil)) 
                 (algebra.Alg.Term algebra.F.id_mark (x11::nil))
             
              (* <a__isNatIListKind(cons(V1_,V2_)),a__isNatKind(V1_)> *)
             | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_15 :
              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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7 
                 (algebra.Alg.Term algebra.F.id_a__isNatKind (x1::nil)) 
                 (algebra.Alg.Term algebra.F.id_a__isNatIListKind (x11::nil))
             
              (* <a__isNatKind(s(V1_)),a__isNatKind(V1_)> *)
             | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_16 :
              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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7 
                 (algebra.Alg.Term algebra.F.id_a__isNatKind (x1::nil)) 
                 (algebra.Alg.Term algebra.F.id_a__isNatKind (x11::nil))
             
              (* <mark(isNatKind(X_)),a__isNatKind(X_)> *)
             | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_17 :
              forall 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_isNatKind (x5::nil)) 
                 x11) ->
                DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7 
                 (algebra.Alg.Term algebra.F.id_a__isNatKind (x5::nil)) 
                 (algebra.Alg.Term algebra.F.id_mark (x11::nil))
             
              (* <mark(s(X_)),mark(X_)> *)
             | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_18 :
              forall 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_s (x5::nil)) x11) ->
                DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7 
                 (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
                 (algebra.Alg.Term algebra.F.id_mark (x11::nil))
           .
           
           
           Module WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7.
            Inductive DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large  :
             algebra.Alg.term ->algebra.Alg.term ->Prop := 
               (* <mark(U12(X_)),mark(X_)> *)
              | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_0 :
               forall 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_U12 (x5::nil)) x11) ->
                 DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large 
                  (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
                  (algebra.Alg.Term algebra.F.id_mark (x11::nil))
              
               (* <a__U21(tt,V1_),a__isNat(V1_)> *)
              | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large 
                   (algebra.Alg.Term algebra.F.id_a__isNat (x1::nil)) 
                   (algebra.Alg.Term algebra.F.id_a__U21 (x12::x11::nil))
              
               (* <a__isNat(s(V1_)),a__U21(a__isNatKind(V1_),V1_)> *)
              | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_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__isNat(s(V1_)),a__isNatKind(V1_)> *)
              | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_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_s (x1::nil)) x11) ->
                 DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large 
                  (algebra.Alg.Term algebra.F.id_a__isNatKind (x1::nil)) 
                  (algebra.Alg.Term algebra.F.id_a__isNat (x11::nil))
              
               (* <a__isNatKind(length(V1_)),a__isNatIListKind(V1_)> *)
              | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_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_length (x1::nil)) x11) ->
                 DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large 
                  (algebra.Alg.Term algebra.F.id_a__isNatIListKind (x1::nil)) 
                  (algebra.Alg.Term algebra.F.id_a__isNatKind (x11::nil))
              
               (* <a__isNatIListKind(cons(V1_,V2_)),a__and(a__isNatKind(V1_),isNatIListKind(V2_))> *)
              | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large 
                  (algebra.Alg.Term algebra.F.id_a__and ((algebra.Alg.Term 
                   algebra.F.id_a__isNatKind (x1::nil))::(algebra.Alg.Term 
                   algebra.F.id_isNatIListKind (x3::nil))::nil)) 
                  (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
                   (x11::nil))
              
               (* <a__and(tt,X_),mark(X_)> *)
              | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_6 :
               forall x12 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_tt nil) 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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large 
                   (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
                   (algebra.Alg.Term algebra.F.id_a__and (x12::x11::nil))
              
               (* <mark(U22(X_)),mark(X_)> *)
              | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_7 :
               forall 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_U22 (x5::nil)) x11) ->
                 DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large 
                  (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
                  (algebra.Alg.Term algebra.F.id_mark (x11::nil))
              
               (* <mark(U53(X_)),mark(X_)> *)
              | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_8 :
               forall 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_U53 (x5::nil)) x11) ->
                 DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large 
                  (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
                  (algebra.Alg.Term algebra.F.id_mark (x11::nil))
              
               (* <mark(and(X1_,X2_)),a__and(mark(X1_),X2_)> *)
              | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_9 :
               forall x8 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_and (x7::x8::nil)) x11) ->
                 DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large 
                  (algebra.Alg.Term algebra.F.id_a__and ((algebra.Alg.Term 
                   algebra.F.id_mark (x7::nil))::x8::nil)) 
                  (algebra.Alg.Term algebra.F.id_mark (x11::nil))
              
               (* <mark(and(X1_,X2_)),mark(X1_)> *)
              | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_10 :
               forall x8 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_and (x7::x8::nil)) x11) ->
                 DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large 
                  (algebra.Alg.Term algebra.F.id_mark (x7::nil)) 
                  (algebra.Alg.Term algebra.F.id_mark (x11::nil))
              
               (* <mark(isNatIListKind(X_)),a__isNatIListKind(X_)> *)
              | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_11 :
               forall 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_isNatIListKind (x5::nil)) 
                  x11) ->
                 DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large 
                  (algebra.Alg.Term algebra.F.id_a__isNatIListKind (x5::nil)) 
                  (algebra.Alg.Term algebra.F.id_mark (x11::nil))
              
               (* <a__isNatIListKind(cons(V1_,V2_)),a__isNatKind(V1_)> *)
              | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_12 :
               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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large 
                  (algebra.Alg.Term algebra.F.id_a__isNatKind (x1::nil)) 
                  (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
                   (x11::nil))
              
               (* <a__isNatKind(s(V1_)),a__isNatKind(V1_)> *)
              | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_13 :
               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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large 
                  (algebra.Alg.Term algebra.F.id_a__isNatKind (x1::nil)) 
                  (algebra.Alg.Term algebra.F.id_a__isNatKind (x11::nil))
              
               (* <mark(isNatKind(X_)),a__isNatKind(X_)> *)
              | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_14 :
               forall 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_isNatKind (x5::nil)) 
                  x11) ->
                 DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large 
                  (algebra.Alg.Term algebra.F.id_a__isNatKind (x5::nil)) 
                  (algebra.Alg.Term algebra.F.id_mark (x11::nil))
              
               (* <mark(s(X_)),mark(X_)> *)
              | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_15 :
               forall 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_s (x5::nil)) x11) ->
                 DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large 
                  (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
                  (algebra.Alg.Term algebra.F.id_mark (x11::nil))
            .
            
            
            Inductive DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_strict  :
             algebra.Alg.term ->algebra.Alg.term ->Prop := 
               (* <mark(U21(X1_,X2_)),a__U21(mark(X1_),X2_)> *)
              | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_strict_0 :
               forall x8 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 (x7::x8::nil)) x11) ->
                 DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_strict 
                  (algebra.Alg.Term algebra.F.id_a__U21 ((algebra.Alg.Term 
                   algebra.F.id_mark (x7::nil))::x8::nil)) 
                  (algebra.Alg.Term algebra.F.id_mark (x11::nil))
              
               (* <mark(U21(X1_,X2_)),mark(X1_)> *)
              | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_strict_1 :
               forall x8 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 (x7::x8::nil)) x11) ->
                 DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_strict 
                  (algebra.Alg.Term algebra.F.id_mark (x7::nil)) 
                  (algebra.Alg.Term algebra.F.id_mark (x11::nil))
              
               (* <mark(isNat(X_)),a__isNat(X_)> *)
              | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_strict_2 :
               forall 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_isNat (x5::nil)) x11) ->
                 DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_strict 
                  (algebra.Alg.Term algebra.F.id_a__isNat (x5::nil)) 
                  (algebra.Alg.Term algebra.F.id_mark (x11::nil))
            .
            
            
            Module WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large.
             Inductive DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1  :
              algebra.Alg.term ->algebra.Alg.term ->Prop := 
                (* <mark(U22(X_)),mark(X_)> *)
               | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_0 :
                forall 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_U22 (x5::nil)) x11) ->
                  DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1 
                   (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
                   (algebra.Alg.Term algebra.F.id_mark (x11::nil))
               
                (* <mark(U12(X_)),mark(X_)> *)
               | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_1 :
                forall 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_U12 (x5::nil)) x11) ->
                  DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1 
                   (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
                   (algebra.Alg.Term algebra.F.id_mark (x11::nil))
               
                (* <mark(U53(X_)),mark(X_)> *)
               | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_2 :
                forall 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_U53 (x5::nil)) x11) ->
                  DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1 
                   (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
                   (algebra.Alg.Term algebra.F.id_mark (x11::nil))
               
                (* <mark(and(X1_,X2_)),a__and(mark(X1_),X2_)> *)
               | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_3 :
                forall x8 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_and (x7::x8::nil)) 
                   x11) ->
                  DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1 
                   (algebra.Alg.Term algebra.F.id_a__and ((algebra.Alg.Term 
                    algebra.F.id_mark (x7::nil))::x8::nil)) 
                   (algebra.Alg.Term algebra.F.id_mark (x11::nil))
               
                (* <a__and(tt,X_),mark(X_)> *)
               | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_4 :
                forall x12 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_tt nil) 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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1 
                    (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
                    (algebra.Alg.Term algebra.F.id_a__and (x12::x11::nil))
               
                (* <mark(and(X1_,X2_)),mark(X1_)> *)
               | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_5 :
                forall x8 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_and (x7::x8::nil)) 
                   x11) ->
                  DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1 
                   (algebra.Alg.Term algebra.F.id_mark (x7::nil)) 
                   (algebra.Alg.Term algebra.F.id_mark (x11::nil))
               
                (* <mark(isNatIListKind(X_)),a__isNatIListKind(X_)> *)
               | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_6 :
                forall 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_isNatIListKind (x5::nil)) 
                   x11) ->
                  DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1 
                   (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
                    (x5::nil)) 
                   (algebra.Alg.Term algebra.F.id_mark (x11::nil))
               
                (* <a__isNatIListKind(cons(V1_,V2_)),a__and(a__isNatKind(V1_),isNatIListKind(V2_))> *)
               | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_7 :
                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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1 
                   (algebra.Alg.Term algebra.F.id_a__and ((algebra.Alg.Term 
                    algebra.F.id_a__isNatKind (x1::nil))::(algebra.Alg.Term 
                    algebra.F.id_isNatIListKind (x3::nil))::nil)) 
                   (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
                    (x11::nil))
               
                (* <a__isNatIListKind(cons(V1_,V2_)),a__isNatKind(V1_)> *)
               | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_8 :
                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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1 
                   (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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_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_length (x1::nil)) x11) ->
                  DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1 
                   (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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1 
                   (algebra.Alg.Term algebra.F.id_a__isNatKind (x1::nil)) 
                   (algebra.Alg.Term algebra.F.id_a__isNatKind (x11::nil))
               
                (* <mark(isNatKind(X_)),a__isNatKind(X_)> *)
               | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_11 :
                forall 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_isNatKind (x5::nil)) 
                   x11) ->
                  DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1 
                   (algebra.Alg.Term algebra.F.id_a__isNatKind (x5::nil)) 
                   (algebra.Alg.Term algebra.F.id_mark (x11::nil))
               
                (* <mark(s(X_)),mark(X_)> *)
               | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_12 :
                forall 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_s (x5::nil)) x11) ->
                  DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1 
                   (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
                   (algebra.Alg.Term algebra.F.id_mark (x11::nil))
             .
             
             
             Module WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1.
              Inductive DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large  :
               algebra.Alg.term ->algebra.Alg.term ->Prop := 
                 (* <mark(U22(X_)),mark(X_)> *)
                | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_0 :
                 forall 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_U22 (x5::nil)) x11) ->
                   DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large 
                    (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
                    (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                
                 (* <mark(U53(X_)),mark(X_)> *)
                | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_1 :
                 forall 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_U53 (x5::nil)) x11) ->
                   DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large 
                    (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
                    (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                
                 (* <mark(and(X1_,X2_)),a__and(mark(X1_),X2_)> *)
                | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_2 :
                 forall x8 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_and (x7::x8::nil)) 
                    x11) ->
                   DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large 
                    (algebra.Alg.Term algebra.F.id_a__and ((algebra.Alg.Term 
                     algebra.F.id_mark (x7::nil))::x8::nil)) 
                    (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                
                 (* <a__and(tt,X_),mark(X_)> *)
                | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_3 :
                 forall x12 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_tt nil) 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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large 
                     (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
                     (algebra.Alg.Term algebra.F.id_a__and (x12::x11::nil))
                
                 (* <mark(and(X1_,X2_)),mark(X1_)> *)
                | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_4 :
                 forall x8 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_and (x7::x8::nil)) 
                    x11) ->
                   DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large 
                    (algebra.Alg.Term algebra.F.id_mark (x7::nil)) 
                    (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                
                 (* <mark(isNatIListKind(X_)),a__isNatIListKind(X_)> *)
                | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_5 :
                 forall 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_isNatIListKind (x5::nil)) 
                    x11) ->
                   DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large 
                    (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
                     (x5::nil)) 
                    (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                
                 (* <a__isNatIListKind(cons(V1_,V2_)),a__and(a__isNatKind(V1_),isNatIListKind(V2_))> *)
                | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_6 :
                 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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large 
                    (algebra.Alg.Term algebra.F.id_a__and ((algebra.Alg.Term 
                     algebra.F.id_a__isNatKind (x1::nil))::(algebra.Alg.Term 
                     algebra.F.id_isNatIListKind (x3::nil))::nil)) 
                    (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
                     (x11::nil))
                
                 (* <a__isNatIListKind(cons(V1_,V2_)),a__isNatKind(V1_)> *)
                | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_7 :
                 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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large 
                    (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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large 
                    (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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large 
                    (algebra.Alg.Term algebra.F.id_a__isNatKind (x1::nil)) 
                    (algebra.Alg.Term algebra.F.id_a__isNatKind (x11::nil))
                
                 (* <mark(isNatKind(X_)),a__isNatKind(X_)> *)
                | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_10 :
                 forall 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_isNatKind (x5::nil)) 
                    x11) ->
                   DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large 
                    (algebra.Alg.Term algebra.F.id_a__isNatKind (x5::nil)) 
                    (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                
                 (* <mark(s(X_)),mark(X_)> *)
                | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_11 :
                 forall 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_s (x5::nil)) x11) ->
                   DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large 
                    (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
                    (algebra.Alg.Term algebra.F.id_mark (x11::nil))
              .
              
              
              Inductive DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_strict  :
               algebra.Alg.term ->algebra.Alg.term ->Prop := 
                 (* <mark(U12(X_)),mark(X_)> *)
                | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_strict_0 :
                 forall 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_U12 (x5::nil)) x11) ->
                   DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_strict 
                    (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
                    (algebra.Alg.Term algebra.F.id_mark (x11::nil))
              .
              
              
              Module WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large.
               Inductive DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large  :
                algebra.Alg.term ->algebra.Alg.term ->Prop := 
                  (* <mark(U22(X_)),mark(X_)> *)
                 | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_0 :
                  forall 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_U22 (x5::nil)) x11) ->
                    DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large 
                     (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
                     (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                 
                  (* <mark(U53(X_)),mark(X_)> *)
                 | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_1 :
                  forall 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_U53 (x5::nil)) x11) ->
                    DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large 
                     (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
                     (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                 
                  (* <mark(and(X1_,X2_)),a__and(mark(X1_),X2_)> *)
                 | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_2 :
                  forall x8 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_and (x7::x8::nil)) 
                     x11) ->
                    DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large 
                     (algebra.Alg.Term algebra.F.id_a__and 
                      ((algebra.Alg.Term algebra.F.id_mark (x7::nil))::
                      x8::nil)) 
                     (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                 
                  (* <a__and(tt,X_),mark(X_)> *)
                 | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_3 :
                  forall x12 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_tt nil) 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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large 
                      (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
                      (algebra.Alg.Term algebra.F.id_a__and (x12::x11::nil))
                 
                  (* <mark(and(X1_,X2_)),mark(X1_)> *)
                 | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_4 :
                  forall x8 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_and (x7::x8::nil)) 
                     x11) ->
                    DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large 
                     (algebra.Alg.Term algebra.F.id_mark (x7::nil)) 
                     (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                 
                  (* <mark(isNatIListKind(X_)),a__isNatIListKind(X_)> *)
                 | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_5 :
                  forall 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_isNatIListKind (x5::nil)) 
                     x11) ->
                    DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large 
                     (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
                      (x5::nil)) 
                     (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                 
                  (* <a__isNatIListKind(cons(V1_,V2_)),a__and(a__isNatKind(V1_),isNatIListKind(V2_))> *)
                 | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_6 :
                  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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large 
                     (algebra.Alg.Term algebra.F.id_a__and 
                      ((algebra.Alg.Term algebra.F.id_a__isNatKind 
                      (x1::nil))::(algebra.Alg.Term 
                      algebra.F.id_isNatIListKind (x3::nil))::nil)) 
                     (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
                      (x11::nil))
                 
                  (* <a__isNatIListKind(cons(V1_,V2_)),a__isNatKind(V1_)> *)
                 | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_7 :
                  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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large 
                     (algebra.Alg.Term algebra.F.id_a__isNatKind (x1::nil)) 
                     (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
                      (x11::nil))
                 
                  (* <a__isNatKind(s(V1_)),a__isNatKind(V1_)> *)
                 | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_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_s (x1::nil)) x11) ->
                    DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large 
                     (algebra.Alg.Term algebra.F.id_a__isNatKind (x1::nil)) 
                     (algebra.Alg.Term algebra.F.id_a__isNatKind (x11::nil))
                 
                  (* <mark(isNatKind(X_)),a__isNatKind(X_)> *)
                 | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_9 :
                  forall 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_isNatKind (x5::nil)) 
                     x11) ->
                    DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large 
                     (algebra.Alg.Term algebra.F.id_a__isNatKind (x5::nil)) 
                     (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                 
                  (* <mark(s(X_)),mark(X_)> *)
                 | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_10 :
                  forall 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_s (x5::nil)) x11) ->
                    DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large 
                     (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
                     (algebra.Alg.Term algebra.F.id_mark (x11::nil))
               .
               
               
               Inductive DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_strict  :
                algebra.Alg.term ->algebra.Alg.term ->Prop := 
                  (* <a__isNatKind(length(V1_)),a__isNatIListKind(V1_)> *)
                 | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_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_length (x1::nil)) 
                     x11) ->
                    DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large.
                Inductive DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_1  :
                 algebra.Alg.term ->algebra.Alg.term ->Prop := 
                   (* <a__isNatKind(s(V1_)),a__isNatKind(V1_)> *)
                  | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_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  := 1.
                 
                 Definition P_id_isNatKind (x11:Z) := 0.
                 
                 Definition P_id_a__isNatIList (x11:Z) := 3.
                 
                 Definition P_id_a__U21 (x11:Z) (x12:Z) := 2.
                 
                 Definition P_id_U41 (x11:Z) (x12:Z) (x13:Z) := 1.
                 
                 Definition P_id_a__and (x11:Z) (x12:Z) := 1* x11 + 2* x12.
                 
                 Definition P_id_a__U11 (x11:Z) (x12:Z) := 1* x11.
                 
                 Definition P_id_U21 (x11:Z) (x12:Z) := 0.
                 
                 Definition P_id_a__U61 (x11:Z) (x12:Z) := 
                   1 + 1* x11 + 1* x12.
                 
                 Definition P_id_a__U32 (x11:Z) := 2.
                 
                 Definition P_id_U51 (x11:Z) (x12:Z) (x13:Z) := 3* x13.
                 
                 Definition P_id_isNatIListKind (x11:Z) := 0.
                 
                 Definition P_id_0  := 0.
                 
                 Definition P_id_U12 (x11:Z) := 2.
                 
                 Definition P_id_a__U52 (x11:Z) (x12:Z) := 1 + 3* x12.
                 
                 Definition P_id_a__isNat (x11:Z) := 2.
                 
                 Definition P_id_U43 (x11:Z) := 3.
                 
                 Definition P_id_a__isNatIListKind (x11:Z) := 2.
                 
                 Definition P_id_a__U12 (x11:Z) := 2.
                 
                 Definition P_id_U31 (x11:Z) (x12:Z) := 1.
                 
                 Definition P_id_a__length (x11:Z) := 1* x11.
                 
                 Definition P_id_a__U42 (x11:Z) (x12:Z) := 3.
                 
                 Definition P_id_U53 (x11:Z) := 1* x11.
                 
                 Definition P_id_and (x11:Z) (x12:Z) := 1* x11 + 2* x12.
                 
                 Definition P_id_cons (x11:Z) (x12:Z) := 1 + 3* x12.
                 
                 Definition P_id_U11 (x11:Z) (x12:Z) := 1* x11.
                 
                 Definition P_id_a__U51 (x11:Z) (x12:Z) (x13:Z) := 1 + 3* x13
                   .
                 
                 Definition P_id_a__U22 (x11:Z) := 2.
                 
                 Definition P_id_U42 (x11:Z) (x12:Z) := 3.
                 
                 Definition P_id_length (x11:Z) := 1* x11.
                 
                 Definition P_id_tt  := 2.
                 
                 Definition P_id_U22 (x11:Z) := 0.
                 
                 Definition P_id_s (x11:Z) := 1 + 1* x11.
                 
                 Definition P_id_a__U41 (x11:Z) (x12:Z) (x13:Z) := 3.
                 
                 Definition P_id_U52 (x11:Z) (x12:Z) := 1 + 3* x12.
                 
                 Definition P_id_nil  := 2.
                 
                 Definition P_id_zeros  := 0.
                 
                 Definition P_id_isNatList (x11:Z) := 2* x11.
                 
                 Definition P_id_a__U53 (x11:Z) := 1* x11.
                 
                 Definition P_id_a__U31 (x11:Z) (x12:Z) := 3.
                 
                 Definition P_id_isNatIList (x11:Z) := 1.
                 
                 Definition P_id_a__isNatKind (x11:Z) := 2.
                 
                 Definition P_id_a__isNatList (x11:Z) := 2* x11.
                 
                 Definition P_id_U32 (x11:Z) := 2.
                 
                 Definition P_id_mark (x11:Z) := 2 + 1* x11.
                 
                 Definition P_id_a__U43 (x11:Z) := 3.
                 
                 Definition P_id_U61 (x11:Z) (x12:Z) := 1 + 1* x11 + 1* x12.
                 
                 Definition P_id_isNat (x11:Z) := 0.
                 
                 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__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__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_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_a__and_monotonic :
                  forall x12 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_a__and x12 x14 <= P_id_a__and x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   
                   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_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__U61_monotonic :
                  forall x12 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_a__U61 x12 x14 <= P_id_a__U61 x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   
                   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 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_a__U32 x12 <= P_id_a__U32 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U51_monotonic :
                  forall x16 x12 x14 x13 x11 x15, 
                   (0 <= x16)/\ (x16 <= x15) ->
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_U51 x12 x14 x16 <= P_id_U51 x11 x13 x15.
                 Proof.
                   intros x16 x15 x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   intros [H_5 H_4].
                   
                   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_U12_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->P_id_U12 x12 <= P_id_U12 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_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 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_a__U52 x12 x14 <= P_id_a__U52 x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_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_U43_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->P_id_U43 x12 <= P_id_U43 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_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__U12_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_a__U12 x12 <= P_id_a__U12 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   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__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__U42_monotonic :
                  forall x12 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_a__U42 x12 x14 <= P_id_a__U42 x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U53_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->P_id_U53 x12 <= P_id_U53 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_and_monotonic :
                  forall x12 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_and x12 x14 <= P_id_and x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   
                   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_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__U51_monotonic :
                  forall x16 x12 x14 x13 x11 x15, 
                   (0 <= x16)/\ (x16 <= x15) ->
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_a__U51 x12 x14 x16 <= P_id_a__U51 x11 x13 x15.
                 Proof.
                   intros x16 x15 x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   intros [H_5 H_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 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_a__U22 x12 <= P_id_a__U22 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 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 x12 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_U42 x12 x14 <= P_id_U42 x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   
                   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_U22_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->P_id_U22 x12 <= P_id_U22 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   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__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_U52_monotonic :
                  forall x12 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_U52 x12 x14 <= P_id_U52 x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 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__U53_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_a__U53 x12 <= P_id_a__U53 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   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_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__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__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_U32_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->P_id_U32 x12 <= P_id_U32 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 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__U43_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_a__U43 x12 <= P_id_a__U43 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   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 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_U61 x12 x14 <= P_id_U61 x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   
                   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__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_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__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_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_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_a__and_bounded :
                  forall x12 x11, 
                   (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__and x12 x11.
                 Proof.
                   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_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__U61_bounded :
                  forall x12 x11, 
                   (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U61 x12 x11.
                 Proof.
                   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 x11, (0 <= x11) ->0 <= P_id_a__U32 x11.
                 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 x13 x11, 
                   (0 <= x11) ->
                    (0 <= x12) ->(0 <= x13) ->0 <= P_id_U51 x13 x12 x11.
                 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_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_U12_bounded :
                  forall x11, (0 <= x11) ->0 <= P_id_U12 x11.
                 Proof.
                   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 x12 x11, 
                   (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U52 x12 x11.
                 Proof.
                   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_U43_bounded :
                  forall x11, (0 <= x11) ->0 <= P_id_U43 x11.
                 Proof.
                   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_a__U12_bounded :
                  forall x11, (0 <= x11) ->0 <= P_id_a__U12 x11.
                 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__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__U42_bounded :
                  forall x12 x11, 
                   (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U42 x12 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U53_bounded :
                  forall x11, (0 <= x11) ->0 <= P_id_U53 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_and_bounded :
                  forall x12 x11, 
                   (0 <= x11) ->(0 <= x12) ->0 <= P_id_and x12 x11.
                 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_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__U51_bounded :
                  forall x12 x13 x11, 
                   (0 <= x11) ->
                    (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U51 x13 x12 x11.
                 Proof.
                   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 x11, (0 <= x11) ->0 <= P_id_a__U22 x11.
                 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 x11, 
                   (0 <= x11) ->(0 <= x12) ->0 <= P_id_U42 x12 x11.
                 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_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_U22_bounded :
                  forall x11, (0 <= x11) ->0 <= P_id_U22 x11.
                 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__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_U52_bounded :
                  forall x12 x11, 
                   (0 <= x11) ->(0 <= x12) ->0 <= P_id_U52 x12 x11.
                 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_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_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__U53_bounded :
                  forall x11, (0 <= x11) ->0 <= P_id_a__U53 x11.
                 Proof.
                   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_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.
                 
                 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_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_U32_bounded :
                  forall x11, (0 <= x11) ->0 <= P_id_U32 x11.
                 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__U43_bounded :
                  forall x11, (0 <= x11) ->0 <= P_id_a__U43 x11.
                 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 x12 x11, 
                   (0 <= x11) ->(0 <= x12) ->0 <= P_id_U61 x12 x11.
                 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.
                 
                 Definition measure  := 
                   InterpZ.measure 0 P_id_a__zeros P_id_isNatKind 
                    P_id_a__isNatIList P_id_a__U21 P_id_U41 P_id_a__and 
                    P_id_a__U11 P_id_U21 P_id_a__U61 P_id_a__U32 P_id_U51 
                    P_id_isNatIListKind P_id_0 P_id_U12 P_id_a__U52 
                    P_id_a__isNat P_id_U43 P_id_a__isNatIListKind 
                    P_id_a__U12 P_id_U31 P_id_a__length P_id_a__U42 P_id_U53 
                    P_id_and P_id_cons P_id_U11 P_id_a__U51 P_id_a__U22 
                    P_id_U42 P_id_length P_id_tt P_id_U22 P_id_s P_id_a__U41 
                    P_id_U52 P_id_nil P_id_zeros P_id_isNatList P_id_a__U53 
                    P_id_a__U31 P_id_isNatIList P_id_a__isNatKind 
                    P_id_a__isNatList P_id_U32 P_id_mark P_id_a__U43 
                    P_id_U61 P_id_isNat.
                 
                 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_isNatKind 
                                    (x11::nil)) =>
                                  P_id_isNatKind (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__U21 
                                    (x12::x11::nil)) =>
                                  P_id_a__U21 (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_a__and 
                                    (x12::x11::nil)) =>
                                  P_id_a__and (measure x12) (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_U21 (x12::
                                    x11::nil)) =>
                                  P_id_U21 (measure x12) (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U61 
                                    (x12::x11::nil)) =>
                                  P_id_a__U61 (measure x12) (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U32 
                                    (x11::nil)) =>
                                  P_id_a__U32 (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_U51 (x13::
                                    x12::x11::nil)) =>
                                  P_id_U51 (measure x13) (measure x12) 
                                   (measure x11)
                                 | (algebra.Alg.Term 
                                    algebra.F.id_isNatIListKind (x11::nil)) =>
                                  P_id_isNatIListKind (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_0 nil) =>
                                  P_id_0 
                                 | (algebra.Alg.Term algebra.F.id_U12 
                                    (x11::nil)) =>
                                  P_id_U12 (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U52 
                                    (x12::x11::nil)) =>
                                  P_id_a__U52 (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_U43 
                                    (x11::nil)) =>
                                  P_id_U43 (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__U12 
                                    (x11::nil)) =>
                                  P_id_a__U12 (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__length 
                                    (x11::nil)) =>
                                  P_id_a__length (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U42 
                                    (x12::x11::nil)) =>
                                  P_id_a__U42 (measure x12) (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_U53 
                                    (x11::nil)) =>
                                  P_id_U53 (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_and (x12::
                                    x11::nil)) =>
                                  P_id_and (measure x12) (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_U11 (x12::
                                    x11::nil)) =>
                                  P_id_U11 (measure x12) (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U51 
                                    (x13::x12::x11::nil)) =>
                                  P_id_a__U51 (measure x13) (measure x12) 
                                   (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U22 
                                    (x11::nil)) =>
                                  P_id_a__U22 (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_U42 (x12::
                                    x11::nil)) =>
                                  P_id_U42 (measure x12) (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_length 
                                    (x11::nil)) =>
                                  P_id_length (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_tt nil) =>
                                  P_id_tt 
                                 | (algebra.Alg.Term algebra.F.id_U22 
                                    (x11::nil)) =>
                                  P_id_U22 (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_s 
                                    (x11::nil)) =>
                                  P_id_s (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_U52 (x12::
                                    x11::nil)) =>
                                  P_id_U52 (measure x12) (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_nil nil) =>
                                  P_id_nil 
                                 | (algebra.Alg.Term algebra.F.id_zeros nil) =>
                                  P_id_zeros 
                                 | (algebra.Alg.Term algebra.F.id_isNatList 
                                    (x11::nil)) =>
                                  P_id_isNatList (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U53 
                                    (x11::nil)) =>
                                  P_id_a__U53 (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_isNatIList 
                                    (x11::nil)) =>
                                  P_id_isNatIList (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__isNatList (x11::nil)) =>
                                  P_id_a__isNatList (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_U32 
                                    (x11::nil)) =>
                                  P_id_U32 (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_mark 
                                    (x11::nil)) =>
                                  P_id_mark (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U43 
                                    (x11::nil)) =>
                                  P_id_a__U43 (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_U61 (x12::
                                    x11::nil)) =>
                                  P_id_U61 (measure x12) (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_isNat 
                                    (x11::nil)) =>
                                  P_id_isNat (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_isNatKind_monotonic;assumption.
                   intros ;apply P_id_a__isNatIList_monotonic;assumption.
                   intros ;apply P_id_a__U21_monotonic;assumption.
                   intros ;apply P_id_U41_monotonic;assumption.
                   intros ;apply P_id_a__and_monotonic;assumption.
                   intros ;apply P_id_a__U11_monotonic;assumption.
                   intros ;apply P_id_U21_monotonic;assumption.
                   intros ;apply P_id_a__U61_monotonic;assumption.
                   intros ;apply P_id_a__U32_monotonic;assumption.
                   intros ;apply P_id_U51_monotonic;assumption.
                   intros ;apply P_id_isNatIListKind_monotonic;assumption.
                   intros ;apply P_id_U12_monotonic;assumption.
                   intros ;apply P_id_a__U52_monotonic;assumption.
                   intros ;apply P_id_a__isNat_monotonic;assumption.
                   intros ;apply P_id_U43_monotonic;assumption.
                   intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
                   intros ;apply P_id_a__U12_monotonic;assumption.
                   intros ;apply P_id_U31_monotonic;assumption.
                   intros ;apply P_id_a__length_monotonic;assumption.
                   intros ;apply P_id_a__U42_monotonic;assumption.
                   intros ;apply P_id_U53_monotonic;assumption.
                   intros ;apply P_id_and_monotonic;assumption.
                   intros ;apply P_id_cons_monotonic;assumption.
                   intros ;apply P_id_U11_monotonic;assumption.
                   intros ;apply P_id_a__U51_monotonic;assumption.
                   intros ;apply P_id_a__U22_monotonic;assumption.
                   intros ;apply P_id_U42_monotonic;assumption.
                   intros ;apply P_id_length_monotonic;assumption.
                   intros ;apply P_id_U22_monotonic;assumption.
                   intros ;apply P_id_s_monotonic;assumption.
                   intros ;apply P_id_a__U41_monotonic;assumption.
                   intros ;apply P_id_U52_monotonic;assumption.
                   intros ;apply P_id_isNatList_monotonic;assumption.
                   intros ;apply P_id_a__U53_monotonic;assumption.
                   intros ;apply P_id_a__U31_monotonic;assumption.
                   intros ;apply P_id_isNatIList_monotonic;assumption.
                   intros ;apply P_id_a__isNatKind_monotonic;assumption.
                   intros ;apply P_id_a__isNatList_monotonic;assumption.
                   intros ;apply P_id_U32_monotonic;assumption.
                   intros ;apply P_id_mark_monotonic;assumption.
                   intros ;apply P_id_a__U43_monotonic;assumption.
                   intros ;apply P_id_U61_monotonic;assumption.
                   intros ;apply P_id_isNat_monotonic;assumption.
                   intros ;apply P_id_a__zeros_bounded;assumption.
                   intros ;apply P_id_isNatKind_bounded;assumption.
                   intros ;apply P_id_a__isNatIList_bounded;assumption.
                   intros ;apply P_id_a__U21_bounded;assumption.
                   intros ;apply P_id_U41_bounded;assumption.
                   intros ;apply P_id_a__and_bounded;assumption.
                   intros ;apply P_id_a__U11_bounded;assumption.
                   intros ;apply P_id_U21_bounded;assumption.
                   intros ;apply P_id_a__U61_bounded;assumption.
                   intros ;apply P_id_a__U32_bounded;assumption.
                   intros ;apply P_id_U51_bounded;assumption.
                   intros ;apply P_id_isNatIListKind_bounded;assumption.
                   intros ;apply P_id_0_bounded;assumption.
                   intros ;apply P_id_U12_bounded;assumption.
                   intros ;apply P_id_a__U52_bounded;assumption.
                   intros ;apply P_id_a__isNat_bounded;assumption.
                   intros ;apply P_id_U43_bounded;assumption.
                   intros ;apply P_id_a__isNatIListKind_bounded;assumption.
                   intros ;apply P_id_a__U12_bounded;assumption.
                   intros ;apply P_id_U31_bounded;assumption.
                   intros ;apply P_id_a__length_bounded;assumption.
                   intros ;apply P_id_a__U42_bounded;assumption.
                   intros ;apply P_id_U53_bounded;assumption.
                   intros ;apply P_id_and_bounded;assumption.
                   intros ;apply P_id_cons_bounded;assumption.
                   intros ;apply P_id_U11_bounded;assumption.
                   intros ;apply P_id_a__U51_bounded;assumption.
                   intros ;apply P_id_a__U22_bounded;assumption.
                   intros ;apply P_id_U42_bounded;assumption.
                   intros ;apply P_id_length_bounded;assumption.
                   intros ;apply P_id_tt_bounded;assumption.
                   intros ;apply P_id_U22_bounded;assumption.
                   intros ;apply P_id_s_bounded;assumption.
                   intros ;apply P_id_a__U41_bounded;assumption.
                   intros ;apply P_id_U52_bounded;assumption.
                   intros ;apply P_id_nil_bounded;assumption.
                   intros ;apply P_id_zeros_bounded;assumption.
                   intros ;apply P_id_isNatList_bounded;assumption.
                   intros ;apply P_id_a__U53_bounded;assumption.
                   intros ;apply P_id_a__U31_bounded;assumption.
                   intros ;apply P_id_isNatIList_bounded;assumption.
                   intros ;apply P_id_a__isNatKind_bounded;assumption.
                   intros ;apply P_id_a__isNatList_bounded;assumption.
                   intros ;apply P_id_U32_bounded;assumption.
                   intros ;apply P_id_mark_bounded;assumption.
                   intros ;apply P_id_a__U43_bounded;assumption.
                   intros ;apply P_id_U61_bounded;assumption.
                   intros ;apply P_id_isNat_bounded;assumption.
                   apply rules_monotonic.
                 Qed.
                 
                 Definition P_id_A__LENGTH (x11:Z) := 0.
                 
                 Definition P_id_A__U11 (x11:Z) (x12:Z) := 0.
                 
                 Definition P_id_A__U41 (x11:Z) (x12:Z) (x13:Z) := 0.
                 
                 Definition P_id_A__ISNATKIND (x11:Z) := 1* x11.
                 
                 Definition P_id_A__U22 (x11:Z) := 0.
                 
                 Definition P_id_A__U51 (x11:Z) (x12:Z) (x13:Z) := 0.
                 
                 Definition P_id_A__AND (x11:Z) (x12:Z) := 0.
                 
                 Definition P_id_A__ISNATLIST (x11:Z) := 0.
                 
                 Definition P_id_A__U43 (x11:Z) := 0.
                 
                 Definition P_id_A__U31 (x11:Z) (x12:Z) := 0.
                 
                 Definition P_id_A__U53 (x11:Z) := 0.
                 
                 Definition P_id_MARK (x11:Z) := 0.
                 
                 Definition P_id_A__U12 (x11:Z) := 0.
                 
                 Definition P_id_A__U42 (x11:Z) (x12:Z) := 0.
                 
                 Definition P_id_A__ZEROS  := 0.
                 
                 Definition P_id_A__ISNAT (x11:Z) := 0.
                 
                 Definition P_id_A__U52 (x11:Z) (x12:Z) := 0.
                 
                 Definition P_id_A__ISNATILISTKIND (x11:Z) := 0.
                 
                 Definition P_id_A__U21 (x11:Z) (x12:Z) := 0.
                 
                 Definition P_id_A__ISNATILIST (x11:Z) := 0.
                 
                 Definition P_id_A__U32 (x11:Z) := 0.
                 
                 Definition P_id_A__U61 (x11:Z) (x12:Z) := 0.
                 
                 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__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__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__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__U22_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_A__U22 x12 <= P_id_A__U22 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_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 x16 x12 x14 x13 x11 x15, 
                   (0 <= x16)/\ (x16 <= x15) ->
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_A__U51 x12 x14 x16 <= P_id_A__U51 x11 x13 x15.
                 Proof.
                   intros x16 x15 x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   intros [H_5 H_4].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_A__AND_monotonic :
                  forall x12 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_A__AND x12 x14 <= P_id_A__AND x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_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__U43_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_A__U43 x12 <= P_id_A__U43 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   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__U53_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_A__U53 x12 <= P_id_A__U53 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 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__U12_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_A__U12 x12 <= P_id_A__U12 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   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 x12 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_A__U42 x12 x14 <= P_id_A__U42 x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_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_A__U52_monotonic :
                  forall x12 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_A__U52 x12 x14 <= P_id_A__U52 x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   
                   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__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__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__U32_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_A__U32 x12 <= P_id_A__U32 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_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 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_A__U61 x12 x14 <= P_id_A__U61 x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   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_isNatKind 
                    P_id_a__isNatIList P_id_a__U21 P_id_U41 P_id_a__and 
                    P_id_a__U11 P_id_U21 P_id_a__U61 P_id_a__U32 P_id_U51 
                    P_id_isNatIListKind P_id_0 P_id_U12 P_id_a__U52 
                    P_id_a__isNat P_id_U43 P_id_a__isNatIListKind 
                    P_id_a__U12 P_id_U31 P_id_a__length P_id_a__U42 P_id_U53 
                    P_id_and P_id_cons P_id_U11 P_id_a__U51 P_id_a__U22 
                    P_id_U42 P_id_length P_id_tt P_id_U22 P_id_s P_id_a__U41 
                    P_id_U52 P_id_nil P_id_zeros P_id_isNatList P_id_a__U53 
                    P_id_a__U31 P_id_isNatIList P_id_a__isNatKind 
                    P_id_a__isNatList P_id_U32 P_id_mark P_id_a__U43 
                    P_id_U61 P_id_isNat P_id_A__LENGTH P_id_A__U11 
                    P_id_A__U41 P_id_A__ISNATKIND P_id_A__U22 P_id_A__U51 
                    P_id_A__AND P_id_A__ISNATLIST P_id_A__U43 P_id_A__U31 
                    P_id_A__U53 P_id_MARK P_id_A__U12 P_id_A__U42 
                    P_id_A__ZEROS P_id_A__ISNAT P_id_A__U52 
                    P_id_A__ISNATILISTKIND P_id_A__U21 P_id_A__ISNATILIST 
                    P_id_A__U32 P_id_A__U61.
                 
                 Lemma marked_measure_equation :
                  forall t, 
                   marked_measure t = match t with
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__length (x11::nil)) =>
                                         P_id_A__LENGTH (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__U41 (x13::x12::
                                           x11::nil)) =>
                                         P_id_A__U41 (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__U22 (x11::nil)) =>
                                         P_id_A__U22 (measure x11)
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__U51 (x13::x12::
                                           x11::nil)) =>
                                         P_id_A__U51 (measure x13) 
                                          (measure x12) (measure x11)
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__and (x12::
                                           x11::nil)) =>
                                         P_id_A__AND (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__U43 (x11::nil)) =>
                                         P_id_A__U43 (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__U53 (x11::nil)) =>
                                         P_id_A__U53 (measure x11)
                                        | (algebra.Alg.Term 
                                           algebra.F.id_mark (x11::nil)) =>
                                         P_id_MARK (measure x11)
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__U12 (x11::nil)) =>
                                         P_id_A__U12 (measure x11)
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__U42 (x12::
                                           x11::nil)) =>
                                         P_id_A__U42 (measure x12) 
                                          (measure x11)
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__zeros nil) =>
                                         P_id_A__ZEROS 
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__isNat (x11::nil)) =>
                                         P_id_A__ISNAT (measure x11)
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__U52 (x12::
                                           x11::nil)) =>
                                         P_id_A__U52 (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_a__U21 (x12::
                                           x11::nil)) =>
                                         P_id_A__U21 (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__U32 (x11::nil)) =>
                                         P_id_A__U32 (measure x11)
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__U61 (x12::
                                           x11::nil)) =>
                                         P_id_A__U61 (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_isNatKind_monotonic;assumption.
                   intros ;apply P_id_a__isNatIList_monotonic;assumption.
                   intros ;apply P_id_a__U21_monotonic;assumption.
                   intros ;apply P_id_U41_monotonic;assumption.
                   intros ;apply P_id_a__and_monotonic;assumption.
                   intros ;apply P_id_a__U11_monotonic;assumption.
                   intros ;apply P_id_U21_monotonic;assumption.
                   intros ;apply P_id_a__U61_monotonic;assumption.
                   intros ;apply P_id_a__U32_monotonic;assumption.
                   intros ;apply P_id_U51_monotonic;assumption.
                   intros ;apply P_id_isNatIListKind_monotonic;assumption.
                   intros ;apply P_id_U12_monotonic;assumption.
                   intros ;apply P_id_a__U52_monotonic;assumption.
                   intros ;apply P_id_a__isNat_monotonic;assumption.
                   intros ;apply P_id_U43_monotonic;assumption.
                   intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
                   intros ;apply P_id_a__U12_monotonic;assumption.
                   intros ;apply P_id_U31_monotonic;assumption.
                   intros ;apply P_id_a__length_monotonic;assumption.
                   intros ;apply P_id_a__U42_monotonic;assumption.
                   intros ;apply P_id_U53_monotonic;assumption.
                   intros ;apply P_id_and_monotonic;assumption.
                   intros ;apply P_id_cons_monotonic;assumption.
                   intros ;apply P_id_U11_monotonic;assumption.
                   intros ;apply P_id_a__U51_monotonic;assumption.
                   intros ;apply P_id_a__U22_monotonic;assumption.
                   intros ;apply P_id_U42_monotonic;assumption.
                   intros ;apply P_id_length_monotonic;assumption.
                   intros ;apply P_id_U22_monotonic;assumption.
                   intros ;apply P_id_s_monotonic;assumption.
                   intros ;apply P_id_a__U41_monotonic;assumption.
                   intros ;apply P_id_U52_monotonic;assumption.
                   intros ;apply P_id_isNatList_monotonic;assumption.
                   intros ;apply P_id_a__U53_monotonic;assumption.
                   intros ;apply P_id_a__U31_monotonic;assumption.
                   intros ;apply P_id_isNatIList_monotonic;assumption.
                   intros ;apply P_id_a__isNatKind_monotonic;assumption.
                   intros ;apply P_id_a__isNatList_monotonic;assumption.
                   intros ;apply P_id_U32_monotonic;assumption.
                   intros ;apply P_id_mark_monotonic;assumption.
                   intros ;apply P_id_a__U43_monotonic;assumption.
                   intros ;apply P_id_U61_monotonic;assumption.
                   intros ;apply P_id_isNat_monotonic;assumption.
                   intros ;apply P_id_a__zeros_bounded;assumption.
                   intros ;apply P_id_isNatKind_bounded;assumption.
                   intros ;apply P_id_a__isNatIList_bounded;assumption.
                   intros ;apply P_id_a__U21_bounded;assumption.
                   intros ;apply P_id_U41_bounded;assumption.
                   intros ;apply P_id_a__and_bounded;assumption.
                   intros ;apply P_id_a__U11_bounded;assumption.
                   intros ;apply P_id_U21_bounded;assumption.
                   intros ;apply P_id_a__U61_bounded;assumption.
                   intros ;apply P_id_a__U32_bounded;assumption.
                   intros ;apply P_id_U51_bounded;assumption.
                   intros ;apply P_id_isNatIListKind_bounded;assumption.
                   intros ;apply P_id_0_bounded;assumption.
                   intros ;apply P_id_U12_bounded;assumption.
                   intros ;apply P_id_a__U52_bounded;assumption.
                   intros ;apply P_id_a__isNat_bounded;assumption.
                   intros ;apply P_id_U43_bounded;assumption.
                   intros ;apply P_id_a__isNatIListKind_bounded;assumption.
                   intros ;apply P_id_a__U12_bounded;assumption.
                   intros ;apply P_id_U31_bounded;assumption.
                   intros ;apply P_id_a__length_bounded;assumption.
                   intros ;apply P_id_a__U42_bounded;assumption.
                   intros ;apply P_id_U53_bounded;assumption.
                   intros ;apply P_id_and_bounded;assumption.
                   intros ;apply P_id_cons_bounded;assumption.
                   intros ;apply P_id_U11_bounded;assumption.
                   intros ;apply P_id_a__U51_bounded;assumption.
                   intros ;apply P_id_a__U22_bounded;assumption.
                   intros ;apply P_id_U42_bounded;assumption.
                   intros ;apply P_id_length_bounded;assumption.
                   intros ;apply P_id_tt_bounded;assumption.
                   intros ;apply P_id_U22_bounded;assumption.
                   intros ;apply P_id_s_bounded;assumption.
                   intros ;apply P_id_a__U41_bounded;assumption.
                   intros ;apply P_id_U52_bounded;assumption.
                   intros ;apply P_id_nil_bounded;assumption.
                   intros ;apply P_id_zeros_bounded;assumption.
                   intros ;apply P_id_isNatList_bounded;assumption.
                   intros ;apply P_id_a__U53_bounded;assumption.
                   intros ;apply P_id_a__U31_bounded;assumption.
                   intros ;apply P_id_isNatIList_bounded;assumption.
                   intros ;apply P_id_a__isNatKind_bounded;assumption.
                   intros ;apply P_id_a__isNatList_bounded;assumption.
                   intros ;apply P_id_U32_bounded;assumption.
                   intros ;apply P_id_mark_bounded;assumption.
                   intros ;apply P_id_a__U43_bounded;assumption.
                   intros ;apply P_id_U61_bounded;assumption.
                   intros ;apply P_id_isNat_bounded;assumption.
                   apply rules_monotonic.
                   intros ;apply P_id_A__LENGTH_monotonic;assumption.
                   intros ;apply P_id_A__U11_monotonic;assumption.
                   intros ;apply P_id_A__U41_monotonic;assumption.
                   intros ;apply P_id_A__ISNATKIND_monotonic;assumption.
                   intros ;apply P_id_A__U22_monotonic;assumption.
                   intros ;apply P_id_A__U51_monotonic;assumption.
                   intros ;apply P_id_A__AND_monotonic;assumption.
                   intros ;apply P_id_A__ISNATLIST_monotonic;assumption.
                   intros ;apply P_id_A__U43_monotonic;assumption.
                   intros ;apply P_id_A__U31_monotonic;assumption.
                   intros ;apply P_id_A__U53_monotonic;assumption.
                   intros ;apply P_id_MARK_monotonic;assumption.
                   intros ;apply P_id_A__U12_monotonic;assumption.
                   intros ;apply P_id_A__U42_monotonic;assumption.
                   intros ;apply P_id_A__ISNAT_monotonic;assumption.
                   intros ;apply P_id_A__U52_monotonic;assumption.
                   intros ;apply P_id_A__ISNATILISTKIND_monotonic;assumption.
                   intros ;apply P_id_A__U21_monotonic;assumption.
                   intros ;apply P_id_A__ISNATILIST_monotonic;assumption.
                   intros ;apply P_id_A__U32_monotonic;assumption.
                   intros ;apply P_id_A__U61_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large.DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_1.
                
                Definition wf_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_1  := 
                  WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_1.wf
                  .
                
                
                Lemma acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_1 :
                 forall x y, 
                  (DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_1 
                    x y) ->
                   Acc WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large.DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large 
                    x.
                Proof.
                  intros x.
                  pattern x.
                  
                  apply (@Acc_ind _ 
                          DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_1.
                Qed.
                
                
                Inductive DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_non_scc_2  :
                 algebra.Alg.term ->algebra.Alg.term ->Prop := 
                   (* <mark(isNatKind(X_)),a__isNatKind(X_)> *)
                  | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_non_scc_2_0 :
                   forall 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_isNatKind (x5::nil)) 
                      x11) ->
                     DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_non_scc_2 
                      (algebra.Alg.Term algebra.F.id_a__isNatKind (x5::nil)) 
                      (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                .
                
                
                Lemma acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_non_scc_2 :
                 forall x y, 
                  (DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_non_scc_2 
                    x y) ->
                   Acc WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large.DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_non_scc_3  :
                 algebra.Alg.term ->algebra.Alg.term ->Prop := 
                   (* <a__isNatIListKind(cons(V1_,V2_)),a__isNatKind(V1_)> *)
                  | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_non_scc_3_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_non_scc_3 
                      (algebra.Alg.Term algebra.F.id_a__isNatKind (x1::nil)) 
                      (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
                       (x11::nil))
                .
                
                
                Lemma acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_non_scc_3 :
                 forall x y, 
                  (DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_non_scc_3 
                    x y) ->
                   Acc WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large.DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4  :
                 algebra.Alg.term ->algebra.Alg.term ->Prop := 
                   (* <mark(U53(X_)),mark(X_)> *)
                  | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_0 :
                   forall 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_U53 (x5::nil)) x11) ->
                     DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4 
                      (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
                      (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                  
                   (* <mark(U22(X_)),mark(X_)> *)
                  | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_1 :
                   forall 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_U22 (x5::nil)) x11) ->
                     DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4 
                      (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
                      (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                  
                   (* <mark(and(X1_,X2_)),a__and(mark(X1_),X2_)> *)
                  | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_2 :
                   forall x8 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_and (x7::x8::nil)) 
                      x11) ->
                     DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4 
                      (algebra.Alg.Term algebra.F.id_a__and 
                       ((algebra.Alg.Term algebra.F.id_mark (x7::nil))::
                       x8::nil)) 
                      (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                  
                   (* <a__and(tt,X_),mark(X_)> *)
                  | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_3 :
                   forall x12 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_tt nil) 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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4 
                       (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
                       (algebra.Alg.Term algebra.F.id_a__and (x12::x11::nil))
                  
                   (* <mark(and(X1_,X2_)),mark(X1_)> *)
                  | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_4 :
                   forall x8 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_and (x7::x8::nil)) 
                      x11) ->
                     DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4 
                      (algebra.Alg.Term algebra.F.id_mark (x7::nil)) 
                      (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                  
                   (* <mark(isNatIListKind(X_)),a__isNatIListKind(X_)> *)
                  | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_5 :
                   forall 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_isNatIListKind 
                       (x5::nil)) x11) ->
                     DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4 
                      (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
                       (x5::nil)) 
                      (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                  
                   (* <a__isNatIListKind(cons(V1_,V2_)),a__and(a__isNatKind(V1_),isNatIListKind(V2_))> *)
                  | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_6 :
                   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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4 
                      (algebra.Alg.Term algebra.F.id_a__and 
                       ((algebra.Alg.Term algebra.F.id_a__isNatKind 
                       (x1::nil))::(algebra.Alg.Term 
                       algebra.F.id_isNatIListKind (x3::nil))::nil)) 
                      (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
                       (x11::nil))
                  
                   (* <mark(s(X_)),mark(X_)> *)
                  | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_7 :
                   forall 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_s (x5::nil)) x11) ->
                     DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4 
                      (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
                      (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                .
                
                
                Module WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4.
                 Inductive DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large  :
                  algebra.Alg.term ->algebra.Alg.term ->Prop := 
                    (* <mark(U22(X_)),mark(X_)> *)
                   | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_0 :
                    forall 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_U22 (x5::nil)) 
                       x11) ->
                      DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large 
                       (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
                       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                   
                    (* <mark(and(X1_,X2_)),a__and(mark(X1_),X2_)> *)
                   | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_1 :
                    forall x8 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_and (x7::x8::nil)) 
                       x11) ->
                      DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large 
                       (algebra.Alg.Term algebra.F.id_a__and 
                        ((algebra.Alg.Term algebra.F.id_mark (x7::nil))::
                        x8::nil)) 
                       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                   
                    (* <a__and(tt,X_),mark(X_)> *)
                   | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_2 :
                    forall x12 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_tt nil) 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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large 
                        (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
                        (algebra.Alg.Term algebra.F.id_a__and (x12::
                         x11::nil))
                   
                    (* <mark(and(X1_,X2_)),mark(X1_)> *)
                   | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_3 :
                    forall x8 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_and (x7::x8::nil)) 
                       x11) ->
                      DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large 
                       (algebra.Alg.Term algebra.F.id_mark (x7::nil)) 
                       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                   
                    (* <mark(isNatIListKind(X_)),a__isNatIListKind(X_)> *)
                   | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_4 :
                    forall 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_isNatIListKind 
                        (x5::nil)) x11) ->
                      DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large 
                       (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
                        (x5::nil)) 
                       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                   
                    (* <a__isNatIListKind(cons(V1_,V2_)),a__and(a__isNatKind(V1_),isNatIListKind(V2_))> *)
                   | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large 
                       (algebra.Alg.Term algebra.F.id_a__and 
                        ((algebra.Alg.Term algebra.F.id_a__isNatKind 
                        (x1::nil))::(algebra.Alg.Term 
                        algebra.F.id_isNatIListKind (x3::nil))::nil)) 
                       (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
                        (x11::nil))
                   
                    (* <mark(s(X_)),mark(X_)> *)
                   | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_6 :
                    forall 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_s (x5::nil)) x11) ->
                      DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large 
                       (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
                       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                 .
                 
                 
                 Inductive DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_strict  :
                  algebra.Alg.term ->algebra.Alg.term ->Prop := 
                    (* <mark(U53(X_)),mark(X_)> *)
                   | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_strict_0 :
                    forall 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_U53 (x5::nil)) 
                       x11) ->
                      DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_strict 
                       (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
                       (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                 .
                 
                 
                 Module WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large.
                  Inductive DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_large  :
                   algebra.Alg.term ->algebra.Alg.term ->Prop := 
                     (* <mark(U22(X_)),mark(X_)> *)
                    | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_large_0 :
                     forall 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_U22 (x5::nil)) 
                        x11) ->
                       DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_large 
                        (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
                        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                    
                     (* <a__and(tt,X_),mark(X_)> *)
                    | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_large_1 :
                     forall x12 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_tt nil) 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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_large 
                         (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
                         (algebra.Alg.Term algebra.F.id_a__and (x12::
                          x11::nil))
                    
                     (* <mark(s(X_)),mark(X_)> *)
                    | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_large_2 :
                     forall 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_s (x5::nil)) x11) ->
                       DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_large 
                        (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
                        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                  .
                  
                  
                  Inductive DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_strict  :
                   algebra.Alg.term ->algebra.Alg.term ->Prop := 
                     (* <mark(and(X1_,X2_)),a__and(mark(X1_),X2_)> *)
                    | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_strict_0 :
                     forall x8 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_and (x7::x8::nil)) 
                        x11) ->
                       DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_strict 
                        (algebra.Alg.Term algebra.F.id_a__and 
                         ((algebra.Alg.Term algebra.F.id_mark (x7::nil))::
                         x8::nil)) 
                        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                    
                     (* <mark(and(X1_,X2_)),mark(X1_)> *)
                    | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_strict_1 :
                     forall x8 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_and (x7::x8::nil)) 
                        x11) ->
                       DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_strict 
                        (algebra.Alg.Term algebra.F.id_mark (x7::nil)) 
                        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                    
                     (* <mark(isNatIListKind(X_)),a__isNatIListKind(X_)> *)
                    | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_strict_2 :
                     forall 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_isNatIListKind 
                         (x5::nil)) x11) ->
                       DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_strict 
                        (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
                         (x5::nil)) 
                        (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                    
                     (* <a__isNatIListKind(cons(V1_,V2_)),a__and(a__isNatKind(V1_),isNatIListKind(V2_))> *)
                    | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_strict_3 :
                     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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_strict 
                        (algebra.Alg.Term algebra.F.id_a__and 
                         ((algebra.Alg.Term algebra.F.id_a__isNatKind 
                         (x1::nil))::(algebra.Alg.Term 
                         algebra.F.id_isNatIListKind (x3::nil))::nil)) 
                        (algebra.Alg.Term algebra.F.id_a__isNatIListKind 
                         (x11::nil))
                  .
                  
                  
                  Module WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_large.
                   Inductive DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_large_scc_1  :
                    algebra.Alg.term ->algebra.Alg.term ->Prop := 
                      (* <mark(s(X_)),mark(X_)> *)
                     | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_large_scc_1_0 :
                      forall 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_s (x5::nil)) 
                         x11) ->
                        DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_large_scc_1 
                         (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
                         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                     
                      (* <mark(U22(X_)),mark(X_)> *)
                     | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_large_scc_1_1 :
                      forall 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_U22 (x5::nil)) 
                         x11) ->
                        DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_large_scc_1 
                         (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
                         (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                   .
                   
                   
                   Module WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_large_scc_1.
                    Inductive DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_large_scc_1_large  :
                     algebra.Alg.term ->algebra.Alg.term ->Prop := 
                       (* <mark(U22(X_)),mark(X_)> *)
                      | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_large_scc_1_large_0 :
                       forall 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_U22 (x5::nil)) 
                          x11) ->
                         DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_large_scc_1_large 
                          (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
                          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                    .
                    
                    
                    Inductive DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_large_scc_1_strict  :
                     algebra.Alg.term ->algebra.Alg.term ->Prop := 
                       (* <mark(s(X_)),mark(X_)> *)
                      | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_large_scc_1_strict_0 :
                       forall 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_s (x5::nil)) 
                          x11) ->
                         DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_large_scc_1_strict 
                          (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
                          (algebra.Alg.Term algebra.F.id_mark (x11::nil))
                    .
                    
                    
                    Module WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_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_isNatKind (x11:Z) := 0.
                     
                     Definition P_id_a__isNatIList (x11:Z) := 1.
                     
                     Definition P_id_a__U21 (x11:Z) (x12:Z) := 3 + 2* x12.
                     
                     Definition P_id_U41 (x11:Z) (x12:Z) (x13:Z) := 0.
                     
                     Definition P_id_a__and (x11:Z) (x12:Z) := 
                       1* x11 + 2* x12.
                     
                     Definition P_id_a__U11 (x11:Z) (x12:Z) := 1.
                     
                     Definition P_id_U21 (x11:Z) (x12:Z) := 1 + 2* x12.
                     
                     Definition P_id_a__U61 (x11:Z) (x12:Z) := 
                       1 + 1* x11 + 2* x12.
                     
                     Definition P_id_a__U32 (x11:Z) := 1.
                     
                     Definition P_id_U51 (x11:Z) (x12:Z) (x13:Z) := 
                       1* x12 + 3* x13.
                     
                     Definition P_id_isNatIListKind (x11:Z) := 0.
                     
                     Definition P_id_0  := 0.
                     
                     Definition P_id_U12 (x11:Z) := 1.
                     
                     Definition P_id_a__U52 (x11:Z) (x12:Z) := 1* x12.
                     
                     Definition P_id_a__isNat (x11:Z) := 1 + 2* x11.
                     
                     Definition P_id_U43 (x11:Z) := 0.
                     
                     Definition P_id_a__isNatIListKind (x11:Z) := 1.
                     
                     Definition P_id_a__U12 (x11:Z) := 1.
                     
                     Definition P_id_U31 (x11:Z) (x12:Z) := 1* x11.
                     
                     Definition P_id_a__length (x11:Z) := 1* x11.
                     
                     Definition P_id_a__U42 (x11:Z) (x12:Z) := 1.
                     
                     Definition P_id_U53 (x11:Z) := 1* x11.
                     
                     Definition P_id_and (x11:Z) (x12:Z) := 1* x11 + 2* x12.
                     
                     Definition P_id_cons (x11:Z) (x12:Z) := 
                       1 + 2* x11 + 3* x12.
                     
                     Definition P_id_U11 (x11:Z) (x12:Z) := 0.
                     
                     Definition P_id_a__U51 (x11:Z) (x12:Z) (x13:Z) := 
                       2* x12 + 3* x13.
                     
                     Definition P_id_a__U22 (x11:Z) := 2 + 1* x11.
                     
                     Definition P_id_U42 (x11:Z) (x12:Z) := 0.
                     
                     Definition P_id_length (x11:Z) := 1* x11.
                     
                     Definition P_id_tt  := 1.
                     
                     Definition P_id_U22 (x11:Z) := 2 + 1* x11.
                     
                     Definition P_id_s (x11:Z) := 1 + 1* x11.
                     
                     Definition P_id_a__U41 (x11:Z) (x12:Z) (x13:Z) := 1.
                     
                     Definition P_id_U52 (x11:Z) (x12:Z) := 1* x12.
                     
                     Definition P_id_nil  := 2.
                     
                     Definition P_id_zeros  := 0.
                     
                     Definition P_id_isNatList (x11:Z) := 1* x11.
                     
                     Definition P_id_a__U53 (x11:Z) := 1* x11.
                     
                     Definition P_id_a__U31 (x11:Z) (x12:Z) := 1* x11.
                     
                     Definition P_id_isNatIList (x11:Z) := 0.
                     
                     Definition P_id_a__isNatKind (x11:Z) := 1.
                     
                     Definition P_id_a__isNatList (x11:Z) := 1* x11.
                     
                     Definition P_id_U32 (x11:Z) := 1.
                     
                     Definition P_id_mark (x11:Z) := 1 + 2* x11.
                     
                     Definition P_id_a__U43 (x11:Z) := 1.
                     
                     Definition P_id_U61 (x11:Z) (x12:Z) := 
                       1 + 1* x11 + 2* x12.
                     
                     Definition P_id_isNat (x11:Z) := 1* x11.
                     
                     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__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__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_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_a__and_monotonic :
                      forall x12 x14 x13 x11, 
                       (0 <= x14)/\ (x14 <= x13) ->
                        (0 <= x12)/\ (x12 <= x11) ->
                         P_id_a__and x12 x14 <= P_id_a__and x11 x13.
                     Proof.
                       intros x14 x13 x12 x11.
                       intros [H_1 H_0].
                       intros [H_3 H_2].
                       
                       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_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__U61_monotonic :
                      forall x12 x14 x13 x11, 
                       (0 <= x14)/\ (x14 <= x13) ->
                        (0 <= x12)/\ (x12 <= x11) ->
                         P_id_a__U61 x12 x14 <= P_id_a__U61 x11 x13.
                     Proof.
                       intros x14 x13 x12 x11.
                       intros [H_1 H_0].
                       intros [H_3 H_2].
                       
                       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 x11, 
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_a__U32 x12 <= P_id_a__U32 x11.
                     Proof.
                       intros x12 x11.
                       intros [H_1 H_0].
                       
                       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                        intuition;
                        (auto with zarith)||
                        (repeat (translate_vars );prove_ineq ).
                     Qed.
                     
                     Lemma P_id_U51_monotonic :
                      forall x16 x12 x14 x13 x11 x15, 
                       (0 <= x16)/\ (x16 <= x15) ->
                        (0 <= x14)/\ (x14 <= x13) ->
                         (0 <= x12)/\ (x12 <= x11) ->
                          P_id_U51 x12 x14 x16 <= P_id_U51 x11 x13 x15.
                     Proof.
                       intros x16 x15 x14 x13 x12 x11.
                       intros [H_1 H_0].
                       intros [H_3 H_2].
                       intros [H_5 H_4].
                       
                       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_U12_monotonic :
                      forall x12 x11, 
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_U12 x12 <= P_id_U12 x11.
                     Proof.
                       intros x12 x11.
                       intros [H_1 H_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 x14 x13 x11, 
                       (0 <= x14)/\ (x14 <= x13) ->
                        (0 <= x12)/\ (x12 <= x11) ->
                         P_id_a__U52 x12 x14 <= P_id_a__U52 x11 x13.
                     Proof.
                       intros x14 x13 x12 x11.
                       intros [H_1 H_0].
                       intros [H_3 H_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_U43_monotonic :
                      forall x12 x11, 
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_U43 x12 <= P_id_U43 x11.
                     Proof.
                       intros x12 x11.
                       intros [H_1 H_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__U12_monotonic :
                      forall x12 x11, 
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_a__U12 x12 <= P_id_a__U12 x11.
                     Proof.
                       intros x12 x11.
                       intros [H_1 H_0].
                       
                       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__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__U42_monotonic :
                      forall x12 x14 x13 x11, 
                       (0 <= x14)/\ (x14 <= x13) ->
                        (0 <= x12)/\ (x12 <= x11) ->
                         P_id_a__U42 x12 x14 <= P_id_a__U42 x11 x13.
                     Proof.
                       intros x14 x13 x12 x11.
                       intros [H_1 H_0].
                       intros [H_3 H_2].
                       
                       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                        intuition;
                        (auto with zarith)||
                        (repeat (translate_vars );prove_ineq ).
                     Qed.
                     
                     Lemma P_id_U53_monotonic :
                      forall x12 x11, 
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_U53 x12 <= P_id_U53 x11.
                     Proof.
                       intros x12 x11.
                       intros [H_1 H_0].
                       
                       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                        intuition;
                        (auto with zarith)||
                        (repeat (translate_vars );prove_ineq ).
                     Qed.
                     
                     Lemma P_id_and_monotonic :
                      forall x12 x14 x13 x11, 
                       (0 <= x14)/\ (x14 <= x13) ->
                        (0 <= x12)/\ (x12 <= x11) ->
                         P_id_and x12 x14 <= P_id_and x11 x13.
                     Proof.
                       intros x14 x13 x12 x11.
                       intros [H_1 H_0].
                       intros [H_3 H_2].
                       
                       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_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__U51_monotonic :
                      forall x16 x12 x14 x13 x11 x15, 
                       (0 <= x16)/\ (x16 <= x15) ->
                        (0 <= x14)/\ (x14 <= x13) ->
                         (0 <= x12)/\ (x12 <= x11) ->
                          P_id_a__U51 x12 x14 x16 <= P_id_a__U51 x11 x13 x15.
                     Proof.
                       intros x16 x15 x14 x13 x12 x11.
                       intros [H_1 H_0].
                       intros [H_3 H_2].
                       intros [H_5 H_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 x11, 
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_a__U22 x12 <= P_id_a__U22 x11.
                     Proof.
                       intros x12 x11.
                       intros [H_1 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 x12 x14 x13 x11, 
                       (0 <= x14)/\ (x14 <= x13) ->
                        (0 <= x12)/\ (x12 <= x11) ->
                         P_id_U42 x12 x14 <= P_id_U42 x11 x13.
                     Proof.
                       intros x14 x13 x12 x11.
                       intros [H_1 H_0].
                       intros [H_3 H_2].
                       
                       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_U22_monotonic :
                      forall x12 x11, 
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_U22 x12 <= P_id_U22 x11.
                     Proof.
                       intros x12 x11.
                       intros [H_1 H_0].
                       
                       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__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_U52_monotonic :
                      forall x12 x14 x13 x11, 
                       (0 <= x14)/\ (x14 <= x13) ->
                        (0 <= x12)/\ (x12 <= x11) ->
                         P_id_U52 x12 x14 <= P_id_U52 x11 x13.
                     Proof.
                       intros x14 x13 x12 x11.
                       intros [H_1 H_0].
                       intros [H_3 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__U53_monotonic :
                      forall x12 x11, 
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_a__U53 x12 <= P_id_a__U53 x11.
                     Proof.
                       intros x12 x11.
                       intros [H_1 H_0].
                       
                       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_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__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__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_U32_monotonic :
                      forall x12 x11, 
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_U32 x12 <= P_id_U32 x11.
                     Proof.
                       intros x12 x11.
                       intros [H_1 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__U43_monotonic :
                      forall x12 x11, 
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_a__U43 x12 <= P_id_a__U43 x11.
                     Proof.
                       intros x12 x11.
                       intros [H_1 H_0].
                       
                       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 x14 x13 x11, 
                       (0 <= x14)/\ (x14 <= x13) ->
                        (0 <= x12)/\ (x12 <= x11) ->
                         P_id_U61 x12 x14 <= P_id_U61 x11 x13.
                     Proof.
                       intros x14 x13 x12 x11.
                       intros [H_1 H_0].
                       intros [H_3 H_2].
                       
                       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__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_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__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_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_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_a__and_bounded :
                      forall x12 x11, 
                       (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__and x12 x11.
                     Proof.
                       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_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__U61_bounded :
                      forall x12 x11, 
                       (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U61 x12 x11.
                     Proof.
                       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 x11, (0 <= x11) ->0 <= P_id_a__U32 x11.
                     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 x13 x11, 
                       (0 <= x11) ->
                        (0 <= x12) ->(0 <= x13) ->0 <= P_id_U51 x13 x12 x11.
                     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_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_U12_bounded :
                      forall x11, (0 <= x11) ->0 <= P_id_U12 x11.
                     Proof.
                       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 x12 x11, 
                       (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U52 x12 x11.
                     Proof.
                       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_U43_bounded :
                      forall x11, (0 <= x11) ->0 <= P_id_U43 x11.
                     Proof.
                       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_a__U12_bounded :
                      forall x11, (0 <= x11) ->0 <= P_id_a__U12 x11.
                     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__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__U42_bounded :
                      forall x12 x11, 
                       (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U42 x12 x11.
                     Proof.
                       intros .
                       
                       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                        intuition;
                        (auto with zarith)||
                        (repeat (translate_vars );prove_ineq ).
                     Qed.
                     
                     Lemma P_id_U53_bounded :
                      forall x11, (0 <= x11) ->0 <= P_id_U53 x11.
                     Proof.
                       intros .
                       
                       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                        intuition;
                        (auto with zarith)||
                        (repeat (translate_vars );prove_ineq ).
                     Qed.
                     
                     Lemma P_id_and_bounded :
                      forall x12 x11, 
                       (0 <= x11) ->(0 <= x12) ->0 <= P_id_and x12 x11.
                     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_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__U51_bounded :
                      forall x12 x13 x11, 
                       (0 <= x11) ->
                        (0 <= x12) ->
                         (0 <= x13) ->0 <= P_id_a__U51 x13 x12 x11.
                     Proof.
                       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 x11, (0 <= x11) ->0 <= P_id_a__U22 x11.
                     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 x11, 
                       (0 <= x11) ->(0 <= x12) ->0 <= P_id_U42 x12 x11.
                     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_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_U22_bounded :
                      forall x11, (0 <= x11) ->0 <= P_id_U22 x11.
                     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__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_U52_bounded :
                      forall x12 x11, 
                       (0 <= x11) ->(0 <= x12) ->0 <= P_id_U52 x12 x11.
                     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_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_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__U53_bounded :
                      forall x11, (0 <= x11) ->0 <= P_id_a__U53 x11.
                     Proof.
                       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_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.
                     
                     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_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_U32_bounded :
                      forall x11, (0 <= x11) ->0 <= P_id_U32 x11.
                     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__U43_bounded :
                      forall x11, (0 <= x11) ->0 <= P_id_a__U43 x11.
                     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 x12 x11, 
                       (0 <= x11) ->(0 <= x12) ->0 <= P_id_U61 x12 x11.
                     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.
                     
                     Definition measure  := 
                       InterpZ.measure 0 P_id_a__zeros P_id_isNatKind 
                        P_id_a__isNatIList P_id_a__U21 P_id_U41 P_id_a__and 
                        P_id_a__U11 P_id_U21 P_id_a__U61 P_id_a__U32 
                        P_id_U51 P_id_isNatIListKind P_id_0 P_id_U12 
                        P_id_a__U52 P_id_a__isNat P_id_U43 
                        P_id_a__isNatIListKind P_id_a__U12 P_id_U31 
                        P_id_a__length P_id_a__U42 P_id_U53 P_id_and 
                        P_id_cons P_id_U11 P_id_a__U51 P_id_a__U22 P_id_U42 
                        P_id_length P_id_tt P_id_U22 P_id_s P_id_a__U41 
                        P_id_U52 P_id_nil P_id_zeros P_id_isNatList 
                        P_id_a__U53 P_id_a__U31 P_id_isNatIList 
                        P_id_a__isNatKind P_id_a__isNatList P_id_U32 
                        P_id_mark P_id_a__U43 P_id_U61 P_id_isNat.
                     
                     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_isNatKind (x11::nil)) =>
                                      P_id_isNatKind (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__U21 
                                        (x12::x11::nil)) =>
                                      P_id_a__U21 (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_a__and 
                                        (x12::x11::nil)) =>
                                      P_id_a__and (measure x12) (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_U21 
                                        (x12::x11::nil)) =>
                                      P_id_U21 (measure x12) (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_a__U61 
                                        (x12::x11::nil)) =>
                                      P_id_a__U61 (measure x12) (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_a__U32 
                                        (x11::nil)) =>
                                      P_id_a__U32 (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_U51 
                                        (x13::x12::x11::nil)) =>
                                      P_id_U51 (measure x13) (measure x12) 
                                       (measure x11)
                                     | (algebra.Alg.Term 
                                        algebra.F.id_isNatIListKind 
                                        (x11::nil)) =>
                                      P_id_isNatIListKind (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_0 nil) =>
                                      P_id_0 
                                     | (algebra.Alg.Term algebra.F.id_U12 
                                        (x11::nil)) =>
                                      P_id_U12 (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_a__U52 
                                        (x12::x11::nil)) =>
                                      P_id_a__U52 (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_U43 
                                        (x11::nil)) =>
                                      P_id_U43 (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__U12 
                                        (x11::nil)) =>
                                      P_id_a__U12 (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__length (x11::nil)) =>
                                      P_id_a__length (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_a__U42 
                                        (x12::x11::nil)) =>
                                      P_id_a__U42 (measure x12) (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_U53 
                                        (x11::nil)) =>
                                      P_id_U53 (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_and 
                                        (x12::x11::nil)) =>
                                      P_id_and (measure x12) (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_U11 
                                        (x12::x11::nil)) =>
                                      P_id_U11 (measure x12) (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_a__U51 
                                        (x13::x12::x11::nil)) =>
                                      P_id_a__U51 (measure x13) 
                                       (measure x12) (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_a__U22 
                                        (x11::nil)) =>
                                      P_id_a__U22 (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_U42 
                                        (x12::x11::nil)) =>
                                      P_id_U42 (measure x12) (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_length 
                                        (x11::nil)) =>
                                      P_id_length (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_tt nil) =>
                                      P_id_tt 
                                     | (algebra.Alg.Term algebra.F.id_U22 
                                        (x11::nil)) =>
                                      P_id_U22 (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_s 
                                        (x11::nil)) =>
                                      P_id_s (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_U52 
                                        (x12::x11::nil)) =>
                                      P_id_U52 (measure x12) (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_nil 
                                        nil) =>
                                      P_id_nil 
                                     | (algebra.Alg.Term algebra.F.id_zeros 
                                        nil) =>
                                      P_id_zeros 
                                     | (algebra.Alg.Term 
                                        algebra.F.id_isNatList (x11::nil)) =>
                                      P_id_isNatList (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_a__U53 
                                        (x11::nil)) =>
                                      P_id_a__U53 (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_isNatIList (x11::nil)) =>
                                      P_id_isNatIList (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__isNatList (x11::nil)) =>
                                      P_id_a__isNatList (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_U32 
                                        (x11::nil)) =>
                                      P_id_U32 (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_mark 
                                        (x11::nil)) =>
                                      P_id_mark (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_a__U43 
                                        (x11::nil)) =>
                                      P_id_a__U43 (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_U61 
                                        (x12::x11::nil)) =>
                                      P_id_U61 (measure x12) (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_isNat 
                                        (x11::nil)) =>
                                      P_id_isNat (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_isNatKind_monotonic;assumption.
                       intros ;apply P_id_a__isNatIList_monotonic;assumption.
                       intros ;apply P_id_a__U21_monotonic;assumption.
                       intros ;apply P_id_U41_monotonic;assumption.
                       intros ;apply P_id_a__and_monotonic;assumption.
                       intros ;apply P_id_a__U11_monotonic;assumption.
                       intros ;apply P_id_U21_monotonic;assumption.
                       intros ;apply P_id_a__U61_monotonic;assumption.
                       intros ;apply P_id_a__U32_monotonic;assumption.
                       intros ;apply P_id_U51_monotonic;assumption.
                       
                       intros ;apply P_id_isNatIListKind_monotonic;assumption.
                       intros ;apply P_id_U12_monotonic;assumption.
                       intros ;apply P_id_a__U52_monotonic;assumption.
                       intros ;apply P_id_a__isNat_monotonic;assumption.
                       intros ;apply P_id_U43_monotonic;assumption.
                       
                       intros ;apply P_id_a__isNatIListKind_monotonic;
                        assumption.
                       intros ;apply P_id_a__U12_monotonic;assumption.
                       intros ;apply P_id_U31_monotonic;assumption.
                       intros ;apply P_id_a__length_monotonic;assumption.
                       intros ;apply P_id_a__U42_monotonic;assumption.
                       intros ;apply P_id_U53_monotonic;assumption.
                       intros ;apply P_id_and_monotonic;assumption.
                       intros ;apply P_id_cons_monotonic;assumption.
                       intros ;apply P_id_U11_monotonic;assumption.
                       intros ;apply P_id_a__U51_monotonic;assumption.
                       intros ;apply P_id_a__U22_monotonic;assumption.
                       intros ;apply P_id_U42_monotonic;assumption.
                       intros ;apply P_id_length_monotonic;assumption.
                       intros ;apply P_id_U22_monotonic;assumption.
                       intros ;apply P_id_s_monotonic;assumption.
                       intros ;apply P_id_a__U41_monotonic;assumption.
                       intros ;apply P_id_U52_monotonic;assumption.
                       intros ;apply P_id_isNatList_monotonic;assumption.
                       intros ;apply P_id_a__U53_monotonic;assumption.
                       intros ;apply P_id_a__U31_monotonic;assumption.
                       intros ;apply P_id_isNatIList_monotonic;assumption.
                       intros ;apply P_id_a__isNatKind_monotonic;assumption.
                       intros ;apply P_id_a__isNatList_monotonic;assumption.
                       intros ;apply P_id_U32_monotonic;assumption.
                       intros ;apply P_id_mark_monotonic;assumption.
                       intros ;apply P_id_a__U43_monotonic;assumption.
                       intros ;apply P_id_U61_monotonic;assumption.
                       intros ;apply P_id_isNat_monotonic;assumption.
                       intros ;apply P_id_a__zeros_bounded;assumption.
                       intros ;apply P_id_isNatKind_bounded;assumption.
                       intros ;apply P_id_a__isNatIList_bounded;assumption.
                       intros ;apply P_id_a__U21_bounded;assumption.
                       intros ;apply P_id_U41_bounded;assumption.
                       intros ;apply P_id_a__and_bounded;assumption.
                       intros ;apply P_id_a__U11_bounded;assumption.
                       intros ;apply P_id_U21_bounded;assumption.
                       intros ;apply P_id_a__U61_bounded;assumption.
                       intros ;apply P_id_a__U32_bounded;assumption.
                       intros ;apply P_id_U51_bounded;assumption.
                       intros ;apply P_id_isNatIListKind_bounded;assumption.
                       intros ;apply P_id_0_bounded;assumption.
                       intros ;apply P_id_U12_bounded;assumption.
                       intros ;apply P_id_a__U52_bounded;assumption.
                       intros ;apply P_id_a__isNat_bounded;assumption.
                       intros ;apply P_id_U43_bounded;assumption.
                       
                       intros ;apply P_id_a__isNatIListKind_bounded;
                        assumption.
                       intros ;apply P_id_a__U12_bounded;assumption.
                       intros ;apply P_id_U31_bounded;assumption.
                       intros ;apply P_id_a__length_bounded;assumption.
                       intros ;apply P_id_a__U42_bounded;assumption.
                       intros ;apply P_id_U53_bounded;assumption.
                       intros ;apply P_id_and_bounded;assumption.
                       intros ;apply P_id_cons_bounded;assumption.
                       intros ;apply P_id_U11_bounded;assumption.
                       intros ;apply P_id_a__U51_bounded;assumption.
                       intros ;apply P_id_a__U22_bounded;assumption.
                       intros ;apply P_id_U42_bounded;assumption.
                       intros ;apply P_id_length_bounded;assumption.
                       intros ;apply P_id_tt_bounded;assumption.
                       intros ;apply P_id_U22_bounded;assumption.
                       intros ;apply P_id_s_bounded;assumption.
                       intros ;apply P_id_a__U41_bounded;assumption.
                       intros ;apply P_id_U52_bounded;assumption.
                       intros ;apply P_id_nil_bounded;assumption.
                       intros ;apply P_id_zeros_bounded;assumption.
                       intros ;apply P_id_isNatList_bounded;assumption.
                       intros ;apply P_id_a__U53_bounded;assumption.
                       intros ;apply P_id_a__U31_bounded;assumption.
                       intros ;apply P_id_isNatIList_bounded;assumption.
                       intros ;apply P_id_a__isNatKind_bounded;assumption.
                       intros ;apply P_id_a__isNatList_bounded;assumption.
                       intros ;apply P_id_U32_bounded;assumption.
                       intros ;apply P_id_mark_bounded;assumption.
                       intros ;apply P_id_a__U43_bounded;assumption.
                       intros ;apply P_id_U61_bounded;assumption.
                       intros ;apply P_id_isNat_bounded;assumption.
                       apply rules_monotonic.
                     Qed.
                     
                     Definition P_id_A__LENGTH (x11:Z) := 0.
                     
                     Definition P_id_A__U11 (x11:Z) (x12:Z) := 0.
                     
                     Definition P_id_A__U41 (x11:Z) (x12:Z) (x13:Z) := 0.
                     
                     Definition P_id_A__ISNATKIND (x11:Z) := 0.
                     
                     Definition P_id_A__U22 (x11:Z) := 0.
                     
                     Definition P_id_A__U51 (x11:Z) (x12:Z) (x13:Z) := 0.
                     
                     Definition P_id_A__AND (x11:Z) (x12:Z) := 0.
                     
                     Definition P_id_A__ISNATLIST (x11:Z) := 0.
                     
                     Definition P_id_A__U43 (x11:Z) := 0.
                     
                     Definition P_id_A__U31 (x11:Z) (x12:Z) := 0.
                     
                     Definition P_id_A__U53 (x11:Z) := 0.
                     
                     Definition P_id_MARK (x11:Z) := 2* x11.
                     
                     Definition P_id_A__U12 (x11:Z) := 0.
                     
                     Definition P_id_A__U42 (x11:Z) (x12:Z) := 0.
                     
                     Definition P_id_A__ZEROS  := 0.
                     
                     Definition P_id_A__ISNAT (x11:Z) := 0.
                     
                     Definition P_id_A__U52 (x11:Z) (x12:Z) := 0.
                     
                     Definition P_id_A__ISNATILISTKIND (x11:Z) := 0.
                     
                     Definition P_id_A__U21 (x11:Z) (x12:Z) := 0.
                     
                     Definition P_id_A__ISNATILIST (x11:Z) := 0.
                     
                     Definition P_id_A__U32 (x11:Z) := 0.
                     
                     Definition P_id_A__U61 (x11:Z) (x12:Z) := 0.
                     
                     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__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__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__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__U22_monotonic :
                      forall x12 x11, 
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_A__U22 x12 <= P_id_A__U22 x11.
                     Proof.
                       intros x12 x11.
                       intros [H_1 H_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 x16 x12 x14 x13 x11 x15, 
                       (0 <= x16)/\ (x16 <= x15) ->
                        (0 <= x14)/\ (x14 <= x13) ->
                         (0 <= x12)/\ (x12 <= x11) ->
                          P_id_A__U51 x12 x14 x16 <= P_id_A__U51 x11 x13 x15.
                     Proof.
                       intros x16 x15 x14 x13 x12 x11.
                       intros [H_1 H_0].
                       intros [H_3 H_2].
                       intros [H_5 H_4].
                       
                       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                        intuition;
                        (auto with zarith)||
                        (repeat (translate_vars );prove_ineq ).
                     Qed.
                     
                     Lemma P_id_A__AND_monotonic :
                      forall x12 x14 x13 x11, 
                       (0 <= x14)/\ (x14 <= x13) ->
                        (0 <= x12)/\ (x12 <= x11) ->
                         P_id_A__AND x12 x14 <= P_id_A__AND x11 x13.
                     Proof.
                       intros x14 x13 x12 x11.
                       intros [H_1 H_0].
                       intros [H_3 H_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__U43_monotonic :
                      forall x12 x11, 
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_A__U43 x12 <= P_id_A__U43 x11.
                     Proof.
                       intros x12 x11.
                       intros [H_1 H_0].
                       
                       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__U53_monotonic :
                      forall x12 x11, 
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_A__U53 x12 <= P_id_A__U53 x11.
                     Proof.
                       intros x12 x11.
                       intros [H_1 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__U12_monotonic :
                      forall x12 x11, 
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_A__U12 x12 <= P_id_A__U12 x11.
                     Proof.
                       intros x12 x11.
                       intros [H_1 H_0].
                       
                       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 x12 x14 x13 x11, 
                       (0 <= x14)/\ (x14 <= x13) ->
                        (0 <= x12)/\ (x12 <= x11) ->
                         P_id_A__U42 x12 x14 <= P_id_A__U42 x11 x13.
                     Proof.
                       intros x14 x13 x12 x11.
                       intros [H_1 H_0].
                       intros [H_3 H_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_A__U52_monotonic :
                      forall x12 x14 x13 x11, 
                       (0 <= x14)/\ (x14 <= x13) ->
                        (0 <= x12)/\ (x12 <= x11) ->
                         P_id_A__U52 x12 x14 <= P_id_A__U52 x11 x13.
                     Proof.
                       intros x14 x13 x12 x11.
                       intros [H_1 H_0].
                       intros [H_3 H_2].
                       
                       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__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__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__U32_monotonic :
                      forall x12 x11, 
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_A__U32 x12 <= P_id_A__U32 x11.
                     Proof.
                       intros x12 x11.
                       intros [H_1 H_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 x14 x13 x11, 
                       (0 <= x14)/\ (x14 <= x13) ->
                        (0 <= x12)/\ (x12 <= x11) ->
                         P_id_A__U61 x12 x14 <= P_id_A__U61 x11 x13.
                     Proof.
                       intros x14 x13 x12 x11.
                       intros [H_1 H_0].
                       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_isNatKind 
                        P_id_a__isNatIList P_id_a__U21 P_id_U41 P_id_a__and 
                        P_id_a__U11 P_id_U21 P_id_a__U61 P_id_a__U32 
                        P_id_U51 P_id_isNatIListKind P_id_0 P_id_U12 
                        P_id_a__U52 P_id_a__isNat P_id_U43 
                        P_id_a__isNatIListKind P_id_a__U12 P_id_U31 
                        P_id_a__length P_id_a__U42 P_id_U53 P_id_and 
                        P_id_cons P_id_U11 P_id_a__U51 P_id_a__U22 P_id_U42 
                        P_id_length P_id_tt P_id_U22 P_id_s P_id_a__U41 
                        P_id_U52 P_id_nil P_id_zeros P_id_isNatList 
                        P_id_a__U53 P_id_a__U31 P_id_isNatIList 
                        P_id_a__isNatKind P_id_a__isNatList P_id_U32 
                        P_id_mark P_id_a__U43 P_id_U61 P_id_isNat 
                        P_id_A__LENGTH P_id_A__U11 P_id_A__U41 
                        P_id_A__ISNATKIND P_id_A__U22 P_id_A__U51 
                        P_id_A__AND P_id_A__ISNATLIST P_id_A__U43 
                        P_id_A__U31 P_id_A__U53 P_id_MARK P_id_A__U12 
                        P_id_A__U42 P_id_A__ZEROS P_id_A__ISNAT P_id_A__U52 
                        P_id_A__ISNATILISTKIND P_id_A__U21 
                        P_id_A__ISNATILIST P_id_A__U32 P_id_A__U61.
                     
                     Lemma marked_measure_equation :
                      forall t, 
                       marked_measure t = match t with
                                            | (algebra.Alg.Term 
                                               algebra.F.id_a__length 
                                               (x11::nil)) =>
                                             P_id_A__LENGTH (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__U41 (x13::
                                               x12::x11::nil)) =>
                                             P_id_A__U41 (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__U22 
                                               (x11::nil)) =>
                                             P_id_A__U22 (measure x11)
                                            | (algebra.Alg.Term 
                                               algebra.F.id_a__U51 (x13::
                                               x12::x11::nil)) =>
                                             P_id_A__U51 (measure x13) 
                                              (measure x12) (measure x11)
                                            | (algebra.Alg.Term 
                                               algebra.F.id_a__and (x12::
                                               x11::nil)) =>
                                             P_id_A__AND (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__U43 
                                               (x11::nil)) =>
                                             P_id_A__U43 (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__U53 
                                               (x11::nil)) =>
                                             P_id_A__U53 (measure x11)
                                            | (algebra.Alg.Term 
                                               algebra.F.id_mark (x11::nil)) =>
                                             P_id_MARK (measure x11)
                                            | (algebra.Alg.Term 
                                               algebra.F.id_a__U12 
                                               (x11::nil)) =>
                                             P_id_A__U12 (measure x11)
                                            | (algebra.Alg.Term 
                                               algebra.F.id_a__U42 (x12::
                                               x11::nil)) =>
                                             P_id_A__U42 (measure x12) 
                                              (measure x11)
                                            | (algebra.Alg.Term 
                                               algebra.F.id_a__zeros nil) =>
                                             P_id_A__ZEROS 
                                            | (algebra.Alg.Term 
                                               algebra.F.id_a__isNat 
                                               (x11::nil)) =>
                                             P_id_A__ISNAT (measure x11)
                                            | (algebra.Alg.Term 
                                               algebra.F.id_a__U52 (x12::
                                               x11::nil)) =>
                                             P_id_A__U52 (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_a__U21 (x12::
                                               x11::nil)) =>
                                             P_id_A__U21 (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__U32 
                                               (x11::nil)) =>
                                             P_id_A__U32 (measure x11)
                                            | (algebra.Alg.Term 
                                               algebra.F.id_a__U61 (x12::
                                               x11::nil)) =>
                                             P_id_A__U61 (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_isNatKind_monotonic;assumption.
                       intros ;apply P_id_a__isNatIList_monotonic;assumption.
                       intros ;apply P_id_a__U21_monotonic;assumption.
                       intros ;apply P_id_U41_monotonic;assumption.
                       intros ;apply P_id_a__and_monotonic;assumption.
                       intros ;apply P_id_a__U11_monotonic;assumption.
                       intros ;apply P_id_U21_monotonic;assumption.
                       intros ;apply P_id_a__U61_monotonic;assumption.
                       intros ;apply P_id_a__U32_monotonic;assumption.
                       intros ;apply P_id_U51_monotonic;assumption.
                       
                       intros ;apply P_id_isNatIListKind_monotonic;assumption.
                       intros ;apply P_id_U12_monotonic;assumption.
                       intros ;apply P_id_a__U52_monotonic;assumption.
                       intros ;apply P_id_a__isNat_monotonic;assumption.
                       intros ;apply P_id_U43_monotonic;assumption.
                       
                       intros ;apply P_id_a__isNatIListKind_monotonic;
                        assumption.
                       intros ;apply P_id_a__U12_monotonic;assumption.
                       intros ;apply P_id_U31_monotonic;assumption.
                       intros ;apply P_id_a__length_monotonic;assumption.
                       intros ;apply P_id_a__U42_monotonic;assumption.
                       intros ;apply P_id_U53_monotonic;assumption.
                       intros ;apply P_id_and_monotonic;assumption.
                       intros ;apply P_id_cons_monotonic;assumption.
                       intros ;apply P_id_U11_monotonic;assumption.
                       intros ;apply P_id_a__U51_monotonic;assumption.
                       intros ;apply P_id_a__U22_monotonic;assumption.
                       intros ;apply P_id_U42_monotonic;assumption.
                       intros ;apply P_id_length_monotonic;assumption.
                       intros ;apply P_id_U22_monotonic;assumption.
                       intros ;apply P_id_s_monotonic;assumption.
                       intros ;apply P_id_a__U41_monotonic;assumption.
                       intros ;apply P_id_U52_monotonic;assumption.
                       intros ;apply P_id_isNatList_monotonic;assumption.
                       intros ;apply P_id_a__U53_monotonic;assumption.
                       intros ;apply P_id_a__U31_monotonic;assumption.
                       intros ;apply P_id_isNatIList_monotonic;assumption.
                       intros ;apply P_id_a__isNatKind_monotonic;assumption.
                       intros ;apply P_id_a__isNatList_monotonic;assumption.
                       intros ;apply P_id_U32_monotonic;assumption.
                       intros ;apply P_id_mark_monotonic;assumption.
                       intros ;apply P_id_a__U43_monotonic;assumption.
                       intros ;apply P_id_U61_monotonic;assumption.
                       intros ;apply P_id_isNat_monotonic;assumption.
                       intros ;apply P_id_a__zeros_bounded;assumption.
                       intros ;apply P_id_isNatKind_bounded;assumption.
                       intros ;apply P_id_a__isNatIList_bounded;assumption.
                       intros ;apply P_id_a__U21_bounded;assumption.
                       intros ;apply P_id_U41_bounded;assumption.
                       intros ;apply P_id_a__and_bounded;assumption.
                       intros ;apply P_id_a__U11_bounded;assumption.
                       intros ;apply P_id_U21_bounded;assumption.
                       intros ;apply P_id_a__U61_bounded;assumption.
                       intros ;apply P_id_a__U32_bounded;assumption.
                       intros ;apply P_id_U51_bounded;assumption.
                       intros ;apply P_id_isNatIListKind_bounded;assumption.
                       intros ;apply P_id_0_bounded;assumption.
                       intros ;apply P_id_U12_bounded;assumption.
                       intros ;apply P_id_a__U52_bounded;assumption.
                       intros ;apply P_id_a__isNat_bounded;assumption.
                       intros ;apply P_id_U43_bounded;assumption.
                       
                       intros ;apply P_id_a__isNatIListKind_bounded;
                        assumption.
                       intros ;apply P_id_a__U12_bounded;assumption.
                       intros ;apply P_id_U31_bounded;assumption.
                       intros ;apply P_id_a__length_bounded;assumption.
                       intros ;apply P_id_a__U42_bounded;assumption.
                       intros ;apply P_id_U53_bounded;assumption.
                       intros ;apply P_id_and_bounded;assumption.
                       intros ;apply P_id_cons_bounded;assumption.
                       intros ;apply P_id_U11_bounded;assumption.
                       intros ;apply P_id_a__U51_bounded;assumption.
                       intros ;apply P_id_a__U22_bounded;assumption.
                       intros ;apply P_id_U42_bounded;assumption.
                       intros ;apply P_id_length_bounded;assumption.
                       intros ;apply P_id_tt_bounded;assumption.
                       intros ;apply P_id_U22_bounded;assumption.
                       intros ;apply P_id_s_bounded;assumption.
                       intros ;apply P_id_a__U41_bounded;assumption.
                       intros ;apply P_id_U52_bounded;assumption.
                       intros ;apply P_id_nil_bounded;assumption.
                       intros ;apply P_id_zeros_bounded;assumption.
                       intros ;apply P_id_isNatList_bounded;assumption.
                       intros ;apply P_id_a__U53_bounded;assumption.
                       intros ;apply P_id_a__U31_bounded;assumption.
                       intros ;apply P_id_isNatIList_bounded;assumption.
                       intros ;apply P_id_a__isNatKind_bounded;assumption.
                       intros ;apply P_id_a__isNatList_bounded;assumption.
                       intros ;apply P_id_U32_bounded;assumption.
                       intros ;apply P_id_mark_bounded;assumption.
                       intros ;apply P_id_a__U43_bounded;assumption.
                       intros ;apply P_id_U61_bounded;assumption.
                       intros ;apply P_id_isNat_bounded;assumption.
                       apply rules_monotonic.
                       intros ;apply P_id_A__LENGTH_monotonic;assumption.
                       intros ;apply P_id_A__U11_monotonic;assumption.
                       intros ;apply P_id_A__U41_monotonic;assumption.
                       intros ;apply P_id_A__ISNATKIND_monotonic;assumption.
                       intros ;apply P_id_A__U22_monotonic;assumption.
                       intros ;apply P_id_A__U51_monotonic;assumption.
                       intros ;apply P_id_A__AND_monotonic;assumption.
                       intros ;apply P_id_A__ISNATLIST_monotonic;assumption.
                       intros ;apply P_id_A__U43_monotonic;assumption.
                       intros ;apply P_id_A__U31_monotonic;assumption.
                       intros ;apply P_id_A__U53_monotonic;assumption.
                       intros ;apply P_id_MARK_monotonic;assumption.
                       intros ;apply P_id_A__U12_monotonic;assumption.
                       intros ;apply P_id_A__U42_monotonic;assumption.
                       intros ;apply P_id_A__ISNAT_monotonic;assumption.
                       intros ;apply P_id_A__U52_monotonic;assumption.
                       
                       intros ;apply P_id_A__ISNATILISTKIND_monotonic;
                        assumption.
                       intros ;apply P_id_A__U21_monotonic;assumption.
                       intros ;apply P_id_A__ISNATILIST_monotonic;assumption.
                       intros ;apply P_id_A__U32_monotonic;assumption.
                       intros ;apply P_id_A__U61_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_large_scc_1.DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_large_scc_1_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_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_isNatKind (x11:Z) := 0.
                    
                    Definition P_id_a__isNatIList (x11:Z) := 3 + 3* x11.
                    
                    Definition P_id_a__U21 (x11:Z) (x12:Z) := 1.
                    
                    Definition P_id_U41 (x11:Z) (x12:Z) (x13:Z) := 
                      2 + 2* x11 + 2* x13.
                    
                    Definition P_id_a__and (x11:Z) (x12:Z) := 1* x11 + 2* x12
                      .
                    
                    Definition P_id_a__U11 (x11:Z) (x12:Z) := 1.
                    
                    Definition P_id_U21 (x11:Z) (x12:Z) := 0.
                    
                    Definition P_id_a__U61 (x11:Z) (x12:Z) := 
                      1 + 1* x11 + 2* x12.
                    
                    Definition P_id_a__U32 (x11:Z) := 1.
                    
                    Definition P_id_U51 (x11:Z) (x12:Z) (x13:Z) := 1* x13.
                    
                    Definition P_id_isNatIListKind (x11:Z) := 0.
                    
                    Definition P_id_0  := 0.
                    
                    Definition P_id_U12 (x11:Z) := 0.
                    
                    Definition P_id_a__U52 (x11:Z) (x12:Z) := 2* x12.
                    
                    Definition P_id_a__isNat (x11:Z) := 1.
                    
                    Definition P_id_U43 (x11:Z) := 2.
                    
                    Definition P_id_a__isNatIListKind (x11:Z) := 1.
                    
                    Definition P_id_a__U12 (x11:Z) := 1.
                    
                    Definition P_id_U31 (x11:Z) (x12:Z) := 1 + 3* x12.
                    
                    Definition P_id_a__length (x11:Z) := 1* x11.
                    
                    Definition P_id_a__U42 (x11:Z) (x12:Z) := 3.
                    
                    Definition P_id_U53 (x11:Z) := 1* x11.
                    
                    Definition P_id_and (x11:Z) (x12:Z) := 1* x11 + 1* x12.
                    
                    Definition P_id_cons (x11:Z) (x12:Z) := 1 + 3* x12.
                    
                    Definition P_id_U11 (x11:Z) (x12:Z) := 0.
                    
                    Definition P_id_a__U51 (x11:Z) (x12:Z) (x13:Z) := 
                      1 + 2* x13.
                    
                    Definition P_id_a__U22 (x11:Z) := 1* x11.
                    
                    Definition P_id_U42 (x11:Z) (x12:Z) := 1.
                    
                    Definition P_id_length (x11:Z) := 1* x11.
                    
                    Definition P_id_tt  := 1.
                    
                    Definition P_id_U22 (x11:Z) := 1* x11.
                    
                    Definition P_id_s (x11:Z) := 1 + 1* x11.
                    
                    Definition P_id_a__U41 (x11:Z) (x12:Z) (x13:Z) := 
                      2 + 2* x11 + 2* x13.
                    
                    Definition P_id_U52 (x11:Z) (x12:Z) := 1* x12.
                    
                    Definition P_id_nil  := 1.
                    
                    Definition P_id_zeros  := 0.
                    
                    Definition P_id_isNatList (x11:Z) := 1* x11.
                    
                    Definition P_id_a__U53 (x11:Z) := 1* x11.
                    
                    Definition P_id_a__U31 (x11:Z) (x12:Z) := 1 + 3* x12.
                    
                    Definition P_id_isNatIList (x11:Z) := 2 + 2* x11.
                    
                    Definition P_id_a__isNatKind (x11:Z) := 1.
                    
                    Definition P_id_a__isNatList (x11:Z) := 1* x11.
                    
                    Definition P_id_U32 (x11:Z) := 0.
                    
                    Definition P_id_mark (x11:Z) := 1 + 2* x11.
                    
                    Definition P_id_a__U43 (x11:Z) := 3.
                    
                    Definition P_id_U61 (x11:Z) (x12:Z) := 
                      1 + 1* x11 + 2* x12.
                    
                    Definition P_id_isNat (x11:Z) := 0.
                    
                    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__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__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_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_a__and_monotonic :
                     forall x12 x14 x13 x11, 
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_a__and x12 x14 <= P_id_a__and x11 x13.
                    Proof.
                      intros x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      
                      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_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__U61_monotonic :
                     forall x12 x14 x13 x11, 
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_a__U61 x12 x14 <= P_id_a__U61 x11 x13.
                    Proof.
                      intros x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      
                      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 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_a__U32 x12 <= P_id_a__U32 x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_0].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U51_monotonic :
                     forall x16 x12 x14 x13 x11 x15, 
                      (0 <= x16)/\ (x16 <= x15) ->
                       (0 <= x14)/\ (x14 <= x13) ->
                        (0 <= x12)/\ (x12 <= x11) ->
                         P_id_U51 x12 x14 x16 <= P_id_U51 x11 x13 x15.
                    Proof.
                      intros x16 x15 x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      intros [H_5 H_4].
                      
                      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_U12_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_U12 x12 <= P_id_U12 x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_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 x14 x13 x11, 
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_a__U52 x12 x14 <= P_id_a__U52 x11 x13.
                    Proof.
                      intros x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_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_U43_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_U43 x12 <= P_id_U43 x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_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__U12_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_a__U12 x12 <= P_id_a__U12 x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_0].
                      
                      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__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__U42_monotonic :
                     forall x12 x14 x13 x11, 
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_a__U42 x12 x14 <= P_id_a__U42 x11 x13.
                    Proof.
                      intros x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U53_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_U53 x12 <= P_id_U53 x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_0].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_and_monotonic :
                     forall x12 x14 x13 x11, 
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_and x12 x14 <= P_id_and x11 x13.
                    Proof.
                      intros x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      
                      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_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__U51_monotonic :
                     forall x16 x12 x14 x13 x11 x15, 
                      (0 <= x16)/\ (x16 <= x15) ->
                       (0 <= x14)/\ (x14 <= x13) ->
                        (0 <= x12)/\ (x12 <= x11) ->
                         P_id_a__U51 x12 x14 x16 <= P_id_a__U51 x11 x13 x15.
                    Proof.
                      intros x16 x15 x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      intros [H_5 H_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 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_a__U22 x12 <= P_id_a__U22 x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 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 x12 x14 x13 x11, 
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_U42 x12 x14 <= P_id_U42 x11 x13.
                    Proof.
                      intros x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      
                      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_U22_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_U22 x12 <= P_id_U22 x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_0].
                      
                      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__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_U52_monotonic :
                     forall x12 x14 x13 x11, 
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_U52 x12 x14 <= P_id_U52 x11 x13.
                    Proof.
                      intros x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 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__U53_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_a__U53 x12 <= P_id_a__U53 x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_0].
                      
                      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_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__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__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_U32_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_U32 x12 <= P_id_U32 x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 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__U43_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_a__U43 x12 <= P_id_a__U43 x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_0].
                      
                      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 x14 x13 x11, 
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_U61 x12 x14 <= P_id_U61 x11 x13.
                    Proof.
                      intros x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      
                      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__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_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__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_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_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_a__and_bounded :
                     forall x12 x11, 
                      (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__and x12 x11.
                    Proof.
                      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_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__U61_bounded :
                     forall x12 x11, 
                      (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U61 x12 x11.
                    Proof.
                      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 x11, (0 <= x11) ->0 <= P_id_a__U32 x11.
                    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 x13 x11, 
                      (0 <= x11) ->
                       (0 <= x12) ->(0 <= x13) ->0 <= P_id_U51 x13 x12 x11.
                    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_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_U12_bounded :
                     forall x11, (0 <= x11) ->0 <= P_id_U12 x11.
                    Proof.
                      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 x12 x11, 
                      (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U52 x12 x11.
                    Proof.
                      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_U43_bounded :
                     forall x11, (0 <= x11) ->0 <= P_id_U43 x11.
                    Proof.
                      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_a__U12_bounded :
                     forall x11, (0 <= x11) ->0 <= P_id_a__U12 x11.
                    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__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__U42_bounded :
                     forall x12 x11, 
                      (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U42 x12 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_U53_bounded :
                     forall x11, (0 <= x11) ->0 <= P_id_U53 x11.
                    Proof.
                      intros .
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_and_bounded :
                     forall x12 x11, 
                      (0 <= x11) ->(0 <= x12) ->0 <= P_id_and x12 x11.
                    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_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__U51_bounded :
                     forall x12 x13 x11, 
                      (0 <= x11) ->
                       (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U51 x13 x12 x11.
                    Proof.
                      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 x11, (0 <= x11) ->0 <= P_id_a__U22 x11.
                    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 x11, 
                      (0 <= x11) ->(0 <= x12) ->0 <= P_id_U42 x12 x11.
                    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_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_U22_bounded :
                     forall x11, (0 <= x11) ->0 <= P_id_U22 x11.
                    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__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_U52_bounded :
                     forall x12 x11, 
                      (0 <= x11) ->(0 <= x12) ->0 <= P_id_U52 x12 x11.
                    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_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_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__U53_bounded :
                     forall x11, (0 <= x11) ->0 <= P_id_a__U53 x11.
                    Proof.
                      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_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.
                    
                    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_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_U32_bounded :
                     forall x11, (0 <= x11) ->0 <= P_id_U32 x11.
                    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__U43_bounded :
                     forall x11, (0 <= x11) ->0 <= P_id_a__U43 x11.
                    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 x12 x11, 
                      (0 <= x11) ->(0 <= x12) ->0 <= P_id_U61 x12 x11.
                    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.
                    
                    Definition measure  := 
                      InterpZ.measure 0 P_id_a__zeros P_id_isNatKind 
                       P_id_a__isNatIList P_id_a__U21 P_id_U41 P_id_a__and 
                       P_id_a__U11 P_id_U21 P_id_a__U61 P_id_a__U32 P_id_U51 
                       P_id_isNatIListKind P_id_0 P_id_U12 P_id_a__U52 
                       P_id_a__isNat P_id_U43 P_id_a__isNatIListKind 
                       P_id_a__U12 P_id_U31 P_id_a__length P_id_a__U42 
                       P_id_U53 P_id_and P_id_cons P_id_U11 P_id_a__U51 
                       P_id_a__U22 P_id_U42 P_id_length P_id_tt P_id_U22 
                       P_id_s P_id_a__U41 P_id_U52 P_id_nil P_id_zeros 
                       P_id_isNatList P_id_a__U53 P_id_a__U31 
                       P_id_isNatIList P_id_a__isNatKind P_id_a__isNatList 
                       P_id_U32 P_id_mark P_id_a__U43 P_id_U61 P_id_isNat.
                    
                    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_isNatKind (x11::nil)) =>
                                     P_id_isNatKind (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__U21 
                                       (x12::x11::nil)) =>
                                     P_id_a__U21 (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_a__and 
                                       (x12::x11::nil)) =>
                                     P_id_a__and (measure x12) (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_U21 
                                       (x12::x11::nil)) =>
                                     P_id_U21 (measure x12) (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U61 
                                       (x12::x11::nil)) =>
                                     P_id_a__U61 (measure x12) (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U32 
                                       (x11::nil)) =>
                                     P_id_a__U32 (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_U51 
                                       (x13::x12::x11::nil)) =>
                                     P_id_U51 (measure x13) (measure x12) 
                                      (measure x11)
                                    | (algebra.Alg.Term 
                                       algebra.F.id_isNatIListKind 
                                       (x11::nil)) =>
                                     P_id_isNatIListKind (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_0 nil) =>
                                     P_id_0 
                                    | (algebra.Alg.Term algebra.F.id_U12 
                                       (x11::nil)) =>
                                     P_id_U12 (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U52 
                                       (x12::x11::nil)) =>
                                     P_id_a__U52 (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_U43 
                                       (x11::nil)) =>
                                     P_id_U43 (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__U12 
                                       (x11::nil)) =>
                                     P_id_a__U12 (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__length (x11::nil)) =>
                                     P_id_a__length (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U42 
                                       (x12::x11::nil)) =>
                                     P_id_a__U42 (measure x12) (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_U53 
                                       (x11::nil)) =>
                                     P_id_U53 (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_and 
                                       (x12::x11::nil)) =>
                                     P_id_and (measure x12) (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_U11 
                                       (x12::x11::nil)) =>
                                     P_id_U11 (measure x12) (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U51 
                                       (x13::x12::x11::nil)) =>
                                     P_id_a__U51 (measure x13) (measure x12) 
                                      (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U22 
                                       (x11::nil)) =>
                                     P_id_a__U22 (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_U42 
                                       (x12::x11::nil)) =>
                                     P_id_U42 (measure x12) (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_length 
                                       (x11::nil)) =>
                                     P_id_length (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_tt nil) =>
                                     P_id_tt 
                                    | (algebra.Alg.Term algebra.F.id_U22 
                                       (x11::nil)) =>
                                     P_id_U22 (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_s 
                                       (x11::nil)) =>
                                     P_id_s (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_U52 
                                       (x12::x11::nil)) =>
                                     P_id_U52 (measure x12) (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_nil nil) =>
                                     P_id_nil 
                                    | (algebra.Alg.Term algebra.F.id_zeros 
                                       nil) =>
                                     P_id_zeros 
                                    | (algebra.Alg.Term 
                                       algebra.F.id_isNatList (x11::nil)) =>
                                     P_id_isNatList (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U53 
                                       (x11::nil)) =>
                                     P_id_a__U53 (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_isNatIList (x11::nil)) =>
                                     P_id_isNatIList (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__isNatList (x11::nil)) =>
                                     P_id_a__isNatList (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_U32 
                                       (x11::nil)) =>
                                     P_id_U32 (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_mark 
                                       (x11::nil)) =>
                                     P_id_mark (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_a__U43 
                                       (x11::nil)) =>
                                     P_id_a__U43 (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_U61 
                                       (x12::x11::nil)) =>
                                     P_id_U61 (measure x12) (measure x11)
                                    | (algebra.Alg.Term algebra.F.id_isNat 
                                       (x11::nil)) =>
                                     P_id_isNat (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_isNatKind_monotonic;assumption.
                      intros ;apply P_id_a__isNatIList_monotonic;assumption.
                      intros ;apply P_id_a__U21_monotonic;assumption.
                      intros ;apply P_id_U41_monotonic;assumption.
                      intros ;apply P_id_a__and_monotonic;assumption.
                      intros ;apply P_id_a__U11_monotonic;assumption.
                      intros ;apply P_id_U21_monotonic;assumption.
                      intros ;apply P_id_a__U61_monotonic;assumption.
                      intros ;apply P_id_a__U32_monotonic;assumption.
                      intros ;apply P_id_U51_monotonic;assumption.
                      intros ;apply P_id_isNatIListKind_monotonic;assumption.
                      intros ;apply P_id_U12_monotonic;assumption.
                      intros ;apply P_id_a__U52_monotonic;assumption.
                      intros ;apply P_id_a__isNat_monotonic;assumption.
                      intros ;apply P_id_U43_monotonic;assumption.
                      
                      intros ;apply P_id_a__isNatIListKind_monotonic;
                       assumption.
                      intros ;apply P_id_a__U12_monotonic;assumption.
                      intros ;apply P_id_U31_monotonic;assumption.
                      intros ;apply P_id_a__length_monotonic;assumption.
                      intros ;apply P_id_a__U42_monotonic;assumption.
                      intros ;apply P_id_U53_monotonic;assumption.
                      intros ;apply P_id_and_monotonic;assumption.
                      intros ;apply P_id_cons_monotonic;assumption.
                      intros ;apply P_id_U11_monotonic;assumption.
                      intros ;apply P_id_a__U51_monotonic;assumption.
                      intros ;apply P_id_a__U22_monotonic;assumption.
                      intros ;apply P_id_U42_monotonic;assumption.
                      intros ;apply P_id_length_monotonic;assumption.
                      intros ;apply P_id_U22_monotonic;assumption.
                      intros ;apply P_id_s_monotonic;assumption.
                      intros ;apply P_id_a__U41_monotonic;assumption.
                      intros ;apply P_id_U52_monotonic;assumption.
                      intros ;apply P_id_isNatList_monotonic;assumption.
                      intros ;apply P_id_a__U53_monotonic;assumption.
                      intros ;apply P_id_a__U31_monotonic;assumption.
                      intros ;apply P_id_isNatIList_monotonic;assumption.
                      intros ;apply P_id_a__isNatKind_monotonic;assumption.
                      intros ;apply P_id_a__isNatList_monotonic;assumption.
                      intros ;apply P_id_U32_monotonic;assumption.
                      intros ;apply P_id_mark_monotonic;assumption.
                      intros ;apply P_id_a__U43_monotonic;assumption.
                      intros ;apply P_id_U61_monotonic;assumption.
                      intros ;apply P_id_isNat_monotonic;assumption.
                      intros ;apply P_id_a__zeros_bounded;assumption.
                      intros ;apply P_id_isNatKind_bounded;assumption.
                      intros ;apply P_id_a__isNatIList_bounded;assumption.
                      intros ;apply P_id_a__U21_bounded;assumption.
                      intros ;apply P_id_U41_bounded;assumption.
                      intros ;apply P_id_a__and_bounded;assumption.
                      intros ;apply P_id_a__U11_bounded;assumption.
                      intros ;apply P_id_U21_bounded;assumption.
                      intros ;apply P_id_a__U61_bounded;assumption.
                      intros ;apply P_id_a__U32_bounded;assumption.
                      intros ;apply P_id_U51_bounded;assumption.
                      intros ;apply P_id_isNatIListKind_bounded;assumption.
                      intros ;apply P_id_0_bounded;assumption.
                      intros ;apply P_id_U12_bounded;assumption.
                      intros ;apply P_id_a__U52_bounded;assumption.
                      intros ;apply P_id_a__isNat_bounded;assumption.
                      intros ;apply P_id_U43_bounded;assumption.
                      
                      intros ;apply P_id_a__isNatIListKind_bounded;assumption.
                      intros ;apply P_id_a__U12_bounded;assumption.
                      intros ;apply P_id_U31_bounded;assumption.
                      intros ;apply P_id_a__length_bounded;assumption.
                      intros ;apply P_id_a__U42_bounded;assumption.
                      intros ;apply P_id_U53_bounded;assumption.
                      intros ;apply P_id_and_bounded;assumption.
                      intros ;apply P_id_cons_bounded;assumption.
                      intros ;apply P_id_U11_bounded;assumption.
                      intros ;apply P_id_a__U51_bounded;assumption.
                      intros ;apply P_id_a__U22_bounded;assumption.
                      intros ;apply P_id_U42_bounded;assumption.
                      intros ;apply P_id_length_bounded;assumption.
                      intros ;apply P_id_tt_bounded;assumption.
                      intros ;apply P_id_U22_bounded;assumption.
                      intros ;apply P_id_s_bounded;assumption.
                      intros ;apply P_id_a__U41_bounded;assumption.
                      intros ;apply P_id_U52_bounded;assumption.
                      intros ;apply P_id_nil_bounded;assumption.
                      intros ;apply P_id_zeros_bounded;assumption.
                      intros ;apply P_id_isNatList_bounded;assumption.
                      intros ;apply P_id_a__U53_bounded;assumption.
                      intros ;apply P_id_a__U31_bounded;assumption.
                      intros ;apply P_id_isNatIList_bounded;assumption.
                      intros ;apply P_id_a__isNatKind_bounded;assumption.
                      intros ;apply P_id_a__isNatList_bounded;assumption.
                      intros ;apply P_id_U32_bounded;assumption.
                      intros ;apply P_id_mark_bounded;assumption.
                      intros ;apply P_id_a__U43_bounded;assumption.
                      intros ;apply P_id_U61_bounded;assumption.
                      intros ;apply P_id_isNat_bounded;assumption.
                      apply rules_monotonic.
                    Qed.
                    
                    Definition P_id_A__LENGTH (x11:Z) := 0.
                    
                    Definition P_id_A__U11 (x11:Z) (x12:Z) := 0.
                    
                    Definition P_id_A__U41 (x11:Z) (x12:Z) (x13:Z) := 0.
                    
                    Definition P_id_A__ISNATKIND (x11:Z) := 0.
                    
                    Definition P_id_A__U22 (x11:Z) := 0.
                    
                    Definition P_id_A__U51 (x11:Z) (x12:Z) (x13:Z) := 0.
                    
                    Definition P_id_A__AND (x11:Z) (x12:Z) := 0.
                    
                    Definition P_id_A__ISNATLIST (x11:Z) := 0.
                    
                    Definition P_id_A__U43 (x11:Z) := 0.
                    
                    Definition P_id_A__U31 (x11:Z) (x12:Z) := 0.
                    
                    Definition P_id_A__U53 (x11:Z) := 0.
                    
                    Definition P_id_MARK (x11:Z) := 1* x11.
                    
                    Definition P_id_A__U12 (x11:Z) := 0.
                    
                    Definition P_id_A__U42 (x11:Z) (x12:Z) := 0.
                    
                    Definition P_id_A__ZEROS  := 0.
                    
                    Definition P_id_A__ISNAT (x11:Z) := 0.
                    
                    Definition P_id_A__U52 (x11:Z) (x12:Z) := 0.
                    
                    Definition P_id_A__ISNATILISTKIND (x11:Z) := 0.
                    
                    Definition P_id_A__U21 (x11:Z) (x12:Z) := 0.
                    
                    Definition P_id_A__ISNATILIST (x11:Z) := 0.
                    
                    Definition P_id_A__U32 (x11:Z) := 0.
                    
                    Definition P_id_A__U61 (x11:Z) (x12:Z) := 0.
                    
                    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__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__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__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__U22_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_A__U22 x12 <= P_id_A__U22 x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_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 x16 x12 x14 x13 x11 x15, 
                      (0 <= x16)/\ (x16 <= x15) ->
                       (0 <= x14)/\ (x14 <= x13) ->
                        (0 <= x12)/\ (x12 <= x11) ->
                         P_id_A__U51 x12 x14 x16 <= P_id_A__U51 x11 x13 x15.
                    Proof.
                      intros x16 x15 x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      intros [H_5 H_4].
                      
                      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                       intuition;
                       (auto with zarith)||
                       (repeat (translate_vars );prove_ineq ).
                    Qed.
                    
                    Lemma P_id_A__AND_monotonic :
                     forall x12 x14 x13 x11, 
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_A__AND x12 x14 <= P_id_A__AND x11 x13.
                    Proof.
                      intros x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_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__U43_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_A__U43 x12 <= P_id_A__U43 x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_0].
                      
                      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__U53_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_A__U53 x12 <= P_id_A__U53 x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 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__U12_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_A__U12 x12 <= P_id_A__U12 x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_0].
                      
                      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 x12 x14 x13 x11, 
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_A__U42 x12 x14 <= P_id_A__U42 x11 x13.
                    Proof.
                      intros x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_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_A__U52_monotonic :
                     forall x12 x14 x13 x11, 
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_A__U52 x12 x14 <= P_id_A__U52 x11 x13.
                    Proof.
                      intros x14 x13 x12 x11.
                      intros [H_1 H_0].
                      intros [H_3 H_2].
                      
                      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__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__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__U32_monotonic :
                     forall x12 x11, 
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_A__U32 x12 <= P_id_A__U32 x11.
                    Proof.
                      intros x12 x11.
                      intros [H_1 H_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 x14 x13 x11, 
                      (0 <= x14)/\ (x14 <= x13) ->
                       (0 <= x12)/\ (x12 <= x11) ->
                        P_id_A__U61 x12 x14 <= P_id_A__U61 x11 x13.
                    Proof.
                      intros x14 x13 x12 x11.
                      intros [H_1 H_0].
                      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_isNatKind 
                       P_id_a__isNatIList P_id_a__U21 P_id_U41 P_id_a__and 
                       P_id_a__U11 P_id_U21 P_id_a__U61 P_id_a__U32 P_id_U51 
                       P_id_isNatIListKind P_id_0 P_id_U12 P_id_a__U52 
                       P_id_a__isNat P_id_U43 P_id_a__isNatIListKind 
                       P_id_a__U12 P_id_U31 P_id_a__length P_id_a__U42 
                       P_id_U53 P_id_and P_id_cons P_id_U11 P_id_a__U51 
                       P_id_a__U22 P_id_U42 P_id_length P_id_tt P_id_U22 
                       P_id_s P_id_a__U41 P_id_U52 P_id_nil P_id_zeros 
                       P_id_isNatList P_id_a__U53 P_id_a__U31 
                       P_id_isNatIList P_id_a__isNatKind P_id_a__isNatList 
                       P_id_U32 P_id_mark P_id_a__U43 P_id_U61 P_id_isNat 
                       P_id_A__LENGTH P_id_A__U11 P_id_A__U41 
                       P_id_A__ISNATKIND P_id_A__U22 P_id_A__U51 P_id_A__AND 
                       P_id_A__ISNATLIST P_id_A__U43 P_id_A__U31 P_id_A__U53 
                       P_id_MARK P_id_A__U12 P_id_A__U42 P_id_A__ZEROS 
                       P_id_A__ISNAT P_id_A__U52 P_id_A__ISNATILISTKIND 
                       P_id_A__U21 P_id_A__ISNATILIST P_id_A__U32 P_id_A__U61
                      .
                    
                    Lemma marked_measure_equation :
                     forall t, 
                      marked_measure t = match t with
                                           | (algebra.Alg.Term 
                                              algebra.F.id_a__length 
                                              (x11::nil)) =>
                                            P_id_A__LENGTH (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__U41 (x13::x12::
                                              x11::nil)) =>
                                            P_id_A__U41 (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__U22 (x11::nil)) =>
                                            P_id_A__U22 (measure x11)
                                           | (algebra.Alg.Term 
                                              algebra.F.id_a__U51 (x13::x12::
                                              x11::nil)) =>
                                            P_id_A__U51 (measure x13) 
                                             (measure x12) (measure x11)
                                           | (algebra.Alg.Term 
                                              algebra.F.id_a__and (x12::
                                              x11::nil)) =>
                                            P_id_A__AND (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__U43 (x11::nil)) =>
                                            P_id_A__U43 (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__U53 (x11::nil)) =>
                                            P_id_A__U53 (measure x11)
                                           | (algebra.Alg.Term 
                                              algebra.F.id_mark (x11::nil)) =>
                                            P_id_MARK (measure x11)
                                           | (algebra.Alg.Term 
                                              algebra.F.id_a__U12 (x11::nil)) =>
                                            P_id_A__U12 (measure x11)
                                           | (algebra.Alg.Term 
                                              algebra.F.id_a__U42 (x12::
                                              x11::nil)) =>
                                            P_id_A__U42 (measure x12) 
                                             (measure x11)
                                           | (algebra.Alg.Term 
                                              algebra.F.id_a__zeros nil) =>
                                            P_id_A__ZEROS 
                                           | (algebra.Alg.Term 
                                              algebra.F.id_a__isNat 
                                              (x11::nil)) =>
                                            P_id_A__ISNAT (measure x11)
                                           | (algebra.Alg.Term 
                                              algebra.F.id_a__U52 (x12::
                                              x11::nil)) =>
                                            P_id_A__U52 (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_a__U21 (x12::
                                              x11::nil)) =>
                                            P_id_A__U21 (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__U32 (x11::nil)) =>
                                            P_id_A__U32 (measure x11)
                                           | (algebra.Alg.Term 
                                              algebra.F.id_a__U61 (x12::
                                              x11::nil)) =>
                                            P_id_A__U61 (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_isNatKind_monotonic;assumption.
                      intros ;apply P_id_a__isNatIList_monotonic;assumption.
                      intros ;apply P_id_a__U21_monotonic;assumption.
                      intros ;apply P_id_U41_monotonic;assumption.
                      intros ;apply P_id_a__and_monotonic;assumption.
                      intros ;apply P_id_a__U11_monotonic;assumption.
                      intros ;apply P_id_U21_monotonic;assumption.
                      intros ;apply P_id_a__U61_monotonic;assumption.
                      intros ;apply P_id_a__U32_monotonic;assumption.
                      intros ;apply P_id_U51_monotonic;assumption.
                      intros ;apply P_id_isNatIListKind_monotonic;assumption.
                      intros ;apply P_id_U12_monotonic;assumption.
                      intros ;apply P_id_a__U52_monotonic;assumption.
                      intros ;apply P_id_a__isNat_monotonic;assumption.
                      intros ;apply P_id_U43_monotonic;assumption.
                      
                      intros ;apply P_id_a__isNatIListKind_monotonic;
                       assumption.
                      intros ;apply P_id_a__U12_monotonic;assumption.
                      intros ;apply P_id_U31_monotonic;assumption.
                      intros ;apply P_id_a__length_monotonic;assumption.
                      intros ;apply P_id_a__U42_monotonic;assumption.
                      intros ;apply P_id_U53_monotonic;assumption.
                      intros ;apply P_id_and_monotonic;assumption.
                      intros ;apply P_id_cons_monotonic;assumption.
                      intros ;apply P_id_U11_monotonic;assumption.
                      intros ;apply P_id_a__U51_monotonic;assumption.
                      intros ;apply P_id_a__U22_monotonic;assumption.
                      intros ;apply P_id_U42_monotonic;assumption.
                      intros ;apply P_id_length_monotonic;assumption.
                      intros ;apply P_id_U22_monotonic;assumption.
                      intros ;apply P_id_s_monotonic;assumption.
                      intros ;apply P_id_a__U41_monotonic;assumption.
                      intros ;apply P_id_U52_monotonic;assumption.
                      intros ;apply P_id_isNatList_monotonic;assumption.
                      intros ;apply P_id_a__U53_monotonic;assumption.
                      intros ;apply P_id_a__U31_monotonic;assumption.
                      intros ;apply P_id_isNatIList_monotonic;assumption.
                      intros ;apply P_id_a__isNatKind_monotonic;assumption.
                      intros ;apply P_id_a__isNatList_monotonic;assumption.
                      intros ;apply P_id_U32_monotonic;assumption.
                      intros ;apply P_id_mark_monotonic;assumption.
                      intros ;apply P_id_a__U43_monotonic;assumption.
                      intros ;apply P_id_U61_monotonic;assumption.
                      intros ;apply P_id_isNat_monotonic;assumption.
                      intros ;apply P_id_a__zeros_bounded;assumption.
                      intros ;apply P_id_isNatKind_bounded;assumption.
                      intros ;apply P_id_a__isNatIList_bounded;assumption.
                      intros ;apply P_id_a__U21_bounded;assumption.
                      intros ;apply P_id_U41_bounded;assumption.
                      intros ;apply P_id_a__and_bounded;assumption.
                      intros ;apply P_id_a__U11_bounded;assumption.
                      intros ;apply P_id_U21_bounded;assumption.
                      intros ;apply P_id_a__U61_bounded;assumption.
                      intros ;apply P_id_a__U32_bounded;assumption.
                      intros ;apply P_id_U51_bounded;assumption.
                      intros ;apply P_id_isNatIListKind_bounded;assumption.
                      intros ;apply P_id_0_bounded;assumption.
                      intros ;apply P_id_U12_bounded;assumption.
                      intros ;apply P_id_a__U52_bounded;assumption.
                      intros ;apply P_id_a__isNat_bounded;assumption.
                      intros ;apply P_id_U43_bounded;assumption.
                      
                      intros ;apply P_id_a__isNatIListKind_bounded;assumption.
                      intros ;apply P_id_a__U12_bounded;assumption.
                      intros ;apply P_id_U31_bounded;assumption.
                      intros ;apply P_id_a__length_bounded;assumption.
                      intros ;apply P_id_a__U42_bounded;assumption.
                      intros ;apply P_id_U53_bounded;assumption.
                      intros ;apply P_id_and_bounded;assumption.
                      intros ;apply P_id_cons_bounded;assumption.
                      intros ;apply P_id_U11_bounded;assumption.
                      intros ;apply P_id_a__U51_bounded;assumption.
                      intros ;apply P_id_a__U22_bounded;assumption.
                      intros ;apply P_id_U42_bounded;assumption.
                      intros ;apply P_id_length_bounded;assumption.
                      intros ;apply P_id_tt_bounded;assumption.
                      intros ;apply P_id_U22_bounded;assumption.
                      intros ;apply P_id_s_bounded;assumption.
                      intros ;apply P_id_a__U41_bounded;assumption.
                      intros ;apply P_id_U52_bounded;assumption.
                      intros ;apply P_id_nil_bounded;assumption.
                      intros ;apply P_id_zeros_bounded;assumption.
                      intros ;apply P_id_isNatList_bounded;assumption.
                      intros ;apply P_id_a__U53_bounded;assumption.
                      intros ;apply P_id_a__U31_bounded;assumption.
                      intros ;apply P_id_isNatIList_bounded;assumption.
                      intros ;apply P_id_a__isNatKind_bounded;assumption.
                      intros ;apply P_id_a__isNatList_bounded;assumption.
                      intros ;apply P_id_U32_bounded;assumption.
                      intros ;apply P_id_mark_bounded;assumption.
                      intros ;apply P_id_a__U43_bounded;assumption.
                      intros ;apply P_id_U61_bounded;assumption.
                      intros ;apply P_id_isNat_bounded;assumption.
                      apply rules_monotonic.
                      intros ;apply P_id_A__LENGTH_monotonic;assumption.
                      intros ;apply P_id_A__U11_monotonic;assumption.
                      intros ;apply P_id_A__U41_monotonic;assumption.
                      intros ;apply P_id_A__ISNATKIND_monotonic;assumption.
                      intros ;apply P_id_A__U22_monotonic;assumption.
                      intros ;apply P_id_A__U51_monotonic;assumption.
                      intros ;apply P_id_A__AND_monotonic;assumption.
                      intros ;apply P_id_A__ISNATLIST_monotonic;assumption.
                      intros ;apply P_id_A__U43_monotonic;assumption.
                      intros ;apply P_id_A__U31_monotonic;assumption.
                      intros ;apply P_id_A__U53_monotonic;assumption.
                      intros ;apply P_id_MARK_monotonic;assumption.
                      intros ;apply P_id_A__U12_monotonic;assumption.
                      intros ;apply P_id_A__U42_monotonic;assumption.
                      intros ;apply P_id_A__ISNAT_monotonic;assumption.
                      intros ;apply P_id_A__U52_monotonic;assumption.
                      
                      intros ;apply P_id_A__ISNATILISTKIND_monotonic;
                       assumption.
                      intros ;apply P_id_A__U21_monotonic;assumption.
                      intros ;apply P_id_A__ISNATILIST_monotonic;assumption.
                      intros ;apply P_id_A__U32_monotonic;assumption.
                      intros ;apply P_id_A__U61_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_large_scc_1_strict_in_lt :
                     Relation_Definitions.inclusion _ 
                      DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_large_scc_1_large_in_le :
                     Relation_Definitions.inclusion _ 
                      DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_large_scc_1_large  := 
                      WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_large_scc_1_large.wf
                      .
                    
                    
                    Lemma wf :
                     well_founded WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_large.DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_large_scc_1_large)
                             .
                      clear x.
                      intros x _ IHx IHx'.
                      constructor.
                      intros y H.
                      
                      destruct H;
                       (apply IHx';
                         apply DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_large_scc_1_large_in_le;
                          econstructor eassumption])).
                      
                      apply wf_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_large_scc_1_large.
                    Qed.
                   End WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_large_scc_1.
                   
                   Definition wf_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_large_scc_1  := 
                     WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_large_scc_1.wf
                     .
                   
                   
                   Lemma acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_large_scc_1 :
                    forall x y, 
                     (DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_large_scc_1 
                       x y) ->
                      Acc WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large.DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_large 
                       x.
                   Proof.
                     intros x.
                     pattern x.
                     
                     apply (@Acc_ind _ 
                             DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_large_scc_1.
                   Qed.
                   
                   
                   Inductive DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_large_non_scc_2  :
                    algebra.Alg.term ->algebra.Alg.term ->Prop := 
                      (* <a__and(tt,X_),mark(X_)> *)
                     | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_large_non_scc_2_0 :
                      forall x12 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_tt nil) 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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_large_non_scc_2 
                          (algebra.Alg.Term algebra.F.id_mark (x5::nil)) 
                          (algebra.Alg.Term algebra.F.id_a__and (x12::
                           x11::nil))
                   .
                   
                   
                   Lemma acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_large_non_scc_2 :
                    forall x y, 
                     (DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_large_non_scc_2 
                       x y) ->
                      Acc WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large.DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_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.
                   
                   
                   Lemma wf :
                    well_founded WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large.DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_large
                     .
                   Proof.
                     constructor;intros _y _h;inversion _h;clear _h;subst;
                      (eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_large_non_scc_2;
                        
                        econstructor 
                        (eassumption)||(algebra.Alg_ext.star_refl' ))||
                      ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_large_non_scc_1;
                         
                         econstructor 
                         (eassumption)||(algebra.Alg_ext.star_refl' ))||
                       ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_large_non_scc_0;
                          
                          econstructor 
                          (eassumption)||(algebra.Alg_ext.star_refl' ))||
                        ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_large_scc_1;
                           
                           econstructor 
                           (eassumption)||(algebra.Alg_ext.star_refl' ))||
                         ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_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_isNatKind (x11:Z) := 1* x11.
                  
                  Definition P_id_a__isNatIList (x11:Z) := 3 + 2* x11.
                  
                  Definition P_id_a__U21 (x11:Z) (x12:Z) := 0.
                  
                  Definition P_id_U41 (x11:Z) (x12:Z) (x13:Z) := 
                    1* x12 + 1* x13.
                  
                  Definition P_id_a__and (x11:Z) (x12:Z) := 
                    2 + 2* x11 + 2* x12.
                  
                  Definition P_id_a__U11 (x11:Z) (x12:Z) := 0.
                  
                  Definition P_id_U21 (x11:Z) (x12:Z) := 0.
                  
                  Definition P_id_a__U61 (x11:Z) (x12:Z) := 3 + 2* x12.
                  
                  Definition P_id_a__U32 (x11:Z) := 2.
                  
                  Definition P_id_U51 (x11:Z) (x12:Z) (x13:Z) := 0.
                  
                  Definition P_id_isNatIListKind (x11:Z) := 1 + 2* x11.
                  
                  Definition P_id_0  := 0.
                  
                  Definition P_id_U12 (x11:Z) := 0.
                  
                  Definition P_id_a__U52 (x11:Z) (x12:Z) := 0.
                  
                  Definition P_id_a__isNat (x11:Z) := 0.
                  
                  Definition P_id_U43 (x11:Z) := 0.
                  
                  Definition P_id_a__isNatIListKind (x11:Z) := 1 + 2* x11.
                  
                  Definition P_id_a__U12 (x11:Z) := 0.
                  
                  Definition P_id_U31 (x11:Z) (x12:Z) := 1* x12.
                  
                  Definition P_id_a__length (x11:Z) := 1 + 1* x11.
                  
                  Definition P_id_a__U42 (x11:Z) (x12:Z) := 0.
                  
                  Definition P_id_U53 (x11:Z) := 0.
                  
                  Definition P_id_and (x11:Z) (x12:Z) := 2 + 2* x11 + 2* x12.
                  
                  Definition P_id_cons (x11:Z) (x12:Z) := 2 + 2* x11 + 2* x12
                    .
                  
                  Definition P_id_U11 (x11:Z) (x12:Z) := 0.
                  
                  Definition P_id_a__U51 (x11:Z) (x12:Z) (x13:Z) := 0.
                  
                  Definition P_id_a__U22 (x11:Z) := 1* x11.
                  
                  Definition P_id_U42 (x11:Z) (x12:Z) := 0.
                  
                  Definition P_id_length (x11:Z) := 1 + 1* x11.
                  
                  Definition P_id_tt  := 0.
                  
                  Definition P_id_U22 (x11:Z) := 1* x11.
                  
                  Definition P_id_s (x11:Z) := 1* x11.
                  
                  Definition P_id_a__U41 (x11:Z) (x12:Z) (x13:Z) := 
                    1* x12 + 1* x13.
                  
                  Definition P_id_U52 (x11:Z) (x12:Z) := 0.
                  
                  Definition P_id_nil  := 0.
                  
                  Definition P_id_zeros  := 0.
                  
                  Definition P_id_isNatList (x11:Z) := 0.
                  
                  Definition P_id_a__U53 (x11:Z) := 0.
                  
                  Definition P_id_a__U31 (x11:Z) (x12:Z) := 2 + 2* x12.
                  
                  Definition P_id_isNatIList (x11:Z) := 1 + 2* x11.
                  
                  Definition P_id_a__isNatKind (x11:Z) := 2* x11.
                  
                  Definition P_id_a__isNatList (x11:Z) := 0.
                  
                  Definition P_id_U32 (x11:Z) := 0.
                  
                  Definition P_id_mark (x11:Z) := 2 + 2* x11.
                  
                  Definition P_id_a__U43 (x11:Z) := 0.
                  
                  Definition P_id_U61 (x11:Z) (x12:Z) := 1 + 2* x12.
                  
                  Definition P_id_isNat (x11:Z) := 0.
                  
                  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__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__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_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_a__and_monotonic :
                   forall x12 x14 x13 x11, 
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_a__and x12 x14 <= P_id_a__and x11 x13.
                  Proof.
                    intros x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    
                    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_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__U61_monotonic :
                   forall x12 x14 x13 x11, 
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_a__U61 x12 x14 <= P_id_a__U61 x11 x13.
                  Proof.
                    intros x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    
                    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 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_a__U32 x12 <= P_id_a__U32 x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_0].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U51_monotonic :
                   forall x16 x12 x14 x13 x11 x15, 
                    (0 <= x16)/\ (x16 <= x15) ->
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_U51 x12 x14 x16 <= P_id_U51 x11 x13 x15.
                  Proof.
                    intros x16 x15 x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    intros [H_5 H_4].
                    
                    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_U12_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->P_id_U12 x12 <= P_id_U12 x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_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 x14 x13 x11, 
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_a__U52 x12 x14 <= P_id_a__U52 x11 x13.
                  Proof.
                    intros x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_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_U43_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->P_id_U43 x12 <= P_id_U43 x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_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__U12_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_a__U12 x12 <= P_id_a__U12 x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_0].
                    
                    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__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__U42_monotonic :
                   forall x12 x14 x13 x11, 
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_a__U42 x12 x14 <= P_id_a__U42 x11 x13.
                  Proof.
                    intros x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U53_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->P_id_U53 x12 <= P_id_U53 x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_0].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_and_monotonic :
                   forall x12 x14 x13 x11, 
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_and x12 x14 <= P_id_and x11 x13.
                  Proof.
                    intros x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    
                    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_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__U51_monotonic :
                   forall x16 x12 x14 x13 x11 x15, 
                    (0 <= x16)/\ (x16 <= x15) ->
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_a__U51 x12 x14 x16 <= P_id_a__U51 x11 x13 x15.
                  Proof.
                    intros x16 x15 x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    intros [H_5 H_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 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_a__U22 x12 <= P_id_a__U22 x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 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 x12 x14 x13 x11, 
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_U42 x12 x14 <= P_id_U42 x11 x13.
                  Proof.
                    intros x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    
                    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_U22_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->P_id_U22 x12 <= P_id_U22 x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_0].
                    
                    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__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_U52_monotonic :
                   forall x12 x14 x13 x11, 
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_U52 x12 x14 <= P_id_U52 x11 x13.
                  Proof.
                    intros x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 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__U53_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_a__U53 x12 <= P_id_a__U53 x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_0].
                    
                    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_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__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__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_U32_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->P_id_U32 x12 <= P_id_U32 x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 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__U43_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_a__U43 x12 <= P_id_a__U43 x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_0].
                    
                    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 x14 x13 x11, 
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_U61 x12 x14 <= P_id_U61 x11 x13.
                  Proof.
                    intros x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    
                    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__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_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__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_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_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_a__and_bounded :
                   forall x12 x11, 
                    (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__and x12 x11.
                  Proof.
                    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_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__U61_bounded :
                   forall x12 x11, 
                    (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U61 x12 x11.
                  Proof.
                    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 x11, (0 <= x11) ->0 <= P_id_a__U32 x11.
                  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 x13 x11, 
                    (0 <= x11) ->
                     (0 <= x12) ->(0 <= x13) ->0 <= P_id_U51 x13 x12 x11.
                  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_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_U12_bounded :
                   forall x11, (0 <= x11) ->0 <= P_id_U12 x11.
                  Proof.
                    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 x12 x11, 
                    (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U52 x12 x11.
                  Proof.
                    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_U43_bounded :
                   forall x11, (0 <= x11) ->0 <= P_id_U43 x11.
                  Proof.
                    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_a__U12_bounded :
                   forall x11, (0 <= x11) ->0 <= P_id_a__U12 x11.
                  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__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__U42_bounded :
                   forall x12 x11, 
                    (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U42 x12 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_U53_bounded :
                   forall x11, (0 <= x11) ->0 <= P_id_U53 x11.
                  Proof.
                    intros .
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_and_bounded :
                   forall x12 x11, 
                    (0 <= x11) ->(0 <= x12) ->0 <= P_id_and x12 x11.
                  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_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__U51_bounded :
                   forall x12 x13 x11, 
                    (0 <= x11) ->
                     (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U51 x13 x12 x11.
                  Proof.
                    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 x11, (0 <= x11) ->0 <= P_id_a__U22 x11.
                  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 x11, 
                    (0 <= x11) ->(0 <= x12) ->0 <= P_id_U42 x12 x11.
                  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_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_U22_bounded :
                   forall x11, (0 <= x11) ->0 <= P_id_U22 x11.
                  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__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_U52_bounded :
                   forall x12 x11, 
                    (0 <= x11) ->(0 <= x12) ->0 <= P_id_U52 x12 x11.
                  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_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_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__U53_bounded :
                   forall x11, (0 <= x11) ->0 <= P_id_a__U53 x11.
                  Proof.
                    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_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.
                  
                  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_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_U32_bounded :
                   forall x11, (0 <= x11) ->0 <= P_id_U32 x11.
                  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__U43_bounded :
                   forall x11, (0 <= x11) ->0 <= P_id_a__U43 x11.
                  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 x12 x11, 
                    (0 <= x11) ->(0 <= x12) ->0 <= P_id_U61 x12 x11.
                  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.
                  
                  Definition measure  := 
                    InterpZ.measure 0 P_id_a__zeros P_id_isNatKind 
                     P_id_a__isNatIList P_id_a__U21 P_id_U41 P_id_a__and 
                     P_id_a__U11 P_id_U21 P_id_a__U61 P_id_a__U32 P_id_U51 
                     P_id_isNatIListKind P_id_0 P_id_U12 P_id_a__U52 
                     P_id_a__isNat P_id_U43 P_id_a__isNatIListKind 
                     P_id_a__U12 P_id_U31 P_id_a__length P_id_a__U42 
                     P_id_U53 P_id_and P_id_cons P_id_U11 P_id_a__U51 
                     P_id_a__U22 P_id_U42 P_id_length P_id_tt P_id_U22 
                     P_id_s P_id_a__U41 P_id_U52 P_id_nil P_id_zeros 
                     P_id_isNatList P_id_a__U53 P_id_a__U31 P_id_isNatIList 
                     P_id_a__isNatKind P_id_a__isNatList P_id_U32 P_id_mark 
                     P_id_a__U43 P_id_U61 P_id_isNat.
                  
                  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_isNatKind 
                                     (x11::nil)) =>
                                   P_id_isNatKind (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__U21 
                                     (x12::x11::nil)) =>
                                   P_id_a__U21 (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_a__and 
                                     (x12::x11::nil)) =>
                                   P_id_a__and (measure x12) (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_U21 (x12::
                                     x11::nil)) =>
                                   P_id_U21 (measure x12) (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U61 
                                     (x12::x11::nil)) =>
                                   P_id_a__U61 (measure x12) (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U32 
                                     (x11::nil)) =>
                                   P_id_a__U32 (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_U51 (x13::
                                     x12::x11::nil)) =>
                                   P_id_U51 (measure x13) (measure x12) 
                                    (measure x11)
                                  | (algebra.Alg.Term 
                                     algebra.F.id_isNatIListKind (x11::nil)) =>
                                   P_id_isNatIListKind (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_0 nil) =>
                                   P_id_0 
                                  | (algebra.Alg.Term algebra.F.id_U12 
                                     (x11::nil)) =>
                                   P_id_U12 (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U52 
                                     (x12::x11::nil)) =>
                                   P_id_a__U52 (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_U43 
                                     (x11::nil)) =>
                                   P_id_U43 (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__U12 
                                     (x11::nil)) =>
                                   P_id_a__U12 (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__length 
                                     (x11::nil)) =>
                                   P_id_a__length (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U42 
                                     (x12::x11::nil)) =>
                                   P_id_a__U42 (measure x12) (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_U53 
                                     (x11::nil)) =>
                                   P_id_U53 (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_and (x12::
                                     x11::nil)) =>
                                   P_id_and (measure x12) (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_U11 (x12::
                                     x11::nil)) =>
                                   P_id_U11 (measure x12) (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U51 
                                     (x13::x12::x11::nil)) =>
                                   P_id_a__U51 (measure x13) (measure x12) 
                                    (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U22 
                                     (x11::nil)) =>
                                   P_id_a__U22 (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_U42 (x12::
                                     x11::nil)) =>
                                   P_id_U42 (measure x12) (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_length 
                                     (x11::nil)) =>
                                   P_id_length (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_tt nil) =>
                                   P_id_tt 
                                  | (algebra.Alg.Term algebra.F.id_U22 
                                     (x11::nil)) =>
                                   P_id_U22 (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_s 
                                     (x11::nil)) =>
                                   P_id_s (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_U52 (x12::
                                     x11::nil)) =>
                                   P_id_U52 (measure x12) (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_nil nil) =>
                                   P_id_nil 
                                  | (algebra.Alg.Term algebra.F.id_zeros nil) =>
                                   P_id_zeros 
                                  | (algebra.Alg.Term algebra.F.id_isNatList 
                                     (x11::nil)) =>
                                   P_id_isNatList (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U53 
                                     (x11::nil)) =>
                                   P_id_a__U53 (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_isNatIList (x11::nil)) =>
                                   P_id_isNatIList (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__isNatList (x11::nil)) =>
                                   P_id_a__isNatList (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_U32 
                                     (x11::nil)) =>
                                   P_id_U32 (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_mark 
                                     (x11::nil)) =>
                                   P_id_mark (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_a__U43 
                                     (x11::nil)) =>
                                   P_id_a__U43 (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_U61 (x12::
                                     x11::nil)) =>
                                   P_id_U61 (measure x12) (measure x11)
                                  | (algebra.Alg.Term algebra.F.id_isNat 
                                     (x11::nil)) =>
                                   P_id_isNat (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_isNatKind_monotonic;assumption.
                    intros ;apply P_id_a__isNatIList_monotonic;assumption.
                    intros ;apply P_id_a__U21_monotonic;assumption.
                    intros ;apply P_id_U41_monotonic;assumption.
                    intros ;apply P_id_a__and_monotonic;assumption.
                    intros ;apply P_id_a__U11_monotonic;assumption.
                    intros ;apply P_id_U21_monotonic;assumption.
                    intros ;apply P_id_a__U61_monotonic;assumption.
                    intros ;apply P_id_a__U32_monotonic;assumption.
                    intros ;apply P_id_U51_monotonic;assumption.
                    intros ;apply P_id_isNatIListKind_monotonic;assumption.
                    intros ;apply P_id_U12_monotonic;assumption.
                    intros ;apply P_id_a__U52_monotonic;assumption.
                    intros ;apply P_id_a__isNat_monotonic;assumption.
                    intros ;apply P_id_U43_monotonic;assumption.
                    
                    intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
                    intros ;apply P_id_a__U12_monotonic;assumption.
                    intros ;apply P_id_U31_monotonic;assumption.
                    intros ;apply P_id_a__length_monotonic;assumption.
                    intros ;apply P_id_a__U42_monotonic;assumption.
                    intros ;apply P_id_U53_monotonic;assumption.
                    intros ;apply P_id_and_monotonic;assumption.
                    intros ;apply P_id_cons_monotonic;assumption.
                    intros ;apply P_id_U11_monotonic;assumption.
                    intros ;apply P_id_a__U51_monotonic;assumption.
                    intros ;apply P_id_a__U22_monotonic;assumption.
                    intros ;apply P_id_U42_monotonic;assumption.
                    intros ;apply P_id_length_monotonic;assumption.
                    intros ;apply P_id_U22_monotonic;assumption.
                    intros ;apply P_id_s_monotonic;assumption.
                    intros ;apply P_id_a__U41_monotonic;assumption.
                    intros ;apply P_id_U52_monotonic;assumption.
                    intros ;apply P_id_isNatList_monotonic;assumption.
                    intros ;apply P_id_a__U53_monotonic;assumption.
                    intros ;apply P_id_a__U31_monotonic;assumption.
                    intros ;apply P_id_isNatIList_monotonic;assumption.
                    intros ;apply P_id_a__isNatKind_monotonic;assumption.
                    intros ;apply P_id_a__isNatList_monotonic;assumption.
                    intros ;apply P_id_U32_monotonic;assumption.
                    intros ;apply P_id_mark_monotonic;assumption.
                    intros ;apply P_id_a__U43_monotonic;assumption.
                    intros ;apply P_id_U61_monotonic;assumption.
                    intros ;apply P_id_isNat_monotonic;assumption.
                    intros ;apply P_id_a__zeros_bounded;assumption.
                    intros ;apply P_id_isNatKind_bounded;assumption.
                    intros ;apply P_id_a__isNatIList_bounded;assumption.
                    intros ;apply P_id_a__U21_bounded;assumption.
                    intros ;apply P_id_U41_bounded;assumption.
                    intros ;apply P_id_a__and_bounded;assumption.
                    intros ;apply P_id_a__U11_bounded;assumption.
                    intros ;apply P_id_U21_bounded;assumption.
                    intros ;apply P_id_a__U61_bounded;assumption.
                    intros ;apply P_id_a__U32_bounded;assumption.
                    intros ;apply P_id_U51_bounded;assumption.
                    intros ;apply P_id_isNatIListKind_bounded;assumption.
                    intros ;apply P_id_0_bounded;assumption.
                    intros ;apply P_id_U12_bounded;assumption.
                    intros ;apply P_id_a__U52_bounded;assumption.
                    intros ;apply P_id_a__isNat_bounded;assumption.
                    intros ;apply P_id_U43_bounded;assumption.
                    intros ;apply P_id_a__isNatIListKind_bounded;assumption.
                    intros ;apply P_id_a__U12_bounded;assumption.
                    intros ;apply P_id_U31_bounded;assumption.
                    intros ;apply P_id_a__length_bounded;assumption.
                    intros ;apply P_id_a__U42_bounded;assumption.
                    intros ;apply P_id_U53_bounded;assumption.
                    intros ;apply P_id_and_bounded;assumption.
                    intros ;apply P_id_cons_bounded;assumption.
                    intros ;apply P_id_U11_bounded;assumption.
                    intros ;apply P_id_a__U51_bounded;assumption.
                    intros ;apply P_id_a__U22_bounded;assumption.
                    intros ;apply P_id_U42_bounded;assumption.
                    intros ;apply P_id_length_bounded;assumption.
                    intros ;apply P_id_tt_bounded;assumption.
                    intros ;apply P_id_U22_bounded;assumption.
                    intros ;apply P_id_s_bounded;assumption.
                    intros ;apply P_id_a__U41_bounded;assumption.
                    intros ;apply P_id_U52_bounded;assumption.
                    intros ;apply P_id_nil_bounded;assumption.
                    intros ;apply P_id_zeros_bounded;assumption.
                    intros ;apply P_id_isNatList_bounded;assumption.
                    intros ;apply P_id_a__U53_bounded;assumption.
                    intros ;apply P_id_a__U31_bounded;assumption.
                    intros ;apply P_id_isNatIList_bounded;assumption.
                    intros ;apply P_id_a__isNatKind_bounded;assumption.
                    intros ;apply P_id_a__isNatList_bounded;assumption.
                    intros ;apply P_id_U32_bounded;assumption.
                    intros ;apply P_id_mark_bounded;assumption.
                    intros ;apply P_id_a__U43_bounded;assumption.
                    intros ;apply P_id_U61_bounded;assumption.
                    intros ;apply P_id_isNat_bounded;assumption.
                    apply rules_monotonic.
                  Qed.
                  
                  Definition P_id_A__LENGTH (x11:Z) := 0.
                  
                  Definition P_id_A__U11 (x11:Z) (x12:Z) := 0.
                  
                  Definition P_id_A__U41 (x11:Z) (x12:Z) (x13:Z) := 0.
                  
                  Definition P_id_A__ISNATKIND (x11:Z) := 0.
                  
                  Definition P_id_A__U22 (x11:Z) := 0.
                  
                  Definition P_id_A__U51 (x11:Z) (x12:Z) (x13:Z) := 0.
                  
                  Definition P_id_A__AND (x11:Z) (x12:Z) := 2 + 1* x12.
                  
                  Definition P_id_A__ISNATLIST (x11:Z) := 0.
                  
                  Definition P_id_A__U43 (x11:Z) := 0.
                  
                  Definition P_id_A__U31 (x11:Z) (x12:Z) := 0.
                  
                  Definition P_id_A__U53 (x11:Z) := 0.
                  
                  Definition P_id_MARK (x11:Z) := 2 + 1* x11.
                  
                  Definition P_id_A__U12 (x11:Z) := 0.
                  
                  Definition P_id_A__U42 (x11:Z) (x12:Z) := 0.
                  
                  Definition P_id_A__ZEROS  := 0.
                  
                  Definition P_id_A__ISNAT (x11:Z) := 0.
                  
                  Definition P_id_A__U52 (x11:Z) (x12:Z) := 0.
                  
                  Definition P_id_A__ISNATILISTKIND (x11:Z) := 2* x11.
                  
                  Definition P_id_A__U21 (x11:Z) (x12:Z) := 0.
                  
                  Definition P_id_A__ISNATILIST (x11:Z) := 0.
                  
                  Definition P_id_A__U32 (x11:Z) := 0.
                  
                  Definition P_id_A__U61 (x11:Z) (x12:Z) := 0.
                  
                  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__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__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__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__U22_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_A__U22 x12 <= P_id_A__U22 x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_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 x16 x12 x14 x13 x11 x15, 
                    (0 <= x16)/\ (x16 <= x15) ->
                     (0 <= x14)/\ (x14 <= x13) ->
                      (0 <= x12)/\ (x12 <= x11) ->
                       P_id_A__U51 x12 x14 x16 <= P_id_A__U51 x11 x13 x15.
                  Proof.
                    intros x16 x15 x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    intros [H_5 H_4].
                    
                    cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                     intuition;
                     (auto with zarith)||
                     (repeat (translate_vars );prove_ineq ).
                  Qed.
                  
                  Lemma P_id_A__AND_monotonic :
                   forall x12 x14 x13 x11, 
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_A__AND x12 x14 <= P_id_A__AND x11 x13.
                  Proof.
                    intros x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_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__U43_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_A__U43 x12 <= P_id_A__U43 x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_0].
                    
                    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__U53_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_A__U53 x12 <= P_id_A__U53 x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 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__U12_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_A__U12 x12 <= P_id_A__U12 x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_0].
                    
                    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 x12 x14 x13 x11, 
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_A__U42 x12 x14 <= P_id_A__U42 x11 x13.
                  Proof.
                    intros x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_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_A__U52_monotonic :
                   forall x12 x14 x13 x11, 
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_A__U52 x12 x14 <= P_id_A__U52 x11 x13.
                  Proof.
                    intros x14 x13 x12 x11.
                    intros [H_1 H_0].
                    intros [H_3 H_2].
                    
                    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__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__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__U32_monotonic :
                   forall x12 x11, 
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_A__U32 x12 <= P_id_A__U32 x11.
                  Proof.
                    intros x12 x11.
                    intros [H_1 H_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 x14 x13 x11, 
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_A__U61 x12 x14 <= P_id_A__U61 x11 x13.
                  Proof.
                    intros x14 x13 x12 x11.
                    intros [H_1 H_0].
                    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_isNatKind 
                     P_id_a__isNatIList P_id_a__U21 P_id_U41 P_id_a__and 
                     P_id_a__U11 P_id_U21 P_id_a__U61 P_id_a__U32 P_id_U51 
                     P_id_isNatIListKind P_id_0 P_id_U12 P_id_a__U52 
                     P_id_a__isNat P_id_U43 P_id_a__isNatIListKind 
                     P_id_a__U12 P_id_U31 P_id_a__length P_id_a__U42 
                     P_id_U53 P_id_and P_id_cons P_id_U11 P_id_a__U51 
                     P_id_a__U22 P_id_U42 P_id_length P_id_tt P_id_U22 
                     P_id_s P_id_a__U41 P_id_U52 P_id_nil P_id_zeros 
                     P_id_isNatList P_id_a__U53 P_id_a__U31 P_id_isNatIList 
                     P_id_a__isNatKind P_id_a__isNatList P_id_U32 P_id_mark 
                     P_id_a__U43 P_id_U61 P_id_isNat P_id_A__LENGTH 
                     P_id_A__U11 P_id_A__U41 P_id_A__ISNATKIND P_id_A__U22 
                     P_id_A__U51 P_id_A__AND P_id_A__ISNATLIST P_id_A__U43 
                     P_id_A__U31 P_id_A__U53 P_id_MARK P_id_A__U12 
                     P_id_A__U42 P_id_A__ZEROS P_id_A__ISNAT P_id_A__U52 
                     P_id_A__ISNATILISTKIND P_id_A__U21 P_id_A__ISNATILIST 
                     P_id_A__U32 P_id_A__U61.
                  
                  Lemma marked_measure_equation :
                   forall t, 
                    marked_measure t = match t with
                                         | (algebra.Alg.Term 
                                            algebra.F.id_a__length 
                                            (x11::nil)) =>
                                          P_id_A__LENGTH (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__U41 (x13::x12::
                                            x11::nil)) =>
                                          P_id_A__U41 (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__U22 (x11::nil)) =>
                                          P_id_A__U22 (measure x11)
                                         | (algebra.Alg.Term 
                                            algebra.F.id_a__U51 (x13::x12::
                                            x11::nil)) =>
                                          P_id_A__U51 (measure x13) 
                                           (measure x12) (measure x11)
                                         | (algebra.Alg.Term 
                                            algebra.F.id_a__and (x12::
                                            x11::nil)) =>
                                          P_id_A__AND (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__U43 (x11::nil)) =>
                                          P_id_A__U43 (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__U53 (x11::nil)) =>
                                          P_id_A__U53 (measure x11)
                                         | (algebra.Alg.Term 
                                            algebra.F.id_mark (x11::nil)) =>
                                          P_id_MARK (measure x11)
                                         | (algebra.Alg.Term 
                                            algebra.F.id_a__U12 (x11::nil)) =>
                                          P_id_A__U12 (measure x11)
                                         | (algebra.Alg.Term 
                                            algebra.F.id_a__U42 (x12::
                                            x11::nil)) =>
                                          P_id_A__U42 (measure x12) 
                                           (measure x11)
                                         | (algebra.Alg.Term 
                                            algebra.F.id_a__zeros nil) =>
                                          P_id_A__ZEROS 
                                         | (algebra.Alg.Term 
                                            algebra.F.id_a__isNat (x11::nil)) =>
                                          P_id_A__ISNAT (measure x11)
                                         | (algebra.Alg.Term 
                                            algebra.F.id_a__U52 (x12::
                                            x11::nil)) =>
                                          P_id_A__U52 (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_a__U21 (x12::
                                            x11::nil)) =>
                                          P_id_A__U21 (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__U32 (x11::nil)) =>
                                          P_id_A__U32 (measure x11)
                                         | (algebra.Alg.Term 
                                            algebra.F.id_a__U61 (x12::
                                            x11::nil)) =>
                                          P_id_A__U61 (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_isNatKind_monotonic;assumption.
                    intros ;apply P_id_a__isNatIList_monotonic;assumption.
                    intros ;apply P_id_a__U21_monotonic;assumption.
                    intros ;apply P_id_U41_monotonic;assumption.
                    intros ;apply P_id_a__and_monotonic;assumption.
                    intros ;apply P_id_a__U11_monotonic;assumption.
                    intros ;apply P_id_U21_monotonic;assumption.
                    intros ;apply P_id_a__U61_monotonic;assumption.
                    intros ;apply P_id_a__U32_monotonic;assumption.
                    intros ;apply P_id_U51_monotonic;assumption.
                    intros ;apply P_id_isNatIListKind_monotonic;assumption.
                    intros ;apply P_id_U12_monotonic;assumption.
                    intros ;apply P_id_a__U52_monotonic;assumption.
                    intros ;apply P_id_a__isNat_monotonic;assumption.
                    intros ;apply P_id_U43_monotonic;assumption.
                    
                    intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
                    intros ;apply P_id_a__U12_monotonic;assumption.
                    intros ;apply P_id_U31_monotonic;assumption.
                    intros ;apply P_id_a__length_monotonic;assumption.
                    intros ;apply P_id_a__U42_monotonic;assumption.
                    intros ;apply P_id_U53_monotonic;assumption.
                    intros ;apply P_id_and_monotonic;assumption.
                    intros ;apply P_id_cons_monotonic;assumption.
                    intros ;apply P_id_U11_monotonic;assumption.
                    intros ;apply P_id_a__U51_monotonic;assumption.
                    intros ;apply P_id_a__U22_monotonic;assumption.
                    intros ;apply P_id_U42_monotonic;assumption.
                    intros ;apply P_id_length_monotonic;assumption.
                    intros ;apply P_id_U22_monotonic;assumption.
                    intros ;apply P_id_s_monotonic;assumption.
                    intros ;apply P_id_a__U41_monotonic;assumption.
                    intros ;apply P_id_U52_monotonic;assumption.
                    intros ;apply P_id_isNatList_monotonic;assumption.
                    intros ;apply P_id_a__U53_monotonic;assumption.
                    intros ;apply P_id_a__U31_monotonic;assumption.
                    intros ;apply P_id_isNatIList_monotonic;assumption.
                    intros ;apply P_id_a__isNatKind_monotonic;assumption.
                    intros ;apply P_id_a__isNatList_monotonic;assumption.
                    intros ;apply P_id_U32_monotonic;assumption.
                    intros ;apply P_id_mark_monotonic;assumption.
                    intros ;apply P_id_a__U43_monotonic;assumption.
                    intros ;apply P_id_U61_monotonic;assumption.
                    intros ;apply P_id_isNat_monotonic;assumption.
                    intros ;apply P_id_a__zeros_bounded;assumption.
                    intros ;apply P_id_isNatKind_bounded;assumption.
                    intros ;apply P_id_a__isNatIList_bounded;assumption.
                    intros ;apply P_id_a__U21_bounded;assumption.
                    intros ;apply P_id_U41_bounded;assumption.
                    intros ;apply P_id_a__and_bounded;assumption.
                    intros ;apply P_id_a__U11_bounded;assumption.
                    intros ;apply P_id_U21_bounded;assumption.
                    intros ;apply P_id_a__U61_bounded;assumption.
                    intros ;apply P_id_a__U32_bounded;assumption.
                    intros ;apply P_id_U51_bounded;assumption.
                    intros ;apply P_id_isNatIListKind_bounded;assumption.
                    intros ;apply P_id_0_bounded;assumption.
                    intros ;apply P_id_U12_bounded;assumption.
                    intros ;apply P_id_a__U52_bounded;assumption.
                    intros ;apply P_id_a__isNat_bounded;assumption.
                    intros ;apply P_id_U43_bounded;assumption.
                    intros ;apply P_id_a__isNatIListKind_bounded;assumption.
                    intros ;apply P_id_a__U12_bounded;assumption.
                    intros ;apply P_id_U31_bounded;assumption.
                    intros ;apply P_id_a__length_bounded;assumption.
                    intros ;apply P_id_a__U42_bounded;assumption.
                    intros ;apply P_id_U53_bounded;assumption.
                    intros ;apply P_id_and_bounded;assumption.
                    intros ;apply P_id_cons_bounded;assumption.
                    intros ;apply P_id_U11_bounded;assumption.
                    intros ;apply P_id_a__U51_bounded;assumption.
                    intros ;apply P_id_a__U22_bounded;assumption.
                    intros ;apply P_id_U42_bounded;assumption.
                    intros ;apply P_id_length_bounded;assumption.
                    intros ;apply P_id_tt_bounded;assumption.
                    intros ;apply P_id_U22_bounded;assumption.
                    intros ;apply P_id_s_bounded;assumption.
                    intros ;apply P_id_a__U41_bounded;assumption.
                    intros ;apply P_id_U52_bounded;assumption.
                    intros ;apply P_id_nil_bounded;assumption.
                    intros ;apply P_id_zeros_bounded;assumption.
                    intros ;apply P_id_isNatList_bounded;assumption.
                    intros ;apply P_id_a__U53_bounded;assumption.
                    intros ;apply P_id_a__U31_bounded;assumption.
                    intros ;apply P_id_isNatIList_bounded;assumption.
                    intros ;apply P_id_a__isNatKind_bounded;assumption.
                    intros ;apply P_id_a__isNatList_bounded;assumption.
                    intros ;apply P_id_U32_bounded;assumption.
                    intros ;apply P_id_mark_bounded;assumption.
                    intros ;apply P_id_a__U43_bounded;assumption.
                    intros ;apply P_id_U61_bounded;assumption.
                    intros ;apply P_id_isNat_bounded;assumption.
                    apply rules_monotonic.
                    intros ;apply P_id_A__LENGTH_monotonic;assumption.
                    intros ;apply P_id_A__U11_monotonic;assumption.
                    intros ;apply P_id_A__U41_monotonic;assumption.
                    intros ;apply P_id_A__ISNATKIND_monotonic;assumption.
                    intros ;apply P_id_A__U22_monotonic;assumption.
                    intros ;apply P_id_A__U51_monotonic;assumption.
                    intros ;apply P_id_A__AND_monotonic;assumption.
                    intros ;apply P_id_A__ISNATLIST_monotonic;assumption.
                    intros ;apply P_id_A__U43_monotonic;assumption.
                    intros ;apply P_id_A__U31_monotonic;assumption.
                    intros ;apply P_id_A__U53_monotonic;assumption.
                    intros ;apply P_id_MARK_monotonic;assumption.
                    intros ;apply P_id_A__U12_monotonic;assumption.
                    intros ;apply P_id_A__U42_monotonic;assumption.
                    intros ;apply P_id_A__ISNAT_monotonic;assumption.
                    intros ;apply P_id_A__U52_monotonic;assumption.
                    
                    intros ;apply P_id_A__ISNATILISTKIND_monotonic;assumption.
                    intros ;apply P_id_A__U21_monotonic;assumption.
                    intros ;apply P_id_A__ISNATILIST_monotonic;assumption.
                    intros ;apply P_id_A__U32_monotonic;assumption.
                    intros ;apply P_id_A__U61_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_strict_in_lt :
                   Relation_Definitions.inclusion _ 
                    DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_large_in_le :
                   Relation_Definitions.inclusion _ 
                    DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_large  := 
                    WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_large.wf
                    .
                  
                  
                  Lemma wf :
                   well_founded WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4.DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large
                    .
                  Proof.
                    intros x.
                    apply (well_founded_ind wf_lt).
                    clear x.
                    intros x.
                    pattern x.
                    
                    apply (@Acc_ind _ 
                            DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_large)
                           .
                    clear x.
                    intros x _ IHx IHx'.
                    constructor.
                    intros y H.
                    
                    destruct H;
                     (apply IHx';
                       apply DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_large_in_le;
                        econstructor eassumption])).
                    
                    apply wf_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_large.
                  Qed.
                 End WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_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_isNatKind (x11:Z) := 0.
                 
                 Definition P_id_a__isNatIList (x11:Z) := 2 + 2* x11.
                 
                 Definition P_id_a__U21 (x11:Z) (x12:Z) := 0.
                 
                 Definition P_id_U41 (x11:Z) (x12:Z) (x13:Z) := 1 + 1* x13.
                 
                 Definition P_id_a__and (x11:Z) (x12:Z) := 2* x11 + 2* x12.
                 
                 Definition P_id_a__U11 (x11:Z) (x12:Z) := 0.
                 
                 Definition P_id_U21 (x11:Z) (x12:Z) := 0.
                 
                 Definition P_id_a__U61 (x11:Z) (x12:Z) := 0.
                 
                 Definition P_id_a__U32 (x11:Z) := 2.
                 
                 Definition P_id_U51 (x11:Z) (x12:Z) (x13:Z) := 2 + 2* x13.
                 
                 Definition P_id_isNatIListKind (x11:Z) := 0.
                 
                 Definition P_id_0  := 0.
                 
                 Definition P_id_U12 (x11:Z) := 0.
                 
                 Definition P_id_a__U52 (x11:Z) (x12:Z) := 2 + 2* x12.
                 
                 Definition P_id_a__isNat (x11:Z) := 0.
                 
                 Definition P_id_U43 (x11:Z) := 1.
                 
                 Definition P_id_a__isNatIListKind (x11:Z) := 0.
                 
                 Definition P_id_a__U12 (x11:Z) := 0.
                 
                 Definition P_id_U31 (x11:Z) (x12:Z) := 1 + 1* x12.
                 
                 Definition P_id_a__length (x11:Z) := 0.
                 
                 Definition P_id_a__U42 (x11:Z) (x12:Z) := 1 + 2* x12.
                 
                 Definition P_id_U53 (x11:Z) := 2 + 1* x11.
                 
                 Definition P_id_and (x11:Z) (x12:Z) := 2* x11 + 1* x12.
                 
                 Definition P_id_cons (x11:Z) (x12:Z) := 1 + 1* x12.
                 
                 Definition P_id_U11 (x11:Z) (x12:Z) := 0.
                 
                 Definition P_id_a__U51 (x11:Z) (x12:Z) (x13:Z) := 2 + 2* x13
                   .
                 
                 Definition P_id_a__U22 (x11:Z) := 1* x11.
                 
                 Definition P_id_U42 (x11:Z) (x12:Z) := 1 + 1* x12.
                 
                 Definition P_id_length (x11:Z) := 0.
                 
                 Definition P_id_tt  := 0.
                 
                 Definition P_id_U22 (x11:Z) := 1* x11.
                 
                 Definition P_id_s (x11:Z) := 2* x11.
                 
                 Definition P_id_a__U41 (x11:Z) (x12:Z) (x13:Z) := 2 + 2* x13
                   .
                 
                 Definition P_id_U52 (x11:Z) (x12:Z) := 1 + 1* x12.
                 
                 Definition P_id_nil  := 0.
                 
                 Definition P_id_zeros  := 1.
                 
                 Definition P_id_isNatList (x11:Z) := 1* x11.
                 
                 Definition P_id_a__U53 (x11:Z) := 2 + 1* x11.
                 
                 Definition P_id_a__U31 (x11:Z) (x12:Z) := 2 + 1* x12.
                 
                 Definition P_id_isNatIList (x11:Z) := 2 + 1* x11.
                 
                 Definition P_id_a__isNatKind (x11:Z) := 0.
                 
                 Definition P_id_a__isNatList (x11:Z) := 2* x11.
                 
                 Definition P_id_U32 (x11:Z) := 1.
                 
                 Definition P_id_mark (x11:Z) := 2* x11.
                 
                 Definition P_id_a__U43 (x11:Z) := 1.
                 
                 Definition P_id_U61 (x11:Z) (x12:Z) := 0.
                 
                 Definition P_id_isNat (x11:Z) := 0.
                 
                 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__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__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_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_a__and_monotonic :
                  forall x12 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_a__and x12 x14 <= P_id_a__and x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   
                   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_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__U61_monotonic :
                  forall x12 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_a__U61 x12 x14 <= P_id_a__U61 x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   
                   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 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_a__U32 x12 <= P_id_a__U32 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U51_monotonic :
                  forall x16 x12 x14 x13 x11 x15, 
                   (0 <= x16)/\ (x16 <= x15) ->
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_U51 x12 x14 x16 <= P_id_U51 x11 x13 x15.
                 Proof.
                   intros x16 x15 x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   intros [H_5 H_4].
                   
                   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_U12_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->P_id_U12 x12 <= P_id_U12 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_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 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_a__U52 x12 x14 <= P_id_a__U52 x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_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_U43_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->P_id_U43 x12 <= P_id_U43 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_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__U12_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_a__U12 x12 <= P_id_a__U12 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   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__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__U42_monotonic :
                  forall x12 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_a__U42 x12 x14 <= P_id_a__U42 x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U53_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->P_id_U53 x12 <= P_id_U53 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_and_monotonic :
                  forall x12 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_and x12 x14 <= P_id_and x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   
                   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_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__U51_monotonic :
                  forall x16 x12 x14 x13 x11 x15, 
                   (0 <= x16)/\ (x16 <= x15) ->
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_a__U51 x12 x14 x16 <= P_id_a__U51 x11 x13 x15.
                 Proof.
                   intros x16 x15 x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   intros [H_5 H_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 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_a__U22 x12 <= P_id_a__U22 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 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 x12 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_U42 x12 x14 <= P_id_U42 x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   
                   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_U22_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->P_id_U22 x12 <= P_id_U22 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   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__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_U52_monotonic :
                  forall x12 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_U52 x12 x14 <= P_id_U52 x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 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__U53_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_a__U53 x12 <= P_id_a__U53 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   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_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__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__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_U32_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->P_id_U32 x12 <= P_id_U32 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 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__U43_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_a__U43 x12 <= P_id_a__U43 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   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 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_U61 x12 x14 <= P_id_U61 x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   
                   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__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_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__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_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_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_a__and_bounded :
                  forall x12 x11, 
                   (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__and x12 x11.
                 Proof.
                   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_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__U61_bounded :
                  forall x12 x11, 
                   (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U61 x12 x11.
                 Proof.
                   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 x11, (0 <= x11) ->0 <= P_id_a__U32 x11.
                 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 x13 x11, 
                   (0 <= x11) ->
                    (0 <= x12) ->(0 <= x13) ->0 <= P_id_U51 x13 x12 x11.
                 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_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_U12_bounded :
                  forall x11, (0 <= x11) ->0 <= P_id_U12 x11.
                 Proof.
                   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 x12 x11, 
                   (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U52 x12 x11.
                 Proof.
                   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_U43_bounded :
                  forall x11, (0 <= x11) ->0 <= P_id_U43 x11.
                 Proof.
                   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_a__U12_bounded :
                  forall x11, (0 <= x11) ->0 <= P_id_a__U12 x11.
                 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__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__U42_bounded :
                  forall x12 x11, 
                   (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U42 x12 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_U53_bounded :
                  forall x11, (0 <= x11) ->0 <= P_id_U53 x11.
                 Proof.
                   intros .
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_and_bounded :
                  forall x12 x11, 
                   (0 <= x11) ->(0 <= x12) ->0 <= P_id_and x12 x11.
                 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_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__U51_bounded :
                  forall x12 x13 x11, 
                   (0 <= x11) ->
                    (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U51 x13 x12 x11.
                 Proof.
                   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 x11, (0 <= x11) ->0 <= P_id_a__U22 x11.
                 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 x11, 
                   (0 <= x11) ->(0 <= x12) ->0 <= P_id_U42 x12 x11.
                 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_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_U22_bounded :
                  forall x11, (0 <= x11) ->0 <= P_id_U22 x11.
                 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__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_U52_bounded :
                  forall x12 x11, 
                   (0 <= x11) ->(0 <= x12) ->0 <= P_id_U52 x12 x11.
                 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_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_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__U53_bounded :
                  forall x11, (0 <= x11) ->0 <= P_id_a__U53 x11.
                 Proof.
                   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_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.
                 
                 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_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_U32_bounded :
                  forall x11, (0 <= x11) ->0 <= P_id_U32 x11.
                 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__U43_bounded :
                  forall x11, (0 <= x11) ->0 <= P_id_a__U43 x11.
                 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 x12 x11, 
                   (0 <= x11) ->(0 <= x12) ->0 <= P_id_U61 x12 x11.
                 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.
                 
                 Definition measure  := 
                   InterpZ.measure 0 P_id_a__zeros P_id_isNatKind 
                    P_id_a__isNatIList P_id_a__U21 P_id_U41 P_id_a__and 
                    P_id_a__U11 P_id_U21 P_id_a__U61 P_id_a__U32 P_id_U51 
                    P_id_isNatIListKind P_id_0 P_id_U12 P_id_a__U52 
                    P_id_a__isNat P_id_U43 P_id_a__isNatIListKind 
                    P_id_a__U12 P_id_U31 P_id_a__length P_id_a__U42 P_id_U53 
                    P_id_and P_id_cons P_id_U11 P_id_a__U51 P_id_a__U22 
                    P_id_U42 P_id_length P_id_tt P_id_U22 P_id_s P_id_a__U41 
                    P_id_U52 P_id_nil P_id_zeros P_id_isNatList P_id_a__U53 
                    P_id_a__U31 P_id_isNatIList P_id_a__isNatKind 
                    P_id_a__isNatList P_id_U32 P_id_mark P_id_a__U43 
                    P_id_U61 P_id_isNat.
                 
                 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_isNatKind 
                                    (x11::nil)) =>
                                  P_id_isNatKind (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__U21 
                                    (x12::x11::nil)) =>
                                  P_id_a__U21 (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_a__and 
                                    (x12::x11::nil)) =>
                                  P_id_a__and (measure x12) (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_U21 (x12::
                                    x11::nil)) =>
                                  P_id_U21 (measure x12) (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U61 
                                    (x12::x11::nil)) =>
                                  P_id_a__U61 (measure x12) (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U32 
                                    (x11::nil)) =>
                                  P_id_a__U32 (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_U51 (x13::
                                    x12::x11::nil)) =>
                                  P_id_U51 (measure x13) (measure x12) 
                                   (measure x11)
                                 | (algebra.Alg.Term 
                                    algebra.F.id_isNatIListKind (x11::nil)) =>
                                  P_id_isNatIListKind (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_0 nil) =>
                                  P_id_0 
                                 | (algebra.Alg.Term algebra.F.id_U12 
                                    (x11::nil)) =>
                                  P_id_U12 (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U52 
                                    (x12::x11::nil)) =>
                                  P_id_a__U52 (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_U43 
                                    (x11::nil)) =>
                                  P_id_U43 (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__U12 
                                    (x11::nil)) =>
                                  P_id_a__U12 (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__length 
                                    (x11::nil)) =>
                                  P_id_a__length (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U42 
                                    (x12::x11::nil)) =>
                                  P_id_a__U42 (measure x12) (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_U53 
                                    (x11::nil)) =>
                                  P_id_U53 (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_and (x12::
                                    x11::nil)) =>
                                  P_id_and (measure x12) (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_U11 (x12::
                                    x11::nil)) =>
                                  P_id_U11 (measure x12) (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U51 
                                    (x13::x12::x11::nil)) =>
                                  P_id_a__U51 (measure x13) (measure x12) 
                                   (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U22 
                                    (x11::nil)) =>
                                  P_id_a__U22 (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_U42 (x12::
                                    x11::nil)) =>
                                  P_id_U42 (measure x12) (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_length 
                                    (x11::nil)) =>
                                  P_id_length (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_tt nil) =>
                                  P_id_tt 
                                 | (algebra.Alg.Term algebra.F.id_U22 
                                    (x11::nil)) =>
                                  P_id_U22 (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_s 
                                    (x11::nil)) =>
                                  P_id_s (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_U52 (x12::
                                    x11::nil)) =>
                                  P_id_U52 (measure x12) (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_nil nil) =>
                                  P_id_nil 
                                 | (algebra.Alg.Term algebra.F.id_zeros nil) =>
                                  P_id_zeros 
                                 | (algebra.Alg.Term algebra.F.id_isNatList 
                                    (x11::nil)) =>
                                  P_id_isNatList (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U53 
                                    (x11::nil)) =>
                                  P_id_a__U53 (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_isNatIList 
                                    (x11::nil)) =>
                                  P_id_isNatIList (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__isNatList (x11::nil)) =>
                                  P_id_a__isNatList (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_U32 
                                    (x11::nil)) =>
                                  P_id_U32 (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_mark 
                                    (x11::nil)) =>
                                  P_id_mark (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U43 
                                    (x11::nil)) =>
                                  P_id_a__U43 (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_U61 (x12::
                                    x11::nil)) =>
                                  P_id_U61 (measure x12) (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_isNat 
                                    (x11::nil)) =>
                                  P_id_isNat (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_isNatKind_monotonic;assumption.
                   intros ;apply P_id_a__isNatIList_monotonic;assumption.
                   intros ;apply P_id_a__U21_monotonic;assumption.
                   intros ;apply P_id_U41_monotonic;assumption.
                   intros ;apply P_id_a__and_monotonic;assumption.
                   intros ;apply P_id_a__U11_monotonic;assumption.
                   intros ;apply P_id_U21_monotonic;assumption.
                   intros ;apply P_id_a__U61_monotonic;assumption.
                   intros ;apply P_id_a__U32_monotonic;assumption.
                   intros ;apply P_id_U51_monotonic;assumption.
                   intros ;apply P_id_isNatIListKind_monotonic;assumption.
                   intros ;apply P_id_U12_monotonic;assumption.
                   intros ;apply P_id_a__U52_monotonic;assumption.
                   intros ;apply P_id_a__isNat_monotonic;assumption.
                   intros ;apply P_id_U43_monotonic;assumption.
                   intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
                   intros ;apply P_id_a__U12_monotonic;assumption.
                   intros ;apply P_id_U31_monotonic;assumption.
                   intros ;apply P_id_a__length_monotonic;assumption.
                   intros ;apply P_id_a__U42_monotonic;assumption.
                   intros ;apply P_id_U53_monotonic;assumption.
                   intros ;apply P_id_and_monotonic;assumption.
                   intros ;apply P_id_cons_monotonic;assumption.
                   intros ;apply P_id_U11_monotonic;assumption.
                   intros ;apply P_id_a__U51_monotonic;assumption.
                   intros ;apply P_id_a__U22_monotonic;assumption.
                   intros ;apply P_id_U42_monotonic;assumption.
                   intros ;apply P_id_length_monotonic;assumption.
                   intros ;apply P_id_U22_monotonic;assumption.
                   intros ;apply P_id_s_monotonic;assumption.
                   intros ;apply P_id_a__U41_monotonic;assumption.
                   intros ;apply P_id_U52_monotonic;assumption.
                   intros ;apply P_id_isNatList_monotonic;assumption.
                   intros ;apply P_id_a__U53_monotonic;assumption.
                   intros ;apply P_id_a__U31_monotonic;assumption.
                   intros ;apply P_id_isNatIList_monotonic;assumption.
                   intros ;apply P_id_a__isNatKind_monotonic;assumption.
                   intros ;apply P_id_a__isNatList_monotonic;assumption.
                   intros ;apply P_id_U32_monotonic;assumption.
                   intros ;apply P_id_mark_monotonic;assumption.
                   intros ;apply P_id_a__U43_monotonic;assumption.
                   intros ;apply P_id_U61_monotonic;assumption.
                   intros ;apply P_id_isNat_monotonic;assumption.
                   intros ;apply P_id_a__zeros_bounded;assumption.
                   intros ;apply P_id_isNatKind_bounded;assumption.
                   intros ;apply P_id_a__isNatIList_bounded;assumption.
                   intros ;apply P_id_a__U21_bounded;assumption.
                   intros ;apply P_id_U41_bounded;assumption.
                   intros ;apply P_id_a__and_bounded;assumption.
                   intros ;apply P_id_a__U11_bounded;assumption.
                   intros ;apply P_id_U21_bounded;assumption.
                   intros ;apply P_id_a__U61_bounded;assumption.
                   intros ;apply P_id_a__U32_bounded;assumption.
                   intros ;apply P_id_U51_bounded;assumption.
                   intros ;apply P_id_isNatIListKind_bounded;assumption.
                   intros ;apply P_id_0_bounded;assumption.
                   intros ;apply P_id_U12_bounded;assumption.
                   intros ;apply P_id_a__U52_bounded;assumption.
                   intros ;apply P_id_a__isNat_bounded;assumption.
                   intros ;apply P_id_U43_bounded;assumption.
                   intros ;apply P_id_a__isNatIListKind_bounded;assumption.
                   intros ;apply P_id_a__U12_bounded;assumption.
                   intros ;apply P_id_U31_bounded;assumption.
                   intros ;apply P_id_a__length_bounded;assumption.
                   intros ;apply P_id_a__U42_bounded;assumption.
                   intros ;apply P_id_U53_bounded;assumption.
                   intros ;apply P_id_and_bounded;assumption.
                   intros ;apply P_id_cons_bounded;assumption.
                   intros ;apply P_id_U11_bounded;assumption.
                   intros ;apply P_id_a__U51_bounded;assumption.
                   intros ;apply P_id_a__U22_bounded;assumption.
                   intros ;apply P_id_U42_bounded;assumption.
                   intros ;apply P_id_length_bounded;assumption.
                   intros ;apply P_id_tt_bounded;assumption.
                   intros ;apply P_id_U22_bounded;assumption.
                   intros ;apply P_id_s_bounded;assumption.
                   intros ;apply P_id_a__U41_bounded;assumption.
                   intros ;apply P_id_U52_bounded;assumption.
                   intros ;apply P_id_nil_bounded;assumption.
                   intros ;apply P_id_zeros_bounded;assumption.
                   intros ;apply P_id_isNatList_bounded;assumption.
                   intros ;apply P_id_a__U53_bounded;assumption.
                   intros ;apply P_id_a__U31_bounded;assumption.
                   intros ;apply P_id_isNatIList_bounded;assumption.
                   intros ;apply P_id_a__isNatKind_bounded;assumption.
                   intros ;apply P_id_a__isNatList_bounded;assumption.
                   intros ;apply P_id_U32_bounded;assumption.
                   intros ;apply P_id_mark_bounded;assumption.
                   intros ;apply P_id_a__U43_bounded;assumption.
                   intros ;apply P_id_U61_bounded;assumption.
                   intros ;apply P_id_isNat_bounded;assumption.
                   apply rules_monotonic.
                 Qed.
                 
                 Definition P_id_A__LENGTH (x11:Z) := 0.
                 
                 Definition P_id_A__U11 (x11:Z) (x12:Z) := 0.
                 
                 Definition P_id_A__U41 (x11:Z) (x12:Z) (x13:Z) := 0.
                 
                 Definition P_id_A__ISNATKIND (x11:Z) := 0.
                 
                 Definition P_id_A__U22 (x11:Z) := 0.
                 
                 Definition P_id_A__U51 (x11:Z) (x12:Z) (x13:Z) := 0.
                 
                 Definition P_id_A__AND (x11:Z) (x12:Z) := 2* x12.
                 
                 Definition P_id_A__ISNATLIST (x11:Z) := 0.
                 
                 Definition P_id_A__U43 (x11:Z) := 0.
                 
                 Definition P_id_A__U31 (x11:Z) (x12:Z) := 0.
                 
                 Definition P_id_A__U53 (x11:Z) := 0.
                 
                 Definition P_id_MARK (x11:Z) := 2* x11.
                 
                 Definition P_id_A__U12 (x11:Z) := 0.
                 
                 Definition P_id_A__U42 (x11:Z) (x12:Z) := 0.
                 
                 Definition P_id_A__ZEROS  := 0.
                 
                 Definition P_id_A__ISNAT (x11:Z) := 0.
                 
                 Definition P_id_A__U52 (x11:Z) (x12:Z) := 0.
                 
                 Definition P_id_A__ISNATILISTKIND (x11:Z) := 0.
                 
                 Definition P_id_A__U21 (x11:Z) (x12:Z) := 0.
                 
                 Definition P_id_A__ISNATILIST (x11:Z) := 0.
                 
                 Definition P_id_A__U32 (x11:Z) := 0.
                 
                 Definition P_id_A__U61 (x11:Z) (x12:Z) := 0.
                 
                 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__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__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__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__U22_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_A__U22 x12 <= P_id_A__U22 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_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 x16 x12 x14 x13 x11 x15, 
                   (0 <= x16)/\ (x16 <= x15) ->
                    (0 <= x14)/\ (x14 <= x13) ->
                     (0 <= x12)/\ (x12 <= x11) ->
                      P_id_A__U51 x12 x14 x16 <= P_id_A__U51 x11 x13 x15.
                 Proof.
                   intros x16 x15 x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   intros [H_5 H_4].
                   
                   cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;
                    intuition;
                    (auto with zarith)||
                    (repeat (translate_vars );prove_ineq ).
                 Qed.
                 
                 Lemma P_id_A__AND_monotonic :
                  forall x12 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_A__AND x12 x14 <= P_id_A__AND x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_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__U43_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_A__U43 x12 <= P_id_A__U43 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   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__U53_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_A__U53 x12 <= P_id_A__U53 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 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__U12_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_A__U12 x12 <= P_id_A__U12 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_0].
                   
                   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 x12 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_A__U42 x12 x14 <= P_id_A__U42 x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_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_A__U52_monotonic :
                  forall x12 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_A__U52 x12 x14 <= P_id_A__U52 x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   intros [H_3 H_2].
                   
                   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__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__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__U32_monotonic :
                  forall x12 x11, 
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_A__U32 x12 <= P_id_A__U32 x11.
                 Proof.
                   intros x12 x11.
                   intros [H_1 H_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 x14 x13 x11, 
                   (0 <= x14)/\ (x14 <= x13) ->
                    (0 <= x12)/\ (x12 <= x11) ->
                     P_id_A__U61 x12 x14 <= P_id_A__U61 x11 x13.
                 Proof.
                   intros x14 x13 x12 x11.
                   intros [H_1 H_0].
                   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_isNatKind 
                    P_id_a__isNatIList P_id_a__U21 P_id_U41 P_id_a__and 
                    P_id_a__U11 P_id_U21 P_id_a__U61 P_id_a__U32 P_id_U51 
                    P_id_isNatIListKind P_id_0 P_id_U12 P_id_a__U52 
                    P_id_a__isNat P_id_U43 P_id_a__isNatIListKind 
                    P_id_a__U12 P_id_U31 P_id_a__length P_id_a__U42 P_id_U53 
                    P_id_and P_id_cons P_id_U11 P_id_a__U51 P_id_a__U22 
                    P_id_U42 P_id_length P_id_tt P_id_U22 P_id_s P_id_a__U41 
                    P_id_U52 P_id_nil P_id_zeros P_id_isNatList P_id_a__U53 
                    P_id_a__U31 P_id_isNatIList P_id_a__isNatKind 
                    P_id_a__isNatList P_id_U32 P_id_mark P_id_a__U43 
                    P_id_U61 P_id_isNat P_id_A__LENGTH P_id_A__U11 
                    P_id_A__U41 P_id_A__ISNATKIND P_id_A__U22 P_id_A__U51 
                    P_id_A__AND P_id_A__ISNATLIST P_id_A__U43 P_id_A__U31 
                    P_id_A__U53 P_id_MARK P_id_A__U12 P_id_A__U42 
                    P_id_A__ZEROS P_id_A__ISNAT P_id_A__U52 
                    P_id_A__ISNATILISTKIND P_id_A__U21 P_id_A__ISNATILIST 
                    P_id_A__U32 P_id_A__U61.
                 
                 Lemma marked_measure_equation :
                  forall t, 
                   marked_measure t = match t with
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__length (x11::nil)) =>
                                         P_id_A__LENGTH (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__U41 (x13::x12::
                                           x11::nil)) =>
                                         P_id_A__U41 (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__U22 (x11::nil)) =>
                                         P_id_A__U22 (measure x11)
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__U51 (x13::x12::
                                           x11::nil)) =>
                                         P_id_A__U51 (measure x13) 
                                          (measure x12) (measure x11)
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__and (x12::
                                           x11::nil)) =>
                                         P_id_A__AND (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__U43 (x11::nil)) =>
                                         P_id_A__U43 (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__U53 (x11::nil)) =>
                                         P_id_A__U53 (measure x11)
                                        | (algebra.Alg.Term 
                                           algebra.F.id_mark (x11::nil)) =>
                                         P_id_MARK (measure x11)
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__U12 (x11::nil)) =>
                                         P_id_A__U12 (measure x11)
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__U42 (x12::
                                           x11::nil)) =>
                                         P_id_A__U42 (measure x12) 
                                          (measure x11)
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__zeros nil) =>
                                         P_id_A__ZEROS 
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__isNat (x11::nil)) =>
                                         P_id_A__ISNAT (measure x11)
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__U52 (x12::
                                           x11::nil)) =>
                                         P_id_A__U52 (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_a__U21 (x12::
                                           x11::nil)) =>
                                         P_id_A__U21 (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__U32 (x11::nil)) =>
                                         P_id_A__U32 (measure x11)
                                        | (algebra.Alg.Term 
                                           algebra.F.id_a__U61 (x12::
                                           x11::nil)) =>
                                         P_id_A__U61 (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_isNatKind_monotonic;assumption.
                   intros ;apply P_id_a__isNatIList_monotonic;assumption.
                   intros ;apply P_id_a__U21_monotonic;assumption.
                   intros ;apply P_id_U41_monotonic;assumption.
                   intros ;apply P_id_a__and_monotonic;assumption.
                   intros ;apply P_id_a__U11_monotonic;assumption.
                   intros ;apply P_id_U21_monotonic;assumption.
                   intros ;apply P_id_a__U61_monotonic;assumption.
                   intros ;apply P_id_a__U32_monotonic;assumption.
                   intros ;apply P_id_U51_monotonic;assumption.
                   intros ;apply P_id_isNatIListKind_monotonic;assumption.
                   intros ;apply P_id_U12_monotonic;assumption.
                   intros ;apply P_id_a__U52_monotonic;assumption.
                   intros ;apply P_id_a__isNat_monotonic;assumption.
                   intros ;apply P_id_U43_monotonic;assumption.
                   intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
                   intros ;apply P_id_a__U12_monotonic;assumption.
                   intros ;apply P_id_U31_monotonic;assumption.
                   intros ;apply P_id_a__length_monotonic;assumption.
                   intros ;apply P_id_a__U42_monotonic;assumption.
                   intros ;apply P_id_U53_monotonic;assumption.
                   intros ;apply P_id_and_monotonic;assumption.
                   intros ;apply P_id_cons_monotonic;assumption.
                   intros ;apply P_id_U11_monotonic;assumption.
                   intros ;apply P_id_a__U51_monotonic;assumption.
                   intros ;apply P_id_a__U22_monotonic;assumption.
                   intros ;apply P_id_U42_monotonic;assumption.
                   intros ;apply P_id_length_monotonic;assumption.
                   intros ;apply P_id_U22_monotonic;assumption.
                   intros ;apply P_id_s_monotonic;assumption.
                   intros ;apply P_id_a__U41_monotonic;assumption.
                   intros ;apply P_id_U52_monotonic;assumption.
                   intros ;apply P_id_isNatList_monotonic;assumption.
                   intros ;apply P_id_a__U53_monotonic;assumption.
                   intros ;apply P_id_a__U31_monotonic;assumption.
                   intros ;apply P_id_isNatIList_monotonic;assumption.
                   intros ;apply P_id_a__isNatKind_monotonic;assumption.
                   intros ;apply P_id_a__isNatList_monotonic;assumption.
                   intros ;apply P_id_U32_monotonic;assumption.
                   intros ;apply P_id_mark_monotonic;assumption.
                   intros ;apply P_id_a__U43_monotonic;assumption.
                   intros ;apply P_id_U61_monotonic;assumption.
                   intros ;apply P_id_isNat_monotonic;assumption.
                   intros ;apply P_id_a__zeros_bounded;assumption.
                   intros ;apply P_id_isNatKind_bounded;assumption.
                   intros ;apply P_id_a__isNatIList_bounded;assumption.
                   intros ;apply P_id_a__U21_bounded;assumption.
                   intros ;apply P_id_U41_bounded;assumption.
                   intros ;apply P_id_a__and_bounded;assumption.
                   intros ;apply P_id_a__U11_bounded;assumption.
                   intros ;apply P_id_U21_bounded;assumption.
                   intros ;apply P_id_a__U61_bounded;assumption.
                   intros ;apply P_id_a__U32_bounded;assumption.
                   intros ;apply P_id_U51_bounded;assumption.
                   intros ;apply P_id_isNatIListKind_bounded;assumption.
                   intros ;apply P_id_0_bounded;assumption.
                   intros ;apply P_id_U12_bounded;assumption.
                   intros ;apply P_id_a__U52_bounded;assumption.
                   intros ;apply P_id_a__isNat_bounded;assumption.
                   intros ;apply P_id_U43_bounded;assumption.
                   intros ;apply P_id_a__isNatIListKind_bounded;assumption.
                   intros ;apply P_id_a__U12_bounded;assumption.
                   intros ;apply P_id_U31_bounded;assumption.
                   intros ;apply P_id_a__length_bounded;assumption.
                   intros ;apply P_id_a__U42_bounded;assumption.
                   intros ;apply P_id_U53_bounded;assumption.
                   intros ;apply P_id_and_bounded;assumption.
                   intros ;apply P_id_cons_bounded;assumption.
                   intros ;apply P_id_U11_bounded;assumption.
                   intros ;apply P_id_a__U51_bounded;assumption.
                   intros ;apply P_id_a__U22_bounded;assumption.
                   intros ;apply P_id_U42_bounded;assumption.
                   intros ;apply P_id_length_bounded;assumption.
                   intros ;apply P_id_tt_bounded;assumption.
                   intros ;apply P_id_U22_bounded;assumption.
                   intros ;apply P_id_s_bounded;assumption.
                   intros ;apply P_id_a__U41_bounded;assumption.
                   intros ;apply P_id_U52_bounded;assumption.
                   intros ;apply P_id_nil_bounded;assumption.
                   intros ;apply P_id_zeros_bounded;assumption.
                   intros ;apply P_id_isNatList_bounded;assumption.
                   intros ;apply P_id_a__U53_bounded;assumption.
                   intros ;apply P_id_a__U31_bounded;assumption.
                   intros ;apply P_id_isNatIList_bounded;assumption.
                   intros ;apply P_id_a__isNatKind_bounded;assumption.
                   intros ;apply P_id_a__isNatList_bounded;assumption.
                   intros ;apply P_id_U32_bounded;assumption.
                   intros ;apply P_id_mark_bounded;assumption.
                   intros ;apply P_id_a__U43_bounded;assumption.
                   intros ;apply P_id_U61_bounded;assumption.
                   intros ;apply P_id_isNat_bounded;assumption.
                   apply rules_monotonic.
                   intros ;apply P_id_A__LENGTH_monotonic;assumption.
                   intros ;apply P_id_A__U11_monotonic;assumption.
                   intros ;apply P_id_A__U41_monotonic;assumption.
                   intros ;apply P_id_A__ISNATKIND_monotonic;assumption.
                   intros ;apply P_id_A__U22_monotonic;assumption.
                   intros ;apply P_id_A__U51_monotonic;assumption.
                   intros ;apply P_id_A__AND_monotonic;assumption.
                   intros ;apply P_id_A__ISNATLIST_monotonic;assumption.
                   intros ;apply P_id_A__U43_monotonic;assumption.
                   intros ;apply P_id_A__U31_monotonic;assumption.
                   intros ;apply P_id_A__U53_monotonic;assumption.
                   intros ;apply P_id_MARK_monotonic;assumption.
                   intros ;apply P_id_A__U12_monotonic;assumption.
                   intros ;apply P_id_A__U42_monotonic;assumption.
                   intros ;apply P_id_A__ISNAT_monotonic;assumption.
                   intros ;apply P_id_A__U52_monotonic;assumption.
                   intros ;apply P_id_A__ISNATILISTKIND_monotonic;assumption.
                   intros ;apply P_id_A__U21_monotonic;assumption.
                   intros ;apply P_id_A__ISNATILIST_monotonic;assumption.
                   intros ;apply P_id_A__U32_monotonic;assumption.
                   intros ;apply P_id_A__U61_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_strict_in_lt :
                  Relation_Definitions.inclusion _ 
                   DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_in_le :
                  Relation_Definitions.inclusion _ 
                   DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large  := 
                   WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large.wf
                   .
                 
                 
                 Lemma wf :
                  well_founded WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large.DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4
                   .
                 Proof.
                   intros x.
                   apply (well_founded_ind wf_lt).
                   clear x.
                   intros x.
                   pattern x.
                   
                   apply (@Acc_ind _ 
                           DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large)
                          .
                   clear x.
                   intros x _ IHx IHx'.
                   constructor.
                   intros y H.
                   
                   destruct H;
                    (apply IHx';
                      apply DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large_in_le;
                       econstructor eassumption])).
                   
                   apply wf_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4_large.
                 Qed.
                End WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4.
                
                Definition wf_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4  := 
                  WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4.wf
                  .
                
                
                Lemma acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4 :
                 forall x y, 
                  (DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4 
                    x y) ->
                   Acc WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large.DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large 
                    x.
                Proof.
                  intros x.
                  pattern x.
                  
                  apply (@Acc_ind _ 
                          DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4)
                         .
                  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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_non_scc_3;
                       
                       econstructor 
                       (eassumption)||(algebra.Alg_ext.star_refl' ))||
                     ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4.
                Qed.
                
                
                Lemma wf :
                 well_founded WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large.DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large
                  .
                Proof.
                  constructor;intros _y _h;inversion _h;clear _h;subst;
                   (eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_non_scc_3;
                     
                     econstructor 
                     (eassumption)||(algebra.Alg_ext.star_refl' ))||
                   ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_non_scc_2;
                      
                      econstructor 
                      (eassumption)||(algebra.Alg_ext.star_refl' ))||
                    ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_non_scc_1;
                       
                       econstructor 
                       (eassumption)||(algebra.Alg_ext.star_refl' ))||
                     ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_non_scc_0;
                        
                        econstructor 
                        (eassumption)||(algebra.Alg_ext.star_refl' ))||
                      ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_4;
                         
                         econstructor 
                         (eassumption)||(algebra.Alg_ext.star_refl' ))||
                       ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_3;
                          
                          econstructor 
                          (eassumption)||(algebra.Alg_ext.star_refl' ))||
                        ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_2;
                           
                           econstructor 
                           (eassumption)||(algebra.Alg_ext.star_refl' ))||
                         ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_scc_1;
                            
                            econstructor 
                            (eassumption)||(algebra.Alg_ext.star_refl' ))||
                          ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_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_isNatKind (x11:Z) := 1* x11.
               
               Definition P_id_a__isNatIList (x11:Z) := 2 + 2* x11.
               
               Definition P_id_a__U21 (x11:Z) (x12:Z) := 0.
               
               Definition P_id_U41 (x11:Z) (x12:Z) (x13:Z) := 1 + 1* x12.
               
               Definition P_id_a__and (x11:Z) (x12:Z) := 1* x11 + 2* x12.
               
               Definition P_id_a__U11 (x11:Z) (x12:Z) := 0.
               
               Definition P_id_U21 (x11:Z) (x12:Z) := 0.
               
               Definition P_id_a__U61 (x11:Z) (x12:Z) := 1 + 2* x12.
               
               Definition P_id_a__U32 (x11:Z) := 0.
               
               Definition P_id_U51 (x11:Z) (x12:Z) (x13:Z) := 0.
               
               Definition P_id_isNatIListKind (x11:Z) := 1* x11.
               
               Definition P_id_0  := 0.
               
               Definition P_id_U12 (x11:Z) := 1* x11.
               
               Definition P_id_a__U52 (x11:Z) (x12:Z) := 0.
               
               Definition P_id_a__isNat (x11:Z) := 0.
               
               Definition P_id_U43 (x11:Z) := 0.
               
               Definition P_id_a__isNatIListKind (x11:Z) := 1* x11.
               
               Definition P_id_a__U12 (x11:Z) := 1* x11.
               
               Definition P_id_U31 (x11:Z) (x12:Z) := 1* x11.
               
               Definition P_id_a__length (x11:Z) := 1 + 1* x11.
               
               Definition P_id_a__U42 (x11:Z) (x12:Z) := 0.
               
               Definition P_id_U53 (x11:Z) := 2* x11.
               
               Definition P_id_and (x11:Z) (x12:Z) := 1* x11 + 1* x12.
               
               Definition P_id_cons (x11:Z) (x12:Z) := 1* x11 + 2* x12.
               
               Definition P_id_U11 (x11:Z) (x12:Z) := 0.
               
               Definition P_id_a__U51 (x11:Z) (x12:Z) (x13:Z) := 0.
               
               Definition P_id_a__U22 (x11:Z) := 1* x11.
               
               Definition P_id_U42 (x11:Z) (x12:Z) := 0.
               
               Definition P_id_length (x11:Z) := 1 + 1* x11.
               
               Definition P_id_tt  := 0.
               
               Definition P_id_U22 (x11:Z) := 1* x11.
               
               Definition P_id_s (x11:Z) := 1* x11.
               
               Definition P_id_a__U41 (x11:Z) (x12:Z) (x13:Z) := 2 + 2* x12.
               
               Definition P_id_U52 (x11:Z) (x12:Z) := 0.
               
               Definition P_id_nil  := 0.
               
               Definition P_id_zeros  := 0.
               
               Definition P_id_isNatList (x11:Z) := 0.
               
               Definition P_id_a__U53 (x11:Z) := 2* x11.
               
               Definition P_id_a__U31 (x11:Z) (x12:Z) := 1* x11.
               
               Definition P_id_isNatIList (x11:Z) := 2 + 1* x11.
               
               Definition P_id_a__isNatKind (x11:Z) := 1* x11.
               
               Definition P_id_a__isNatList (x11:Z) := 0.
               
               Definition P_id_U32 (x11:Z) := 0.
               
               Definition P_id_mark (x11:Z) := 2* x11.
               
               Definition P_id_a__U43 (x11:Z) := 0.
               
               Definition P_id_U61 (x11:Z) (x12:Z) := 1 + 2* x12.
               
               Definition P_id_isNat (x11:Z) := 0.
               
               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__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__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_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_a__and_monotonic :
                forall x12 x14 x13 x11, 
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_a__and x12 x14 <= P_id_a__and x11 x13.
               Proof.
                 intros x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 
                 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_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__U61_monotonic :
                forall x12 x14 x13 x11, 
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_a__U61 x12 x14 <= P_id_a__U61 x11 x13.
               Proof.
                 intros x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 
                 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 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_a__U32 x12 <= P_id_a__U32 x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_0].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U51_monotonic :
                forall x16 x12 x14 x13 x11 x15, 
                 (0 <= x16)/\ (x16 <= x15) ->
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_U51 x12 x14 x16 <= P_id_U51 x11 x13 x15.
               Proof.
                 intros x16 x15 x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 intros [H_5 H_4].
                 
                 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_U12_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->P_id_U12 x12 <= P_id_U12 x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_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 x14 x13 x11, 
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_a__U52 x12 x14 <= P_id_a__U52 x11 x13.
               Proof.
                 intros x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_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_U43_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->P_id_U43 x12 <= P_id_U43 x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_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__U12_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_a__U12 x12 <= P_id_a__U12 x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_0].
                 
                 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__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__U42_monotonic :
                forall x12 x14 x13 x11, 
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_a__U42 x12 x14 <= P_id_a__U42 x11 x13.
               Proof.
                 intros x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U53_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->P_id_U53 x12 <= P_id_U53 x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_0].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_and_monotonic :
                forall x12 x14 x13 x11, 
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_and x12 x14 <= P_id_and x11 x13.
               Proof.
                 intros x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 
                 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_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__U51_monotonic :
                forall x16 x12 x14 x13 x11 x15, 
                 (0 <= x16)/\ (x16 <= x15) ->
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_a__U51 x12 x14 x16 <= P_id_a__U51 x11 x13 x15.
               Proof.
                 intros x16 x15 x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 intros [H_5 H_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 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_a__U22 x12 <= P_id_a__U22 x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 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 x12 x14 x13 x11, 
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_U42 x12 x14 <= P_id_U42 x11 x13.
               Proof.
                 intros x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 
                 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_U22_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->P_id_U22 x12 <= P_id_U22 x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_0].
                 
                 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__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_U52_monotonic :
                forall x12 x14 x13 x11, 
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_U52 x12 x14 <= P_id_U52 x11 x13.
               Proof.
                 intros x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 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__U53_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_a__U53 x12 <= P_id_a__U53 x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_0].
                 
                 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_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__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__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_U32_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->P_id_U32 x12 <= P_id_U32 x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 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__U43_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_a__U43 x12 <= P_id_a__U43 x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_0].
                 
                 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 x14 x13 x11, 
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_U61 x12 x14 <= P_id_U61 x11 x13.
               Proof.
                 intros x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 
                 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__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_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__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_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_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_a__and_bounded :
                forall x12 x11, 
                 (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__and x12 x11.
               Proof.
                 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_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__U61_bounded :
                forall x12 x11, 
                 (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U61 x12 x11.
               Proof.
                 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 x11, (0 <= x11) ->0 <= P_id_a__U32 x11.
               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 x13 x11, 
                 (0 <= x11) ->
                  (0 <= x12) ->(0 <= x13) ->0 <= P_id_U51 x13 x12 x11.
               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_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_U12_bounded :
                forall x11, (0 <= x11) ->0 <= P_id_U12 x11.
               Proof.
                 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 x12 x11, 
                 (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U52 x12 x11.
               Proof.
                 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_U43_bounded :
                forall x11, (0 <= x11) ->0 <= P_id_U43 x11.
               Proof.
                 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_a__U12_bounded :
                forall x11, (0 <= x11) ->0 <= P_id_a__U12 x11.
               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__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__U42_bounded :
                forall x12 x11, 
                 (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U42 x12 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_U53_bounded :
                forall x11, (0 <= x11) ->0 <= P_id_U53 x11.
               Proof.
                 intros .
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_and_bounded :
                forall x12 x11, 
                 (0 <= x11) ->(0 <= x12) ->0 <= P_id_and x12 x11.
               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_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__U51_bounded :
                forall x12 x13 x11, 
                 (0 <= x11) ->
                  (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U51 x13 x12 x11.
               Proof.
                 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 x11, (0 <= x11) ->0 <= P_id_a__U22 x11.
               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 x11, 
                 (0 <= x11) ->(0 <= x12) ->0 <= P_id_U42 x12 x11.
               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_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_U22_bounded :
                forall x11, (0 <= x11) ->0 <= P_id_U22 x11.
               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__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_U52_bounded :
                forall x12 x11, 
                 (0 <= x11) ->(0 <= x12) ->0 <= P_id_U52 x12 x11.
               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_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_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__U53_bounded :
                forall x11, (0 <= x11) ->0 <= P_id_a__U53 x11.
               Proof.
                 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_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.
               
               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_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_U32_bounded :
                forall x11, (0 <= x11) ->0 <= P_id_U32 x11.
               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__U43_bounded :
                forall x11, (0 <= x11) ->0 <= P_id_a__U43 x11.
               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 x12 x11, 
                 (0 <= x11) ->(0 <= x12) ->0 <= P_id_U61 x12 x11.
               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.
               
               Definition measure  := 
                 InterpZ.measure 0 P_id_a__zeros P_id_isNatKind 
                  P_id_a__isNatIList P_id_a__U21 P_id_U41 P_id_a__and 
                  P_id_a__U11 P_id_U21 P_id_a__U61 P_id_a__U32 P_id_U51 
                  P_id_isNatIListKind P_id_0 P_id_U12 P_id_a__U52 
                  P_id_a__isNat P_id_U43 P_id_a__isNatIListKind P_id_a__U12 
                  P_id_U31 P_id_a__length P_id_a__U42 P_id_U53 P_id_and 
                  P_id_cons P_id_U11 P_id_a__U51 P_id_a__U22 P_id_U42 
                  P_id_length P_id_tt P_id_U22 P_id_s P_id_a__U41 P_id_U52 
                  P_id_nil P_id_zeros P_id_isNatList P_id_a__U53 P_id_a__U31 
                  P_id_isNatIList P_id_a__isNatKind P_id_a__isNatList 
                  P_id_U32 P_id_mark P_id_a__U43 P_id_U61 P_id_isNat.
               
               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_isNatKind 
                                  (x11::nil)) =>
                                P_id_isNatKind (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__U21 (x12::
                                  x11::nil)) =>
                                P_id_a__U21 (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_a__and (x12::
                                  x11::nil)) =>
                                P_id_a__and (measure x12) (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_U21 (x12::
                                  x11::nil)) =>
                                P_id_U21 (measure x12) (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U61 (x12::
                                  x11::nil)) =>
                                P_id_a__U61 (measure x12) (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U32 
                                  (x11::nil)) =>
                                P_id_a__U32 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_U51 (x13::
                                  x12::x11::nil)) =>
                                P_id_U51 (measure x13) (measure x12) 
                                 (measure x11)
                               | (algebra.Alg.Term 
                                  algebra.F.id_isNatIListKind (x11::nil)) =>
                                P_id_isNatIListKind (measure x11)
                               | (algebra.Alg.Term algebra.F.id_0 nil) =>
                                P_id_0 
                               | (algebra.Alg.Term algebra.F.id_U12 
                                  (x11::nil)) =>
                                P_id_U12 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U52 (x12::
                                  x11::nil)) =>
                                P_id_a__U52 (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_U43 
                                  (x11::nil)) =>
                                P_id_U43 (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__U12 
                                  (x11::nil)) =>
                                P_id_a__U12 (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__length 
                                  (x11::nil)) =>
                                P_id_a__length (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U42 (x12::
                                  x11::nil)) =>
                                P_id_a__U42 (measure x12) (measure x11)
                               | (algebra.Alg.Term algebra.F.id_U53 
                                  (x11::nil)) =>
                                P_id_U53 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_and (x12::
                                  x11::nil)) =>
                                P_id_and (measure x12) (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_U11 (x12::
                                  x11::nil)) =>
                                P_id_U11 (measure x12) (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U51 (x13::
                                  x12::x11::nil)) =>
                                P_id_a__U51 (measure x13) (measure x12) 
                                 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U22 
                                  (x11::nil)) =>
                                P_id_a__U22 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_U42 (x12::
                                  x11::nil)) =>
                                P_id_U42 (measure x12) (measure x11)
                               | (algebra.Alg.Term algebra.F.id_length 
                                  (x11::nil)) =>
                                P_id_length (measure x11)
                               | (algebra.Alg.Term algebra.F.id_tt nil) =>
                                P_id_tt 
                               | (algebra.Alg.Term algebra.F.id_U22 
                                  (x11::nil)) =>
                                P_id_U22 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_s (x11::nil)) =>
                                P_id_s (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_U52 (x12::
                                  x11::nil)) =>
                                P_id_U52 (measure x12) (measure x11)
                               | (algebra.Alg.Term algebra.F.id_nil nil) =>
                                P_id_nil 
                               | (algebra.Alg.Term algebra.F.id_zeros nil) =>
                                P_id_zeros 
                               | (algebra.Alg.Term algebra.F.id_isNatList 
                                  (x11::nil)) =>
                                P_id_isNatList (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U53 
                                  (x11::nil)) =>
                                P_id_a__U53 (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_isNatIList 
                                  (x11::nil)) =>
                                P_id_isNatIList (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__isNatList 
                                  (x11::nil)) =>
                                P_id_a__isNatList (measure x11)
                               | (algebra.Alg.Term algebra.F.id_U32 
                                  (x11::nil)) =>
                                P_id_U32 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_mark 
                                  (x11::nil)) =>
                                P_id_mark (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U43 
                                  (x11::nil)) =>
                                P_id_a__U43 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_U61 (x12::
                                  x11::nil)) =>
                                P_id_U61 (measure x12) (measure x11)
                               | (algebra.Alg.Term algebra.F.id_isNat 
                                  (x11::nil)) =>
                                P_id_isNat (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_isNatKind_monotonic;assumption.
                 intros ;apply P_id_a__isNatIList_monotonic;assumption.
                 intros ;apply P_id_a__U21_monotonic;assumption.
                 intros ;apply P_id_U41_monotonic;assumption.
                 intros ;apply P_id_a__and_monotonic;assumption.
                 intros ;apply P_id_a__U11_monotonic;assumption.
                 intros ;apply P_id_U21_monotonic;assumption.
                 intros ;apply P_id_a__U61_monotonic;assumption.
                 intros ;apply P_id_a__U32_monotonic;assumption.
                 intros ;apply P_id_U51_monotonic;assumption.
                 intros ;apply P_id_isNatIListKind_monotonic;assumption.
                 intros ;apply P_id_U12_monotonic;assumption.
                 intros ;apply P_id_a__U52_monotonic;assumption.
                 intros ;apply P_id_a__isNat_monotonic;assumption.
                 intros ;apply P_id_U43_monotonic;assumption.
                 intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
                 intros ;apply P_id_a__U12_monotonic;assumption.
                 intros ;apply P_id_U31_monotonic;assumption.
                 intros ;apply P_id_a__length_monotonic;assumption.
                 intros ;apply P_id_a__U42_monotonic;assumption.
                 intros ;apply P_id_U53_monotonic;assumption.
                 intros ;apply P_id_and_monotonic;assumption.
                 intros ;apply P_id_cons_monotonic;assumption.
                 intros ;apply P_id_U11_monotonic;assumption.
                 intros ;apply P_id_a__U51_monotonic;assumption.
                 intros ;apply P_id_a__U22_monotonic;assumption.
                 intros ;apply P_id_U42_monotonic;assumption.
                 intros ;apply P_id_length_monotonic;assumption.
                 intros ;apply P_id_U22_monotonic;assumption.
                 intros ;apply P_id_s_monotonic;assumption.
                 intros ;apply P_id_a__U41_monotonic;assumption.
                 intros ;apply P_id_U52_monotonic;assumption.
                 intros ;apply P_id_isNatList_monotonic;assumption.
                 intros ;apply P_id_a__U53_monotonic;assumption.
                 intros ;apply P_id_a__U31_monotonic;assumption.
                 intros ;apply P_id_isNatIList_monotonic;assumption.
                 intros ;apply P_id_a__isNatKind_monotonic;assumption.
                 intros ;apply P_id_a__isNatList_monotonic;assumption.
                 intros ;apply P_id_U32_monotonic;assumption.
                 intros ;apply P_id_mark_monotonic;assumption.
                 intros ;apply P_id_a__U43_monotonic;assumption.
                 intros ;apply P_id_U61_monotonic;assumption.
                 intros ;apply P_id_isNat_monotonic;assumption.
                 intros ;apply P_id_a__zeros_bounded;assumption.
                 intros ;apply P_id_isNatKind_bounded;assumption.
                 intros ;apply P_id_a__isNatIList_bounded;assumption.
                 intros ;apply P_id_a__U21_bounded;assumption.
                 intros ;apply P_id_U41_bounded;assumption.
                 intros ;apply P_id_a__and_bounded;assumption.
                 intros ;apply P_id_a__U11_bounded;assumption.
                 intros ;apply P_id_U21_bounded;assumption.
                 intros ;apply P_id_a__U61_bounded;assumption.
                 intros ;apply P_id_a__U32_bounded;assumption.
                 intros ;apply P_id_U51_bounded;assumption.
                 intros ;apply P_id_isNatIListKind_bounded;assumption.
                 intros ;apply P_id_0_bounded;assumption.
                 intros ;apply P_id_U12_bounded;assumption.
                 intros ;apply P_id_a__U52_bounded;assumption.
                 intros ;apply P_id_a__isNat_bounded;assumption.
                 intros ;apply P_id_U43_bounded;assumption.
                 intros ;apply P_id_a__isNatIListKind_bounded;assumption.
                 intros ;apply P_id_a__U12_bounded;assumption.
                 intros ;apply P_id_U31_bounded;assumption.
                 intros ;apply P_id_a__length_bounded;assumption.
                 intros ;apply P_id_a__U42_bounded;assumption.
                 intros ;apply P_id_U53_bounded;assumption.
                 intros ;apply P_id_and_bounded;assumption.
                 intros ;apply P_id_cons_bounded;assumption.
                 intros ;apply P_id_U11_bounded;assumption.
                 intros ;apply P_id_a__U51_bounded;assumption.
                 intros ;apply P_id_a__U22_bounded;assumption.
                 intros ;apply P_id_U42_bounded;assumption.
                 intros ;apply P_id_length_bounded;assumption.
                 intros ;apply P_id_tt_bounded;assumption.
                 intros ;apply P_id_U22_bounded;assumption.
                 intros ;apply P_id_s_bounded;assumption.
                 intros ;apply P_id_a__U41_bounded;assumption.
                 intros ;apply P_id_U52_bounded;assumption.
                 intros ;apply P_id_nil_bounded;assumption.
                 intros ;apply P_id_zeros_bounded;assumption.
                 intros ;apply P_id_isNatList_bounded;assumption.
                 intros ;apply P_id_a__U53_bounded;assumption.
                 intros ;apply P_id_a__U31_bounded;assumption.
                 intros ;apply P_id_isNatIList_bounded;assumption.
                 intros ;apply P_id_a__isNatKind_bounded;assumption.
                 intros ;apply P_id_a__isNatList_bounded;assumption.
                 intros ;apply P_id_U32_bounded;assumption.
                 intros ;apply P_id_mark_bounded;assumption.
                 intros ;apply P_id_a__U43_bounded;assumption.
                 intros ;apply P_id_U61_bounded;assumption.
                 intros ;apply P_id_isNat_bounded;assumption.
                 apply rules_monotonic.
               Qed.
               
               Definition P_id_A__LENGTH (x11:Z) := 0.
               
               Definition P_id_A__U11 (x11:Z) (x12:Z) := 0.
               
               Definition P_id_A__U41 (x11:Z) (x12:Z) (x13:Z) := 0.
               
               Definition P_id_A__ISNATKIND (x11:Z) := 1* x11.
               
               Definition P_id_A__U22 (x11:Z) := 0.
               
               Definition P_id_A__U51 (x11:Z) (x12:Z) (x13:Z) := 0.
               
               Definition P_id_A__AND (x11:Z) (x12:Z) := 1* x12.
               
               Definition P_id_A__ISNATLIST (x11:Z) := 0.
               
               Definition P_id_A__U43 (x11:Z) := 0.
               
               Definition P_id_A__U31 (x11:Z) (x12:Z) := 0.
               
               Definition P_id_A__U53 (x11:Z) := 0.
               
               Definition P_id_MARK (x11:Z) := 1* x11.
               
               Definition P_id_A__U12 (x11:Z) := 0.
               
               Definition P_id_A__U42 (x11:Z) (x12:Z) := 0.
               
               Definition P_id_A__ZEROS  := 0.
               
               Definition P_id_A__ISNAT (x11:Z) := 0.
               
               Definition P_id_A__U52 (x11:Z) (x12:Z) := 0.
               
               Definition P_id_A__ISNATILISTKIND (x11:Z) := 1* x11.
               
               Definition P_id_A__U21 (x11:Z) (x12:Z) := 0.
               
               Definition P_id_A__ISNATILIST (x11:Z) := 0.
               
               Definition P_id_A__U32 (x11:Z) := 0.
               
               Definition P_id_A__U61 (x11:Z) (x12:Z) := 0.
               
               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__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__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__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__U22_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_A__U22 x12 <= P_id_A__U22 x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_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 x16 x12 x14 x13 x11 x15, 
                 (0 <= x16)/\ (x16 <= x15) ->
                  (0 <= x14)/\ (x14 <= x13) ->
                   (0 <= x12)/\ (x12 <= x11) ->
                    P_id_A__U51 x12 x14 x16 <= P_id_A__U51 x11 x13 x15.
               Proof.
                 intros x16 x15 x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 intros [H_5 H_4].
                 
                 cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                  (auto with zarith)||(repeat (translate_vars );prove_ineq ).
               Qed.
               
               Lemma P_id_A__AND_monotonic :
                forall x12 x14 x13 x11, 
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_A__AND x12 x14 <= P_id_A__AND x11 x13.
               Proof.
                 intros x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_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__U43_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_A__U43 x12 <= P_id_A__U43 x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_0].
                 
                 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__U53_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_A__U53 x12 <= P_id_A__U53 x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 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__U12_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_A__U12 x12 <= P_id_A__U12 x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_0].
                 
                 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 x12 x14 x13 x11, 
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_A__U42 x12 x14 <= P_id_A__U42 x11 x13.
               Proof.
                 intros x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_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_A__U52_monotonic :
                forall x12 x14 x13 x11, 
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_A__U52 x12 x14 <= P_id_A__U52 x11 x13.
               Proof.
                 intros x14 x13 x12 x11.
                 intros [H_1 H_0].
                 intros [H_3 H_2].
                 
                 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__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__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__U32_monotonic :
                forall x12 x11, 
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_A__U32 x12 <= P_id_A__U32 x11.
               Proof.
                 intros x12 x11.
                 intros [H_1 H_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 x14 x13 x11, 
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_A__U61 x12 x14 <= P_id_A__U61 x11 x13.
               Proof.
                 intros x14 x13 x12 x11.
                 intros [H_1 H_0].
                 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_isNatKind 
                  P_id_a__isNatIList P_id_a__U21 P_id_U41 P_id_a__and 
                  P_id_a__U11 P_id_U21 P_id_a__U61 P_id_a__U32 P_id_U51 
                  P_id_isNatIListKind P_id_0 P_id_U12 P_id_a__U52 
                  P_id_a__isNat P_id_U43 P_id_a__isNatIListKind P_id_a__U12 
                  P_id_U31 P_id_a__length P_id_a__U42 P_id_U53 P_id_and 
                  P_id_cons P_id_U11 P_id_a__U51 P_id_a__U22 P_id_U42 
                  P_id_length P_id_tt P_id_U22 P_id_s P_id_a__U41 P_id_U52 
                  P_id_nil P_id_zeros P_id_isNatList P_id_a__U53 P_id_a__U31 
                  P_id_isNatIList P_id_a__isNatKind P_id_a__isNatList 
                  P_id_U32 P_id_mark P_id_a__U43 P_id_U61 P_id_isNat 
                  P_id_A__LENGTH P_id_A__U11 P_id_A__U41 P_id_A__ISNATKIND 
                  P_id_A__U22 P_id_A__U51 P_id_A__AND P_id_A__ISNATLIST 
                  P_id_A__U43 P_id_A__U31 P_id_A__U53 P_id_MARK P_id_A__U12 
                  P_id_A__U42 P_id_A__ZEROS P_id_A__ISNAT P_id_A__U52 
                  P_id_A__ISNATILISTKIND P_id_A__U21 P_id_A__ISNATILIST 
                  P_id_A__U32 P_id_A__U61.
               
               Lemma marked_measure_equation :
                forall t, 
                 marked_measure t = match t with
                                      | (algebra.Alg.Term 
                                         algebra.F.id_a__length (x11::nil)) =>
                                       P_id_A__LENGTH (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__U41 (x13::x12::
                                         x11::nil)) =>
                                       P_id_A__U41 (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__U22 (x11::nil)) =>
                                       P_id_A__U22 (measure x11)
                                      | (algebra.Alg.Term 
                                         algebra.F.id_a__U51 (x13::x12::
                                         x11::nil)) =>
                                       P_id_A__U51 (measure x13) 
                                        (measure x12) (measure x11)
                                      | (algebra.Alg.Term 
                                         algebra.F.id_a__and (x12::x11::nil)) =>
                                       P_id_A__AND (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__U43 (x11::nil)) =>
                                       P_id_A__U43 (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__U53 (x11::nil)) =>
                                       P_id_A__U53 (measure x11)
                                      | (algebra.Alg.Term algebra.F.id_mark 
                                         (x11::nil)) =>
                                       P_id_MARK (measure x11)
                                      | (algebra.Alg.Term 
                                         algebra.F.id_a__U12 (x11::nil)) =>
                                       P_id_A__U12 (measure x11)
                                      | (algebra.Alg.Term 
                                         algebra.F.id_a__U42 (x12::x11::nil)) =>
                                       P_id_A__U42 (measure x12) 
                                        (measure x11)
                                      | (algebra.Alg.Term 
                                         algebra.F.id_a__zeros nil) =>
                                       P_id_A__ZEROS 
                                      | (algebra.Alg.Term 
                                         algebra.F.id_a__isNat (x11::nil)) =>
                                       P_id_A__ISNAT (measure x11)
                                      | (algebra.Alg.Term 
                                         algebra.F.id_a__U52 (x12::x11::nil)) =>
                                       P_id_A__U52 (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_a__U21 (x12::x11::nil)) =>
                                       P_id_A__U21 (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__U32 (x11::nil)) =>
                                       P_id_A__U32 (measure x11)
                                      | (algebra.Alg.Term 
                                         algebra.F.id_a__U61 (x12::x11::nil)) =>
                                       P_id_A__U61 (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_isNatKind_monotonic;assumption.
                 intros ;apply P_id_a__isNatIList_monotonic;assumption.
                 intros ;apply P_id_a__U21_monotonic;assumption.
                 intros ;apply P_id_U41_monotonic;assumption.
                 intros ;apply P_id_a__and_monotonic;assumption.
                 intros ;apply P_id_a__U11_monotonic;assumption.
                 intros ;apply P_id_U21_monotonic;assumption.
                 intros ;apply P_id_a__U61_monotonic;assumption.
                 intros ;apply P_id_a__U32_monotonic;assumption.
                 intros ;apply P_id_U51_monotonic;assumption.
                 intros ;apply P_id_isNatIListKind_monotonic;assumption.
                 intros ;apply P_id_U12_monotonic;assumption.
                 intros ;apply P_id_a__U52_monotonic;assumption.
                 intros ;apply P_id_a__isNat_monotonic;assumption.
                 intros ;apply P_id_U43_monotonic;assumption.
                 intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
                 intros ;apply P_id_a__U12_monotonic;assumption.
                 intros ;apply P_id_U31_monotonic;assumption.
                 intros ;apply P_id_a__length_monotonic;assumption.
                 intros ;apply P_id_a__U42_monotonic;assumption.
                 intros ;apply P_id_U53_monotonic;assumption.
                 intros ;apply P_id_and_monotonic;assumption.
                 intros ;apply P_id_cons_monotonic;assumption.
                 intros ;apply P_id_U11_monotonic;assumption.
                 intros ;apply P_id_a__U51_monotonic;assumption.
                 intros ;apply P_id_a__U22_monotonic;assumption.
                 intros ;apply P_id_U42_monotonic;assumption.
                 intros ;apply P_id_length_monotonic;assumption.
                 intros ;apply P_id_U22_monotonic;assumption.
                 intros ;apply P_id_s_monotonic;assumption.
                 intros ;apply P_id_a__U41_monotonic;assumption.
                 intros ;apply P_id_U52_monotonic;assumption.
                 intros ;apply P_id_isNatList_monotonic;assumption.
                 intros ;apply P_id_a__U53_monotonic;assumption.
                 intros ;apply P_id_a__U31_monotonic;assumption.
                 intros ;apply P_id_isNatIList_monotonic;assumption.
                 intros ;apply P_id_a__isNatKind_monotonic;assumption.
                 intros ;apply P_id_a__isNatList_monotonic;assumption.
                 intros ;apply P_id_U32_monotonic;assumption.
                 intros ;apply P_id_mark_monotonic;assumption.
                 intros ;apply P_id_a__U43_monotonic;assumption.
                 intros ;apply P_id_U61_monotonic;assumption.
                 intros ;apply P_id_isNat_monotonic;assumption.
                 intros ;apply P_id_a__zeros_bounded;assumption.
                 intros ;apply P_id_isNatKind_bounded;assumption.
                 intros ;apply P_id_a__isNatIList_bounded;assumption.
                 intros ;apply P_id_a__U21_bounded;assumption.
                 intros ;apply P_id_U41_bounded;assumption.
                 intros ;apply P_id_a__and_bounded;assumption.
                 intros ;apply P_id_a__U11_bounded;assumption.
                 intros ;apply P_id_U21_bounded;assumption.
                 intros ;apply P_id_a__U61_bounded;assumption.
                 intros ;apply P_id_a__U32_bounded;assumption.
                 intros ;apply P_id_U51_bounded;assumption.
                 intros ;apply P_id_isNatIListKind_bounded;assumption.
                 intros ;apply P_id_0_bounded;assumption.
                 intros ;apply P_id_U12_bounded;assumption.
                 intros ;apply P_id_a__U52_bounded;assumption.
                 intros ;apply P_id_a__isNat_bounded;assumption.
                 intros ;apply P_id_U43_bounded;assumption.
                 intros ;apply P_id_a__isNatIListKind_bounded;assumption.
                 intros ;apply P_id_a__U12_bounded;assumption.
                 intros ;apply P_id_U31_bounded;assumption.
                 intros ;apply P_id_a__length_bounded;assumption.
                 intros ;apply P_id_a__U42_bounded;assumption.
                 intros ;apply P_id_U53_bounded;assumption.
                 intros ;apply P_id_and_bounded;assumption.
                 intros ;apply P_id_cons_bounded;assumption.
                 intros ;apply P_id_U11_bounded;assumption.
                 intros ;apply P_id_a__U51_bounded;assumption.
                 intros ;apply P_id_a__U22_bounded;assumption.
                 intros ;apply P_id_U42_bounded;assumption.
                 intros ;apply P_id_length_bounded;assumption.
                 intros ;apply P_id_tt_bounded;assumption.
                 intros ;apply P_id_U22_bounded;assumption.
                 intros ;apply P_id_s_bounded;assumption.
                 intros ;apply P_id_a__U41_bounded;assumption.
                 intros ;apply P_id_U52_bounded;assumption.
                 intros ;apply P_id_nil_bounded;assumption.
                 intros ;apply P_id_zeros_bounded;assumption.
                 intros ;apply P_id_isNatList_bounded;assumption.
                 intros ;apply P_id_a__U53_bounded;assumption.
                 intros ;apply P_id_a__U31_bounded;assumption.
                 intros ;apply P_id_isNatIList_bounded;assumption.
                 intros ;apply P_id_a__isNatKind_bounded;assumption.
                 intros ;apply P_id_a__isNatList_bounded;assumption.
                 intros ;apply P_id_U32_bounded;assumption.
                 intros ;apply P_id_mark_bounded;assumption.
                 intros ;apply P_id_a__U43_bounded;assumption.
                 intros ;apply P_id_U61_bounded;assumption.
                 intros ;apply P_id_isNat_bounded;assumption.
                 apply rules_monotonic.
                 intros ;apply P_id_A__LENGTH_monotonic;assumption.
                 intros ;apply P_id_A__U11_monotonic;assumption.
                 intros ;apply P_id_A__U41_monotonic;assumption.
                 intros ;apply P_id_A__ISNATKIND_monotonic;assumption.
                 intros ;apply P_id_A__U22_monotonic;assumption.
                 intros ;apply P_id_A__U51_monotonic;assumption.
                 intros ;apply P_id_A__AND_monotonic;assumption.
                 intros ;apply P_id_A__ISNATLIST_monotonic;assumption.
                 intros ;apply P_id_A__U43_monotonic;assumption.
                 intros ;apply P_id_A__U31_monotonic;assumption.
                 intros ;apply P_id_A__U53_monotonic;assumption.
                 intros ;apply P_id_MARK_monotonic;assumption.
                 intros ;apply P_id_A__U12_monotonic;assumption.
                 intros ;apply P_id_A__U42_monotonic;assumption.
                 intros ;apply P_id_A__ISNAT_monotonic;assumption.
                 intros ;apply P_id_A__U52_monotonic;assumption.
                 intros ;apply P_id_A__ISNATILISTKIND_monotonic;assumption.
                 intros ;apply P_id_A__U21_monotonic;assumption.
                 intros ;apply P_id_A__ISNATILIST_monotonic;assumption.
                 intros ;apply P_id_A__U32_monotonic;assumption.
                 intros ;apply P_id_A__U61_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_strict_in_lt :
                Relation_Definitions.inclusion _ 
                 DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_in_le :
                Relation_Definitions.inclusion _ 
                 DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large  := 
                 WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large.wf
                 .
               
               
               Lemma wf :
                well_founded WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1.DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large_in_le;
                     econstructor eassumption])).
                 
                 apply wf_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_large.
               Qed.
              End WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_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  := 0.
              
              Definition P_id_isNatKind (x11:Z) := 0.
              
              Definition P_id_a__isNatIList (x11:Z) := 0.
              
              Definition P_id_a__U21 (x11:Z) (x12:Z) := 2.
              
              Definition P_id_U41 (x11:Z) (x12:Z) (x13:Z) := 0.
              
              Definition P_id_a__and (x11:Z) (x12:Z) := 2* x11 + 2* x12.
              
              Definition P_id_a__U11 (x11:Z) (x12:Z) := 1.
              
              Definition P_id_U21 (x11:Z) (x12:Z) := 2.
              
              Definition P_id_a__U61 (x11:Z) (x12:Z) := 0.
              
              Definition P_id_a__U32 (x11:Z) := 0.
              
              Definition P_id_U51 (x11:Z) (x12:Z) (x13:Z) := 0.
              
              Definition P_id_isNatIListKind (x11:Z) := 0.
              
              Definition P_id_0  := 0.
              
              Definition P_id_U12 (x11:Z) := 1 + 1* x11.
              
              Definition P_id_a__U52 (x11:Z) (x12:Z) := 0.
              
              Definition P_id_a__isNat (x11:Z) := 2.
              
              Definition P_id_U43 (x11:Z) := 0.
              
              Definition P_id_a__isNatIListKind (x11:Z) := 0.
              
              Definition P_id_a__U12 (x11:Z) := 1 + 1* x11.
              
              Definition P_id_U31 (x11:Z) (x12:Z) := 0.
              
              Definition P_id_a__length (x11:Z) := 0.
              
              Definition P_id_a__U42 (x11:Z) (x12:Z) := 0.
              
              Definition P_id_U53 (x11:Z) := 2* x11.
              
              Definition P_id_and (x11:Z) (x12:Z) := 2* x11 + 2* x12.
              
              Definition P_id_cons (x11:Z) (x12:Z) := 0.
              
              Definition P_id_U11 (x11:Z) (x12:Z) := 1.
              
              Definition P_id_a__U51 (x11:Z) (x12:Z) (x13:Z) := 0.
              
              Definition P_id_a__U22 (x11:Z) := 1* x11.
              
              Definition P_id_U42 (x11:Z) (x12:Z) := 0.
              
              Definition P_id_length (x11:Z) := 0.
              
              Definition P_id_tt  := 0.
              
              Definition P_id_U22 (x11:Z) := 1* x11.
              
              Definition P_id_s (x11:Z) := 1* x11.
              
              Definition P_id_a__U41 (x11:Z) (x12:Z) (x13:Z) := 0.
              
              Definition P_id_U52 (x11:Z) (x12:Z) := 0.
              
              Definition P_id_nil  := 1.
              
              Definition P_id_zeros  := 0.
              
              Definition P_id_isNatList (x11:Z) := 0.
              
              Definition P_id_a__U53 (x11:Z) := 2* x11.
              
              Definition P_id_a__U31 (x11:Z) (x12:Z) := 0.
              
              Definition P_id_isNatIList (x11:Z) := 0.
              
              Definition P_id_a__isNatKind (x11:Z) := 0.
              
              Definition P_id_a__isNatList (x11:Z) := 0.
              
              Definition P_id_U32 (x11:Z) := 0.
              
              Definition P_id_mark (x11:Z) := 1* x11.
              
              Definition P_id_a__U43 (x11:Z) := 0.
              
              Definition P_id_U61 (x11:Z) (x12:Z) := 0.
              
              Definition P_id_isNat (x11:Z) := 2.
              
              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__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__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_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_a__and_monotonic :
               forall x12 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_a__and x12 x14 <= P_id_a__and x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                
                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_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__U61_monotonic :
               forall x12 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_a__U61 x12 x14 <= P_id_a__U61 x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                
                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 x11, 
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_a__U32 x12 <= P_id_a__U32 x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U51_monotonic :
               forall x16 x12 x14 x13 x11 x15, 
                (0 <= x16)/\ (x16 <= x15) ->
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_U51 x12 x14 x16 <= P_id_U51 x11 x13 x15.
              Proof.
                intros x16 x15 x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                intros [H_5 H_4].
                
                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_U12_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->P_id_U12 x12 <= P_id_U12 x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_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 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_a__U52 x12 x14 <= P_id_a__U52 x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_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_U43_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->P_id_U43 x12 <= P_id_U43 x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_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__U12_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_a__U12 x12 <= P_id_a__U12 x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                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__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__U42_monotonic :
               forall x12 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_a__U42 x12 x14 <= P_id_a__U42 x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U53_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->P_id_U53 x12 <= P_id_U53 x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_and_monotonic :
               forall x12 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_and x12 x14 <= P_id_and x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                
                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_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__U51_monotonic :
               forall x16 x12 x14 x13 x11 x15, 
                (0 <= x16)/\ (x16 <= x15) ->
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_a__U51 x12 x14 x16 <= P_id_a__U51 x11 x13 x15.
              Proof.
                intros x16 x15 x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                intros [H_5 H_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 x11, 
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_a__U22 x12 <= P_id_a__U22 x11.
              Proof.
                intros x12 x11.
                intros [H_1 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 x12 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_U42 x12 x14 <= P_id_U42 x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                
                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_U22_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->P_id_U22 x12 <= P_id_U22 x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                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__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_U52_monotonic :
               forall x12 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_U52 x12 x14 <= P_id_U52 x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 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__U53_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_a__U53 x12 <= P_id_a__U53 x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                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_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__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__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_U32_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->P_id_U32 x12 <= P_id_U32 x11.
              Proof.
                intros x12 x11.
                intros [H_1 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__U43_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_a__U43 x12 <= P_id_a__U43 x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                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 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_U61 x12 x14 <= P_id_U61 x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                
                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__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_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__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_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_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_a__and_bounded :
               forall x12 x11, 
                (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__and x12 x11.
              Proof.
                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_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__U61_bounded :
               forall x12 x11, 
                (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U61 x12 x11.
              Proof.
                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 x11, (0 <= x11) ->0 <= P_id_a__U32 x11.
              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 x13 x11, 
                (0 <= x11) ->
                 (0 <= x12) ->(0 <= x13) ->0 <= P_id_U51 x13 x12 x11.
              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_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_U12_bounded :
               forall x11, (0 <= x11) ->0 <= P_id_U12 x11.
              Proof.
                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 x12 x11, 
                (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U52 x12 x11.
              Proof.
                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_U43_bounded :
               forall x11, (0 <= x11) ->0 <= P_id_U43 x11.
              Proof.
                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_a__U12_bounded :
               forall x11, (0 <= x11) ->0 <= P_id_a__U12 x11.
              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__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__U42_bounded :
               forall x12 x11, 
                (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U42 x12 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U53_bounded :
               forall x11, (0 <= x11) ->0 <= P_id_U53 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_and_bounded :
               forall x12 x11, 
                (0 <= x11) ->(0 <= x12) ->0 <= P_id_and x12 x11.
              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_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__U51_bounded :
               forall x12 x13 x11, 
                (0 <= x11) ->
                 (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U51 x13 x12 x11.
              Proof.
                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 x11, (0 <= x11) ->0 <= P_id_a__U22 x11.
              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 x11, 
                (0 <= x11) ->(0 <= x12) ->0 <= P_id_U42 x12 x11.
              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_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_U22_bounded :
               forall x11, (0 <= x11) ->0 <= P_id_U22 x11.
              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__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_U52_bounded :
               forall x12 x11, 
                (0 <= x11) ->(0 <= x12) ->0 <= P_id_U52 x12 x11.
              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_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_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__U53_bounded :
               forall x11, (0 <= x11) ->0 <= P_id_a__U53 x11.
              Proof.
                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_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.
              
              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_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_U32_bounded :
               forall x11, (0 <= x11) ->0 <= P_id_U32 x11.
              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__U43_bounded :
               forall x11, (0 <= x11) ->0 <= P_id_a__U43 x11.
              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 x12 x11, 
                (0 <= x11) ->(0 <= x12) ->0 <= P_id_U61 x12 x11.
              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.
              
              Definition measure  := 
                InterpZ.measure 0 P_id_a__zeros P_id_isNatKind 
                 P_id_a__isNatIList P_id_a__U21 P_id_U41 P_id_a__and 
                 P_id_a__U11 P_id_U21 P_id_a__U61 P_id_a__U32 P_id_U51 
                 P_id_isNatIListKind P_id_0 P_id_U12 P_id_a__U52 
                 P_id_a__isNat P_id_U43 P_id_a__isNatIListKind P_id_a__U12 
                 P_id_U31 P_id_a__length P_id_a__U42 P_id_U53 P_id_and 
                 P_id_cons P_id_U11 P_id_a__U51 P_id_a__U22 P_id_U42 
                 P_id_length P_id_tt P_id_U22 P_id_s P_id_a__U41 P_id_U52 
                 P_id_nil P_id_zeros P_id_isNatList P_id_a__U53 P_id_a__U31 
                 P_id_isNatIList P_id_a__isNatKind P_id_a__isNatList 
                 P_id_U32 P_id_mark P_id_a__U43 P_id_U61 P_id_isNat.
              
              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_isNatKind 
                                 (x11::nil)) =>
                               P_id_isNatKind (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__U21 (x12::
                                 x11::nil)) =>
                               P_id_a__U21 (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_a__and (x12::
                                 x11::nil)) =>
                               P_id_a__and (measure x12) (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_U21 (x12::
                                 x11::nil)) =>
                               P_id_U21 (measure x12) (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U61 (x12::
                                 x11::nil)) =>
                               P_id_a__U61 (measure x12) (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U32 
                                 (x11::nil)) =>
                               P_id_a__U32 (measure x11)
                              | (algebra.Alg.Term algebra.F.id_U51 (x13::
                                 x12::x11::nil)) =>
                               P_id_U51 (measure x13) (measure x12) 
                                (measure x11)
                              | (algebra.Alg.Term 
                                 algebra.F.id_isNatIListKind (x11::nil)) =>
                               P_id_isNatIListKind (measure x11)
                              | (algebra.Alg.Term algebra.F.id_0 nil) =>
                               P_id_0 
                              | (algebra.Alg.Term algebra.F.id_U12 
                                 (x11::nil)) =>
                               P_id_U12 (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U52 (x12::
                                 x11::nil)) =>
                               P_id_a__U52 (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_U43 
                                 (x11::nil)) =>
                               P_id_U43 (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__U12 
                                 (x11::nil)) =>
                               P_id_a__U12 (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__length 
                                 (x11::nil)) =>
                               P_id_a__length (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U42 (x12::
                                 x11::nil)) =>
                               P_id_a__U42 (measure x12) (measure x11)
                              | (algebra.Alg.Term algebra.F.id_U53 
                                 (x11::nil)) =>
                               P_id_U53 (measure x11)
                              | (algebra.Alg.Term algebra.F.id_and (x12::
                                 x11::nil)) =>
                               P_id_and (measure x12) (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_U11 (x12::
                                 x11::nil)) =>
                               P_id_U11 (measure x12) (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U51 (x13::
                                 x12::x11::nil)) =>
                               P_id_a__U51 (measure x13) (measure x12) 
                                (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U22 
                                 (x11::nil)) =>
                               P_id_a__U22 (measure x11)
                              | (algebra.Alg.Term algebra.F.id_U42 (x12::
                                 x11::nil)) =>
                               P_id_U42 (measure x12) (measure x11)
                              | (algebra.Alg.Term algebra.F.id_length 
                                 (x11::nil)) =>
                               P_id_length (measure x11)
                              | (algebra.Alg.Term algebra.F.id_tt nil) =>
                               P_id_tt 
                              | (algebra.Alg.Term algebra.F.id_U22 
                                 (x11::nil)) =>
                               P_id_U22 (measure x11)
                              | (algebra.Alg.Term algebra.F.id_s (x11::nil)) =>
                               P_id_s (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_U52 (x12::
                                 x11::nil)) =>
                               P_id_U52 (measure x12) (measure x11)
                              | (algebra.Alg.Term algebra.F.id_nil nil) =>
                               P_id_nil 
                              | (algebra.Alg.Term algebra.F.id_zeros nil) =>
                               P_id_zeros 
                              | (algebra.Alg.Term algebra.F.id_isNatList 
                                 (x11::nil)) =>
                               P_id_isNatList (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U53 
                                 (x11::nil)) =>
                               P_id_a__U53 (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_isNatIList 
                                 (x11::nil)) =>
                               P_id_isNatIList (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__isNatList 
                                 (x11::nil)) =>
                               P_id_a__isNatList (measure x11)
                              | (algebra.Alg.Term algebra.F.id_U32 
                                 (x11::nil)) =>
                               P_id_U32 (measure x11)
                              | (algebra.Alg.Term algebra.F.id_mark 
                                 (x11::nil)) =>
                               P_id_mark (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U43 
                                 (x11::nil)) =>
                               P_id_a__U43 (measure x11)
                              | (algebra.Alg.Term algebra.F.id_U61 (x12::
                                 x11::nil)) =>
                               P_id_U61 (measure x12) (measure x11)
                              | (algebra.Alg.Term algebra.F.id_isNat 
                                 (x11::nil)) =>
                               P_id_isNat (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_isNatKind_monotonic;assumption.
                intros ;apply P_id_a__isNatIList_monotonic;assumption.
                intros ;apply P_id_a__U21_monotonic;assumption.
                intros ;apply P_id_U41_monotonic;assumption.
                intros ;apply P_id_a__and_monotonic;assumption.
                intros ;apply P_id_a__U11_monotonic;assumption.
                intros ;apply P_id_U21_monotonic;assumption.
                intros ;apply P_id_a__U61_monotonic;assumption.
                intros ;apply P_id_a__U32_monotonic;assumption.
                intros ;apply P_id_U51_monotonic;assumption.
                intros ;apply P_id_isNatIListKind_monotonic;assumption.
                intros ;apply P_id_U12_monotonic;assumption.
                intros ;apply P_id_a__U52_monotonic;assumption.
                intros ;apply P_id_a__isNat_monotonic;assumption.
                intros ;apply P_id_U43_monotonic;assumption.
                intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
                intros ;apply P_id_a__U12_monotonic;assumption.
                intros ;apply P_id_U31_monotonic;assumption.
                intros ;apply P_id_a__length_monotonic;assumption.
                intros ;apply P_id_a__U42_monotonic;assumption.
                intros ;apply P_id_U53_monotonic;assumption.
                intros ;apply P_id_and_monotonic;assumption.
                intros ;apply P_id_cons_monotonic;assumption.
                intros ;apply P_id_U11_monotonic;assumption.
                intros ;apply P_id_a__U51_monotonic;assumption.
                intros ;apply P_id_a__U22_monotonic;assumption.
                intros ;apply P_id_U42_monotonic;assumption.
                intros ;apply P_id_length_monotonic;assumption.
                intros ;apply P_id_U22_monotonic;assumption.
                intros ;apply P_id_s_monotonic;assumption.
                intros ;apply P_id_a__U41_monotonic;assumption.
                intros ;apply P_id_U52_monotonic;assumption.
                intros ;apply P_id_isNatList_monotonic;assumption.
                intros ;apply P_id_a__U53_monotonic;assumption.
                intros ;apply P_id_a__U31_monotonic;assumption.
                intros ;apply P_id_isNatIList_monotonic;assumption.
                intros ;apply P_id_a__isNatKind_monotonic;assumption.
                intros ;apply P_id_a__isNatList_monotonic;assumption.
                intros ;apply P_id_U32_monotonic;assumption.
                intros ;apply P_id_mark_monotonic;assumption.
                intros ;apply P_id_a__U43_monotonic;assumption.
                intros ;apply P_id_U61_monotonic;assumption.
                intros ;apply P_id_isNat_monotonic;assumption.
                intros ;apply P_id_a__zeros_bounded;assumption.
                intros ;apply P_id_isNatKind_bounded;assumption.
                intros ;apply P_id_a__isNatIList_bounded;assumption.
                intros ;apply P_id_a__U21_bounded;assumption.
                intros ;apply P_id_U41_bounded;assumption.
                intros ;apply P_id_a__and_bounded;assumption.
                intros ;apply P_id_a__U11_bounded;assumption.
                intros ;apply P_id_U21_bounded;assumption.
                intros ;apply P_id_a__U61_bounded;assumption.
                intros ;apply P_id_a__U32_bounded;assumption.
                intros ;apply P_id_U51_bounded;assumption.
                intros ;apply P_id_isNatIListKind_bounded;assumption.
                intros ;apply P_id_0_bounded;assumption.
                intros ;apply P_id_U12_bounded;assumption.
                intros ;apply P_id_a__U52_bounded;assumption.
                intros ;apply P_id_a__isNat_bounded;assumption.
                intros ;apply P_id_U43_bounded;assumption.
                intros ;apply P_id_a__isNatIListKind_bounded;assumption.
                intros ;apply P_id_a__U12_bounded;assumption.
                intros ;apply P_id_U31_bounded;assumption.
                intros ;apply P_id_a__length_bounded;assumption.
                intros ;apply P_id_a__U42_bounded;assumption.
                intros ;apply P_id_U53_bounded;assumption.
                intros ;apply P_id_and_bounded;assumption.
                intros ;apply P_id_cons_bounded;assumption.
                intros ;apply P_id_U11_bounded;assumption.
                intros ;apply P_id_a__U51_bounded;assumption.
                intros ;apply P_id_a__U22_bounded;assumption.
                intros ;apply P_id_U42_bounded;assumption.
                intros ;apply P_id_length_bounded;assumption.
                intros ;apply P_id_tt_bounded;assumption.
                intros ;apply P_id_U22_bounded;assumption.
                intros ;apply P_id_s_bounded;assumption.
                intros ;apply P_id_a__U41_bounded;assumption.
                intros ;apply P_id_U52_bounded;assumption.
                intros ;apply P_id_nil_bounded;assumption.
                intros ;apply P_id_zeros_bounded;assumption.
                intros ;apply P_id_isNatList_bounded;assumption.
                intros ;apply P_id_a__U53_bounded;assumption.
                intros ;apply P_id_a__U31_bounded;assumption.
                intros ;apply P_id_isNatIList_bounded;assumption.
                intros ;apply P_id_a__isNatKind_bounded;assumption.
                intros ;apply P_id_a__isNatList_bounded;assumption.
                intros ;apply P_id_U32_bounded;assumption.
                intros ;apply P_id_mark_bounded;assumption.
                intros ;apply P_id_a__U43_bounded;assumption.
                intros ;apply P_id_U61_bounded;assumption.
                intros ;apply P_id_isNat_bounded;assumption.
                apply rules_monotonic.
              Qed.
              
              Definition P_id_A__LENGTH (x11:Z) := 0.
              
              Definition P_id_A__U11 (x11:Z) (x12:Z) := 0.
              
              Definition P_id_A__U41 (x11:Z) (x12:Z) (x13:Z) := 0.
              
              Definition P_id_A__ISNATKIND (x11:Z) := 0.
              
              Definition P_id_A__U22 (x11:Z) := 0.
              
              Definition P_id_A__U51 (x11:Z) (x12:Z) (x13:Z) := 0.
              
              Definition P_id_A__AND (x11:Z) (x12:Z) := 2* x12.
              
              Definition P_id_A__ISNATLIST (x11:Z) := 0.
              
              Definition P_id_A__U43 (x11:Z) := 0.
              
              Definition P_id_A__U31 (x11:Z) (x12:Z) := 0.
              
              Definition P_id_A__U53 (x11:Z) := 0.
              
              Definition P_id_MARK (x11:Z) := 2* x11.
              
              Definition P_id_A__U12 (x11:Z) := 0.
              
              Definition P_id_A__U42 (x11:Z) (x12:Z) := 0.
              
              Definition P_id_A__ZEROS  := 0.
              
              Definition P_id_A__ISNAT (x11:Z) := 0.
              
              Definition P_id_A__U52 (x11:Z) (x12:Z) := 0.
              
              Definition P_id_A__ISNATILISTKIND (x11:Z) := 0.
              
              Definition P_id_A__U21 (x11:Z) (x12:Z) := 0.
              
              Definition P_id_A__ISNATILIST (x11:Z) := 0.
              
              Definition P_id_A__U32 (x11:Z) := 0.
              
              Definition P_id_A__U61 (x11:Z) (x12:Z) := 0.
              
              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__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__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__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__U22_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_A__U22 x12 <= P_id_A__U22 x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_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 x16 x12 x14 x13 x11 x15, 
                (0 <= x16)/\ (x16 <= x15) ->
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_A__U51 x12 x14 x16 <= P_id_A__U51 x11 x13 x15.
              Proof.
                intros x16 x15 x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                intros [H_5 H_4].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_A__AND_monotonic :
               forall x12 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_A__AND x12 x14 <= P_id_A__AND x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_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__U43_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_A__U43 x12 <= P_id_A__U43 x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                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__U53_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_A__U53 x12 <= P_id_A__U53 x11.
              Proof.
                intros x12 x11.
                intros [H_1 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__U12_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_A__U12 x12 <= P_id_A__U12 x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                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 x12 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_A__U42 x12 x14 <= P_id_A__U42 x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_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_A__U52_monotonic :
               forall x12 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_A__U52 x12 x14 <= P_id_A__U52 x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                
                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__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__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__U32_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_A__U32 x12 <= P_id_A__U32 x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_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 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_A__U61 x12 x14 <= P_id_A__U61 x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                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_isNatKind 
                 P_id_a__isNatIList P_id_a__U21 P_id_U41 P_id_a__and 
                 P_id_a__U11 P_id_U21 P_id_a__U61 P_id_a__U32 P_id_U51 
                 P_id_isNatIListKind P_id_0 P_id_U12 P_id_a__U52 
                 P_id_a__isNat P_id_U43 P_id_a__isNatIListKind P_id_a__U12 
                 P_id_U31 P_id_a__length P_id_a__U42 P_id_U53 P_id_and 
                 P_id_cons P_id_U11 P_id_a__U51 P_id_a__U22 P_id_U42 
                 P_id_length P_id_tt P_id_U22 P_id_s P_id_a__U41 P_id_U52 
                 P_id_nil P_id_zeros P_id_isNatList P_id_a__U53 P_id_a__U31 
                 P_id_isNatIList P_id_a__isNatKind P_id_a__isNatList 
                 P_id_U32 P_id_mark P_id_a__U43 P_id_U61 P_id_isNat 
                 P_id_A__LENGTH P_id_A__U11 P_id_A__U41 P_id_A__ISNATKIND 
                 P_id_A__U22 P_id_A__U51 P_id_A__AND P_id_A__ISNATLIST 
                 P_id_A__U43 P_id_A__U31 P_id_A__U53 P_id_MARK P_id_A__U12 
                 P_id_A__U42 P_id_A__ZEROS P_id_A__ISNAT P_id_A__U52 
                 P_id_A__ISNATILISTKIND P_id_A__U21 P_id_A__ISNATILIST 
                 P_id_A__U32 P_id_A__U61.
              
              Lemma marked_measure_equation :
               forall t, 
                marked_measure t = match t with
                                     | (algebra.Alg.Term 
                                        algebra.F.id_a__length (x11::nil)) =>
                                      P_id_A__LENGTH (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__U41 
                                        (x13::x12::x11::nil)) =>
                                      P_id_A__U41 (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__U22 
                                        (x11::nil)) =>
                                      P_id_A__U22 (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_a__U51 
                                        (x13::x12::x11::nil)) =>
                                      P_id_A__U51 (measure x13) 
                                       (measure x12) (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_a__and 
                                        (x12::x11::nil)) =>
                                      P_id_A__AND (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__U43 
                                        (x11::nil)) =>
                                      P_id_A__U43 (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__U53 
                                        (x11::nil)) =>
                                      P_id_A__U53 (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_mark 
                                        (x11::nil)) =>
                                      P_id_MARK (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_a__U12 
                                        (x11::nil)) =>
                                      P_id_A__U12 (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_a__U42 
                                        (x12::x11::nil)) =>
                                      P_id_A__U42 (measure x12) (measure x11)
                                     | (algebra.Alg.Term 
                                        algebra.F.id_a__zeros nil) =>
                                      P_id_A__ZEROS 
                                     | (algebra.Alg.Term 
                                        algebra.F.id_a__isNat (x11::nil)) =>
                                      P_id_A__ISNAT (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_a__U52 
                                        (x12::x11::nil)) =>
                                      P_id_A__U52 (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_a__U21 
                                        (x12::x11::nil)) =>
                                      P_id_A__U21 (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__U32 
                                        (x11::nil)) =>
                                      P_id_A__U32 (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_a__U61 
                                        (x12::x11::nil)) =>
                                      P_id_A__U61 (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_isNatKind_monotonic;assumption.
                intros ;apply P_id_a__isNatIList_monotonic;assumption.
                intros ;apply P_id_a__U21_monotonic;assumption.
                intros ;apply P_id_U41_monotonic;assumption.
                intros ;apply P_id_a__and_monotonic;assumption.
                intros ;apply P_id_a__U11_monotonic;assumption.
                intros ;apply P_id_U21_monotonic;assumption.
                intros ;apply P_id_a__U61_monotonic;assumption.
                intros ;apply P_id_a__U32_monotonic;assumption.
                intros ;apply P_id_U51_monotonic;assumption.
                intros ;apply P_id_isNatIListKind_monotonic;assumption.
                intros ;apply P_id_U12_monotonic;assumption.
                intros ;apply P_id_a__U52_monotonic;assumption.
                intros ;apply P_id_a__isNat_monotonic;assumption.
                intros ;apply P_id_U43_monotonic;assumption.
                intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
                intros ;apply P_id_a__U12_monotonic;assumption.
                intros ;apply P_id_U31_monotonic;assumption.
                intros ;apply P_id_a__length_monotonic;assumption.
                intros ;apply P_id_a__U42_monotonic;assumption.
                intros ;apply P_id_U53_monotonic;assumption.
                intros ;apply P_id_and_monotonic;assumption.
                intros ;apply P_id_cons_monotonic;assumption.
                intros ;apply P_id_U11_monotonic;assumption.
                intros ;apply P_id_a__U51_monotonic;assumption.
                intros ;apply P_id_a__U22_monotonic;assumption.
                intros ;apply P_id_U42_monotonic;assumption.
                intros ;apply P_id_length_monotonic;assumption.
                intros ;apply P_id_U22_monotonic;assumption.
                intros ;apply P_id_s_monotonic;assumption.
                intros ;apply P_id_a__U41_monotonic;assumption.
                intros ;apply P_id_U52_monotonic;assumption.
                intros ;apply P_id_isNatList_monotonic;assumption.
                intros ;apply P_id_a__U53_monotonic;assumption.
                intros ;apply P_id_a__U31_monotonic;assumption.
                intros ;apply P_id_isNatIList_monotonic;assumption.
                intros ;apply P_id_a__isNatKind_monotonic;assumption.
                intros ;apply P_id_a__isNatList_monotonic;assumption.
                intros ;apply P_id_U32_monotonic;assumption.
                intros ;apply P_id_mark_monotonic;assumption.
                intros ;apply P_id_a__U43_monotonic;assumption.
                intros ;apply P_id_U61_monotonic;assumption.
                intros ;apply P_id_isNat_monotonic;assumption.
                intros ;apply P_id_a__zeros_bounded;assumption.
                intros ;apply P_id_isNatKind_bounded;assumption.
                intros ;apply P_id_a__isNatIList_bounded;assumption.
                intros ;apply P_id_a__U21_bounded;assumption.
                intros ;apply P_id_U41_bounded;assumption.
                intros ;apply P_id_a__and_bounded;assumption.
                intros ;apply P_id_a__U11_bounded;assumption.
                intros ;apply P_id_U21_bounded;assumption.
                intros ;apply P_id_a__U61_bounded;assumption.
                intros ;apply P_id_a__U32_bounded;assumption.
                intros ;apply P_id_U51_bounded;assumption.
                intros ;apply P_id_isNatIListKind_bounded;assumption.
                intros ;apply P_id_0_bounded;assumption.
                intros ;apply P_id_U12_bounded;assumption.
                intros ;apply P_id_a__U52_bounded;assumption.
                intros ;apply P_id_a__isNat_bounded;assumption.
                intros ;apply P_id_U43_bounded;assumption.
                intros ;apply P_id_a__isNatIListKind_bounded;assumption.
                intros ;apply P_id_a__U12_bounded;assumption.
                intros ;apply P_id_U31_bounded;assumption.
                intros ;apply P_id_a__length_bounded;assumption.
                intros ;apply P_id_a__U42_bounded;assumption.
                intros ;apply P_id_U53_bounded;assumption.
                intros ;apply P_id_and_bounded;assumption.
                intros ;apply P_id_cons_bounded;assumption.
                intros ;apply P_id_U11_bounded;assumption.
                intros ;apply P_id_a__U51_bounded;assumption.
                intros ;apply P_id_a__U22_bounded;assumption.
                intros ;apply P_id_U42_bounded;assumption.
                intros ;apply P_id_length_bounded;assumption.
                intros ;apply P_id_tt_bounded;assumption.
                intros ;apply P_id_U22_bounded;assumption.
                intros ;apply P_id_s_bounded;assumption.
                intros ;apply P_id_a__U41_bounded;assumption.
                intros ;apply P_id_U52_bounded;assumption.
                intros ;apply P_id_nil_bounded;assumption.
                intros ;apply P_id_zeros_bounded;assumption.
                intros ;apply P_id_isNatList_bounded;assumption.
                intros ;apply P_id_a__U53_bounded;assumption.
                intros ;apply P_id_a__U31_bounded;assumption.
                intros ;apply P_id_isNatIList_bounded;assumption.
                intros ;apply P_id_a__isNatKind_bounded;assumption.
                intros ;apply P_id_a__isNatList_bounded;assumption.
                intros ;apply P_id_U32_bounded;assumption.
                intros ;apply P_id_mark_bounded;assumption.
                intros ;apply P_id_a__U43_bounded;assumption.
                intros ;apply P_id_U61_bounded;assumption.
                intros ;apply P_id_isNat_bounded;assumption.
                apply rules_monotonic.
                intros ;apply P_id_A__LENGTH_monotonic;assumption.
                intros ;apply P_id_A__U11_monotonic;assumption.
                intros ;apply P_id_A__U41_monotonic;assumption.
                intros ;apply P_id_A__ISNATKIND_monotonic;assumption.
                intros ;apply P_id_A__U22_monotonic;assumption.
                intros ;apply P_id_A__U51_monotonic;assumption.
                intros ;apply P_id_A__AND_monotonic;assumption.
                intros ;apply P_id_A__ISNATLIST_monotonic;assumption.
                intros ;apply P_id_A__U43_monotonic;assumption.
                intros ;apply P_id_A__U31_monotonic;assumption.
                intros ;apply P_id_A__U53_monotonic;assumption.
                intros ;apply P_id_MARK_monotonic;assumption.
                intros ;apply P_id_A__U12_monotonic;assumption.
                intros ;apply P_id_A__U42_monotonic;assumption.
                intros ;apply P_id_A__ISNAT_monotonic;assumption.
                intros ;apply P_id_A__U52_monotonic;assumption.
                intros ;apply P_id_A__ISNATILISTKIND_monotonic;assumption.
                intros ;apply P_id_A__U21_monotonic;assumption.
                intros ;apply P_id_A__ISNATILIST_monotonic;assumption.
                intros ;apply P_id_A__U32_monotonic;assumption.
                intros ;apply P_id_A__U61_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_strict_in_lt :
               Relation_Definitions.inclusion _ 
                DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_in_le :
               Relation_Definitions.inclusion _ 
                DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large  := 
                WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large.wf
                .
              
              
              Lemma wf :
               well_founded WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large.DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large)
                       .
                clear x.
                intros x _ IHx IHx'.
                constructor.
                intros y H.
                
                destruct H;
                 (apply IHx';
                   apply DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large_in_le;
                    econstructor eassumption])).
                
                apply wf_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1_large.
              Qed.
             End WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1.
             
             Definition wf_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1  := 
               WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1.wf
               .
             
             
             Lemma acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1 :
              forall x y, 
               (DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1 
                 x y) ->
                Acc WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7.DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large 
                 x.
             Proof.
               intros x.
               pattern x.
               
               apply (@Acc_ind _ 
                       DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1.
             Qed.
             
             
             Inductive DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_non_scc_2  :
              algebra.Alg.term ->algebra.Alg.term ->Prop := 
                (* <a__isNat(s(V1_)),a__isNatKind(V1_)> *)
               | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_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_s (x1::nil)) x11) ->
                  DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_non_scc_2 
                   (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_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_non_scc_2 :
              forall x y, 
               (DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_non_scc_2 
                 x y) ->
                Acc WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7.DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_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__isNat(V1_)> *)
               | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_3 
                    (algebra.Alg.Term algebra.F.id_a__isNat (x1::nil)) 
                    (algebra.Alg.Term algebra.F.id_a__U21 (x12::x11::nil))
             .
             
             
             Module WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_3.
              Inductive DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_3_large  :
               algebra.Alg.term ->algebra.Alg.term ->Prop := 
                 (* <a__U21(tt,V1_),a__isNat(V1_)> *)
                | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_3_large 
                     (algebra.Alg.Term algebra.F.id_a__isNat (x1::nil)) 
                     (algebra.Alg.Term algebra.F.id_a__U21 (x12::x11::nil))
              .
              
              
              Inductive DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_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))
              .
              
              
              Module WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_3_large.
               Inductive DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_3_large_non_scc_1  :
                algebra.Alg.term ->algebra.Alg.term ->Prop := 
                  (* <a__U21(tt,V1_),a__isNat(V1_)> *)
                 | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_3_large_non_scc_1 
                      (algebra.Alg.Term algebra.F.id_a__isNat (x1::nil)) 
                      (algebra.Alg.Term algebra.F.id_a__U21 (x12::x11::nil))
               .
               
               
               Lemma acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_3_large_non_scc_1 :
                forall x y, 
                 (DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_3_large_non_scc_1 
                   x y) ->
                  Acc WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_3.DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_3.DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_3_large
                 .
               Proof.
                 constructor;intros _y _h;inversion _h;clear _h;subst;
                  (eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_3_large_non_scc_1;
                    econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
                  ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_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  := 1.
              
              Definition P_id_isNatKind (x11:Z) := 0.
              
              Definition P_id_a__isNatIList (x11:Z) := 1 + 3* x11.
              
              Definition P_id_a__U21 (x11:Z) (x12:Z) := 1.
              
              Definition P_id_U41 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
              
              Definition P_id_a__and (x11:Z) (x12:Z) := 1* x11 + 1* x12.
              
              Definition P_id_a__U11 (x11:Z) (x12:Z) := 1.
              
              Definition P_id_U21 (x11:Z) (x12:Z) := 1.
              
              Definition P_id_a__U61 (x11:Z) (x12:Z) := 1 + 1* x11 + 2* x12.
              
              Definition P_id_a__U32 (x11:Z) := 1* x11.
              
              Definition P_id_U51 (x11:Z) (x12:Z) (x13:Z) := 3* x13.
              
              Definition P_id_isNatIListKind (x11:Z) := 0.
              
              Definition P_id_0  := 0.
              
              Definition P_id_U12 (x11:Z) := 0.
              
              Definition P_id_a__U52 (x11:Z) (x12:Z) := 2* x12.
              
              Definition P_id_a__isNat (x11:Z) := 1.
              
              Definition P_id_U43 (x11:Z) := 0.
              
              Definition P_id_a__isNatIListKind (x11:Z) := 1.
              
              Definition P_id_a__U12 (x11:Z) := 1.
              
              Definition P_id_U31 (x11:Z) (x12:Z) := 1* x12.
              
              Definition P_id_a__length (x11:Z) := 1* x11.
              
              Definition P_id_a__U42 (x11:Z) (x12:Z) := 1.
              
              Definition P_id_U53 (x11:Z) := 1* x11.
              
              Definition P_id_and (x11:Z) (x12:Z) := 1* x11 + 1* x12.
              
              Definition P_id_cons (x11:Z) (x12:Z) := 1 + 3* x12.
              
              Definition P_id_U11 (x11:Z) (x12:Z) := 1.
              
              Definition P_id_a__U51 (x11:Z) (x12:Z) (x13:Z) := 3* x13.
              
              Definition P_id_a__U22 (x11:Z) := 1.
              
              Definition P_id_U42 (x11:Z) (x12:Z) := 0.
              
              Definition P_id_length (x11:Z) := 1* x11.
              
              Definition P_id_tt  := 1.
              
              Definition P_id_U22 (x11:Z) := 0.
              
              Definition P_id_s (x11:Z) := 1 + 1* x11.
              
              Definition P_id_a__U41 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
              
              Definition P_id_U52 (x11:Z) (x12:Z) := 2* x12.
              
              Definition P_id_nil  := 1.
              
              Definition P_id_zeros  := 0.
              
              Definition P_id_isNatList (x11:Z) := 1* x11.
              
              Definition P_id_a__U53 (x11:Z) := 1* x11.
              
              Definition P_id_a__U31 (x11:Z) (x12:Z) := 1* x12.
              
              Definition P_id_isNatIList (x11:Z) := 3* x11.
              
              Definition P_id_a__isNatKind (x11:Z) := 1.
              
              Definition P_id_a__isNatList (x11:Z) := 1* x11.
              
              Definition P_id_U32 (x11:Z) := 1* x11.
              
              Definition P_id_mark (x11:Z) := 1 + 1* x11.
              
              Definition P_id_a__U43 (x11:Z) := 1.
              
              Definition P_id_U61 (x11:Z) (x12:Z) := 1 + 1* x11 + 2* x12.
              
              Definition P_id_isNat (x11:Z) := 0.
              
              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__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__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_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_a__and_monotonic :
               forall x12 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_a__and x12 x14 <= P_id_a__and x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                
                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_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__U61_monotonic :
               forall x12 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_a__U61 x12 x14 <= P_id_a__U61 x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                
                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 x11, 
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_a__U32 x12 <= P_id_a__U32 x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U51_monotonic :
               forall x16 x12 x14 x13 x11 x15, 
                (0 <= x16)/\ (x16 <= x15) ->
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_U51 x12 x14 x16 <= P_id_U51 x11 x13 x15.
              Proof.
                intros x16 x15 x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                intros [H_5 H_4].
                
                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_U12_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->P_id_U12 x12 <= P_id_U12 x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_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 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_a__U52 x12 x14 <= P_id_a__U52 x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_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_U43_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->P_id_U43 x12 <= P_id_U43 x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_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__U12_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_a__U12 x12 <= P_id_a__U12 x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                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__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__U42_monotonic :
               forall x12 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_a__U42 x12 x14 <= P_id_a__U42 x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U53_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->P_id_U53 x12 <= P_id_U53 x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_and_monotonic :
               forall x12 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_and x12 x14 <= P_id_and x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                
                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_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__U51_monotonic :
               forall x16 x12 x14 x13 x11 x15, 
                (0 <= x16)/\ (x16 <= x15) ->
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_a__U51 x12 x14 x16 <= P_id_a__U51 x11 x13 x15.
              Proof.
                intros x16 x15 x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                intros [H_5 H_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 x11, 
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_a__U22 x12 <= P_id_a__U22 x11.
              Proof.
                intros x12 x11.
                intros [H_1 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 x12 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_U42 x12 x14 <= P_id_U42 x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                
                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_U22_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->P_id_U22 x12 <= P_id_U22 x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                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__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_U52_monotonic :
               forall x12 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_U52 x12 x14 <= P_id_U52 x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 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__U53_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_a__U53 x12 <= P_id_a__U53 x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                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_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__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__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_U32_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->P_id_U32 x12 <= P_id_U32 x11.
              Proof.
                intros x12 x11.
                intros [H_1 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__U43_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_a__U43 x12 <= P_id_a__U43 x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                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 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_U61 x12 x14 <= P_id_U61 x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                
                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__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_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__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_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_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_a__and_bounded :
               forall x12 x11, 
                (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__and x12 x11.
              Proof.
                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_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__U61_bounded :
               forall x12 x11, 
                (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U61 x12 x11.
              Proof.
                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 x11, (0 <= x11) ->0 <= P_id_a__U32 x11.
              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 x13 x11, 
                (0 <= x11) ->
                 (0 <= x12) ->(0 <= x13) ->0 <= P_id_U51 x13 x12 x11.
              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_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_U12_bounded :
               forall x11, (0 <= x11) ->0 <= P_id_U12 x11.
              Proof.
                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 x12 x11, 
                (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U52 x12 x11.
              Proof.
                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_U43_bounded :
               forall x11, (0 <= x11) ->0 <= P_id_U43 x11.
              Proof.
                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_a__U12_bounded :
               forall x11, (0 <= x11) ->0 <= P_id_a__U12 x11.
              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__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__U42_bounded :
               forall x12 x11, 
                (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U42 x12 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_U53_bounded :
               forall x11, (0 <= x11) ->0 <= P_id_U53 x11.
              Proof.
                intros .
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_and_bounded :
               forall x12 x11, 
                (0 <= x11) ->(0 <= x12) ->0 <= P_id_and x12 x11.
              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_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__U51_bounded :
               forall x12 x13 x11, 
                (0 <= x11) ->
                 (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U51 x13 x12 x11.
              Proof.
                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 x11, (0 <= x11) ->0 <= P_id_a__U22 x11.
              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 x11, 
                (0 <= x11) ->(0 <= x12) ->0 <= P_id_U42 x12 x11.
              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_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_U22_bounded :
               forall x11, (0 <= x11) ->0 <= P_id_U22 x11.
              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__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_U52_bounded :
               forall x12 x11, 
                (0 <= x11) ->(0 <= x12) ->0 <= P_id_U52 x12 x11.
              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_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_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__U53_bounded :
               forall x11, (0 <= x11) ->0 <= P_id_a__U53 x11.
              Proof.
                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_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.
              
              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_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_U32_bounded :
               forall x11, (0 <= x11) ->0 <= P_id_U32 x11.
              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__U43_bounded :
               forall x11, (0 <= x11) ->0 <= P_id_a__U43 x11.
              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 x12 x11, 
                (0 <= x11) ->(0 <= x12) ->0 <= P_id_U61 x12 x11.
              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.
              
              Definition measure  := 
                InterpZ.measure 0 P_id_a__zeros P_id_isNatKind 
                 P_id_a__isNatIList P_id_a__U21 P_id_U41 P_id_a__and 
                 P_id_a__U11 P_id_U21 P_id_a__U61 P_id_a__U32 P_id_U51 
                 P_id_isNatIListKind P_id_0 P_id_U12 P_id_a__U52 
                 P_id_a__isNat P_id_U43 P_id_a__isNatIListKind P_id_a__U12 
                 P_id_U31 P_id_a__length P_id_a__U42 P_id_U53 P_id_and 
                 P_id_cons P_id_U11 P_id_a__U51 P_id_a__U22 P_id_U42 
                 P_id_length P_id_tt P_id_U22 P_id_s P_id_a__U41 P_id_U52 
                 P_id_nil P_id_zeros P_id_isNatList P_id_a__U53 P_id_a__U31 
                 P_id_isNatIList P_id_a__isNatKind P_id_a__isNatList 
                 P_id_U32 P_id_mark P_id_a__U43 P_id_U61 P_id_isNat.
              
              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_isNatKind 
                                 (x11::nil)) =>
                               P_id_isNatKind (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__U21 (x12::
                                 x11::nil)) =>
                               P_id_a__U21 (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_a__and (x12::
                                 x11::nil)) =>
                               P_id_a__and (measure x12) (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_U21 (x12::
                                 x11::nil)) =>
                               P_id_U21 (measure x12) (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U61 (x12::
                                 x11::nil)) =>
                               P_id_a__U61 (measure x12) (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U32 
                                 (x11::nil)) =>
                               P_id_a__U32 (measure x11)
                              | (algebra.Alg.Term algebra.F.id_U51 (x13::
                                 x12::x11::nil)) =>
                               P_id_U51 (measure x13) (measure x12) 
                                (measure x11)
                              | (algebra.Alg.Term 
                                 algebra.F.id_isNatIListKind (x11::nil)) =>
                               P_id_isNatIListKind (measure x11)
                              | (algebra.Alg.Term algebra.F.id_0 nil) =>
                               P_id_0 
                              | (algebra.Alg.Term algebra.F.id_U12 
                                 (x11::nil)) =>
                               P_id_U12 (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U52 (x12::
                                 x11::nil)) =>
                               P_id_a__U52 (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_U43 
                                 (x11::nil)) =>
                               P_id_U43 (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__U12 
                                 (x11::nil)) =>
                               P_id_a__U12 (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__length 
                                 (x11::nil)) =>
                               P_id_a__length (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U42 (x12::
                                 x11::nil)) =>
                               P_id_a__U42 (measure x12) (measure x11)
                              | (algebra.Alg.Term algebra.F.id_U53 
                                 (x11::nil)) =>
                               P_id_U53 (measure x11)
                              | (algebra.Alg.Term algebra.F.id_and (x12::
                                 x11::nil)) =>
                               P_id_and (measure x12) (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_U11 (x12::
                                 x11::nil)) =>
                               P_id_U11 (measure x12) (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U51 (x13::
                                 x12::x11::nil)) =>
                               P_id_a__U51 (measure x13) (measure x12) 
                                (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U22 
                                 (x11::nil)) =>
                               P_id_a__U22 (measure x11)
                              | (algebra.Alg.Term algebra.F.id_U42 (x12::
                                 x11::nil)) =>
                               P_id_U42 (measure x12) (measure x11)
                              | (algebra.Alg.Term algebra.F.id_length 
                                 (x11::nil)) =>
                               P_id_length (measure x11)
                              | (algebra.Alg.Term algebra.F.id_tt nil) =>
                               P_id_tt 
                              | (algebra.Alg.Term algebra.F.id_U22 
                                 (x11::nil)) =>
                               P_id_U22 (measure x11)
                              | (algebra.Alg.Term algebra.F.id_s (x11::nil)) =>
                               P_id_s (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_U52 (x12::
                                 x11::nil)) =>
                               P_id_U52 (measure x12) (measure x11)
                              | (algebra.Alg.Term algebra.F.id_nil nil) =>
                               P_id_nil 
                              | (algebra.Alg.Term algebra.F.id_zeros nil) =>
                               P_id_zeros 
                              | (algebra.Alg.Term algebra.F.id_isNatList 
                                 (x11::nil)) =>
                               P_id_isNatList (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U53 
                                 (x11::nil)) =>
                               P_id_a__U53 (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_isNatIList 
                                 (x11::nil)) =>
                               P_id_isNatIList (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__isNatList 
                                 (x11::nil)) =>
                               P_id_a__isNatList (measure x11)
                              | (algebra.Alg.Term algebra.F.id_U32 
                                 (x11::nil)) =>
                               P_id_U32 (measure x11)
                              | (algebra.Alg.Term algebra.F.id_mark 
                                 (x11::nil)) =>
                               P_id_mark (measure x11)
                              | (algebra.Alg.Term algebra.F.id_a__U43 
                                 (x11::nil)) =>
                               P_id_a__U43 (measure x11)
                              | (algebra.Alg.Term algebra.F.id_U61 (x12::
                                 x11::nil)) =>
                               P_id_U61 (measure x12) (measure x11)
                              | (algebra.Alg.Term algebra.F.id_isNat 
                                 (x11::nil)) =>
                               P_id_isNat (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_isNatKind_monotonic;assumption.
                intros ;apply P_id_a__isNatIList_monotonic;assumption.
                intros ;apply P_id_a__U21_monotonic;assumption.
                intros ;apply P_id_U41_monotonic;assumption.
                intros ;apply P_id_a__and_monotonic;assumption.
                intros ;apply P_id_a__U11_monotonic;assumption.
                intros ;apply P_id_U21_monotonic;assumption.
                intros ;apply P_id_a__U61_monotonic;assumption.
                intros ;apply P_id_a__U32_monotonic;assumption.
                intros ;apply P_id_U51_monotonic;assumption.
                intros ;apply P_id_isNatIListKind_monotonic;assumption.
                intros ;apply P_id_U12_monotonic;assumption.
                intros ;apply P_id_a__U52_monotonic;assumption.
                intros ;apply P_id_a__isNat_monotonic;assumption.
                intros ;apply P_id_U43_monotonic;assumption.
                intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
                intros ;apply P_id_a__U12_monotonic;assumption.
                intros ;apply P_id_U31_monotonic;assumption.
                intros ;apply P_id_a__length_monotonic;assumption.
                intros ;apply P_id_a__U42_monotonic;assumption.
                intros ;apply P_id_U53_monotonic;assumption.
                intros ;apply P_id_and_monotonic;assumption.
                intros ;apply P_id_cons_monotonic;assumption.
                intros ;apply P_id_U11_monotonic;assumption.
                intros ;apply P_id_a__U51_monotonic;assumption.
                intros ;apply P_id_a__U22_monotonic;assumption.
                intros ;apply P_id_U42_monotonic;assumption.
                intros ;apply P_id_length_monotonic;assumption.
                intros ;apply P_id_U22_monotonic;assumption.
                intros ;apply P_id_s_monotonic;assumption.
                intros ;apply P_id_a__U41_monotonic;assumption.
                intros ;apply P_id_U52_monotonic;assumption.
                intros ;apply P_id_isNatList_monotonic;assumption.
                intros ;apply P_id_a__U53_monotonic;assumption.
                intros ;apply P_id_a__U31_monotonic;assumption.
                intros ;apply P_id_isNatIList_monotonic;assumption.
                intros ;apply P_id_a__isNatKind_monotonic;assumption.
                intros ;apply P_id_a__isNatList_monotonic;assumption.
                intros ;apply P_id_U32_monotonic;assumption.
                intros ;apply P_id_mark_monotonic;assumption.
                intros ;apply P_id_a__U43_monotonic;assumption.
                intros ;apply P_id_U61_monotonic;assumption.
                intros ;apply P_id_isNat_monotonic;assumption.
                intros ;apply P_id_a__zeros_bounded;assumption.
                intros ;apply P_id_isNatKind_bounded;assumption.
                intros ;apply P_id_a__isNatIList_bounded;assumption.
                intros ;apply P_id_a__U21_bounded;assumption.
                intros ;apply P_id_U41_bounded;assumption.
                intros ;apply P_id_a__and_bounded;assumption.
                intros ;apply P_id_a__U11_bounded;assumption.
                intros ;apply P_id_U21_bounded;assumption.
                intros ;apply P_id_a__U61_bounded;assumption.
                intros ;apply P_id_a__U32_bounded;assumption.
                intros ;apply P_id_U51_bounded;assumption.
                intros ;apply P_id_isNatIListKind_bounded;assumption.
                intros ;apply P_id_0_bounded;assumption.
                intros ;apply P_id_U12_bounded;assumption.
                intros ;apply P_id_a__U52_bounded;assumption.
                intros ;apply P_id_a__isNat_bounded;assumption.
                intros ;apply P_id_U43_bounded;assumption.
                intros ;apply P_id_a__isNatIListKind_bounded;assumption.
                intros ;apply P_id_a__U12_bounded;assumption.
                intros ;apply P_id_U31_bounded;assumption.
                intros ;apply P_id_a__length_bounded;assumption.
                intros ;apply P_id_a__U42_bounded;assumption.
                intros ;apply P_id_U53_bounded;assumption.
                intros ;apply P_id_and_bounded;assumption.
                intros ;apply P_id_cons_bounded;assumption.
                intros ;apply P_id_U11_bounded;assumption.
                intros ;apply P_id_a__U51_bounded;assumption.
                intros ;apply P_id_a__U22_bounded;assumption.
                intros ;apply P_id_U42_bounded;assumption.
                intros ;apply P_id_length_bounded;assumption.
                intros ;apply P_id_tt_bounded;assumption.
                intros ;apply P_id_U22_bounded;assumption.
                intros ;apply P_id_s_bounded;assumption.
                intros ;apply P_id_a__U41_bounded;assumption.
                intros ;apply P_id_U52_bounded;assumption.
                intros ;apply P_id_nil_bounded;assumption.
                intros ;apply P_id_zeros_bounded;assumption.
                intros ;apply P_id_isNatList_bounded;assumption.
                intros ;apply P_id_a__U53_bounded;assumption.
                intros ;apply P_id_a__U31_bounded;assumption.
                intros ;apply P_id_isNatIList_bounded;assumption.
                intros ;apply P_id_a__isNatKind_bounded;assumption.
                intros ;apply P_id_a__isNatList_bounded;assumption.
                intros ;apply P_id_U32_bounded;assumption.
                intros ;apply P_id_mark_bounded;assumption.
                intros ;apply P_id_a__U43_bounded;assumption.
                intros ;apply P_id_U61_bounded;assumption.
                intros ;apply P_id_isNat_bounded;assumption.
                apply rules_monotonic.
              Qed.
              
              Definition P_id_A__LENGTH (x11:Z) := 0.
              
              Definition P_id_A__U11 (x11:Z) (x12:Z) := 0.
              
              Definition P_id_A__U41 (x11:Z) (x12:Z) (x13:Z) := 0.
              
              Definition P_id_A__ISNATKIND (x11:Z) := 0.
              
              Definition P_id_A__U22 (x11:Z) := 0.
              
              Definition P_id_A__U51 (x11:Z) (x12:Z) (x13:Z) := 0.
              
              Definition P_id_A__AND (x11:Z) (x12:Z) := 0.
              
              Definition P_id_A__ISNATLIST (x11:Z) := 0.
              
              Definition P_id_A__U43 (x11:Z) := 0.
              
              Definition P_id_A__U31 (x11:Z) (x12:Z) := 0.
              
              Definition P_id_A__U53 (x11:Z) := 0.
              
              Definition P_id_MARK (x11:Z) := 0.
              
              Definition P_id_A__U12 (x11:Z) := 0.
              
              Definition P_id_A__U42 (x11:Z) (x12:Z) := 0.
              
              Definition P_id_A__ZEROS  := 0.
              
              Definition P_id_A__ISNAT (x11:Z) := 1* x11.
              
              Definition P_id_A__U52 (x11:Z) (x12:Z) := 0.
              
              Definition P_id_A__ISNATILISTKIND (x11:Z) := 0.
              
              Definition P_id_A__U21 (x11:Z) (x12:Z) := 1* x12.
              
              Definition P_id_A__ISNATILIST (x11:Z) := 0.
              
              Definition P_id_A__U32 (x11:Z) := 0.
              
              Definition P_id_A__U61 (x11:Z) (x12:Z) := 0.
              
              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__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__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__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__U22_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_A__U22 x12 <= P_id_A__U22 x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_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 x16 x12 x14 x13 x11 x15, 
                (0 <= x16)/\ (x16 <= x15) ->
                 (0 <= x14)/\ (x14 <= x13) ->
                  (0 <= x12)/\ (x12 <= x11) ->
                   P_id_A__U51 x12 x14 x16 <= P_id_A__U51 x11 x13 x15.
              Proof.
                intros x16 x15 x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                intros [H_5 H_4].
                
                cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
                 (auto with zarith)||(repeat (translate_vars );prove_ineq ).
              Qed.
              
              Lemma P_id_A__AND_monotonic :
               forall x12 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_A__AND x12 x14 <= P_id_A__AND x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_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__U43_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_A__U43 x12 <= P_id_A__U43 x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                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__U53_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_A__U53 x12 <= P_id_A__U53 x11.
              Proof.
                intros x12 x11.
                intros [H_1 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__U12_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_A__U12 x12 <= P_id_A__U12 x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_0].
                
                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 x12 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_A__U42 x12 x14 <= P_id_A__U42 x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_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_A__U52_monotonic :
               forall x12 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_A__U52 x12 x14 <= P_id_A__U52 x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                intros [H_3 H_2].
                
                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__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__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__U32_monotonic :
               forall x12 x11, 
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_A__U32 x12 <= P_id_A__U32 x11.
              Proof.
                intros x12 x11.
                intros [H_1 H_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 x14 x13 x11, 
                (0 <= x14)/\ (x14 <= x13) ->
                 (0 <= x12)/\ (x12 <= x11) ->
                  P_id_A__U61 x12 x14 <= P_id_A__U61 x11 x13.
              Proof.
                intros x14 x13 x12 x11.
                intros [H_1 H_0].
                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_isNatKind 
                 P_id_a__isNatIList P_id_a__U21 P_id_U41 P_id_a__and 
                 P_id_a__U11 P_id_U21 P_id_a__U61 P_id_a__U32 P_id_U51 
                 P_id_isNatIListKind P_id_0 P_id_U12 P_id_a__U52 
                 P_id_a__isNat P_id_U43 P_id_a__isNatIListKind P_id_a__U12 
                 P_id_U31 P_id_a__length P_id_a__U42 P_id_U53 P_id_and 
                 P_id_cons P_id_U11 P_id_a__U51 P_id_a__U22 P_id_U42 
                 P_id_length P_id_tt P_id_U22 P_id_s P_id_a__U41 P_id_U52 
                 P_id_nil P_id_zeros P_id_isNatList P_id_a__U53 P_id_a__U31 
                 P_id_isNatIList P_id_a__isNatKind P_id_a__isNatList 
                 P_id_U32 P_id_mark P_id_a__U43 P_id_U61 P_id_isNat 
                 P_id_A__LENGTH P_id_A__U11 P_id_A__U41 P_id_A__ISNATKIND 
                 P_id_A__U22 P_id_A__U51 P_id_A__AND P_id_A__ISNATLIST 
                 P_id_A__U43 P_id_A__U31 P_id_A__U53 P_id_MARK P_id_A__U12 
                 P_id_A__U42 P_id_A__ZEROS P_id_A__ISNAT P_id_A__U52 
                 P_id_A__ISNATILISTKIND P_id_A__U21 P_id_A__ISNATILIST 
                 P_id_A__U32 P_id_A__U61.
              
              Lemma marked_measure_equation :
               forall t, 
                marked_measure t = match t with
                                     | (algebra.Alg.Term 
                                        algebra.F.id_a__length (x11::nil)) =>
                                      P_id_A__LENGTH (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__U41 
                                        (x13::x12::x11::nil)) =>
                                      P_id_A__U41 (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__U22 
                                        (x11::nil)) =>
                                      P_id_A__U22 (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_a__U51 
                                        (x13::x12::x11::nil)) =>
                                      P_id_A__U51 (measure x13) 
                                       (measure x12) (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_a__and 
                                        (x12::x11::nil)) =>
                                      P_id_A__AND (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__U43 
                                        (x11::nil)) =>
                                      P_id_A__U43 (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__U53 
                                        (x11::nil)) =>
                                      P_id_A__U53 (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_mark 
                                        (x11::nil)) =>
                                      P_id_MARK (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_a__U12 
                                        (x11::nil)) =>
                                      P_id_A__U12 (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_a__U42 
                                        (x12::x11::nil)) =>
                                      P_id_A__U42 (measure x12) (measure x11)
                                     | (algebra.Alg.Term 
                                        algebra.F.id_a__zeros nil) =>
                                      P_id_A__ZEROS 
                                     | (algebra.Alg.Term 
                                        algebra.F.id_a__isNat (x11::nil)) =>
                                      P_id_A__ISNAT (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_a__U52 
                                        (x12::x11::nil)) =>
                                      P_id_A__U52 (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_a__U21 
                                        (x12::x11::nil)) =>
                                      P_id_A__U21 (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__U32 
                                        (x11::nil)) =>
                                      P_id_A__U32 (measure x11)
                                     | (algebra.Alg.Term algebra.F.id_a__U61 
                                        (x12::x11::nil)) =>
                                      P_id_A__U61 (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_isNatKind_monotonic;assumption.
                intros ;apply P_id_a__isNatIList_monotonic;assumption.
                intros ;apply P_id_a__U21_monotonic;assumption.
                intros ;apply P_id_U41_monotonic;assumption.
                intros ;apply P_id_a__and_monotonic;assumption.
                intros ;apply P_id_a__U11_monotonic;assumption.
                intros ;apply P_id_U21_monotonic;assumption.
                intros ;apply P_id_a__U61_monotonic;assumption.
                intros ;apply P_id_a__U32_monotonic;assumption.
                intros ;apply P_id_U51_monotonic;assumption.
                intros ;apply P_id_isNatIListKind_monotonic;assumption.
                intros ;apply P_id_U12_monotonic;assumption.
                intros ;apply P_id_a__U52_monotonic;assumption.
                intros ;apply P_id_a__isNat_monotonic;assumption.
                intros ;apply P_id_U43_monotonic;assumption.
                intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
                intros ;apply P_id_a__U12_monotonic;assumption.
                intros ;apply P_id_U31_monotonic;assumption.
                intros ;apply P_id_a__length_monotonic;assumption.
                intros ;apply P_id_a__U42_monotonic;assumption.
                intros ;apply P_id_U53_monotonic;assumption.
                intros ;apply P_id_and_monotonic;assumption.
                intros ;apply P_id_cons_monotonic;assumption.
                intros ;apply P_id_U11_monotonic;assumption.
                intros ;apply P_id_a__U51_monotonic;assumption.
                intros ;apply P_id_a__U22_monotonic;assumption.
                intros ;apply P_id_U42_monotonic;assumption.
                intros ;apply P_id_length_monotonic;assumption.
                intros ;apply P_id_U22_monotonic;assumption.
                intros ;apply P_id_s_monotonic;assumption.
                intros ;apply P_id_a__U41_monotonic;assumption.
                intros ;apply P_id_U52_monotonic;assumption.
                intros ;apply P_id_isNatList_monotonic;assumption.
                intros ;apply P_id_a__U53_monotonic;assumption.
                intros ;apply P_id_a__U31_monotonic;assumption.
                intros ;apply P_id_isNatIList_monotonic;assumption.
                intros ;apply P_id_a__isNatKind_monotonic;assumption.
                intros ;apply P_id_a__isNatList_monotonic;assumption.
                intros ;apply P_id_U32_monotonic;assumption.
                intros ;apply P_id_mark_monotonic;assumption.
                intros ;apply P_id_a__U43_monotonic;assumption.
                intros ;apply P_id_U61_monotonic;assumption.
                intros ;apply P_id_isNat_monotonic;assumption.
                intros ;apply P_id_a__zeros_bounded;assumption.
                intros ;apply P_id_isNatKind_bounded;assumption.
                intros ;apply P_id_a__isNatIList_bounded;assumption.
                intros ;apply P_id_a__U21_bounded;assumption.
                intros ;apply P_id_U41_bounded;assumption.
                intros ;apply P_id_a__and_bounded;assumption.
                intros ;apply P_id_a__U11_bounded;assumption.
                intros ;apply P_id_U21_bounded;assumption.
                intros ;apply P_id_a__U61_bounded;assumption.
                intros ;apply P_id_a__U32_bounded;assumption.
                intros ;apply P_id_U51_bounded;assumption.
                intros ;apply P_id_isNatIListKind_bounded;assumption.
                intros ;apply P_id_0_bounded;assumption.
                intros ;apply P_id_U12_bounded;assumption.
                intros ;apply P_id_a__U52_bounded;assumption.
                intros ;apply P_id_a__isNat_bounded;assumption.
                intros ;apply P_id_U43_bounded;assumption.
                intros ;apply P_id_a__isNatIListKind_bounded;assumption.
                intros ;apply P_id_a__U12_bounded;assumption.
                intros ;apply P_id_U31_bounded;assumption.
                intros ;apply P_id_a__length_bounded;assumption.
                intros ;apply P_id_a__U42_bounded;assumption.
                intros ;apply P_id_U53_bounded;assumption.
                intros ;apply P_id_and_bounded;assumption.
                intros ;apply P_id_cons_bounded;assumption.
                intros ;apply P_id_U11_bounded;assumption.
                intros ;apply P_id_a__U51_bounded;assumption.
                intros ;apply P_id_a__U22_bounded;assumption.
                intros ;apply P_id_U42_bounded;assumption.
                intros ;apply P_id_length_bounded;assumption.
                intros ;apply P_id_tt_bounded;assumption.
                intros ;apply P_id_U22_bounded;assumption.
                intros ;apply P_id_s_bounded;assumption.
                intros ;apply P_id_a__U41_bounded;assumption.
                intros ;apply P_id_U52_bounded;assumption.
                intros ;apply P_id_nil_bounded;assumption.
                intros ;apply P_id_zeros_bounded;assumption.
                intros ;apply P_id_isNatList_bounded;assumption.
                intros ;apply P_id_a__U53_bounded;assumption.
                intros ;apply P_id_a__U31_bounded;assumption.
                intros ;apply P_id_isNatIList_bounded;assumption.
                intros ;apply P_id_a__isNatKind_bounded;assumption.
                intros ;apply P_id_a__isNatList_bounded;assumption.
                intros ;apply P_id_U32_bounded;assumption.
                intros ;apply P_id_mark_bounded;assumption.
                intros ;apply P_id_a__U43_bounded;assumption.
                intros ;apply P_id_U61_bounded;assumption.
                intros ;apply P_id_isNat_bounded;assumption.
                apply rules_monotonic.
                intros ;apply P_id_A__LENGTH_monotonic;assumption.
                intros ;apply P_id_A__U11_monotonic;assumption.
                intros ;apply P_id_A__U41_monotonic;assumption.
                intros ;apply P_id_A__ISNATKIND_monotonic;assumption.
                intros ;apply P_id_A__U22_monotonic;assumption.
                intros ;apply P_id_A__U51_monotonic;assumption.
                intros ;apply P_id_A__AND_monotonic;assumption.
                intros ;apply P_id_A__ISNATLIST_monotonic;assumption.
                intros ;apply P_id_A__U43_monotonic;assumption.
                intros ;apply P_id_A__U31_monotonic;assumption.
                intros ;apply P_id_A__U53_monotonic;assumption.
                intros ;apply P_id_MARK_monotonic;assumption.
                intros ;apply P_id_A__U12_monotonic;assumption.
                intros ;apply P_id_A__U42_monotonic;assumption.
                intros ;apply P_id_A__ISNAT_monotonic;assumption.
                intros ;apply P_id_A__U52_monotonic;assumption.
                intros ;apply P_id_A__ISNATILISTKIND_monotonic;assumption.
                intros ;apply P_id_A__U21_monotonic;assumption.
                intros ;apply P_id_A__ISNATILIST_monotonic;assumption.
                intros ;apply P_id_A__U32_monotonic;assumption.
                intros ;apply P_id_A__U61_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_3_strict_in_lt :
               Relation_Definitions.inclusion _ 
                DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_3_large_in_le :
               Relation_Definitions.inclusion _ 
                DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_3_large  := 
                WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_3_large.wf
                .
              
              
              Lemma wf :
               well_founded WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large.DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_3_large)
                       .
                clear x.
                intros x _ IHx IHx'.
                constructor.
                intros y H.
                
                destruct H;
                 (apply IHx';
                   apply DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_3_large_in_le;
                    econstructor eassumption])).
                
                apply wf_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_3_large.
              Qed.
             End WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_3.
             
             Definition wf_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_3  := 
               WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_3.wf
               .
             
             
             Lemma acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_3 :
              forall x y, 
               (DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_3 
                 x y) ->
                Acc WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7.DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large 
                 x.
             Proof.
               intros x.
               pattern x.
               
               apply (@Acc_ind _ 
                       DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_3.
             Qed.
             
             
             Lemma wf :
              well_founded WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7.DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large
               .
             Proof.
               constructor;intros _y _h;inversion _h;clear _h;subst;
                (eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_non_scc_2;
                  econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
                ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_non_scc_1;
                   econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
                 ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_non_scc_0;
                    econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
                  ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_3;
                     
                     econstructor 
                     (eassumption)||(algebra.Alg_ext.star_refl' ))||
                   ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_2;
                      
                      econstructor 
                      (eassumption)||(algebra.Alg_ext.star_refl' ))||
                    ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_scc_1;
                       
                       econstructor 
                       (eassumption)||(algebra.Alg_ext.star_refl' ))||
                     ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_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_isNatKind (x11:Z) := 0.
            
            Definition P_id_a__isNatIList (x11:Z) := 0.
            
            Definition P_id_a__U21 (x11:Z) (x12:Z) := 1 + 2* x11.
            
            Definition P_id_U41 (x11:Z) (x12:Z) (x13:Z) := 0.
            
            Definition P_id_a__and (x11:Z) (x12:Z) := 2* x11 + 2* x12.
            
            Definition P_id_a__U11 (x11:Z) (x12:Z) := 0.
            
            Definition P_id_U21 (x11:Z) (x12:Z) := 1 + 2* x11.
            
            Definition P_id_a__U61 (x11:Z) (x12:Z) := 0.
            
            Definition P_id_a__U32 (x11:Z) := 0.
            
            Definition P_id_U51 (x11:Z) (x12:Z) (x13:Z) := 0.
            
            Definition P_id_isNatIListKind (x11:Z) := 0.
            
            Definition P_id_0  := 0.
            
            Definition P_id_U12 (x11:Z) := 1* x11.
            
            Definition P_id_a__U52 (x11:Z) (x12:Z) := 0.
            
            Definition P_id_a__isNat (x11:Z) := 1.
            
            Definition P_id_U43 (x11:Z) := 0.
            
            Definition P_id_a__isNatIListKind (x11:Z) := 0.
            
            Definition P_id_a__U12 (x11:Z) := 1* x11.
            
            Definition P_id_U31 (x11:Z) (x12:Z) := 0.
            
            Definition P_id_a__length (x11:Z) := 0.
            
            Definition P_id_a__U42 (x11:Z) (x12:Z) := 0.
            
            Definition P_id_U53 (x11:Z) := 2* x11.
            
            Definition P_id_and (x11:Z) (x12:Z) := 2* x11 + 2* x12.
            
            Definition P_id_cons (x11:Z) (x12:Z) := 0.
            
            Definition P_id_U11 (x11:Z) (x12:Z) := 0.
            
            Definition P_id_a__U51 (x11:Z) (x12:Z) (x13:Z) := 0.
            
            Definition P_id_a__U22 (x11:Z) := 1* x11.
            
            Definition P_id_U42 (x11:Z) (x12:Z) := 0.
            
            Definition P_id_length (x11:Z) := 0.
            
            Definition P_id_tt  := 0.
            
            Definition P_id_U22 (x11:Z) := 1* x11.
            
            Definition P_id_s (x11:Z) := 2* x11.
            
            Definition P_id_a__U41 (x11:Z) (x12:Z) (x13:Z) := 0.
            
            Definition P_id_U52 (x11:Z) (x12:Z) := 0.
            
            Definition P_id_nil  := 0.
            
            Definition P_id_zeros  := 0.
            
            Definition P_id_isNatList (x11:Z) := 0.
            
            Definition P_id_a__U53 (x11:Z) := 2* x11.
            
            Definition P_id_a__U31 (x11:Z) (x12:Z) := 0.
            
            Definition P_id_isNatIList (x11:Z) := 0.
            
            Definition P_id_a__isNatKind (x11:Z) := 0.
            
            Definition P_id_a__isNatList (x11:Z) := 0.
            
            Definition P_id_U32 (x11:Z) := 0.
            
            Definition P_id_mark (x11:Z) := 1* x11.
            
            Definition P_id_a__U43 (x11:Z) := 0.
            
            Definition P_id_U61 (x11:Z) (x12:Z) := 0.
            
            Definition P_id_isNat (x11:Z) := 1.
            
            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__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__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_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_a__and_monotonic :
             forall x12 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_a__and x12 x14 <= P_id_a__and x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              
              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_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__U61_monotonic :
             forall x12 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_a__U61 x12 x14 <= P_id_a__U61 x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              
              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 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_a__U32 x12 <= P_id_a__U32 x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U51_monotonic :
             forall x16 x12 x14 x13 x11 x15, 
              (0 <= x16)/\ (x16 <= x15) ->
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_U51 x12 x14 x16 <= P_id_U51 x11 x13 x15.
            Proof.
              intros x16 x15 x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              intros [H_5 H_4].
              
              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_U12_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_U12 x12 <= P_id_U12 x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_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 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_a__U52 x12 x14 <= P_id_a__U52 x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_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_U43_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_U43 x12 <= P_id_U43 x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_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__U12_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_a__U12 x12 <= P_id_a__U12 x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              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__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__U42_monotonic :
             forall x12 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_a__U42 x12 x14 <= P_id_a__U42 x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U53_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_U53 x12 <= P_id_U53 x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_and_monotonic :
             forall x12 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_and x12 x14 <= P_id_and x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              
              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_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__U51_monotonic :
             forall x16 x12 x14 x13 x11 x15, 
              (0 <= x16)/\ (x16 <= x15) ->
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_a__U51 x12 x14 x16 <= P_id_a__U51 x11 x13 x15.
            Proof.
              intros x16 x15 x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              intros [H_5 H_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 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_a__U22 x12 <= P_id_a__U22 x11.
            Proof.
              intros x12 x11.
              intros [H_1 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 x12 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_U42 x12 x14 <= P_id_U42 x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              
              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_U22_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_U22 x12 <= P_id_U22 x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              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__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_U52_monotonic :
             forall x12 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_U52 x12 x14 <= P_id_U52 x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 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__U53_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_a__U53 x12 <= P_id_a__U53 x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              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_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__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__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_U32_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_U32 x12 <= P_id_U32 x11.
            Proof.
              intros x12 x11.
              intros [H_1 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__U43_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_a__U43 x12 <= P_id_a__U43 x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              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 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_U61 x12 x14 <= P_id_U61 x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              
              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__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_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__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_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_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_a__and_bounded :
             forall x12 x11, 
              (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__and x12 x11.
            Proof.
              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_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__U61_bounded :
             forall x12 x11, 
              (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U61 x12 x11.
            Proof.
              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 x11, (0 <= x11) ->0 <= P_id_a__U32 x11.
            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 x13 x11, 
              (0 <= x11) ->
               (0 <= x12) ->(0 <= x13) ->0 <= P_id_U51 x13 x12 x11.
            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_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_U12_bounded :
             forall x11, (0 <= x11) ->0 <= P_id_U12 x11.
            Proof.
              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 x12 x11, 
              (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U52 x12 x11.
            Proof.
              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_U43_bounded :
             forall x11, (0 <= x11) ->0 <= P_id_U43 x11.
            Proof.
              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_a__U12_bounded :
             forall x11, (0 <= x11) ->0 <= P_id_a__U12 x11.
            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__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__U42_bounded :
             forall x12 x11, 
              (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U42 x12 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_U53_bounded :
             forall x11, (0 <= x11) ->0 <= P_id_U53 x11.
            Proof.
              intros .
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_and_bounded :
             forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_and x12 x11.
            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_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__U51_bounded :
             forall x12 x13 x11, 
              (0 <= x11) ->
               (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U51 x13 x12 x11.
            Proof.
              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 x11, (0 <= x11) ->0 <= P_id_a__U22 x11.
            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 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U42 x12 x11.
            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_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_U22_bounded :
             forall x11, (0 <= x11) ->0 <= P_id_U22 x11.
            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__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_U52_bounded :
             forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U52 x12 x11.
            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_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_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__U53_bounded :
             forall x11, (0 <= x11) ->0 <= P_id_a__U53 x11.
            Proof.
              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_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.
            
            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_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_U32_bounded :
             forall x11, (0 <= x11) ->0 <= P_id_U32 x11.
            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__U43_bounded :
             forall x11, (0 <= x11) ->0 <= P_id_a__U43 x11.
            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 x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U61 x12 x11.
            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.
            
            Definition measure  := 
              InterpZ.measure 0 P_id_a__zeros P_id_isNatKind 
               P_id_a__isNatIList P_id_a__U21 P_id_U41 P_id_a__and 
               P_id_a__U11 P_id_U21 P_id_a__U61 P_id_a__U32 P_id_U51 
               P_id_isNatIListKind P_id_0 P_id_U12 P_id_a__U52 P_id_a__isNat 
               P_id_U43 P_id_a__isNatIListKind P_id_a__U12 P_id_U31 
               P_id_a__length P_id_a__U42 P_id_U53 P_id_and P_id_cons 
               P_id_U11 P_id_a__U51 P_id_a__U22 P_id_U42 P_id_length 
               P_id_tt P_id_U22 P_id_s P_id_a__U41 P_id_U52 P_id_nil 
               P_id_zeros P_id_isNatList P_id_a__U53 P_id_a__U31 
               P_id_isNatIList P_id_a__isNatKind P_id_a__isNatList P_id_U32 
               P_id_mark P_id_a__U43 P_id_U61 P_id_isNat.
            
            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_isNatKind 
                               (x11::nil)) =>
                             P_id_isNatKind (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__U21 (x12::
                               x11::nil)) =>
                             P_id_a__U21 (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_a__and (x12::
                               x11::nil)) =>
                             P_id_a__and (measure x12) (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_U21 (x12::
                               x11::nil)) =>
                             P_id_U21 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U61 (x12::
                               x11::nil)) =>
                             P_id_a__U61 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U32 
                               (x11::nil)) =>
                             P_id_a__U32 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_U51 (x13::x12::
                               x11::nil)) =>
                             P_id_U51 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_isNatIListKind 
                               (x11::nil)) =>
                             P_id_isNatIListKind (measure x11)
                            | (algebra.Alg.Term algebra.F.id_0 nil) =>
                             P_id_0 
                            | (algebra.Alg.Term algebra.F.id_U12 (x11::nil)) =>
                             P_id_U12 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U52 (x12::
                               x11::nil)) =>
                             P_id_a__U52 (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_U43 (x11::nil)) =>
                             P_id_U43 (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__U12 
                               (x11::nil)) =>
                             P_id_a__U12 (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__length 
                               (x11::nil)) =>
                             P_id_a__length (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U42 (x12::
                               x11::nil)) =>
                             P_id_a__U42 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_U53 (x11::nil)) =>
                             P_id_U53 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_and (x12::
                               x11::nil)) =>
                             P_id_and (measure x12) (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_U11 (x12::
                               x11::nil)) =>
                             P_id_U11 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U51 (x13::
                               x12::x11::nil)) =>
                             P_id_a__U51 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U22 
                               (x11::nil)) =>
                             P_id_a__U22 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_U42 (x12::
                               x11::nil)) =>
                             P_id_U42 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_length 
                               (x11::nil)) =>
                             P_id_length (measure x11)
                            | (algebra.Alg.Term algebra.F.id_tt nil) =>
                             P_id_tt 
                            | (algebra.Alg.Term algebra.F.id_U22 (x11::nil)) =>
                             P_id_U22 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_s (x11::nil)) =>
                             P_id_s (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_U52 (x12::
                               x11::nil)) =>
                             P_id_U52 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_nil nil) =>
                             P_id_nil 
                            | (algebra.Alg.Term algebra.F.id_zeros nil) =>
                             P_id_zeros 
                            | (algebra.Alg.Term algebra.F.id_isNatList 
                               (x11::nil)) =>
                             P_id_isNatList (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U53 
                               (x11::nil)) =>
                             P_id_a__U53 (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_isNatIList 
                               (x11::nil)) =>
                             P_id_isNatIList (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__isNatList 
                               (x11::nil)) =>
                             P_id_a__isNatList (measure x11)
                            | (algebra.Alg.Term algebra.F.id_U32 (x11::nil)) =>
                             P_id_U32 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_mark (x11::nil)) =>
                             P_id_mark (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U43 
                               (x11::nil)) =>
                             P_id_a__U43 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_U61 (x12::
                               x11::nil)) =>
                             P_id_U61 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_isNat 
                               (x11::nil)) =>
                             P_id_isNat (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_isNatKind_monotonic;assumption.
              intros ;apply P_id_a__isNatIList_monotonic;assumption.
              intros ;apply P_id_a__U21_monotonic;assumption.
              intros ;apply P_id_U41_monotonic;assumption.
              intros ;apply P_id_a__and_monotonic;assumption.
              intros ;apply P_id_a__U11_monotonic;assumption.
              intros ;apply P_id_U21_monotonic;assumption.
              intros ;apply P_id_a__U61_monotonic;assumption.
              intros ;apply P_id_a__U32_monotonic;assumption.
              intros ;apply P_id_U51_monotonic;assumption.
              intros ;apply P_id_isNatIListKind_monotonic;assumption.
              intros ;apply P_id_U12_monotonic;assumption.
              intros ;apply P_id_a__U52_monotonic;assumption.
              intros ;apply P_id_a__isNat_monotonic;assumption.
              intros ;apply P_id_U43_monotonic;assumption.
              intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
              intros ;apply P_id_a__U12_monotonic;assumption.
              intros ;apply P_id_U31_monotonic;assumption.
              intros ;apply P_id_a__length_monotonic;assumption.
              intros ;apply P_id_a__U42_monotonic;assumption.
              intros ;apply P_id_U53_monotonic;assumption.
              intros ;apply P_id_and_monotonic;assumption.
              intros ;apply P_id_cons_monotonic;assumption.
              intros ;apply P_id_U11_monotonic;assumption.
              intros ;apply P_id_a__U51_monotonic;assumption.
              intros ;apply P_id_a__U22_monotonic;assumption.
              intros ;apply P_id_U42_monotonic;assumption.
              intros ;apply P_id_length_monotonic;assumption.
              intros ;apply P_id_U22_monotonic;assumption.
              intros ;apply P_id_s_monotonic;assumption.
              intros ;apply P_id_a__U41_monotonic;assumption.
              intros ;apply P_id_U52_monotonic;assumption.
              intros ;apply P_id_isNatList_monotonic;assumption.
              intros ;apply P_id_a__U53_monotonic;assumption.
              intros ;apply P_id_a__U31_monotonic;assumption.
              intros ;apply P_id_isNatIList_monotonic;assumption.
              intros ;apply P_id_a__isNatKind_monotonic;assumption.
              intros ;apply P_id_a__isNatList_monotonic;assumption.
              intros ;apply P_id_U32_monotonic;assumption.
              intros ;apply P_id_mark_monotonic;assumption.
              intros ;apply P_id_a__U43_monotonic;assumption.
              intros ;apply P_id_U61_monotonic;assumption.
              intros ;apply P_id_isNat_monotonic;assumption.
              intros ;apply P_id_a__zeros_bounded;assumption.
              intros ;apply P_id_isNatKind_bounded;assumption.
              intros ;apply P_id_a__isNatIList_bounded;assumption.
              intros ;apply P_id_a__U21_bounded;assumption.
              intros ;apply P_id_U41_bounded;assumption.
              intros ;apply P_id_a__and_bounded;assumption.
              intros ;apply P_id_a__U11_bounded;assumption.
              intros ;apply P_id_U21_bounded;assumption.
              intros ;apply P_id_a__U61_bounded;assumption.
              intros ;apply P_id_a__U32_bounded;assumption.
              intros ;apply P_id_U51_bounded;assumption.
              intros ;apply P_id_isNatIListKind_bounded;assumption.
              intros ;apply P_id_0_bounded;assumption.
              intros ;apply P_id_U12_bounded;assumption.
              intros ;apply P_id_a__U52_bounded;assumption.
              intros ;apply P_id_a__isNat_bounded;assumption.
              intros ;apply P_id_U43_bounded;assumption.
              intros ;apply P_id_a__isNatIListKind_bounded;assumption.
              intros ;apply P_id_a__U12_bounded;assumption.
              intros ;apply P_id_U31_bounded;assumption.
              intros ;apply P_id_a__length_bounded;assumption.
              intros ;apply P_id_a__U42_bounded;assumption.
              intros ;apply P_id_U53_bounded;assumption.
              intros ;apply P_id_and_bounded;assumption.
              intros ;apply P_id_cons_bounded;assumption.
              intros ;apply P_id_U11_bounded;assumption.
              intros ;apply P_id_a__U51_bounded;assumption.
              intros ;apply P_id_a__U22_bounded;assumption.
              intros ;apply P_id_U42_bounded;assumption.
              intros ;apply P_id_length_bounded;assumption.
              intros ;apply P_id_tt_bounded;assumption.
              intros ;apply P_id_U22_bounded;assumption.
              intros ;apply P_id_s_bounded;assumption.
              intros ;apply P_id_a__U41_bounded;assumption.
              intros ;apply P_id_U52_bounded;assumption.
              intros ;apply P_id_nil_bounded;assumption.
              intros ;apply P_id_zeros_bounded;assumption.
              intros ;apply P_id_isNatList_bounded;assumption.
              intros ;apply P_id_a__U53_bounded;assumption.
              intros ;apply P_id_a__U31_bounded;assumption.
              intros ;apply P_id_isNatIList_bounded;assumption.
              intros ;apply P_id_a__isNatKind_bounded;assumption.
              intros ;apply P_id_a__isNatList_bounded;assumption.
              intros ;apply P_id_U32_bounded;assumption.
              intros ;apply P_id_mark_bounded;assumption.
              intros ;apply P_id_a__U43_bounded;assumption.
              intros ;apply P_id_U61_bounded;assumption.
              intros ;apply P_id_isNat_bounded;assumption.
              apply rules_monotonic.
            Qed.
            
            Definition P_id_A__LENGTH (x11:Z) := 0.
            
            Definition P_id_A__U11 (x11:Z) (x12:Z) := 0.
            
            Definition P_id_A__U41 (x11:Z) (x12:Z) (x13:Z) := 0.
            
            Definition P_id_A__ISNATKIND (x11:Z) := 0.
            
            Definition P_id_A__U22 (x11:Z) := 0.
            
            Definition P_id_A__U51 (x11:Z) (x12:Z) (x13:Z) := 0.
            
            Definition P_id_A__AND (x11:Z) (x12:Z) := 1* x12.
            
            Definition P_id_A__ISNATLIST (x11:Z) := 0.
            
            Definition P_id_A__U43 (x11:Z) := 0.
            
            Definition P_id_A__U31 (x11:Z) (x12:Z) := 0.
            
            Definition P_id_A__U53 (x11:Z) := 0.
            
            Definition P_id_MARK (x11:Z) := 1* x11.
            
            Definition P_id_A__U12 (x11:Z) := 0.
            
            Definition P_id_A__U42 (x11:Z) (x12:Z) := 0.
            
            Definition P_id_A__ZEROS  := 0.
            
            Definition P_id_A__ISNAT (x11:Z) := 0.
            
            Definition P_id_A__U52 (x11:Z) (x12:Z) := 0.
            
            Definition P_id_A__ISNATILISTKIND (x11:Z) := 0.
            
            Definition P_id_A__U21 (x11:Z) (x12:Z) := 0.
            
            Definition P_id_A__ISNATILIST (x11:Z) := 0.
            
            Definition P_id_A__U32 (x11:Z) := 0.
            
            Definition P_id_A__U61 (x11:Z) (x12:Z) := 0.
            
            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__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__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__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__U22_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_A__U22 x12 <= P_id_A__U22 x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_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 x16 x12 x14 x13 x11 x15, 
              (0 <= x16)/\ (x16 <= x15) ->
               (0 <= x14)/\ (x14 <= x13) ->
                (0 <= x12)/\ (x12 <= x11) ->
                 P_id_A__U51 x12 x14 x16 <= P_id_A__U51 x11 x13 x15.
            Proof.
              intros x16 x15 x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              intros [H_5 H_4].
              
              cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
               (auto with zarith)||(repeat (translate_vars );prove_ineq ).
            Qed.
            
            Lemma P_id_A__AND_monotonic :
             forall x12 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_A__AND x12 x14 <= P_id_A__AND x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_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__U43_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_A__U43 x12 <= P_id_A__U43 x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              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__U53_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_A__U53 x12 <= P_id_A__U53 x11.
            Proof.
              intros x12 x11.
              intros [H_1 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__U12_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_A__U12 x12 <= P_id_A__U12 x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_0].
              
              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 x12 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_A__U42 x12 x14 <= P_id_A__U42 x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_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_A__U52_monotonic :
             forall x12 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_A__U52 x12 x14 <= P_id_A__U52 x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              intros [H_3 H_2].
              
              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__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__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__U32_monotonic :
             forall x12 x11, 
              (0 <= x12)/\ (x12 <= x11) ->P_id_A__U32 x12 <= P_id_A__U32 x11.
            Proof.
              intros x12 x11.
              intros [H_1 H_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 x14 x13 x11, 
              (0 <= x14)/\ (x14 <= x13) ->
               (0 <= x12)/\ (x12 <= x11) ->
                P_id_A__U61 x12 x14 <= P_id_A__U61 x11 x13.
            Proof.
              intros x14 x13 x12 x11.
              intros [H_1 H_0].
              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_isNatKind 
               P_id_a__isNatIList P_id_a__U21 P_id_U41 P_id_a__and 
               P_id_a__U11 P_id_U21 P_id_a__U61 P_id_a__U32 P_id_U51 
               P_id_isNatIListKind P_id_0 P_id_U12 P_id_a__U52 P_id_a__isNat 
               P_id_U43 P_id_a__isNatIListKind P_id_a__U12 P_id_U31 
               P_id_a__length P_id_a__U42 P_id_U53 P_id_and P_id_cons 
               P_id_U11 P_id_a__U51 P_id_a__U22 P_id_U42 P_id_length 
               P_id_tt P_id_U22 P_id_s P_id_a__U41 P_id_U52 P_id_nil 
               P_id_zeros P_id_isNatList P_id_a__U53 P_id_a__U31 
               P_id_isNatIList P_id_a__isNatKind P_id_a__isNatList P_id_U32 
               P_id_mark P_id_a__U43 P_id_U61 P_id_isNat P_id_A__LENGTH 
               P_id_A__U11 P_id_A__U41 P_id_A__ISNATKIND P_id_A__U22 
               P_id_A__U51 P_id_A__AND P_id_A__ISNATLIST P_id_A__U43 
               P_id_A__U31 P_id_A__U53 P_id_MARK P_id_A__U12 P_id_A__U42 
               P_id_A__ZEROS P_id_A__ISNAT P_id_A__U52 
               P_id_A__ISNATILISTKIND P_id_A__U21 P_id_A__ISNATILIST 
               P_id_A__U32 P_id_A__U61.
            
            Lemma marked_measure_equation :
             forall t, 
              marked_measure t = match t with
                                   | (algebra.Alg.Term 
                                      algebra.F.id_a__length (x11::nil)) =>
                                    P_id_A__LENGTH (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__U41 
                                      (x13::x12::x11::nil)) =>
                                    P_id_A__U41 (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__U22 
                                      (x11::nil)) =>
                                    P_id_A__U22 (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U51 
                                      (x13::x12::x11::nil)) =>
                                    P_id_A__U51 (measure x13) (measure x12) 
                                     (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__and 
                                      (x12::x11::nil)) =>
                                    P_id_A__AND (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__U43 
                                      (x11::nil)) =>
                                    P_id_A__U43 (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__U53 
                                      (x11::nil)) =>
                                    P_id_A__U53 (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_mark 
                                      (x11::nil)) =>
                                    P_id_MARK (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U12 
                                      (x11::nil)) =>
                                    P_id_A__U12 (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U42 
                                      (x12::x11::nil)) =>
                                    P_id_A__U42 (measure x12) (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__zeros 
                                      nil) =>
                                    P_id_A__ZEROS 
                                   | (algebra.Alg.Term algebra.F.id_a__isNat 
                                      (x11::nil)) =>
                                    P_id_A__ISNAT (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U52 
                                      (x12::x11::nil)) =>
                                    P_id_A__U52 (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_a__U21 
                                      (x12::x11::nil)) =>
                                    P_id_A__U21 (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__U32 
                                      (x11::nil)) =>
                                    P_id_A__U32 (measure x11)
                                   | (algebra.Alg.Term algebra.F.id_a__U61 
                                      (x12::x11::nil)) =>
                                    P_id_A__U61 (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_isNatKind_monotonic;assumption.
              intros ;apply P_id_a__isNatIList_monotonic;assumption.
              intros ;apply P_id_a__U21_monotonic;assumption.
              intros ;apply P_id_U41_monotonic;assumption.
              intros ;apply P_id_a__and_monotonic;assumption.
              intros ;apply P_id_a__U11_monotonic;assumption.
              intros ;apply P_id_U21_monotonic;assumption.
              intros ;apply P_id_a__U61_monotonic;assumption.
              intros ;apply P_id_a__U32_monotonic;assumption.
              intros ;apply P_id_U51_monotonic;assumption.
              intros ;apply P_id_isNatIListKind_monotonic;assumption.
              intros ;apply P_id_U12_monotonic;assumption.
              intros ;apply P_id_a__U52_monotonic;assumption.
              intros ;apply P_id_a__isNat_monotonic;assumption.
              intros ;apply P_id_U43_monotonic;assumption.
              intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
              intros ;apply P_id_a__U12_monotonic;assumption.
              intros ;apply P_id_U31_monotonic;assumption.
              intros ;apply P_id_a__length_monotonic;assumption.
              intros ;apply P_id_a__U42_monotonic;assumption.
              intros ;apply P_id_U53_monotonic;assumption.
              intros ;apply P_id_and_monotonic;assumption.
              intros ;apply P_id_cons_monotonic;assumption.
              intros ;apply P_id_U11_monotonic;assumption.
              intros ;apply P_id_a__U51_monotonic;assumption.
              intros ;apply P_id_a__U22_monotonic;assumption.
              intros ;apply P_id_U42_monotonic;assumption.
              intros ;apply P_id_length_monotonic;assumption.
              intros ;apply P_id_U22_monotonic;assumption.
              intros ;apply P_id_s_monotonic;assumption.
              intros ;apply P_id_a__U41_monotonic;assumption.
              intros ;apply P_id_U52_monotonic;assumption.
              intros ;apply P_id_isNatList_monotonic;assumption.
              intros ;apply P_id_a__U53_monotonic;assumption.
              intros ;apply P_id_a__U31_monotonic;assumption.
              intros ;apply P_id_isNatIList_monotonic;assumption.
              intros ;apply P_id_a__isNatKind_monotonic;assumption.
              intros ;apply P_id_a__isNatList_monotonic;assumption.
              intros ;apply P_id_U32_monotonic;assumption.
              intros ;apply P_id_mark_monotonic;assumption.
              intros ;apply P_id_a__U43_monotonic;assumption.
              intros ;apply P_id_U61_monotonic;assumption.
              intros ;apply P_id_isNat_monotonic;assumption.
              intros ;apply P_id_a__zeros_bounded;assumption.
              intros ;apply P_id_isNatKind_bounded;assumption.
              intros ;apply P_id_a__isNatIList_bounded;assumption.
              intros ;apply P_id_a__U21_bounded;assumption.
              intros ;apply P_id_U41_bounded;assumption.
              intros ;apply P_id_a__and_bounded;assumption.
              intros ;apply P_id_a__U11_bounded;assumption.
              intros ;apply P_id_U21_bounded;assumption.
              intros ;apply P_id_a__U61_bounded;assumption.
              intros ;apply P_id_a__U32_bounded;assumption.
              intros ;apply P_id_U51_bounded;assumption.
              intros ;apply P_id_isNatIListKind_bounded;assumption.
              intros ;apply P_id_0_bounded;assumption.
              intros ;apply P_id_U12_bounded;assumption.
              intros ;apply P_id_a__U52_bounded;assumption.
              intros ;apply P_id_a__isNat_bounded;assumption.
              intros ;apply P_id_U43_bounded;assumption.
              intros ;apply P_id_a__isNatIListKind_bounded;assumption.
              intros ;apply P_id_a__U12_bounded;assumption.
              intros ;apply P_id_U31_bounded;assumption.
              intros ;apply P_id_a__length_bounded;assumption.
              intros ;apply P_id_a__U42_bounded;assumption.
              intros ;apply P_id_U53_bounded;assumption.
              intros ;apply P_id_and_bounded;assumption.
              intros ;apply P_id_cons_bounded;assumption.
              intros ;apply P_id_U11_bounded;assumption.
              intros ;apply P_id_a__U51_bounded;assumption.
              intros ;apply P_id_a__U22_bounded;assumption.
              intros ;apply P_id_U42_bounded;assumption.
              intros ;apply P_id_length_bounded;assumption.
              intros ;apply P_id_tt_bounded;assumption.
              intros ;apply P_id_U22_bounded;assumption.
              intros ;apply P_id_s_bounded;assumption.
              intros ;apply P_id_a__U41_bounded;assumption.
              intros ;apply P_id_U52_bounded;assumption.
              intros ;apply P_id_nil_bounded;assumption.
              intros ;apply P_id_zeros_bounded;assumption.
              intros ;apply P_id_isNatList_bounded;assumption.
              intros ;apply P_id_a__U53_bounded;assumption.
              intros ;apply P_id_a__U31_bounded;assumption.
              intros ;apply P_id_isNatIList_bounded;assumption.
              intros ;apply P_id_a__isNatKind_bounded;assumption.
              intros ;apply P_id_a__isNatList_bounded;assumption.
              intros ;apply P_id_U32_bounded;assumption.
              intros ;apply P_id_mark_bounded;assumption.
              intros ;apply P_id_a__U43_bounded;assumption.
              intros ;apply P_id_U61_bounded;assumption.
              intros ;apply P_id_isNat_bounded;assumption.
              apply rules_monotonic.
              intros ;apply P_id_A__LENGTH_monotonic;assumption.
              intros ;apply P_id_A__U11_monotonic;assumption.
              intros ;apply P_id_A__U41_monotonic;assumption.
              intros ;apply P_id_A__ISNATKIND_monotonic;assumption.
              intros ;apply P_id_A__U22_monotonic;assumption.
              intros ;apply P_id_A__U51_monotonic;assumption.
              intros ;apply P_id_A__AND_monotonic;assumption.
              intros ;apply P_id_A__ISNATLIST_monotonic;assumption.
              intros ;apply P_id_A__U43_monotonic;assumption.
              intros ;apply P_id_A__U31_monotonic;assumption.
              intros ;apply P_id_A__U53_monotonic;assumption.
              intros ;apply P_id_MARK_monotonic;assumption.
              intros ;apply P_id_A__U12_monotonic;assumption.
              intros ;apply P_id_A__U42_monotonic;assumption.
              intros ;apply P_id_A__ISNAT_monotonic;assumption.
              intros ;apply P_id_A__U52_monotonic;assumption.
              intros ;apply P_id_A__ISNATILISTKIND_monotonic;assumption.
              intros ;apply P_id_A__U21_monotonic;assumption.
              intros ;apply P_id_A__ISNATILIST_monotonic;assumption.
              intros ;apply P_id_A__U32_monotonic;assumption.
              intros ;apply P_id_A__U61_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_strict_in_lt :
             Relation_Definitions.inclusion _ 
              DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_in_le :
             Relation_Definitions.inclusion _ 
              DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large  := 
              WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large.wf
              .
            
            
            Lemma wf :
             well_founded WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large.DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7
              .
            Proof.
              intros x.
              apply (well_founded_ind wf_lt).
              clear x.
              intros x.
              pattern x.
              
              apply (@Acc_ind _ 
                      DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large)
                     .
              clear x.
              intros x _ IHx IHx'.
              constructor.
              intros y H.
              
              destruct H;
               (apply IHx';
                 apply DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large_in_le;
                  econstructor eassumption])).
              
              apply wf_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7_large.
            Qed.
           End WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7.
           
           Definition wf_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7  := 
             WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7.wf
             .
           
           
           Lemma acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7 :
            forall x y, 
             (DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7 
               x y) ->
              Acc WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large.DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large 
               x.
           Proof.
             intros x.
             pattern x.
             
             apply (@Acc_ind _ 
                     DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7)
                    .
             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_12_large_scc_2_large_large_scc_1_large_large_large_non_scc_6;
                  econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
                ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_non_scc_5;
                   econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
                 ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_non_scc_3;
                    econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
                  ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_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_12_large_scc_2_large_large_scc_1_large_large_large_scc_7.
           Qed.
           
           
           Lemma wf :
            well_founded WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large.DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large
             .
           Proof.
             constructor;intros _y _h;inversion _h;clear _h;subst;
              (eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_non_scc_6;
                econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
              ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_non_scc_5;
                 econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
               ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_non_scc_4;
                  econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
                ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_non_scc_3;
                   econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
                 ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_non_scc_2;
                    econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
                  ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_non_scc_1;
                     
                     econstructor 
                     (eassumption)||(algebra.Alg_ext.star_refl' ))||
                   ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_non_scc_0;
                      
                      econstructor 
                      (eassumption)||(algebra.Alg_ext.star_refl' ))||
                    ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_7;
                       
                       econstructor 
                       (eassumption)||(algebra.Alg_ext.star_refl' ))||
                     ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_6;
                        
                        econstructor 
                        (eassumption)||(algebra.Alg_ext.star_refl' ))||
                      ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_5;
                         
                         econstructor 
                         (eassumption)||(algebra.Alg_ext.star_refl' ))||
                       ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_4;
                          
                          econstructor 
                          (eassumption)||(algebra.Alg_ext.star_refl' ))||
                        ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_3;
                           
                           econstructor 
                           (eassumption)||(algebra.Alg_ext.star_refl' ))||
                         ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_2;
                            
                            econstructor 
                            (eassumption)||(algebra.Alg_ext.star_refl' ))||
                          ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large_scc_1;
                             
                             econstructor 
                             (eassumption)||(algebra.Alg_ext.star_refl' ))||
                           ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_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_12_large_scc_2_large_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  := 0.
          
          Definition P_id_isNatKind (x11:Z) := 0.
          
          Definition P_id_a__isNatIList (x11:Z) := 0.
          
          Definition P_id_a__U21 (x11:Z) (x12:Z) := 1* x11 + 1* x12.
          
          Definition P_id_U41 (x11:Z) (x12:Z) (x13:Z) := 0.
          
          Definition P_id_a__and (x11:Z) (x12:Z) := 1* x11 + 2* x12.
          
          Definition P_id_a__U11 (x11:Z) (x12:Z) := 1 + 2* x12.
          
          Definition P_id_U21 (x11:Z) (x12:Z) := 1* x11 + 1* x12.
          
          Definition P_id_a__U61 (x11:Z) (x12:Z) := 1 + 2* x12.
          
          Definition P_id_a__U32 (x11:Z) := 0.
          
          Definition P_id_U51 (x11:Z) (x12:Z) (x13:Z) := 
            1 + 1* x11 + 2* x12 + 2* x13.
          
          Definition P_id_isNatIListKind (x11:Z) := 0.
          
          Definition P_id_0  := 0.
          
          Definition P_id_U12 (x11:Z) := 1* x11.
          
          Definition P_id_a__U52 (x11:Z) (x12:Z) := 1 + 2* x11 + 2* x12.
          
          Definition P_id_a__isNat (x11:Z) := 1* x11.
          
          Definition P_id_U43 (x11:Z) := 0.
          
          Definition P_id_a__isNatIListKind (x11:Z) := 0.
          
          Definition P_id_a__U12 (x11:Z) := 1* x11.
          
          Definition P_id_U31 (x11:Z) (x12:Z) := 1* x11.
          
          Definition P_id_a__length (x11:Z) := 1 + 2* x11.
          
          Definition P_id_a__U42 (x11:Z) (x12:Z) := 0.
          
          Definition P_id_U53 (x11:Z) := 1* x11.
          
          Definition P_id_and (x11:Z) (x12:Z) := 1* x11 + 2* x12.
          
          Definition P_id_cons (x11:Z) (x12:Z) := 1* x11 + 1* x12.
          
          Definition P_id_U11 (x11:Z) (x12:Z) := 1 + 2* x12.
          
          Definition P_id_a__U51 (x11:Z) (x12:Z) (x13:Z) := 
            1 + 1* x11 + 2* x12 + 2* x13.
          
          Definition P_id_a__U22 (x11:Z) := 1* x11.
          
          Definition P_id_U42 (x11:Z) (x12:Z) := 0.
          
          Definition P_id_length (x11:Z) := 1 + 2* x11.
          
          Definition P_id_tt  := 0.
          
          Definition P_id_U22 (x11:Z) := 1* x11.
          
          Definition P_id_s (x11:Z) := 1* x11.
          
          Definition P_id_a__U41 (x11:Z) (x12:Z) (x13:Z) := 0.
          
          Definition P_id_U52 (x11:Z) (x12:Z) := 1 + 2* x11 + 2* x12.
          
          Definition P_id_nil  := 1.
          
          Definition P_id_zeros  := 0.
          
          Definition P_id_isNatList (x11:Z) := 1 + 2* x11.
          
          Definition P_id_a__U53 (x11:Z) := 1* x11.
          
          Definition P_id_a__U31 (x11:Z) (x12:Z) := 1* x11.
          
          Definition P_id_isNatIList (x11:Z) := 0.
          
          Definition P_id_a__isNatKind (x11:Z) := 0.
          
          Definition P_id_a__isNatList (x11:Z) := 1 + 2* x11.
          
          Definition P_id_U32 (x11:Z) := 0.
          
          Definition P_id_mark (x11:Z) := 1* x11.
          
          Definition P_id_a__U43 (x11:Z) := 0.
          
          Definition P_id_U61 (x11:Z) (x12:Z) := 1 + 2* x12.
          
          Definition P_id_isNat (x11:Z) := 1* x11.
          
          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__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__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_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_a__and_monotonic :
           forall x12 x14 x13 x11, 
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_a__and x12 x14 <= P_id_a__and x11 x13.
          Proof.
            intros x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            
            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_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__U61_monotonic :
           forall x12 x14 x13 x11, 
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_a__U61 x12 x14 <= P_id_a__U61 x11 x13.
          Proof.
            intros x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            
            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 x11, 
            (0 <= x12)/\ (x12 <= x11) ->P_id_a__U32 x12 <= P_id_a__U32 x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_0].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U51_monotonic :
           forall x16 x12 x14 x13 x11 x15, 
            (0 <= x16)/\ (x16 <= x15) ->
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_U51 x12 x14 x16 <= P_id_U51 x11 x13 x15.
          Proof.
            intros x16 x15 x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            intros [H_5 H_4].
            
            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_U12_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->P_id_U12 x12 <= P_id_U12 x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_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 x14 x13 x11, 
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_a__U52 x12 x14 <= P_id_a__U52 x11 x13.
          Proof.
            intros x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_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_U43_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->P_id_U43 x12 <= P_id_U43 x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_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__U12_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->P_id_a__U12 x12 <= P_id_a__U12 x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_0].
            
            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__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__U42_monotonic :
           forall x12 x14 x13 x11, 
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_a__U42 x12 x14 <= P_id_a__U42 x11 x13.
          Proof.
            intros x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U53_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->P_id_U53 x12 <= P_id_U53 x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_0].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_and_monotonic :
           forall x12 x14 x13 x11, 
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->P_id_and x12 x14 <= P_id_and x11 x13.
          Proof.
            intros x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            
            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_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__U51_monotonic :
           forall x16 x12 x14 x13 x11 x15, 
            (0 <= x16)/\ (x16 <= x15) ->
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_a__U51 x12 x14 x16 <= P_id_a__U51 x11 x13 x15.
          Proof.
            intros x16 x15 x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            intros [H_5 H_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 x11, 
            (0 <= x12)/\ (x12 <= x11) ->P_id_a__U22 x12 <= P_id_a__U22 x11.
          Proof.
            intros x12 x11.
            intros [H_1 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 x12 x14 x13 x11, 
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->P_id_U42 x12 x14 <= P_id_U42 x11 x13.
          Proof.
            intros x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            
            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_U22_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->P_id_U22 x12 <= P_id_U22 x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_0].
            
            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__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_U52_monotonic :
           forall x12 x14 x13 x11, 
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->P_id_U52 x12 x14 <= P_id_U52 x11 x13.
          Proof.
            intros x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 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__U53_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->P_id_a__U53 x12 <= P_id_a__U53 x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_0].
            
            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_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__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__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_U32_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->P_id_U32 x12 <= P_id_U32 x11.
          Proof.
            intros x12 x11.
            intros [H_1 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__U43_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->P_id_a__U43 x12 <= P_id_a__U43 x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_0].
            
            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 x14 x13 x11, 
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->P_id_U61 x12 x14 <= P_id_U61 x11 x13.
          Proof.
            intros x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            
            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__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_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__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_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_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_a__and_bounded :
           forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__and x12 x11.
          Proof.
            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_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__U61_bounded :
           forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U61 x12 x11.
          Proof.
            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 x11, (0 <= x11) ->0 <= P_id_a__U32 x11.
          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 x13 x11, 
            (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U51 x13 x12 x11.
          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_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_U12_bounded :
           forall x11, (0 <= x11) ->0 <= P_id_U12 x11.
          Proof.
            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 x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U52 x12 x11.
          Proof.
            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_U43_bounded :
           forall x11, (0 <= x11) ->0 <= P_id_U43 x11.
          Proof.
            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_a__U12_bounded :
           forall x11, (0 <= x11) ->0 <= P_id_a__U12 x11.
          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__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__U42_bounded :
           forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U42 x12 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_U53_bounded :
           forall x11, (0 <= x11) ->0 <= P_id_U53 x11.
          Proof.
            intros .
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_and_bounded :
           forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_and x12 x11.
          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_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__U51_bounded :
           forall x12 x13 x11, 
            (0 <= x11) ->
             (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U51 x13 x12 x11.
          Proof.
            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 x11, (0 <= x11) ->0 <= P_id_a__U22 x11.
          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 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U42 x12 x11.
          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_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_U22_bounded :
           forall x11, (0 <= x11) ->0 <= P_id_U22 x11.
          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__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_U52_bounded :
           forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U52 x12 x11.
          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_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_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__U53_bounded :
           forall x11, (0 <= x11) ->0 <= P_id_a__U53 x11.
          Proof.
            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_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.
          
          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_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_U32_bounded :
           forall x11, (0 <= x11) ->0 <= P_id_U32 x11.
          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__U43_bounded :
           forall x11, (0 <= x11) ->0 <= P_id_a__U43 x11.
          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 x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U61 x12 x11.
          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.
          
          Definition measure  := 
            InterpZ.measure 0 P_id_a__zeros P_id_isNatKind 
             P_id_a__isNatIList P_id_a__U21 P_id_U41 P_id_a__and P_id_a__U11 
             P_id_U21 P_id_a__U61 P_id_a__U32 P_id_U51 P_id_isNatIListKind 
             P_id_0 P_id_U12 P_id_a__U52 P_id_a__isNat P_id_U43 
             P_id_a__isNatIListKind P_id_a__U12 P_id_U31 P_id_a__length 
             P_id_a__U42 P_id_U53 P_id_and P_id_cons P_id_U11 P_id_a__U51 
             P_id_a__U22 P_id_U42 P_id_length P_id_tt P_id_U22 P_id_s 
             P_id_a__U41 P_id_U52 P_id_nil P_id_zeros P_id_isNatList 
             P_id_a__U53 P_id_a__U31 P_id_isNatIList P_id_a__isNatKind 
             P_id_a__isNatList P_id_U32 P_id_mark P_id_a__U43 P_id_U61 
             P_id_isNat.
          
          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_isNatKind 
                             (x11::nil)) =>
                           P_id_isNatKind (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__U21 (x12::
                             x11::nil)) =>
                           P_id_a__U21 (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_a__and (x12::
                             x11::nil)) =>
                           P_id_a__and (measure x12) (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_U21 (x12::
                             x11::nil)) =>
                           P_id_U21 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U61 (x12::
                             x11::nil)) =>
                           P_id_a__U61 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U32 (x11::nil)) =>
                           P_id_a__U32 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_U51 (x13::x12::
                             x11::nil)) =>
                           P_id_U51 (measure x13) (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_isNatIListKind 
                             (x11::nil)) =>
                           P_id_isNatIListKind (measure x11)
                          | (algebra.Alg.Term algebra.F.id_0 nil) => 
                          P_id_0 
                          | (algebra.Alg.Term algebra.F.id_U12 (x11::nil)) =>
                           P_id_U12 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U52 (x12::
                             x11::nil)) =>
                           P_id_a__U52 (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_U43 (x11::nil)) =>
                           P_id_U43 (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__U12 (x11::nil)) =>
                           P_id_a__U12 (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__length 
                             (x11::nil)) =>
                           P_id_a__length (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U42 (x12::
                             x11::nil)) =>
                           P_id_a__U42 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_U53 (x11::nil)) =>
                           P_id_U53 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_and (x12::
                             x11::nil)) =>
                           P_id_and (measure x12) (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_U11 (x12::
                             x11::nil)) =>
                           P_id_U11 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U51 (x13::x12::
                             x11::nil)) =>
                           P_id_a__U51 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U22 (x11::nil)) =>
                           P_id_a__U22 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_U42 (x12::
                             x11::nil)) =>
                           P_id_U42 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_length (x11::nil)) =>
                           P_id_length (measure x11)
                          | (algebra.Alg.Term algebra.F.id_tt nil) =>
                           P_id_tt 
                          | (algebra.Alg.Term algebra.F.id_U22 (x11::nil)) =>
                           P_id_U22 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_s (x11::nil)) =>
                           P_id_s (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_U52 (x12::
                             x11::nil)) =>
                           P_id_U52 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_nil nil) =>
                           P_id_nil 
                          | (algebra.Alg.Term algebra.F.id_zeros nil) =>
                           P_id_zeros 
                          | (algebra.Alg.Term algebra.F.id_isNatList 
                             (x11::nil)) =>
                           P_id_isNatList (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U53 (x11::nil)) =>
                           P_id_a__U53 (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_isNatIList 
                             (x11::nil)) =>
                           P_id_isNatIList (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__isNatList 
                             (x11::nil)) =>
                           P_id_a__isNatList (measure x11)
                          | (algebra.Alg.Term algebra.F.id_U32 (x11::nil)) =>
                           P_id_U32 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_mark (x11::nil)) =>
                           P_id_mark (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U43 (x11::nil)) =>
                           P_id_a__U43 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_U61 (x12::
                             x11::nil)) =>
                           P_id_U61 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_isNat (x11::nil)) =>
                           P_id_isNat (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_isNatKind_monotonic;assumption.
            intros ;apply P_id_a__isNatIList_monotonic;assumption.
            intros ;apply P_id_a__U21_monotonic;assumption.
            intros ;apply P_id_U41_monotonic;assumption.
            intros ;apply P_id_a__and_monotonic;assumption.
            intros ;apply P_id_a__U11_monotonic;assumption.
            intros ;apply P_id_U21_monotonic;assumption.
            intros ;apply P_id_a__U61_monotonic;assumption.
            intros ;apply P_id_a__U32_monotonic;assumption.
            intros ;apply P_id_U51_monotonic;assumption.
            intros ;apply P_id_isNatIListKind_monotonic;assumption.
            intros ;apply P_id_U12_monotonic;assumption.
            intros ;apply P_id_a__U52_monotonic;assumption.
            intros ;apply P_id_a__isNat_monotonic;assumption.
            intros ;apply P_id_U43_monotonic;assumption.
            intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
            intros ;apply P_id_a__U12_monotonic;assumption.
            intros ;apply P_id_U31_monotonic;assumption.
            intros ;apply P_id_a__length_monotonic;assumption.
            intros ;apply P_id_a__U42_monotonic;assumption.
            intros ;apply P_id_U53_monotonic;assumption.
            intros ;apply P_id_and_monotonic;assumption.
            intros ;apply P_id_cons_monotonic;assumption.
            intros ;apply P_id_U11_monotonic;assumption.
            intros ;apply P_id_a__U51_monotonic;assumption.
            intros ;apply P_id_a__U22_monotonic;assumption.
            intros ;apply P_id_U42_monotonic;assumption.
            intros ;apply P_id_length_monotonic;assumption.
            intros ;apply P_id_U22_monotonic;assumption.
            intros ;apply P_id_s_monotonic;assumption.
            intros ;apply P_id_a__U41_monotonic;assumption.
            intros ;apply P_id_U52_monotonic;assumption.
            intros ;apply P_id_isNatList_monotonic;assumption.
            intros ;apply P_id_a__U53_monotonic;assumption.
            intros ;apply P_id_a__U31_monotonic;assumption.
            intros ;apply P_id_isNatIList_monotonic;assumption.
            intros ;apply P_id_a__isNatKind_monotonic;assumption.
            intros ;apply P_id_a__isNatList_monotonic;assumption.
            intros ;apply P_id_U32_monotonic;assumption.
            intros ;apply P_id_mark_monotonic;assumption.
            intros ;apply P_id_a__U43_monotonic;assumption.
            intros ;apply P_id_U61_monotonic;assumption.
            intros ;apply P_id_isNat_monotonic;assumption.
            intros ;apply P_id_a__zeros_bounded;assumption.
            intros ;apply P_id_isNatKind_bounded;assumption.
            intros ;apply P_id_a__isNatIList_bounded;assumption.
            intros ;apply P_id_a__U21_bounded;assumption.
            intros ;apply P_id_U41_bounded;assumption.
            intros ;apply P_id_a__and_bounded;assumption.
            intros ;apply P_id_a__U11_bounded;assumption.
            intros ;apply P_id_U21_bounded;assumption.
            intros ;apply P_id_a__U61_bounded;assumption.
            intros ;apply P_id_a__U32_bounded;assumption.
            intros ;apply P_id_U51_bounded;assumption.
            intros ;apply P_id_isNatIListKind_bounded;assumption.
            intros ;apply P_id_0_bounded;assumption.
            intros ;apply P_id_U12_bounded;assumption.
            intros ;apply P_id_a__U52_bounded;assumption.
            intros ;apply P_id_a__isNat_bounded;assumption.
            intros ;apply P_id_U43_bounded;assumption.
            intros ;apply P_id_a__isNatIListKind_bounded;assumption.
            intros ;apply P_id_a__U12_bounded;assumption.
            intros ;apply P_id_U31_bounded;assumption.
            intros ;apply P_id_a__length_bounded;assumption.
            intros ;apply P_id_a__U42_bounded;assumption.
            intros ;apply P_id_U53_bounded;assumption.
            intros ;apply P_id_and_bounded;assumption.
            intros ;apply P_id_cons_bounded;assumption.
            intros ;apply P_id_U11_bounded;assumption.
            intros ;apply P_id_a__U51_bounded;assumption.
            intros ;apply P_id_a__U22_bounded;assumption.
            intros ;apply P_id_U42_bounded;assumption.
            intros ;apply P_id_length_bounded;assumption.
            intros ;apply P_id_tt_bounded;assumption.
            intros ;apply P_id_U22_bounded;assumption.
            intros ;apply P_id_s_bounded;assumption.
            intros ;apply P_id_a__U41_bounded;assumption.
            intros ;apply P_id_U52_bounded;assumption.
            intros ;apply P_id_nil_bounded;assumption.
            intros ;apply P_id_zeros_bounded;assumption.
            intros ;apply P_id_isNatList_bounded;assumption.
            intros ;apply P_id_a__U53_bounded;assumption.
            intros ;apply P_id_a__U31_bounded;assumption.
            intros ;apply P_id_isNatIList_bounded;assumption.
            intros ;apply P_id_a__isNatKind_bounded;assumption.
            intros ;apply P_id_a__isNatList_bounded;assumption.
            intros ;apply P_id_U32_bounded;assumption.
            intros ;apply P_id_mark_bounded;assumption.
            intros ;apply P_id_a__U43_bounded;assumption.
            intros ;apply P_id_U61_bounded;assumption.
            intros ;apply P_id_isNat_bounded;assumption.
            apply rules_monotonic.
          Qed.
          
          Definition P_id_A__LENGTH (x11:Z) := 0.
          
          Definition P_id_A__U11 (x11:Z) (x12:Z) := 1 + 2* x12.
          
          Definition P_id_A__U41 (x11:Z) (x12:Z) (x13:Z) := 0.
          
          Definition P_id_A__ISNATKIND (x11:Z) := 0.
          
          Definition P_id_A__U22 (x11:Z) := 0.
          
          Definition P_id_A__U51 (x11:Z) (x12:Z) (x13:Z) := 
            1 + 1* x12 + 2* x13.
          
          Definition P_id_A__AND (x11:Z) (x12:Z) := 1* x12.
          
          Definition P_id_A__ISNATLIST (x11:Z) := 1 + 2* x11.
          
          Definition P_id_A__U43 (x11:Z) := 0.
          
          Definition P_id_A__U31 (x11:Z) (x12:Z) := 0.
          
          Definition P_id_A__U53 (x11:Z) := 0.
          
          Definition P_id_MARK (x11:Z) := 1* x11.
          
          Definition P_id_A__U12 (x11:Z) := 0.
          
          Definition P_id_A__U42 (x11:Z) (x12:Z) := 0.
          
          Definition P_id_A__ZEROS  := 0.
          
          Definition P_id_A__ISNAT (x11:Z) := 1* x11.
          
          Definition P_id_A__U52 (x11:Z) (x12:Z) := 1 + 2* x12.
          
          Definition P_id_A__ISNATILISTKIND (x11:Z) := 0.
          
          Definition P_id_A__U21 (x11:Z) (x12:Z) := 1* x12.
          
          Definition P_id_A__ISNATILIST (x11:Z) := 0.
          
          Definition P_id_A__U32 (x11:Z) := 0.
          
          Definition P_id_A__U61 (x11:Z) (x12:Z) := 0.
          
          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__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__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__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__U22_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->P_id_A__U22 x12 <= P_id_A__U22 x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_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 x16 x12 x14 x13 x11 x15, 
            (0 <= x16)/\ (x16 <= x15) ->
             (0 <= x14)/\ (x14 <= x13) ->
              (0 <= x12)/\ (x12 <= x11) ->
               P_id_A__U51 x12 x14 x16 <= P_id_A__U51 x11 x13 x15.
          Proof.
            intros x16 x15 x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            intros [H_5 H_4].
            
            cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
             (auto with zarith)||(repeat (translate_vars );prove_ineq ).
          Qed.
          
          Lemma P_id_A__AND_monotonic :
           forall x12 x14 x13 x11, 
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_A__AND x12 x14 <= P_id_A__AND x11 x13.
          Proof.
            intros x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_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__U43_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->P_id_A__U43 x12 <= P_id_A__U43 x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_0].
            
            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__U53_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->P_id_A__U53 x12 <= P_id_A__U53 x11.
          Proof.
            intros x12 x11.
            intros [H_1 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__U12_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->P_id_A__U12 x12 <= P_id_A__U12 x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_0].
            
            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 x12 x14 x13 x11, 
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_A__U42 x12 x14 <= P_id_A__U42 x11 x13.
          Proof.
            intros x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_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_A__U52_monotonic :
           forall x12 x14 x13 x11, 
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_A__U52 x12 x14 <= P_id_A__U52 x11 x13.
          Proof.
            intros x14 x13 x12 x11.
            intros [H_1 H_0].
            intros [H_3 H_2].
            
            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__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__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__U32_monotonic :
           forall x12 x11, 
            (0 <= x12)/\ (x12 <= x11) ->P_id_A__U32 x12 <= P_id_A__U32 x11.
          Proof.
            intros x12 x11.
            intros [H_1 H_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 x14 x13 x11, 
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_A__U61 x12 x14 <= P_id_A__U61 x11 x13.
          Proof.
            intros x14 x13 x12 x11.
            intros [H_1 H_0].
            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_isNatKind 
             P_id_a__isNatIList P_id_a__U21 P_id_U41 P_id_a__and P_id_a__U11 
             P_id_U21 P_id_a__U61 P_id_a__U32 P_id_U51 P_id_isNatIListKind 
             P_id_0 P_id_U12 P_id_a__U52 P_id_a__isNat P_id_U43 
             P_id_a__isNatIListKind P_id_a__U12 P_id_U31 P_id_a__length 
             P_id_a__U42 P_id_U53 P_id_and P_id_cons P_id_U11 P_id_a__U51 
             P_id_a__U22 P_id_U42 P_id_length P_id_tt P_id_U22 P_id_s 
             P_id_a__U41 P_id_U52 P_id_nil P_id_zeros P_id_isNatList 
             P_id_a__U53 P_id_a__U31 P_id_isNatIList P_id_a__isNatKind 
             P_id_a__isNatList P_id_U32 P_id_mark P_id_a__U43 P_id_U61 
             P_id_isNat P_id_A__LENGTH P_id_A__U11 P_id_A__U41 
             P_id_A__ISNATKIND P_id_A__U22 P_id_A__U51 P_id_A__AND 
             P_id_A__ISNATLIST P_id_A__U43 P_id_A__U31 P_id_A__U53 P_id_MARK 
             P_id_A__U12 P_id_A__U42 P_id_A__ZEROS P_id_A__ISNAT P_id_A__U52 
             P_id_A__ISNATILISTKIND P_id_A__U21 P_id_A__ISNATILIST 
             P_id_A__U32 P_id_A__U61.
          
          Lemma marked_measure_equation :
           forall t, 
            marked_measure t = match t with
                                 | (algebra.Alg.Term algebra.F.id_a__length 
                                    (x11::nil)) =>
                                  P_id_A__LENGTH (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__U41 
                                    (x13::x12::x11::nil)) =>
                                  P_id_A__U41 (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__U22 
                                    (x11::nil)) =>
                                  P_id_A__U22 (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U51 
                                    (x13::x12::x11::nil)) =>
                                  P_id_A__U51 (measure x13) (measure x12) 
                                   (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__and 
                                    (x12::x11::nil)) =>
                                  P_id_A__AND (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__U43 
                                    (x11::nil)) =>
                                  P_id_A__U43 (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__U53 
                                    (x11::nil)) =>
                                  P_id_A__U53 (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_mark 
                                    (x11::nil)) =>
                                  P_id_MARK (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U12 
                                    (x11::nil)) =>
                                  P_id_A__U12 (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U42 
                                    (x12::x11::nil)) =>
                                  P_id_A__U42 (measure x12) (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__zeros 
                                    nil) =>
                                  P_id_A__ZEROS 
                                 | (algebra.Alg.Term algebra.F.id_a__isNat 
                                    (x11::nil)) =>
                                  P_id_A__ISNAT (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U52 
                                    (x12::x11::nil)) =>
                                  P_id_A__U52 (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_a__U21 
                                    (x12::x11::nil)) =>
                                  P_id_A__U21 (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__U32 
                                    (x11::nil)) =>
                                  P_id_A__U32 (measure x11)
                                 | (algebra.Alg.Term algebra.F.id_a__U61 
                                    (x12::x11::nil)) =>
                                  P_id_A__U61 (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_isNatKind_monotonic;assumption.
            intros ;apply P_id_a__isNatIList_monotonic;assumption.
            intros ;apply P_id_a__U21_monotonic;assumption.
            intros ;apply P_id_U41_monotonic;assumption.
            intros ;apply P_id_a__and_monotonic;assumption.
            intros ;apply P_id_a__U11_monotonic;assumption.
            intros ;apply P_id_U21_monotonic;assumption.
            intros ;apply P_id_a__U61_monotonic;assumption.
            intros ;apply P_id_a__U32_monotonic;assumption.
            intros ;apply P_id_U51_monotonic;assumption.
            intros ;apply P_id_isNatIListKind_monotonic;assumption.
            intros ;apply P_id_U12_monotonic;assumption.
            intros ;apply P_id_a__U52_monotonic;assumption.
            intros ;apply P_id_a__isNat_monotonic;assumption.
            intros ;apply P_id_U43_monotonic;assumption.
            intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
            intros ;apply P_id_a__U12_monotonic;assumption.
            intros ;apply P_id_U31_monotonic;assumption.
            intros ;apply P_id_a__length_monotonic;assumption.
            intros ;apply P_id_a__U42_monotonic;assumption.
            intros ;apply P_id_U53_monotonic;assumption.
            intros ;apply P_id_and_monotonic;assumption.
            intros ;apply P_id_cons_monotonic;assumption.
            intros ;apply P_id_U11_monotonic;assumption.
            intros ;apply P_id_a__U51_monotonic;assumption.
            intros ;apply P_id_a__U22_monotonic;assumption.
            intros ;apply P_id_U42_monotonic;assumption.
            intros ;apply P_id_length_monotonic;assumption.
            intros ;apply P_id_U22_monotonic;assumption.
            intros ;apply P_id_s_monotonic;assumption.
            intros ;apply P_id_a__U41_monotonic;assumption.
            intros ;apply P_id_U52_monotonic;assumption.
            intros ;apply P_id_isNatList_monotonic;assumption.
            intros ;apply P_id_a__U53_monotonic;assumption.
            intros ;apply P_id_a__U31_monotonic;assumption.
            intros ;apply P_id_isNatIList_monotonic;assumption.
            intros ;apply P_id_a__isNatKind_monotonic;assumption.
            intros ;apply P_id_a__isNatList_monotonic;assumption.
            intros ;apply P_id_U32_monotonic;assumption.
            intros ;apply P_id_mark_monotonic;assumption.
            intros ;apply P_id_a__U43_monotonic;assumption.
            intros ;apply P_id_U61_monotonic;assumption.
            intros ;apply P_id_isNat_monotonic;assumption.
            intros ;apply P_id_a__zeros_bounded;assumption.
            intros ;apply P_id_isNatKind_bounded;assumption.
            intros ;apply P_id_a__isNatIList_bounded;assumption.
            intros ;apply P_id_a__U21_bounded;assumption.
            intros ;apply P_id_U41_bounded;assumption.
            intros ;apply P_id_a__and_bounded;assumption.
            intros ;apply P_id_a__U11_bounded;assumption.
            intros ;apply P_id_U21_bounded;assumption.
            intros ;apply P_id_a__U61_bounded;assumption.
            intros ;apply P_id_a__U32_bounded;assumption.
            intros ;apply P_id_U51_bounded;assumption.
            intros ;apply P_id_isNatIListKind_bounded;assumption.
            intros ;apply P_id_0_bounded;assumption.
            intros ;apply P_id_U12_bounded;assumption.
            intros ;apply P_id_a__U52_bounded;assumption.
            intros ;apply P_id_a__isNat_bounded;assumption.
            intros ;apply P_id_U43_bounded;assumption.
            intros ;apply P_id_a__isNatIListKind_bounded;assumption.
            intros ;apply P_id_a__U12_bounded;assumption.
            intros ;apply P_id_U31_bounded;assumption.
            intros ;apply P_id_a__length_bounded;assumption.
            intros ;apply P_id_a__U42_bounded;assumption.
            intros ;apply P_id_U53_bounded;assumption.
            intros ;apply P_id_and_bounded;assumption.
            intros ;apply P_id_cons_bounded;assumption.
            intros ;apply P_id_U11_bounded;assumption.
            intros ;apply P_id_a__U51_bounded;assumption.
            intros ;apply P_id_a__U22_bounded;assumption.
            intros ;apply P_id_U42_bounded;assumption.
            intros ;apply P_id_length_bounded;assumption.
            intros ;apply P_id_tt_bounded;assumption.
            intros ;apply P_id_U22_bounded;assumption.
            intros ;apply P_id_s_bounded;assumption.
            intros ;apply P_id_a__U41_bounded;assumption.
            intros ;apply P_id_U52_bounded;assumption.
            intros ;apply P_id_nil_bounded;assumption.
            intros ;apply P_id_zeros_bounded;assumption.
            intros ;apply P_id_isNatList_bounded;assumption.
            intros ;apply P_id_a__U53_bounded;assumption.
            intros ;apply P_id_a__U31_bounded;assumption.
            intros ;apply P_id_isNatIList_bounded;assumption.
            intros ;apply P_id_a__isNatKind_bounded;assumption.
            intros ;apply P_id_a__isNatList_bounded;assumption.
            intros ;apply P_id_U32_bounded;assumption.
            intros ;apply P_id_mark_bounded;assumption.
            intros ;apply P_id_a__U43_bounded;assumption.
            intros ;apply P_id_U61_bounded;assumption.
            intros ;apply P_id_isNat_bounded;assumption.
            apply rules_monotonic.
            intros ;apply P_id_A__LENGTH_monotonic;assumption.
            intros ;apply P_id_A__U11_monotonic;assumption.
            intros ;apply P_id_A__U41_monotonic;assumption.
            intros ;apply P_id_A__ISNATKIND_monotonic;assumption.
            intros ;apply P_id_A__U22_monotonic;assumption.
            intros ;apply P_id_A__U51_monotonic;assumption.
            intros ;apply P_id_A__AND_monotonic;assumption.
            intros ;apply P_id_A__ISNATLIST_monotonic;assumption.
            intros ;apply P_id_A__U43_monotonic;assumption.
            intros ;apply P_id_A__U31_monotonic;assumption.
            intros ;apply P_id_A__U53_monotonic;assumption.
            intros ;apply P_id_MARK_monotonic;assumption.
            intros ;apply P_id_A__U12_monotonic;assumption.
            intros ;apply P_id_A__U42_monotonic;assumption.
            intros ;apply P_id_A__ISNAT_monotonic;assumption.
            intros ;apply P_id_A__U52_monotonic;assumption.
            intros ;apply P_id_A__ISNATILISTKIND_monotonic;assumption.
            intros ;apply P_id_A__U21_monotonic;assumption.
            intros ;apply P_id_A__ISNATILIST_monotonic;assumption.
            intros ;apply P_id_A__U32_monotonic;assumption.
            intros ;apply P_id_A__U61_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_12_large_scc_2_large_large_scc_1_large_large_strict_in_lt :
           Relation_Definitions.inclusion _ 
            DP_R_xml_0_scc_12_large_scc_2_large_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_12_large_scc_2_large_large_scc_1_large_large_large_in_le :
           Relation_Definitions.inclusion _ 
            DP_R_xml_0_scc_12_large_scc_2_large_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_12_large_scc_2_large_large_scc_1_large_large_large  := 
            WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large.wf
            .
          
          
          Lemma wf :
           well_founded WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large.DP_R_xml_0_scc_12_large_scc_2_large_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_12_large_scc_2_large_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_12_large_scc_2_large_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_12_large_scc_2_large_large_scc_1_large_large_large_in_le;
                econstructor eassumption])).
            
            apply wf_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large_large.
          Qed.
         End WF_DP_R_xml_0_scc_12_large_scc_2_large_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  := 3.
         
         Definition P_id_isNatKind (x11:Z) := 0.
         
         Definition P_id_a__isNatIList (x11:Z) := 1* x11.
         
         Definition P_id_a__U21 (x11:Z) (x12:Z) := 2* x11.
         
         Definition P_id_U41 (x11:Z) (x12:Z) (x13:Z) := 1 + 1* x13.
         
         Definition P_id_a__and (x11:Z) (x12:Z) := 2* x11 + 2* x12.
         
         Definition P_id_a__U11 (x11:Z) (x12:Z) := 0.
         
         Definition P_id_U21 (x11:Z) (x12:Z) := 2* x11.
         
         Definition P_id_a__U61 (x11:Z) (x12:Z) := 0.
         
         Definition P_id_a__U32 (x11:Z) := 0.
         
         Definition P_id_U51 (x11:Z) (x12:Z) (x13:Z) := 2* x11.
         
         Definition P_id_isNatIListKind (x11:Z) := 0.
         
         Definition P_id_0  := 0.
         
         Definition P_id_U12 (x11:Z) := 2* x11.
         
         Definition P_id_a__U52 (x11:Z) (x12:Z) := 2* x11.
         
         Definition P_id_a__isNat (x11:Z) := 0.
         
         Definition P_id_U43 (x11:Z) := 1 + 1* x11.
         
         Definition P_id_a__isNatIListKind (x11:Z) := 0.
         
         Definition P_id_a__U12 (x11:Z) := 2* x11.
         
         Definition P_id_U31 (x11:Z) (x12:Z) := 2* x11 + 1* x12.
         
         Definition P_id_a__length (x11:Z) := 0.
         
         Definition P_id_a__U42 (x11:Z) (x12:Z) := 1 + 1* x12.
         
         Definition P_id_U53 (x11:Z) := 1* x11.
         
         Definition P_id_and (x11:Z) (x12:Z) := 2* x11 + 1* x12.
         
         Definition P_id_cons (x11:Z) (x12:Z) := 1 + 1* x12.
         
         Definition P_id_U11 (x11:Z) (x12:Z) := 0.
         
         Definition P_id_a__U51 (x11:Z) (x12:Z) (x13:Z) := 2* x11.
         
         Definition P_id_a__U22 (x11:Z) := 2* x11.
         
         Definition P_id_U42 (x11:Z) (x12:Z) := 1 + 1* x12.
         
         Definition P_id_length (x11:Z) := 0.
         
         Definition P_id_tt  := 0.
         
         Definition P_id_U22 (x11:Z) := 2* x11.
         
         Definition P_id_s (x11:Z) := 2* x11.
         
         Definition P_id_a__U41 (x11:Z) (x12:Z) (x13:Z) := 1 + 1* x13.
         
         Definition P_id_U52 (x11:Z) (x12:Z) := 2* x11.
         
         Definition P_id_nil  := 2.
         
         Definition P_id_zeros  := 2.
         
         Definition P_id_isNatList (x11:Z) := 0.
         
         Definition P_id_a__U53 (x11:Z) := 1* x11.
         
         Definition P_id_a__U31 (x11:Z) (x12:Z) := 2* x11 + 1* x12.
         
         Definition P_id_isNatIList (x11:Z) := 1* x11.
         
         Definition P_id_a__isNatKind (x11:Z) := 0.
         
         Definition P_id_a__isNatList (x11:Z) := 0.
         
         Definition P_id_U32 (x11:Z) := 0.
         
         Definition P_id_mark (x11:Z) := 2* x11.
         
         Definition P_id_a__U43 (x11:Z) := 1 + 1* x11.
         
         Definition P_id_U61 (x11:Z) (x12:Z) := 0.
         
         Definition P_id_isNat (x11:Z) := 0.
         
         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__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__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_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_a__and_monotonic :
          forall x12 x14 x13 x11, 
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_a__and x12 x14 <= P_id_a__and x11 x13.
         Proof.
           intros x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           
           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_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__U61_monotonic :
          forall x12 x14 x13 x11, 
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_a__U61 x12 x14 <= P_id_a__U61 x11 x13.
         Proof.
           intros x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           
           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 x11, 
           (0 <= x12)/\ (x12 <= x11) ->P_id_a__U32 x12 <= P_id_a__U32 x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_0].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U51_monotonic :
          forall x16 x12 x14 x13 x11 x15, 
           (0 <= x16)/\ (x16 <= x15) ->
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_U51 x12 x14 x16 <= P_id_U51 x11 x13 x15.
         Proof.
           intros x16 x15 x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           intros [H_5 H_4].
           
           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_U12_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->P_id_U12 x12 <= P_id_U12 x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_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 x14 x13 x11, 
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_a__U52 x12 x14 <= P_id_a__U52 x11 x13.
         Proof.
           intros x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_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_U43_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->P_id_U43 x12 <= P_id_U43 x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_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__U12_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->P_id_a__U12 x12 <= P_id_a__U12 x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_0].
           
           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__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__U42_monotonic :
          forall x12 x14 x13 x11, 
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_a__U42 x12 x14 <= P_id_a__U42 x11 x13.
         Proof.
           intros x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U53_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->P_id_U53 x12 <= P_id_U53 x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_0].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_and_monotonic :
          forall x12 x14 x13 x11, 
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->P_id_and x12 x14 <= P_id_and x11 x13.
         Proof.
           intros x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           
           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_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__U51_monotonic :
          forall x16 x12 x14 x13 x11 x15, 
           (0 <= x16)/\ (x16 <= x15) ->
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_a__U51 x12 x14 x16 <= P_id_a__U51 x11 x13 x15.
         Proof.
           intros x16 x15 x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           intros [H_5 H_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 x11, 
           (0 <= x12)/\ (x12 <= x11) ->P_id_a__U22 x12 <= P_id_a__U22 x11.
         Proof.
           intros x12 x11.
           intros [H_1 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 x12 x14 x13 x11, 
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->P_id_U42 x12 x14 <= P_id_U42 x11 x13.
         Proof.
           intros x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           
           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_U22_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->P_id_U22 x12 <= P_id_U22 x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_0].
           
           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__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_U52_monotonic :
          forall x12 x14 x13 x11, 
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->P_id_U52 x12 x14 <= P_id_U52 x11 x13.
         Proof.
           intros x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 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__U53_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->P_id_a__U53 x12 <= P_id_a__U53 x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_0].
           
           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_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__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__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_U32_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->P_id_U32 x12 <= P_id_U32 x11.
         Proof.
           intros x12 x11.
           intros [H_1 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__U43_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->P_id_a__U43 x12 <= P_id_a__U43 x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_0].
           
           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 x14 x13 x11, 
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->P_id_U61 x12 x14 <= P_id_U61 x11 x13.
         Proof.
           intros x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           
           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__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_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__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_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_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_a__and_bounded :
          forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__and x12 x11.
         Proof.
           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_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__U61_bounded :
          forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U61 x12 x11.
         Proof.
           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 x11, (0 <= x11) ->0 <= P_id_a__U32 x11.
         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 x13 x11, 
           (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U51 x13 x12 x11.
         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_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_U12_bounded : forall x11, (0 <= x11) ->0 <= P_id_U12 x11.
         Proof.
           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 x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U52 x12 x11.
         Proof.
           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_U43_bounded : forall x11, (0 <= x11) ->0 <= P_id_U43 x11.
         Proof.
           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_a__U12_bounded :
          forall x11, (0 <= x11) ->0 <= P_id_a__U12 x11.
         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__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__U42_bounded :
          forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U42 x12 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_U53_bounded : forall x11, (0 <= x11) ->0 <= P_id_U53 x11.
         Proof.
           intros .
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_and_bounded :
          forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_and x12 x11.
         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_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__U51_bounded :
          forall x12 x13 x11, 
           (0 <= x11) ->
            (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U51 x13 x12 x11.
         Proof.
           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 x11, (0 <= x11) ->0 <= P_id_a__U22 x11.
         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 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U42 x12 x11.
         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_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_U22_bounded : forall x11, (0 <= x11) ->0 <= P_id_U22 x11.
         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__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_U52_bounded :
          forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U52 x12 x11.
         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_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_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__U53_bounded :
          forall x11, (0 <= x11) ->0 <= P_id_a__U53 x11.
         Proof.
           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_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.
         
         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_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_U32_bounded : forall x11, (0 <= x11) ->0 <= P_id_U32 x11.
         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__U43_bounded :
          forall x11, (0 <= x11) ->0 <= P_id_a__U43 x11.
         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 x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U61 x12 x11.
         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.
         
         Definition measure  := 
           InterpZ.measure 0 P_id_a__zeros P_id_isNatKind P_id_a__isNatIList 
            P_id_a__U21 P_id_U41 P_id_a__and P_id_a__U11 P_id_U21 
            P_id_a__U61 P_id_a__U32 P_id_U51 P_id_isNatIListKind P_id_0 
            P_id_U12 P_id_a__U52 P_id_a__isNat P_id_U43 
            P_id_a__isNatIListKind P_id_a__U12 P_id_U31 P_id_a__length 
            P_id_a__U42 P_id_U53 P_id_and P_id_cons P_id_U11 P_id_a__U51 
            P_id_a__U22 P_id_U42 P_id_length P_id_tt P_id_U22 P_id_s 
            P_id_a__U41 P_id_U52 P_id_nil P_id_zeros P_id_isNatList 
            P_id_a__U53 P_id_a__U31 P_id_isNatIList P_id_a__isNatKind 
            P_id_a__isNatList P_id_U32 P_id_mark P_id_a__U43 P_id_U61 
            P_id_isNat.
         
         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_isNatKind 
                            (x11::nil)) =>
                          P_id_isNatKind (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__U21 (x12::
                            x11::nil)) =>
                          P_id_a__U21 (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_a__and (x12::
                            x11::nil)) =>
                          P_id_a__and (measure x12) (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_U21 (x12::
                            x11::nil)) =>
                          P_id_U21 (measure x12) (measure x11)
                         | (algebra.Alg.Term algebra.F.id_a__U61 (x12::
                            x11::nil)) =>
                          P_id_a__U61 (measure x12) (measure x11)
                         | (algebra.Alg.Term algebra.F.id_a__U32 (x11::nil)) =>
                          P_id_a__U32 (measure x11)
                         | (algebra.Alg.Term algebra.F.id_U51 (x13::x12::
                            x11::nil)) =>
                          P_id_U51 (measure x13) (measure x12) (measure x11)
                         | (algebra.Alg.Term algebra.F.id_isNatIListKind 
                            (x11::nil)) =>
                          P_id_isNatIListKind (measure x11)
                         | (algebra.Alg.Term algebra.F.id_0 nil) => P_id_0 
                         | (algebra.Alg.Term algebra.F.id_U12 (x11::nil)) =>
                          P_id_U12 (measure x11)
                         | (algebra.Alg.Term algebra.F.id_a__U52 (x12::
                            x11::nil)) =>
                          P_id_a__U52 (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_U43 (x11::nil)) =>
                          P_id_U43 (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__U12 (x11::nil)) =>
                          P_id_a__U12 (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__length 
                            (x11::nil)) =>
                          P_id_a__length (measure x11)
                         | (algebra.Alg.Term algebra.F.id_a__U42 (x12::
                            x11::nil)) =>
                          P_id_a__U42 (measure x12) (measure x11)
                         | (algebra.Alg.Term algebra.F.id_U53 (x11::nil)) =>
                          P_id_U53 (measure x11)
                         | (algebra.Alg.Term algebra.F.id_and (x12::
                            x11::nil)) =>
                          P_id_and (measure x12) (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_U11 (x12::
                            x11::nil)) =>
                          P_id_U11 (measure x12) (measure x11)
                         | (algebra.Alg.Term algebra.F.id_a__U51 (x13::x12::
                            x11::nil)) =>
                          P_id_a__U51 (measure x13) (measure x12) 
                           (measure x11)
                         | (algebra.Alg.Term algebra.F.id_a__U22 (x11::nil)) =>
                          P_id_a__U22 (measure x11)
                         | (algebra.Alg.Term algebra.F.id_U42 (x12::
                            x11::nil)) =>
                          P_id_U42 (measure x12) (measure x11)
                         | (algebra.Alg.Term algebra.F.id_length (x11::nil)) =>
                          P_id_length (measure x11)
                         | (algebra.Alg.Term algebra.F.id_tt nil) => 
                         P_id_tt 
                         | (algebra.Alg.Term algebra.F.id_U22 (x11::nil)) =>
                          P_id_U22 (measure x11)
                         | (algebra.Alg.Term algebra.F.id_s (x11::nil)) =>
                          P_id_s (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_U52 (x12::
                            x11::nil)) =>
                          P_id_U52 (measure x12) (measure x11)
                         | (algebra.Alg.Term algebra.F.id_nil nil) =>
                          P_id_nil 
                         | (algebra.Alg.Term algebra.F.id_zeros nil) =>
                          P_id_zeros 
                         | (algebra.Alg.Term algebra.F.id_isNatList 
                            (x11::nil)) =>
                          P_id_isNatList (measure x11)
                         | (algebra.Alg.Term algebra.F.id_a__U53 (x11::nil)) =>
                          P_id_a__U53 (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_isNatIList 
                            (x11::nil)) =>
                          P_id_isNatIList (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__isNatList 
                            (x11::nil)) =>
                          P_id_a__isNatList (measure x11)
                         | (algebra.Alg.Term algebra.F.id_U32 (x11::nil)) =>
                          P_id_U32 (measure x11)
                         | (algebra.Alg.Term algebra.F.id_mark (x11::nil)) =>
                          P_id_mark (measure x11)
                         | (algebra.Alg.Term algebra.F.id_a__U43 (x11::nil)) =>
                          P_id_a__U43 (measure x11)
                         | (algebra.Alg.Term algebra.F.id_U61 (x12::
                            x11::nil)) =>
                          P_id_U61 (measure x12) (measure x11)
                         | (algebra.Alg.Term algebra.F.id_isNat (x11::nil)) =>
                          P_id_isNat (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_isNatKind_monotonic;assumption.
           intros ;apply P_id_a__isNatIList_monotonic;assumption.
           intros ;apply P_id_a__U21_monotonic;assumption.
           intros ;apply P_id_U41_monotonic;assumption.
           intros ;apply P_id_a__and_monotonic;assumption.
           intros ;apply P_id_a__U11_monotonic;assumption.
           intros ;apply P_id_U21_monotonic;assumption.
           intros ;apply P_id_a__U61_monotonic;assumption.
           intros ;apply P_id_a__U32_monotonic;assumption.
           intros ;apply P_id_U51_monotonic;assumption.
           intros ;apply P_id_isNatIListKind_monotonic;assumption.
           intros ;apply P_id_U12_monotonic;assumption.
           intros ;apply P_id_a__U52_monotonic;assumption.
           intros ;apply P_id_a__isNat_monotonic;assumption.
           intros ;apply P_id_U43_monotonic;assumption.
           intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
           intros ;apply P_id_a__U12_monotonic;assumption.
           intros ;apply P_id_U31_monotonic;assumption.
           intros ;apply P_id_a__length_monotonic;assumption.
           intros ;apply P_id_a__U42_monotonic;assumption.
           intros ;apply P_id_U53_monotonic;assumption.
           intros ;apply P_id_and_monotonic;assumption.
           intros ;apply P_id_cons_monotonic;assumption.
           intros ;apply P_id_U11_monotonic;assumption.
           intros ;apply P_id_a__U51_monotonic;assumption.
           intros ;apply P_id_a__U22_monotonic;assumption.
           intros ;apply P_id_U42_monotonic;assumption.
           intros ;apply P_id_length_monotonic;assumption.
           intros ;apply P_id_U22_monotonic;assumption.
           intros ;apply P_id_s_monotonic;assumption.
           intros ;apply P_id_a__U41_monotonic;assumption.
           intros ;apply P_id_U52_monotonic;assumption.
           intros ;apply P_id_isNatList_monotonic;assumption.
           intros ;apply P_id_a__U53_monotonic;assumption.
           intros ;apply P_id_a__U31_monotonic;assumption.
           intros ;apply P_id_isNatIList_monotonic;assumption.
           intros ;apply P_id_a__isNatKind_monotonic;assumption.
           intros ;apply P_id_a__isNatList_monotonic;assumption.
           intros ;apply P_id_U32_monotonic;assumption.
           intros ;apply P_id_mark_monotonic;assumption.
           intros ;apply P_id_a__U43_monotonic;assumption.
           intros ;apply P_id_U61_monotonic;assumption.
           intros ;apply P_id_isNat_monotonic;assumption.
           intros ;apply P_id_a__zeros_bounded;assumption.
           intros ;apply P_id_isNatKind_bounded;assumption.
           intros ;apply P_id_a__isNatIList_bounded;assumption.
           intros ;apply P_id_a__U21_bounded;assumption.
           intros ;apply P_id_U41_bounded;assumption.
           intros ;apply P_id_a__and_bounded;assumption.
           intros ;apply P_id_a__U11_bounded;assumption.
           intros ;apply P_id_U21_bounded;assumption.
           intros ;apply P_id_a__U61_bounded;assumption.
           intros ;apply P_id_a__U32_bounded;assumption.
           intros ;apply P_id_U51_bounded;assumption.
           intros ;apply P_id_isNatIListKind_bounded;assumption.
           intros ;apply P_id_0_bounded;assumption.
           intros ;apply P_id_U12_bounded;assumption.
           intros ;apply P_id_a__U52_bounded;assumption.
           intros ;apply P_id_a__isNat_bounded;assumption.
           intros ;apply P_id_U43_bounded;assumption.
           intros ;apply P_id_a__isNatIListKind_bounded;assumption.
           intros ;apply P_id_a__U12_bounded;assumption.
           intros ;apply P_id_U31_bounded;assumption.
           intros ;apply P_id_a__length_bounded;assumption.
           intros ;apply P_id_a__U42_bounded;assumption.
           intros ;apply P_id_U53_bounded;assumption.
           intros ;apply P_id_and_bounded;assumption.
           intros ;apply P_id_cons_bounded;assumption.
           intros ;apply P_id_U11_bounded;assumption.
           intros ;apply P_id_a__U51_bounded;assumption.
           intros ;apply P_id_a__U22_bounded;assumption.
           intros ;apply P_id_U42_bounded;assumption.
           intros ;apply P_id_length_bounded;assumption.
           intros ;apply P_id_tt_bounded;assumption.
           intros ;apply P_id_U22_bounded;assumption.
           intros ;apply P_id_s_bounded;assumption.
           intros ;apply P_id_a__U41_bounded;assumption.
           intros ;apply P_id_U52_bounded;assumption.
           intros ;apply P_id_nil_bounded;assumption.
           intros ;apply P_id_zeros_bounded;assumption.
           intros ;apply P_id_isNatList_bounded;assumption.
           intros ;apply P_id_a__U53_bounded;assumption.
           intros ;apply P_id_a__U31_bounded;assumption.
           intros ;apply P_id_isNatIList_bounded;assumption.
           intros ;apply P_id_a__isNatKind_bounded;assumption.
           intros ;apply P_id_a__isNatList_bounded;assumption.
           intros ;apply P_id_U32_bounded;assumption.
           intros ;apply P_id_mark_bounded;assumption.
           intros ;apply P_id_a__U43_bounded;assumption.
           intros ;apply P_id_U61_bounded;assumption.
           intros ;apply P_id_isNat_bounded;assumption.
           apply rules_monotonic.
         Qed.
         
         Definition P_id_A__LENGTH (x11:Z) := 0.
         
         Definition P_id_A__U11 (x11:Z) (x12:Z) := 0.
         
         Definition P_id_A__U41 (x11:Z) (x12:Z) (x13:Z) := 0.
         
         Definition P_id_A__ISNATKIND (x11:Z) := 0.
         
         Definition P_id_A__U22 (x11:Z) := 0.
         
         Definition P_id_A__U51 (x11:Z) (x12:Z) (x13:Z) := 0.
         
         Definition P_id_A__AND (x11:Z) (x12:Z) := 2* x12.
         
         Definition P_id_A__ISNATLIST (x11:Z) := 0.
         
         Definition P_id_A__U43 (x11:Z) := 0.
         
         Definition P_id_A__U31 (x11:Z) (x12:Z) := 0.
         
         Definition P_id_A__U53 (x11:Z) := 0.
         
         Definition P_id_MARK (x11:Z) := 2* x11.
         
         Definition P_id_A__U12 (x11:Z) := 0.
         
         Definition P_id_A__U42 (x11:Z) (x12:Z) := 0.
         
         Definition P_id_A__ZEROS  := 0.
         
         Definition P_id_A__ISNAT (x11:Z) := 0.
         
         Definition P_id_A__U52 (x11:Z) (x12:Z) := 0.
         
         Definition P_id_A__ISNATILISTKIND (x11:Z) := 0.
         
         Definition P_id_A__U21 (x11:Z) (x12:Z) := 0.
         
         Definition P_id_A__ISNATILIST (x11:Z) := 0.
         
         Definition P_id_A__U32 (x11:Z) := 0.
         
         Definition P_id_A__U61 (x11:Z) (x12:Z) := 0.
         
         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__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__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__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__U22_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->P_id_A__U22 x12 <= P_id_A__U22 x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_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 x16 x12 x14 x13 x11 x15, 
           (0 <= x16)/\ (x16 <= x15) ->
            (0 <= x14)/\ (x14 <= x13) ->
             (0 <= x12)/\ (x12 <= x11) ->
              P_id_A__U51 x12 x14 x16 <= P_id_A__U51 x11 x13 x15.
         Proof.
           intros x16 x15 x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           intros [H_5 H_4].
           
           cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
            (auto with zarith)||(repeat (translate_vars );prove_ineq ).
         Qed.
         
         Lemma P_id_A__AND_monotonic :
          forall x12 x14 x13 x11, 
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_A__AND x12 x14 <= P_id_A__AND x11 x13.
         Proof.
           intros x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_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__U43_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->P_id_A__U43 x12 <= P_id_A__U43 x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_0].
           
           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__U53_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->P_id_A__U53 x12 <= P_id_A__U53 x11.
         Proof.
           intros x12 x11.
           intros [H_1 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__U12_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->P_id_A__U12 x12 <= P_id_A__U12 x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_0].
           
           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 x12 x14 x13 x11, 
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_A__U42 x12 x14 <= P_id_A__U42 x11 x13.
         Proof.
           intros x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_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_A__U52_monotonic :
          forall x12 x14 x13 x11, 
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_A__U52 x12 x14 <= P_id_A__U52 x11 x13.
         Proof.
           intros x14 x13 x12 x11.
           intros [H_1 H_0].
           intros [H_3 H_2].
           
           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__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__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__U32_monotonic :
          forall x12 x11, 
           (0 <= x12)/\ (x12 <= x11) ->P_id_A__U32 x12 <= P_id_A__U32 x11.
         Proof.
           intros x12 x11.
           intros [H_1 H_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 x14 x13 x11, 
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_A__U61 x12 x14 <= P_id_A__U61 x11 x13.
         Proof.
           intros x14 x13 x12 x11.
           intros [H_1 H_0].
           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_isNatKind 
            P_id_a__isNatIList P_id_a__U21 P_id_U41 P_id_a__and P_id_a__U11 
            P_id_U21 P_id_a__U61 P_id_a__U32 P_id_U51 P_id_isNatIListKind 
            P_id_0 P_id_U12 P_id_a__U52 P_id_a__isNat P_id_U43 
            P_id_a__isNatIListKind P_id_a__U12 P_id_U31 P_id_a__length 
            P_id_a__U42 P_id_U53 P_id_and P_id_cons P_id_U11 P_id_a__U51 
            P_id_a__U22 P_id_U42 P_id_length P_id_tt P_id_U22 P_id_s 
            P_id_a__U41 P_id_U52 P_id_nil P_id_zeros P_id_isNatList 
            P_id_a__U53 P_id_a__U31 P_id_isNatIList P_id_a__isNatKind 
            P_id_a__isNatList P_id_U32 P_id_mark P_id_a__U43 P_id_U61 
            P_id_isNat P_id_A__LENGTH P_id_A__U11 P_id_A__U41 
            P_id_A__ISNATKIND P_id_A__U22 P_id_A__U51 P_id_A__AND 
            P_id_A__ISNATLIST P_id_A__U43 P_id_A__U31 P_id_A__U53 P_id_MARK 
            P_id_A__U12 P_id_A__U42 P_id_A__ZEROS P_id_A__ISNAT P_id_A__U52 
            P_id_A__ISNATILISTKIND P_id_A__U21 P_id_A__ISNATILIST 
            P_id_A__U32 P_id_A__U61.
         
         Lemma marked_measure_equation :
          forall t, 
           marked_measure t = match t with
                                | (algebra.Alg.Term algebra.F.id_a__length 
                                   (x11::nil)) =>
                                 P_id_A__LENGTH (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__U41 
                                   (x13::x12::x11::nil)) =>
                                 P_id_A__U41 (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__U22 
                                   (x11::nil)) =>
                                 P_id_A__U22 (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U51 
                                   (x13::x12::x11::nil)) =>
                                 P_id_A__U51 (measure x13) (measure x12) 
                                  (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__and 
                                   (x12::x11::nil)) =>
                                 P_id_A__AND (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__U43 
                                   (x11::nil)) =>
                                 P_id_A__U43 (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__U53 
                                   (x11::nil)) =>
                                 P_id_A__U53 (measure x11)
                                | (algebra.Alg.Term algebra.F.id_mark 
                                   (x11::nil)) =>
                                 P_id_MARK (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U12 
                                   (x11::nil)) =>
                                 P_id_A__U12 (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U42 
                                   (x12::x11::nil)) =>
                                 P_id_A__U42 (measure x12) (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__zeros 
                                   nil) =>
                                 P_id_A__ZEROS 
                                | (algebra.Alg.Term algebra.F.id_a__isNat 
                                   (x11::nil)) =>
                                 P_id_A__ISNAT (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U52 
                                   (x12::x11::nil)) =>
                                 P_id_A__U52 (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_a__U21 
                                   (x12::x11::nil)) =>
                                 P_id_A__U21 (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__U32 
                                   (x11::nil)) =>
                                 P_id_A__U32 (measure x11)
                                | (algebra.Alg.Term algebra.F.id_a__U61 
                                   (x12::x11::nil)) =>
                                 P_id_A__U61 (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_isNatKind_monotonic;assumption.
           intros ;apply P_id_a__isNatIList_monotonic;assumption.
           intros ;apply P_id_a__U21_monotonic;assumption.
           intros ;apply P_id_U41_monotonic;assumption.
           intros ;apply P_id_a__and_monotonic;assumption.
           intros ;apply P_id_a__U11_monotonic;assumption.
           intros ;apply P_id_U21_monotonic;assumption.
           intros ;apply P_id_a__U61_monotonic;assumption.
           intros ;apply P_id_a__U32_monotonic;assumption.
           intros ;apply P_id_U51_monotonic;assumption.
           intros ;apply P_id_isNatIListKind_monotonic;assumption.
           intros ;apply P_id_U12_monotonic;assumption.
           intros ;apply P_id_a__U52_monotonic;assumption.
           intros ;apply P_id_a__isNat_monotonic;assumption.
           intros ;apply P_id_U43_monotonic;assumption.
           intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
           intros ;apply P_id_a__U12_monotonic;assumption.
           intros ;apply P_id_U31_monotonic;assumption.
           intros ;apply P_id_a__length_monotonic;assumption.
           intros ;apply P_id_a__U42_monotonic;assumption.
           intros ;apply P_id_U53_monotonic;assumption.
           intros ;apply P_id_and_monotonic;assumption.
           intros ;apply P_id_cons_monotonic;assumption.
           intros ;apply P_id_U11_monotonic;assumption.
           intros ;apply P_id_a__U51_monotonic;assumption.
           intros ;apply P_id_a__U22_monotonic;assumption.
           intros ;apply P_id_U42_monotonic;assumption.
           intros ;apply P_id_length_monotonic;assumption.
           intros ;apply P_id_U22_monotonic;assumption.
           intros ;apply P_id_s_monotonic;assumption.
           intros ;apply P_id_a__U41_monotonic;assumption.
           intros ;apply P_id_U52_monotonic;assumption.
           intros ;apply P_id_isNatList_monotonic;assumption.
           intros ;apply P_id_a__U53_monotonic;assumption.
           intros ;apply P_id_a__U31_monotonic;assumption.
           intros ;apply P_id_isNatIList_monotonic;assumption.
           intros ;apply P_id_a__isNatKind_monotonic;assumption.
           intros ;apply P_id_a__isNatList_monotonic;assumption.
           intros ;apply P_id_U32_monotonic;assumption.
           intros ;apply P_id_mark_monotonic;assumption.
           intros ;apply P_id_a__U43_monotonic;assumption.
           intros ;apply P_id_U61_monotonic;assumption.
           intros ;apply P_id_isNat_monotonic;assumption.
           intros ;apply P_id_a__zeros_bounded;assumption.
           intros ;apply P_id_isNatKind_bounded;assumption.
           intros ;apply P_id_a__isNatIList_bounded;assumption.
           intros ;apply P_id_a__U21_bounded;assumption.
           intros ;apply P_id_U41_bounded;assumption.
           intros ;apply P_id_a__and_bounded;assumption.
           intros ;apply P_id_a__U11_bounded;assumption.
           intros ;apply P_id_U21_bounded;assumption.
           intros ;apply P_id_a__U61_bounded;assumption.
           intros ;apply P_id_a__U32_bounded;assumption.
           intros ;apply P_id_U51_bounded;assumption.
           intros ;apply P_id_isNatIListKind_bounded;assumption.
           intros ;apply P_id_0_bounded;assumption.
           intros ;apply P_id_U12_bounded;assumption.
           intros ;apply P_id_a__U52_bounded;assumption.
           intros ;apply P_id_a__isNat_bounded;assumption.
           intros ;apply P_id_U43_bounded;assumption.
           intros ;apply P_id_a__isNatIListKind_bounded;assumption.
           intros ;apply P_id_a__U12_bounded;assumption.
           intros ;apply P_id_U31_bounded;assumption.
           intros ;apply P_id_a__length_bounded;assumption.
           intros ;apply P_id_a__U42_bounded;assumption.
           intros ;apply P_id_U53_bounded;assumption.
           intros ;apply P_id_and_bounded;assumption.
           intros ;apply P_id_cons_bounded;assumption.
           intros ;apply P_id_U11_bounded;assumption.
           intros ;apply P_id_a__U51_bounded;assumption.
           intros ;apply P_id_a__U22_bounded;assumption.
           intros ;apply P_id_U42_bounded;assumption.
           intros ;apply P_id_length_bounded;assumption.
           intros ;apply P_id_tt_bounded;assumption.
           intros ;apply P_id_U22_bounded;assumption.
           intros ;apply P_id_s_bounded;assumption.
           intros ;apply P_id_a__U41_bounded;assumption.
           intros ;apply P_id_U52_bounded;assumption.
           intros ;apply P_id_nil_bounded;assumption.
           intros ;apply P_id_zeros_bounded;assumption.
           intros ;apply P_id_isNatList_bounded;assumption.
           intros ;apply P_id_a__U53_bounded;assumption.
           intros ;apply P_id_a__U31_bounded;assumption.
           intros ;apply P_id_isNatIList_bounded;assumption.
           intros ;apply P_id_a__isNatKind_bounded;assumption.
           intros ;apply P_id_a__isNatList_bounded;assumption.
           intros ;apply P_id_U32_bounded;assumption.
           intros ;apply P_id_mark_bounded;assumption.
           intros ;apply P_id_a__U43_bounded;assumption.
           intros ;apply P_id_U61_bounded;assumption.
           intros ;apply P_id_isNat_bounded;assumption.
           apply rules_monotonic.
           intros ;apply P_id_A__LENGTH_monotonic;assumption.
           intros ;apply P_id_A__U11_monotonic;assumption.
           intros ;apply P_id_A__U41_monotonic;assumption.
           intros ;apply P_id_A__ISNATKIND_monotonic;assumption.
           intros ;apply P_id_A__U22_monotonic;assumption.
           intros ;apply P_id_A__U51_monotonic;assumption.
           intros ;apply P_id_A__AND_monotonic;assumption.
           intros ;apply P_id_A__ISNATLIST_monotonic;assumption.
           intros ;apply P_id_A__U43_monotonic;assumption.
           intros ;apply P_id_A__U31_monotonic;assumption.
           intros ;apply P_id_A__U53_monotonic;assumption.
           intros ;apply P_id_MARK_monotonic;assumption.
           intros ;apply P_id_A__U12_monotonic;assumption.
           intros ;apply P_id_A__U42_monotonic;assumption.
           intros ;apply P_id_A__ISNAT_monotonic;assumption.
           intros ;apply P_id_A__U52_monotonic;assumption.
           intros ;apply P_id_A__ISNATILISTKIND_monotonic;assumption.
           intros ;apply P_id_A__U21_monotonic;assumption.
           intros ;apply P_id_A__ISNATILIST_monotonic;assumption.
           intros ;apply P_id_A__U32_monotonic;assumption.
           intros ;apply P_id_A__U61_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_12_large_scc_2_large_large_scc_1_large_strict_in_lt :
          Relation_Definitions.inclusion _ 
           DP_R_xml_0_scc_12_large_scc_2_large_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_12_large_scc_2_large_large_scc_1_large_large_in_le :
          Relation_Definitions.inclusion _ 
           DP_R_xml_0_scc_12_large_scc_2_large_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_12_large_scc_2_large_large_scc_1_large_large  := 
           WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large.wf.
         
         
         Lemma wf :
          well_founded WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1.DP_R_xml_0_scc_12_large_scc_2_large_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_12_large_scc_2_large_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_12_large_scc_2_large_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_12_large_scc_2_large_large_scc_1_large_large_in_le;
               econstructor eassumption])).
           
           apply wf_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large_large.
         Qed.
        End WF_DP_R_xml_0_scc_12_large_scc_2_large_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  := 0.
        
        Definition P_id_isNatKind (x11:Z) := 0.
        
        Definition P_id_a__isNatIList (x11:Z) := 0.
        
        Definition P_id_a__U21 (x11:Z) (x12:Z) := 2* x11 + 1* x12.
        
        Definition P_id_U41 (x11:Z) (x12:Z) (x13:Z) := 3* x11.
        
        Definition P_id_a__and (x11:Z) (x12:Z) := 1* x11 + 2* x12.
        
        Definition P_id_a__U11 (x11:Z) (x12:Z) := 1 + 2* x11 + 1* x12.
        
        Definition P_id_U21 (x11:Z) (x12:Z) := 2* x11 + 1* x12.
        
        Definition P_id_a__U61 (x11:Z) (x12:Z) := 1 + 2* x12.
        
        Definition P_id_a__U32 (x11:Z) := 0.
        
        Definition P_id_U51 (x11:Z) (x12:Z) (x13:Z) := 
          1* x11 + 1* x12 + 1* x13.
        
        Definition P_id_isNatIListKind (x11:Z) := 0.
        
        Definition P_id_0  := 0.
        
        Definition P_id_U12 (x11:Z) := 1* x11.
        
        Definition P_id_a__U52 (x11:Z) (x12:Z) := 2* x11 + 2* x12.
        
        Definition P_id_a__isNat (x11:Z) := 1* x11.
        
        Definition P_id_U43 (x11:Z) := 1* x11.
        
        Definition P_id_a__isNatIListKind (x11:Z) := 0.
        
        Definition P_id_a__U12 (x11:Z) := 1* x11.
        
        Definition P_id_U31 (x11:Z) (x12:Z) := 0.
        
        Definition P_id_a__length (x11:Z) := 1 + 1* x11.
        
        Definition P_id_a__U42 (x11:Z) (x12:Z) := 0.
        
        Definition P_id_U53 (x11:Z) := 1* x11.
        
        Definition P_id_and (x11:Z) (x12:Z) := 1* x11 + 2* x12.
        
        Definition P_id_cons (x11:Z) (x12:Z) := 2* x11 + 2* x12.
        
        Definition P_id_U11 (x11:Z) (x12:Z) := 1 + 2* x11 + 1* x12.
        
        Definition P_id_a__U51 (x11:Z) (x12:Z) (x13:Z) := 
          1* x11 + 2* x12 + 2* x13.
        
        Definition P_id_a__U22 (x11:Z) := 1* x11.
        
        Definition P_id_U42 (x11:Z) (x12:Z) := 0.
        
        Definition P_id_length (x11:Z) := 1 + 1* x11.
        
        Definition P_id_tt  := 0.
        
        Definition P_id_U22 (x11:Z) := 1* x11.
        
        Definition P_id_s (x11:Z) := 1* x11.
        
        Definition P_id_a__U41 (x11:Z) (x12:Z) (x13:Z) := 3* x11.
        
        Definition P_id_U52 (x11:Z) (x12:Z) := 2* x11 + 2* x12.
        
        Definition P_id_nil  := 1.
        
        Definition P_id_zeros  := 0.
        
        Definition P_id_isNatList (x11:Z) := 1* x11.
        
        Definition P_id_a__U53 (x11:Z) := 1* x11.
        
        Definition P_id_a__U31 (x11:Z) (x12:Z) := 0.
        
        Definition P_id_isNatIList (x11:Z) := 0.
        
        Definition P_id_a__isNatKind (x11:Z) := 0.
        
        Definition P_id_a__isNatList (x11:Z) := 1* x11.
        
        Definition P_id_U32 (x11:Z) := 0.
        
        Definition P_id_mark (x11:Z) := 2* x11.
        
        Definition P_id_a__U43 (x11:Z) := 1* x11.
        
        Definition P_id_U61 (x11:Z) (x12:Z) := 1 + 2* x12.
        
        Definition P_id_isNat (x11:Z) := 1* x11.
        
        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__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__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_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_a__and_monotonic :
         forall x12 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_a__and x12 x14 <= P_id_a__and x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          
          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_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__U61_monotonic :
         forall x12 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_a__U61 x12 x14 <= P_id_a__U61 x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          
          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 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_a__U32 x12 <= P_id_a__U32 x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U51_monotonic :
         forall x16 x12 x14 x13 x11 x15, 
          (0 <= x16)/\ (x16 <= x15) ->
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_U51 x12 x14 x16 <= P_id_U51 x11 x13 x15.
        Proof.
          intros x16 x15 x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          intros [H_5 H_4].
          
          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_U12_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_U12 x12 <= P_id_U12 x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_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 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_a__U52 x12 x14 <= P_id_a__U52 x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_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_U43_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_U43 x12 <= P_id_U43 x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_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__U12_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_a__U12 x12 <= P_id_a__U12 x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          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__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__U42_monotonic :
         forall x12 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_a__U42 x12 x14 <= P_id_a__U42 x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U53_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_U53 x12 <= P_id_U53 x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_and_monotonic :
         forall x12 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->P_id_and x12 x14 <= P_id_and x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          
          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_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__U51_monotonic :
         forall x16 x12 x14 x13 x11 x15, 
          (0 <= x16)/\ (x16 <= x15) ->
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_a__U51 x12 x14 x16 <= P_id_a__U51 x11 x13 x15.
        Proof.
          intros x16 x15 x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          intros [H_5 H_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 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_a__U22 x12 <= P_id_a__U22 x11.
        Proof.
          intros x12 x11.
          intros [H_1 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 x12 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->P_id_U42 x12 x14 <= P_id_U42 x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          
          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_U22_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_U22 x12 <= P_id_U22 x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          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__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_U52_monotonic :
         forall x12 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->P_id_U52 x12 x14 <= P_id_U52 x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 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__U53_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_a__U53 x12 <= P_id_a__U53 x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          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_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__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__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_U32_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_U32 x12 <= P_id_U32 x11.
        Proof.
          intros x12 x11.
          intros [H_1 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__U43_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_a__U43 x12 <= P_id_a__U43 x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          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 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->P_id_U61 x12 x14 <= P_id_U61 x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          
          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__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_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__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_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_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_a__and_bounded :
         forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__and x12 x11.
        Proof.
          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_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__U61_bounded :
         forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U61 x12 x11.
        Proof.
          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 x11, (0 <= x11) ->0 <= P_id_a__U32 x11.
        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 x13 x11, 
          (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U51 x13 x12 x11.
        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_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_U12_bounded : forall x11, (0 <= x11) ->0 <= P_id_U12 x11.
        Proof.
          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 x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U52 x12 x11.
        Proof.
          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_U43_bounded : forall x11, (0 <= x11) ->0 <= P_id_U43 x11.
        Proof.
          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_a__U12_bounded :
         forall x11, (0 <= x11) ->0 <= P_id_a__U12 x11.
        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__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__U42_bounded :
         forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U42 x12 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U53_bounded : forall x11, (0 <= x11) ->0 <= P_id_U53 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_and_bounded :
         forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_and x12 x11.
        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_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__U51_bounded :
         forall x12 x13 x11, 
          (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U51 x13 x12 x11.
        Proof.
          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 x11, (0 <= x11) ->0 <= P_id_a__U22 x11.
        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 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U42 x12 x11.
        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_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_U22_bounded : forall x11, (0 <= x11) ->0 <= P_id_U22 x11.
        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__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_U52_bounded :
         forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U52 x12 x11.
        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_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_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__U53_bounded :
         forall x11, (0 <= x11) ->0 <= P_id_a__U53 x11.
        Proof.
          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_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.
        
        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_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_U32_bounded : forall x11, (0 <= x11) ->0 <= P_id_U32 x11.
        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__U43_bounded :
         forall x11, (0 <= x11) ->0 <= P_id_a__U43 x11.
        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 x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U61 x12 x11.
        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.
        
        Definition measure  := 
          InterpZ.measure 0 P_id_a__zeros P_id_isNatKind P_id_a__isNatIList 
           P_id_a__U21 P_id_U41 P_id_a__and P_id_a__U11 P_id_U21 P_id_a__U61 
           P_id_a__U32 P_id_U51 P_id_isNatIListKind P_id_0 P_id_U12 
           P_id_a__U52 P_id_a__isNat P_id_U43 P_id_a__isNatIListKind 
           P_id_a__U12 P_id_U31 P_id_a__length P_id_a__U42 P_id_U53 P_id_and 
           P_id_cons P_id_U11 P_id_a__U51 P_id_a__U22 P_id_U42 P_id_length 
           P_id_tt P_id_U22 P_id_s P_id_a__U41 P_id_U52 P_id_nil P_id_zeros 
           P_id_isNatList P_id_a__U53 P_id_a__U31 P_id_isNatIList 
           P_id_a__isNatKind P_id_a__isNatList P_id_U32 P_id_mark 
           P_id_a__U43 P_id_U61 P_id_isNat.
        
        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_isNatKind 
                           (x11::nil)) =>
                         P_id_isNatKind (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__U21 (x12::
                           x11::nil)) =>
                         P_id_a__U21 (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_a__and (x12::
                           x11::nil)) =>
                         P_id_a__and (measure x12) (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_U21 (x12::x11::nil)) =>
                         P_id_U21 (measure x12) (measure x11)
                        | (algebra.Alg.Term algebra.F.id_a__U61 (x12::
                           x11::nil)) =>
                         P_id_a__U61 (measure x12) (measure x11)
                        | (algebra.Alg.Term algebra.F.id_a__U32 (x11::nil)) =>
                         P_id_a__U32 (measure x11)
                        | (algebra.Alg.Term algebra.F.id_U51 (x13::x12::
                           x11::nil)) =>
                         P_id_U51 (measure x13) (measure x12) (measure x11)
                        | (algebra.Alg.Term algebra.F.id_isNatIListKind 
                           (x11::nil)) =>
                         P_id_isNatIListKind (measure x11)
                        | (algebra.Alg.Term algebra.F.id_0 nil) => P_id_0 
                        | (algebra.Alg.Term algebra.F.id_U12 (x11::nil)) =>
                         P_id_U12 (measure x11)
                        | (algebra.Alg.Term algebra.F.id_a__U52 (x12::
                           x11::nil)) =>
                         P_id_a__U52 (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_U43 (x11::nil)) =>
                         P_id_U43 (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__U12 (x11::nil)) =>
                         P_id_a__U12 (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__length 
                           (x11::nil)) =>
                         P_id_a__length (measure x11)
                        | (algebra.Alg.Term algebra.F.id_a__U42 (x12::
                           x11::nil)) =>
                         P_id_a__U42 (measure x12) (measure x11)
                        | (algebra.Alg.Term algebra.F.id_U53 (x11::nil)) =>
                         P_id_U53 (measure x11)
                        | (algebra.Alg.Term algebra.F.id_and (x12::x11::nil)) =>
                         P_id_and (measure x12) (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_U11 (x12::x11::nil)) =>
                         P_id_U11 (measure x12) (measure x11)
                        | (algebra.Alg.Term algebra.F.id_a__U51 (x13::x12::
                           x11::nil)) =>
                         P_id_a__U51 (measure x13) (measure x12) 
                          (measure x11)
                        | (algebra.Alg.Term algebra.F.id_a__U22 (x11::nil)) =>
                         P_id_a__U22 (measure x11)
                        | (algebra.Alg.Term algebra.F.id_U42 (x12::x11::nil)) =>
                         P_id_U42 (measure x12) (measure x11)
                        | (algebra.Alg.Term algebra.F.id_length (x11::nil)) =>
                         P_id_length (measure x11)
                        | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt 
                        | (algebra.Alg.Term algebra.F.id_U22 (x11::nil)) =>
                         P_id_U22 (measure x11)
                        | (algebra.Alg.Term algebra.F.id_s (x11::nil)) =>
                         P_id_s (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_U52 (x12::x11::nil)) =>
                         P_id_U52 (measure x12) (measure x11)
                        | (algebra.Alg.Term algebra.F.id_nil nil) =>
                         P_id_nil 
                        | (algebra.Alg.Term algebra.F.id_zeros nil) =>
                         P_id_zeros 
                        | (algebra.Alg.Term algebra.F.id_isNatList 
                           (x11::nil)) =>
                         P_id_isNatList (measure x11)
                        | (algebra.Alg.Term algebra.F.id_a__U53 (x11::nil)) =>
                         P_id_a__U53 (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_isNatIList 
                           (x11::nil)) =>
                         P_id_isNatIList (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__isNatList 
                           (x11::nil)) =>
                         P_id_a__isNatList (measure x11)
                        | (algebra.Alg.Term algebra.F.id_U32 (x11::nil)) =>
                         P_id_U32 (measure x11)
                        | (algebra.Alg.Term algebra.F.id_mark (x11::nil)) =>
                         P_id_mark (measure x11)
                        | (algebra.Alg.Term algebra.F.id_a__U43 (x11::nil)) =>
                         P_id_a__U43 (measure x11)
                        | (algebra.Alg.Term algebra.F.id_U61 (x12::x11::nil)) =>
                         P_id_U61 (measure x12) (measure x11)
                        | (algebra.Alg.Term algebra.F.id_isNat (x11::nil)) =>
                         P_id_isNat (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_isNatKind_monotonic;assumption.
          intros ;apply P_id_a__isNatIList_monotonic;assumption.
          intros ;apply P_id_a__U21_monotonic;assumption.
          intros ;apply P_id_U41_monotonic;assumption.
          intros ;apply P_id_a__and_monotonic;assumption.
          intros ;apply P_id_a__U11_monotonic;assumption.
          intros ;apply P_id_U21_monotonic;assumption.
          intros ;apply P_id_a__U61_monotonic;assumption.
          intros ;apply P_id_a__U32_monotonic;assumption.
          intros ;apply P_id_U51_monotonic;assumption.
          intros ;apply P_id_isNatIListKind_monotonic;assumption.
          intros ;apply P_id_U12_monotonic;assumption.
          intros ;apply P_id_a__U52_monotonic;assumption.
          intros ;apply P_id_a__isNat_monotonic;assumption.
          intros ;apply P_id_U43_monotonic;assumption.
          intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
          intros ;apply P_id_a__U12_monotonic;assumption.
          intros ;apply P_id_U31_monotonic;assumption.
          intros ;apply P_id_a__length_monotonic;assumption.
          intros ;apply P_id_a__U42_monotonic;assumption.
          intros ;apply P_id_U53_monotonic;assumption.
          intros ;apply P_id_and_monotonic;assumption.
          intros ;apply P_id_cons_monotonic;assumption.
          intros ;apply P_id_U11_monotonic;assumption.
          intros ;apply P_id_a__U51_monotonic;assumption.
          intros ;apply P_id_a__U22_monotonic;assumption.
          intros ;apply P_id_U42_monotonic;assumption.
          intros ;apply P_id_length_monotonic;assumption.
          intros ;apply P_id_U22_monotonic;assumption.
          intros ;apply P_id_s_monotonic;assumption.
          intros ;apply P_id_a__U41_monotonic;assumption.
          intros ;apply P_id_U52_monotonic;assumption.
          intros ;apply P_id_isNatList_monotonic;assumption.
          intros ;apply P_id_a__U53_monotonic;assumption.
          intros ;apply P_id_a__U31_monotonic;assumption.
          intros ;apply P_id_isNatIList_monotonic;assumption.
          intros ;apply P_id_a__isNatKind_monotonic;assumption.
          intros ;apply P_id_a__isNatList_monotonic;assumption.
          intros ;apply P_id_U32_monotonic;assumption.
          intros ;apply P_id_mark_monotonic;assumption.
          intros ;apply P_id_a__U43_monotonic;assumption.
          intros ;apply P_id_U61_monotonic;assumption.
          intros ;apply P_id_isNat_monotonic;assumption.
          intros ;apply P_id_a__zeros_bounded;assumption.
          intros ;apply P_id_isNatKind_bounded;assumption.
          intros ;apply P_id_a__isNatIList_bounded;assumption.
          intros ;apply P_id_a__U21_bounded;assumption.
          intros ;apply P_id_U41_bounded;assumption.
          intros ;apply P_id_a__and_bounded;assumption.
          intros ;apply P_id_a__U11_bounded;assumption.
          intros ;apply P_id_U21_bounded;assumption.
          intros ;apply P_id_a__U61_bounded;assumption.
          intros ;apply P_id_a__U32_bounded;assumption.
          intros ;apply P_id_U51_bounded;assumption.
          intros ;apply P_id_isNatIListKind_bounded;assumption.
          intros ;apply P_id_0_bounded;assumption.
          intros ;apply P_id_U12_bounded;assumption.
          intros ;apply P_id_a__U52_bounded;assumption.
          intros ;apply P_id_a__isNat_bounded;assumption.
          intros ;apply P_id_U43_bounded;assumption.
          intros ;apply P_id_a__isNatIListKind_bounded;assumption.
          intros ;apply P_id_a__U12_bounded;assumption.
          intros ;apply P_id_U31_bounded;assumption.
          intros ;apply P_id_a__length_bounded;assumption.
          intros ;apply P_id_a__U42_bounded;assumption.
          intros ;apply P_id_U53_bounded;assumption.
          intros ;apply P_id_and_bounded;assumption.
          intros ;apply P_id_cons_bounded;assumption.
          intros ;apply P_id_U11_bounded;assumption.
          intros ;apply P_id_a__U51_bounded;assumption.
          intros ;apply P_id_a__U22_bounded;assumption.
          intros ;apply P_id_U42_bounded;assumption.
          intros ;apply P_id_length_bounded;assumption.
          intros ;apply P_id_tt_bounded;assumption.
          intros ;apply P_id_U22_bounded;assumption.
          intros ;apply P_id_s_bounded;assumption.
          intros ;apply P_id_a__U41_bounded;assumption.
          intros ;apply P_id_U52_bounded;assumption.
          intros ;apply P_id_nil_bounded;assumption.
          intros ;apply P_id_zeros_bounded;assumption.
          intros ;apply P_id_isNatList_bounded;assumption.
          intros ;apply P_id_a__U53_bounded;assumption.
          intros ;apply P_id_a__U31_bounded;assumption.
          intros ;apply P_id_isNatIList_bounded;assumption.
          intros ;apply P_id_a__isNatKind_bounded;assumption.
          intros ;apply P_id_a__isNatList_bounded;assumption.
          intros ;apply P_id_U32_bounded;assumption.
          intros ;apply P_id_mark_bounded;assumption.
          intros ;apply P_id_a__U43_bounded;assumption.
          intros ;apply P_id_U61_bounded;assumption.
          intros ;apply P_id_isNat_bounded;assumption.
          apply rules_monotonic.
        Qed.
        
        Definition P_id_A__LENGTH (x11:Z) := 0.
        
        Definition P_id_A__U11 (x11:Z) (x12:Z) := 2.
        
        Definition P_id_A__U41 (x11:Z) (x12:Z) (x13:Z) := 0.
        
        Definition P_id_A__ISNATKIND (x11:Z) := 2.
        
        Definition P_id_A__U22 (x11:Z) := 0.
        
        Definition P_id_A__U51 (x11:Z) (x12:Z) (x13:Z) := 2.
        
        Definition P_id_A__AND (x11:Z) (x12:Z) := 2 + 2* x12.
        
        Definition P_id_A__ISNATLIST (x11:Z) := 2.
        
        Definition P_id_A__U43 (x11:Z) := 0.
        
        Definition P_id_A__U31 (x11:Z) (x12:Z) := 0.
        
        Definition P_id_A__U53 (x11:Z) := 0.
        
        Definition P_id_MARK (x11:Z) := 2 + 2* x11.
        
        Definition P_id_A__U12 (x11:Z) := 0.
        
        Definition P_id_A__U42 (x11:Z) (x12:Z) := 0.
        
        Definition P_id_A__ZEROS  := 0.
        
        Definition P_id_A__ISNAT (x11:Z) := 2.
        
        Definition P_id_A__U52 (x11:Z) (x12:Z) := 2.
        
        Definition P_id_A__ISNATILISTKIND (x11:Z) := 2.
        
        Definition P_id_A__U21 (x11:Z) (x12:Z) := 2.
        
        Definition P_id_A__ISNATILIST (x11:Z) := 0.
        
        Definition P_id_A__U32 (x11:Z) := 0.
        
        Definition P_id_A__U61 (x11:Z) (x12:Z) := 0.
        
        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__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__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__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__U22_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_A__U22 x12 <= P_id_A__U22 x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_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 x16 x12 x14 x13 x11 x15, 
          (0 <= x16)/\ (x16 <= x15) ->
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_A__U51 x12 x14 x16 <= P_id_A__U51 x11 x13 x15.
        Proof.
          intros x16 x15 x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          intros [H_5 H_4].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_A__AND_monotonic :
         forall x12 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_A__AND x12 x14 <= P_id_A__AND x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_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__U43_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_A__U43 x12 <= P_id_A__U43 x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          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__U53_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_A__U53 x12 <= P_id_A__U53 x11.
        Proof.
          intros x12 x11.
          intros [H_1 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__U12_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_A__U12 x12 <= P_id_A__U12 x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          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 x12 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_A__U42 x12 x14 <= P_id_A__U42 x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_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_A__U52_monotonic :
         forall x12 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_A__U52 x12 x14 <= P_id_A__U52 x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          
          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__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__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__U32_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_A__U32 x12 <= P_id_A__U32 x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_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 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_A__U61 x12 x14 <= P_id_A__U61 x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          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_isNatKind 
           P_id_a__isNatIList P_id_a__U21 P_id_U41 P_id_a__and P_id_a__U11 
           P_id_U21 P_id_a__U61 P_id_a__U32 P_id_U51 P_id_isNatIListKind 
           P_id_0 P_id_U12 P_id_a__U52 P_id_a__isNat P_id_U43 
           P_id_a__isNatIListKind P_id_a__U12 P_id_U31 P_id_a__length 
           P_id_a__U42 P_id_U53 P_id_and P_id_cons P_id_U11 P_id_a__U51 
           P_id_a__U22 P_id_U42 P_id_length P_id_tt P_id_U22 P_id_s 
           P_id_a__U41 P_id_U52 P_id_nil P_id_zeros P_id_isNatList 
           P_id_a__U53 P_id_a__U31 P_id_isNatIList P_id_a__isNatKind 
           P_id_a__isNatList P_id_U32 P_id_mark P_id_a__U43 P_id_U61 
           P_id_isNat P_id_A__LENGTH P_id_A__U11 P_id_A__U41 
           P_id_A__ISNATKIND P_id_A__U22 P_id_A__U51 P_id_A__AND 
           P_id_A__ISNATLIST P_id_A__U43 P_id_A__U31 P_id_A__U53 P_id_MARK 
           P_id_A__U12 P_id_A__U42 P_id_A__ZEROS P_id_A__ISNAT P_id_A__U52 
           P_id_A__ISNATILISTKIND P_id_A__U21 P_id_A__ISNATILIST P_id_A__U32 
           P_id_A__U61.
        
        Lemma marked_measure_equation :
         forall t, 
          marked_measure t = match t with
                               | (algebra.Alg.Term algebra.F.id_a__length 
                                  (x11::nil)) =>
                                P_id_A__LENGTH (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__U41 (x13::
                                  x12::x11::nil)) =>
                                P_id_A__U41 (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__U22 
                                  (x11::nil)) =>
                                P_id_A__U22 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U51 (x13::
                                  x12::x11::nil)) =>
                                P_id_A__U51 (measure x13) (measure x12) 
                                 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__and (x12::
                                  x11::nil)) =>
                                P_id_A__AND (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__U43 
                                  (x11::nil)) =>
                                P_id_A__U43 (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__U53 
                                  (x11::nil)) =>
                                P_id_A__U53 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_mark 
                                  (x11::nil)) =>
                                P_id_MARK (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U12 
                                  (x11::nil)) =>
                                P_id_A__U12 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U42 (x12::
                                  x11::nil)) =>
                                P_id_A__U42 (measure x12) (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__zeros nil) =>
                                P_id_A__ZEROS 
                               | (algebra.Alg.Term algebra.F.id_a__isNat 
                                  (x11::nil)) =>
                                P_id_A__ISNAT (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U52 (x12::
                                  x11::nil)) =>
                                P_id_A__U52 (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_a__U21 (x12::
                                  x11::nil)) =>
                                P_id_A__U21 (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__U32 
                                  (x11::nil)) =>
                                P_id_A__U32 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U61 (x12::
                                  x11::nil)) =>
                                P_id_A__U61 (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_isNatKind_monotonic;assumption.
          intros ;apply P_id_a__isNatIList_monotonic;assumption.
          intros ;apply P_id_a__U21_monotonic;assumption.
          intros ;apply P_id_U41_monotonic;assumption.
          intros ;apply P_id_a__and_monotonic;assumption.
          intros ;apply P_id_a__U11_monotonic;assumption.
          intros ;apply P_id_U21_monotonic;assumption.
          intros ;apply P_id_a__U61_monotonic;assumption.
          intros ;apply P_id_a__U32_monotonic;assumption.
          intros ;apply P_id_U51_monotonic;assumption.
          intros ;apply P_id_isNatIListKind_monotonic;assumption.
          intros ;apply P_id_U12_monotonic;assumption.
          intros ;apply P_id_a__U52_monotonic;assumption.
          intros ;apply P_id_a__isNat_monotonic;assumption.
          intros ;apply P_id_U43_monotonic;assumption.
          intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
          intros ;apply P_id_a__U12_monotonic;assumption.
          intros ;apply P_id_U31_monotonic;assumption.
          intros ;apply P_id_a__length_monotonic;assumption.
          intros ;apply P_id_a__U42_monotonic;assumption.
          intros ;apply P_id_U53_monotonic;assumption.
          intros ;apply P_id_and_monotonic;assumption.
          intros ;apply P_id_cons_monotonic;assumption.
          intros ;apply P_id_U11_monotonic;assumption.
          intros ;apply P_id_a__U51_monotonic;assumption.
          intros ;apply P_id_a__U22_monotonic;assumption.
          intros ;apply P_id_U42_monotonic;assumption.
          intros ;apply P_id_length_monotonic;assumption.
          intros ;apply P_id_U22_monotonic;assumption.
          intros ;apply P_id_s_monotonic;assumption.
          intros ;apply P_id_a__U41_monotonic;assumption.
          intros ;apply P_id_U52_monotonic;assumption.
          intros ;apply P_id_isNatList_monotonic;assumption.
          intros ;apply P_id_a__U53_monotonic;assumption.
          intros ;apply P_id_a__U31_monotonic;assumption.
          intros ;apply P_id_isNatIList_monotonic;assumption.
          intros ;apply P_id_a__isNatKind_monotonic;assumption.
          intros ;apply P_id_a__isNatList_monotonic;assumption.
          intros ;apply P_id_U32_monotonic;assumption.
          intros ;apply P_id_mark_monotonic;assumption.
          intros ;apply P_id_a__U43_monotonic;assumption.
          intros ;apply P_id_U61_monotonic;assumption.
          intros ;apply P_id_isNat_monotonic;assumption.
          intros ;apply P_id_a__zeros_bounded;assumption.
          intros ;apply P_id_isNatKind_bounded;assumption.
          intros ;apply P_id_a__isNatIList_bounded;assumption.
          intros ;apply P_id_a__U21_bounded;assumption.
          intros ;apply P_id_U41_bounded;assumption.
          intros ;apply P_id_a__and_bounded;assumption.
          intros ;apply P_id_a__U11_bounded;assumption.
          intros ;apply P_id_U21_bounded;assumption.
          intros ;apply P_id_a__U61_bounded;assumption.
          intros ;apply P_id_a__U32_bounded;assumption.
          intros ;apply P_id_U51_bounded;assumption.
          intros ;apply P_id_isNatIListKind_bounded;assumption.
          intros ;apply P_id_0_bounded;assumption.
          intros ;apply P_id_U12_bounded;assumption.
          intros ;apply P_id_a__U52_bounded;assumption.
          intros ;apply P_id_a__isNat_bounded;assumption.
          intros ;apply P_id_U43_bounded;assumption.
          intros ;apply P_id_a__isNatIListKind_bounded;assumption.
          intros ;apply P_id_a__U12_bounded;assumption.
          intros ;apply P_id_U31_bounded;assumption.
          intros ;apply P_id_a__length_bounded;assumption.
          intros ;apply P_id_a__U42_bounded;assumption.
          intros ;apply P_id_U53_bounded;assumption.
          intros ;apply P_id_and_bounded;assumption.
          intros ;apply P_id_cons_bounded;assumption.
          intros ;apply P_id_U11_bounded;assumption.
          intros ;apply P_id_a__U51_bounded;assumption.
          intros ;apply P_id_a__U22_bounded;assumption.
          intros ;apply P_id_U42_bounded;assumption.
          intros ;apply P_id_length_bounded;assumption.
          intros ;apply P_id_tt_bounded;assumption.
          intros ;apply P_id_U22_bounded;assumption.
          intros ;apply P_id_s_bounded;assumption.
          intros ;apply P_id_a__U41_bounded;assumption.
          intros ;apply P_id_U52_bounded;assumption.
          intros ;apply P_id_nil_bounded;assumption.
          intros ;apply P_id_zeros_bounded;assumption.
          intros ;apply P_id_isNatList_bounded;assumption.
          intros ;apply P_id_a__U53_bounded;assumption.
          intros ;apply P_id_a__U31_bounded;assumption.
          intros ;apply P_id_isNatIList_bounded;assumption.
          intros ;apply P_id_a__isNatKind_bounded;assumption.
          intros ;apply P_id_a__isNatList_bounded;assumption.
          intros ;apply P_id_U32_bounded;assumption.
          intros ;apply P_id_mark_bounded;assumption.
          intros ;apply P_id_a__U43_bounded;assumption.
          intros ;apply P_id_U61_bounded;assumption.
          intros ;apply P_id_isNat_bounded;assumption.
          apply rules_monotonic.
          intros ;apply P_id_A__LENGTH_monotonic;assumption.
          intros ;apply P_id_A__U11_monotonic;assumption.
          intros ;apply P_id_A__U41_monotonic;assumption.
          intros ;apply P_id_A__ISNATKIND_monotonic;assumption.
          intros ;apply P_id_A__U22_monotonic;assumption.
          intros ;apply P_id_A__U51_monotonic;assumption.
          intros ;apply P_id_A__AND_monotonic;assumption.
          intros ;apply P_id_A__ISNATLIST_monotonic;assumption.
          intros ;apply P_id_A__U43_monotonic;assumption.
          intros ;apply P_id_A__U31_monotonic;assumption.
          intros ;apply P_id_A__U53_monotonic;assumption.
          intros ;apply P_id_MARK_monotonic;assumption.
          intros ;apply P_id_A__U12_monotonic;assumption.
          intros ;apply P_id_A__U42_monotonic;assumption.
          intros ;apply P_id_A__ISNAT_monotonic;assumption.
          intros ;apply P_id_A__U52_monotonic;assumption.
          intros ;apply P_id_A__ISNATILISTKIND_monotonic;assumption.
          intros ;apply P_id_A__U21_monotonic;assumption.
          intros ;apply P_id_A__ISNATILIST_monotonic;assumption.
          intros ;apply P_id_A__U32_monotonic;assumption.
          intros ;apply P_id_A__U61_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_12_large_scc_2_large_large_scc_1_strict_in_lt :
         Relation_Definitions.inclusion _ 
          DP_R_xml_0_scc_12_large_scc_2_large_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_12_large_scc_2_large_large_scc_1_large_in_le :
         Relation_Definitions.inclusion _ 
          DP_R_xml_0_scc_12_large_scc_2_large_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_12_large_scc_2_large_large_scc_1_large  := 
          WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large.wf.
        
        
        Lemma wf :
         well_founded WF_DP_R_xml_0_scc_12_large_scc_2_large_large.DP_R_xml_0_scc_12_large_scc_2_large_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_12_large_scc_2_large_large_scc_1_large).
          clear x.
          intros x _ IHx IHx'.
          constructor.
          intros y H.
          
          destruct H;
           (apply IHx';
             apply DP_R_xml_0_scc_12_large_scc_2_large_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_12_large_scc_2_large_large_scc_1_large_in_le;
              econstructor eassumption])).
          apply wf_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1_large.
        Qed.
       End WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1.
       
       Definition wf_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1  := 
         WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1.wf.
       
       
       Lemma acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1 :
        forall x y, 
         (DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1 x y) ->
          Acc WF_DP_R_xml_0_scc_12_large_scc_2_large.DP_R_xml_0_scc_12_large_scc_2_large_large 
           x.
       Proof.
         intros x.
         pattern x.
         apply (@Acc_ind _ DP_R_xml_0_scc_12_large_scc_2_large_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_12_large_scc_2_large_large_scc_1.
       Qed.
       
       
       Inductive DP_R_xml_0_scc_12_large_scc_2_large_large_non_scc_2  :
        algebra.Alg.term ->algebra.Alg.term ->Prop := 
          (* <a__isNatIList(cons(V1_,V2_)),a__and(a__isNatKind(V1_),isNatIListKind(V2_))> *)
         | DP_R_xml_0_scc_12_large_scc_2_large_large_non_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_12_large_scc_2_large_large_non_scc_2 (algebra.Alg.Term 
                                                                 algebra.F.id_a__and 
                                                                 ((algebra.Alg.Term 
                                                                 algebra.F.id_a__isNatKind 
                                                                 (x1::nil))::
                                                                 (algebra.Alg.Term 
                                                                 algebra.F.id_isNatIListKind 
                                                                 (x3::nil))::nil))
                                                                  
             (algebra.Alg.Term algebra.F.id_a__isNatIList (x11::nil))
       .
       
       
       Lemma acc_DP_R_xml_0_scc_12_large_scc_2_large_large_non_scc_2 :
        forall x y, 
         (DP_R_xml_0_scc_12_large_scc_2_large_large_non_scc_2 x y) ->
          Acc WF_DP_R_xml_0_scc_12_large_scc_2_large.DP_R_xml_0_scc_12_large_scc_2_large_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_12_large_scc_2_large_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_12_large_scc_2_large_large_non_scc_3  :
        algebra.Alg.term ->algebra.Alg.term ->Prop := 
          (* <a__isNatIList(cons(V1_,V2_)),a__isNatKind(V1_)> *)
         | DP_R_xml_0_scc_12_large_scc_2_large_large_non_scc_3_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_12_large_scc_2_large_large_non_scc_3 (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_scc_12_large_scc_2_large_large_non_scc_3 :
        forall x y, 
         (DP_R_xml_0_scc_12_large_scc_2_large_large_non_scc_3 x y) ->
          Acc WF_DP_R_xml_0_scc_12_large_scc_2_large.DP_R_xml_0_scc_12_large_scc_2_large_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_12_large_scc_2_large_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_12_large_scc_2_large_large_non_scc_4  :
        algebra.Alg.term ->algebra.Alg.term ->Prop := 
          (* <a__isNatIList(V_),a__isNatIListKind(V_)> *)
         | DP_R_xml_0_scc_12_large_scc_2_large_large_non_scc_4_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_scc_12_large_scc_2_large_large_non_scc_4 (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_scc_12_large_scc_2_large_large_non_scc_4 :
        forall x y, 
         (DP_R_xml_0_scc_12_large_scc_2_large_large_non_scc_4 x y) ->
          Acc WF_DP_R_xml_0_scc_12_large_scc_2_large.DP_R_xml_0_scc_12_large_scc_2_large_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_12_large_scc_2_large_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_12_large_scc_2_large_large_non_scc_5  :
        algebra.Alg.term ->algebra.Alg.term ->Prop := 
          (* <a__U31(tt,V_),a__isNatList(V_)> *)
         | DP_R_xml_0_scc_12_large_scc_2_large_large_non_scc_5_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_scc_12_large_scc_2_large_large_non_scc_5 (algebra.Alg.Term 
                                                                  algebra.F.id_a__isNatList 
                                                                  (x2::nil)) 
              (algebra.Alg.Term algebra.F.id_a__U31 (x12::x11::nil))
       .
       
       
       Lemma acc_DP_R_xml_0_scc_12_large_scc_2_large_large_non_scc_5 :
        forall x y, 
         (DP_R_xml_0_scc_12_large_scc_2_large_large_non_scc_5 x y) ->
          Acc WF_DP_R_xml_0_scc_12_large_scc_2_large.DP_R_xml_0_scc_12_large_scc_2_large_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_12_large_scc_2_large_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_12_large_scc_2_large_large_non_scc_6  :
        algebra.Alg.term ->algebra.Alg.term ->Prop := 
          (* <a__isNatIList(V_),a__U31(a__isNatIListKind(V_),V_)> *)
         | DP_R_xml_0_scc_12_large_scc_2_large_large_non_scc_6_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_scc_12_large_scc_2_large_large_non_scc_6 (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_scc_12_large_scc_2_large_large_non_scc_6 :
        forall x y, 
         (DP_R_xml_0_scc_12_large_scc_2_large_large_non_scc_6 x y) ->
          Acc WF_DP_R_xml_0_scc_12_large_scc_2_large.DP_R_xml_0_scc_12_large_scc_2_large_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_12_large_scc_2_large_large_non_scc_5;
             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_12_large_scc_2_large_large_non_scc_7  :
        algebra.Alg.term ->algebra.Alg.term ->Prop := 
          (* <a__U41(tt,V1_,V2_),a__isNat(V1_)> *)
         | DP_R_xml_0_scc_12_large_scc_2_large_large_non_scc_7_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_12_large_scc_2_large_large_non_scc_7 (algebra.Alg.Term 
                                                                   algebra.F.id_a__isNat 
                                                                   (x1::nil)) 
               (algebra.Alg.Term algebra.F.id_a__U41 (x13::x12::x11::nil))
       .
       
       
       Lemma acc_DP_R_xml_0_scc_12_large_scc_2_large_large_non_scc_7 :
        forall x y, 
         (DP_R_xml_0_scc_12_large_scc_2_large_large_non_scc_7 x y) ->
          Acc WF_DP_R_xml_0_scc_12_large_scc_2_large.DP_R_xml_0_scc_12_large_scc_2_large_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_12_large_scc_2_large_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_12_large_scc_2_large_large_scc_8  :
        algebra.Alg.term ->algebra.Alg.term ->Prop := 
          (* <a__U41(tt,V1_,V2_),a__U42(a__isNat(V1_),V2_)> *)
         | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_8_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_12_large_scc_2_large_large_scc_8 (algebra.Alg.Term 
                                                               algebra.F.id_a__U42 
                                                               ((algebra.Alg.Term 
                                                               algebra.F.id_a__isNat 
                                                               (x1::nil))::
                                                               x3::nil)) 
               (algebra.Alg.Term algebra.F.id_a__U41 (x13::x12::x11::nil))
         
          (* <a__U42(tt,V2_),a__isNatIList(V2_)> *)
         | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_8_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_12_large_scc_2_large_large_scc_8 (algebra.Alg.Term 
                                                              algebra.F.id_a__isNatIList 
                                                              (x3::nil)) 
              (algebra.Alg.Term algebra.F.id_a__U42 (x12::x11::nil))
         
          (* <a__isNatIList(cons(V1_,V2_)),a__U41(a__and(a__isNatKind(V1_),isNatIListKind(V2_)),V1_,V2_)> *)
         | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_8_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_12_large_scc_2_large_large_scc_8 (algebra.Alg.Term 
                                                             algebra.F.id_a__U41 
                                                             ((algebra.Alg.Term 
                                                             algebra.F.id_a__and 
                                                             ((algebra.Alg.Term 
                                                             algebra.F.id_a__isNatKind 
                                                             (x1::nil))::
                                                             (algebra.Alg.Term 
                                                             algebra.F.id_isNatIListKind 
                                                             (x3::nil))::nil))::
                                                             x1::x3::nil)) 
             (algebra.Alg.Term algebra.F.id_a__isNatIList (x11::nil))
       .
       
       
       Module WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_8.
        Inductive DP_R_xml_0_scc_12_large_scc_2_large_large_scc_8_large  :
         algebra.Alg.term ->algebra.Alg.term ->Prop := 
           (* <a__U41(tt,V1_,V2_),a__U42(a__isNat(V1_),V2_)> *)
          | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_8_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_12_large_scc_2_large_large_scc_8_large 
                (algebra.Alg.Term algebra.F.id_a__U42 ((algebra.Alg.Term 
                 algebra.F.id_a__isNat (x1::nil))::x3::nil)) 
                (algebra.Alg.Term algebra.F.id_a__U41 (x13::x12::x11::nil))
          
           (* <a__isNatIList(cons(V1_,V2_)),a__U41(a__and(a__isNatKind(V1_),isNatIListKind(V2_)),V1_,V2_)> *)
          | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_8_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_12_large_scc_2_large_large_scc_8_large (algebra.Alg.Term 
                                                                    algebra.F.id_a__U41 
                                                                    ((
                                                                    algebra.Alg.Term 
                                                                    algebra.F.id_a__and 
                                                                    ((
                                                                    algebra.Alg.Term 
                                                                    algebra.F.id_a__isNatKind 
                                                                    (x1::nil))::
                                                                    
                                                                    (
                                                                    algebra.Alg.Term 
                                                                    algebra.F.id_isNatIListKind 
                                                                    (x3::nil))::nil))::
                                                                    x1::
                                                                    x3::nil)) 
              (algebra.Alg.Term algebra.F.id_a__isNatIList (x11::nil))
        .
        
        
        Inductive DP_R_xml_0_scc_12_large_scc_2_large_large_scc_8_strict  :
         algebra.Alg.term ->algebra.Alg.term ->Prop := 
           (* <a__U42(tt,V2_),a__isNatIList(V2_)> *)
          | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_8_strict_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_12_large_scc_2_large_large_scc_8_strict 
               (algebra.Alg.Term algebra.F.id_a__isNatIList (x3::nil)) 
               (algebra.Alg.Term algebra.F.id_a__U42 (x12::x11::nil))
        .
        
        
        Module WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_8_large.
         Inductive DP_R_xml_0_scc_12_large_scc_2_large_large_scc_8_large_non_scc_1  :
          algebra.Alg.term ->algebra.Alg.term ->Prop := 
            (* <a__U41(tt,V1_,V2_),a__U42(a__isNat(V1_),V2_)> *)
           | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_8_large_non_scc_1_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_12_large_scc_2_large_large_scc_8_large_non_scc_1 
                 (algebra.Alg.Term algebra.F.id_a__U42 ((algebra.Alg.Term 
                  algebra.F.id_a__isNat (x1::nil))::x3::nil)) 
                 (algebra.Alg.Term algebra.F.id_a__U41 (x13::x12::x11::nil))
         .
         
         
         Lemma acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_8_large_non_scc_1 :
          forall x y, 
           (DP_R_xml_0_scc_12_large_scc_2_large_large_scc_8_large_non_scc_1 
             x y) ->
            Acc WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_8.DP_R_xml_0_scc_12_large_scc_2_large_large_scc_8_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_12_large_scc_2_large_large_scc_8_large_non_scc_2  :
          algebra.Alg.term ->algebra.Alg.term ->Prop := 
            (* <a__isNatIList(cons(V1_,V2_)),a__U41(a__and(a__isNatKind(V1_),isNatIListKind(V2_)),V1_,V2_)> *)
           | DP_R_xml_0_scc_12_large_scc_2_large_large_scc_8_large_non_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_12_large_scc_2_large_large_scc_8_large_non_scc_2 
               (algebra.Alg.Term algebra.F.id_a__U41 ((algebra.Alg.Term 
                algebra.F.id_a__and ((algebra.Alg.Term 
                algebra.F.id_a__isNatKind (x1::nil))::(algebra.Alg.Term 
                algebra.F.id_isNatIListKind (x3::nil))::nil))::x1::x3::nil)) 
               (algebra.Alg.Term algebra.F.id_a__isNatIList (x11::nil))
         .
         
         
         Lemma acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_8_large_non_scc_2 :
          forall x y, 
           (DP_R_xml_0_scc_12_large_scc_2_large_large_scc_8_large_non_scc_2 
             x y) ->
            Acc WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_8.DP_R_xml_0_scc_12_large_scc_2_large_large_scc_8_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_12_large_scc_2_large_large_scc_8_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.
         
         
         Lemma wf :
          well_founded WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_8.DP_R_xml_0_scc_12_large_scc_2_large_large_scc_8_large
           .
         Proof.
           constructor;intros _y _h;inversion _h;clear _h;subst;
            (eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_8_large_non_scc_2;
              econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
            ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_8_large_non_scc_1;
               econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
             ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_8_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_12_large_scc_2_large_large_scc_8_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_isNatKind (x11:Z) := 1.
        
        Definition P_id_a__isNatIList (x11:Z) := 2.
        
        Definition P_id_a__U21 (x11:Z) (x12:Z) := 1.
        
        Definition P_id_U41 (x11:Z) (x12:Z) (x13:Z) := 1.
        
        Definition P_id_a__and (x11:Z) (x12:Z) := 1 + 2* x12.
        
        Definition P_id_a__U11 (x11:Z) (x12:Z) := 1.
        
        Definition P_id_U21 (x11:Z) (x12:Z) := 0.
        
        Definition P_id_a__U61 (x11:Z) (x12:Z) := 1.
        
        Definition P_id_a__U32 (x11:Z) := 2.
        
        Definition P_id_U51 (x11:Z) (x12:Z) (x13:Z) := 0.
        
        Definition P_id_isNatIListKind (x11:Z) := 0.
        
        Definition P_id_0  := 1.
        
        Definition P_id_U12 (x11:Z) := 1.
        
        Definition P_id_a__U52 (x11:Z) (x12:Z) := 1.
        
        Definition P_id_a__isNat (x11:Z) := 1* x11.
        
        Definition P_id_U43 (x11:Z) := 0.
        
        Definition P_id_a__isNatIListKind (x11:Z) := 1.
        
        Definition P_id_a__U12 (x11:Z) := 1.
        
        Definition P_id_U31 (x11:Z) (x12:Z) := 2.
        
        Definition P_id_a__length (x11:Z) := 1.
        
        Definition P_id_a__U42 (x11:Z) (x12:Z) := 1.
        
        Definition P_id_U53 (x11:Z) := 1.
        
        Definition P_id_and (x11:Z) (x12:Z) := 2* x12.
        
        Definition P_id_cons (x11:Z) (x12:Z) := 1* x11 + 1* x12.
        
        Definition P_id_U11 (x11:Z) (x12:Z) := 1.
        
        Definition P_id_a__U51 (x11:Z) (x12:Z) (x13:Z) := 1.
        
        Definition P_id_a__U22 (x11:Z) := 1.
        
        Definition P_id_U42 (x11:Z) (x12:Z) := 0.
        
        Definition P_id_length (x11:Z) := 1.
        
        Definition P_id_tt  := 1.
        
        Definition P_id_U22 (x11:Z) := 1.
        
        Definition P_id_s (x11:Z) := 1.
        
        Definition P_id_a__U41 (x11:Z) (x12:Z) (x13:Z) := 1.
        
        Definition P_id_U52 (x11:Z) (x12:Z) := 0.
        
        Definition P_id_nil  := 0.
        
        Definition P_id_zeros  := 0.
        
        Definition P_id_isNatList (x11:Z) := 0.
        
        Definition P_id_a__U53 (x11:Z) := 1.
        
        Definition P_id_a__U31 (x11:Z) (x12:Z) := 2.
        
        Definition P_id_isNatIList (x11:Z) := 2.
        
        Definition P_id_a__isNatKind (x11:Z) := 2.
        
        Definition P_id_a__isNatList (x11:Z) := 1.
        
        Definition P_id_U32 (x11:Z) := 2.
        
        Definition P_id_mark (x11:Z) := 1 + 1* x11.
        
        Definition P_id_a__U43 (x11:Z) := 1.
        
        Definition P_id_U61 (x11:Z) (x12:Z) := 0.
        
        Definition P_id_isNat (x11:Z) := 1* x11.
        
        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__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__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_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_a__and_monotonic :
         forall x12 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_a__and x12 x14 <= P_id_a__and x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          
          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_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__U61_monotonic :
         forall x12 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_a__U61 x12 x14 <= P_id_a__U61 x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          
          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 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_a__U32 x12 <= P_id_a__U32 x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U51_monotonic :
         forall x16 x12 x14 x13 x11 x15, 
          (0 <= x16)/\ (x16 <= x15) ->
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_U51 x12 x14 x16 <= P_id_U51 x11 x13 x15.
        Proof.
          intros x16 x15 x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          intros [H_5 H_4].
          
          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_U12_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_U12 x12 <= P_id_U12 x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_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 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_a__U52 x12 x14 <= P_id_a__U52 x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_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_U43_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_U43 x12 <= P_id_U43 x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_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__U12_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_a__U12 x12 <= P_id_a__U12 x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          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__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__U42_monotonic :
         forall x12 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_a__U42 x12 x14 <= P_id_a__U42 x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U53_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_U53 x12 <= P_id_U53 x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_and_monotonic :
         forall x12 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->P_id_and x12 x14 <= P_id_and x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          
          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_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__U51_monotonic :
         forall x16 x12 x14 x13 x11 x15, 
          (0 <= x16)/\ (x16 <= x15) ->
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_a__U51 x12 x14 x16 <= P_id_a__U51 x11 x13 x15.
        Proof.
          intros x16 x15 x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          intros [H_5 H_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 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_a__U22 x12 <= P_id_a__U22 x11.
        Proof.
          intros x12 x11.
          intros [H_1 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 x12 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->P_id_U42 x12 x14 <= P_id_U42 x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          
          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_U22_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_U22 x12 <= P_id_U22 x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          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__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_U52_monotonic :
         forall x12 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->P_id_U52 x12 x14 <= P_id_U52 x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 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__U53_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_a__U53 x12 <= P_id_a__U53 x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          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_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__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__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_U32_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_U32 x12 <= P_id_U32 x11.
        Proof.
          intros x12 x11.
          intros [H_1 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__U43_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_a__U43 x12 <= P_id_a__U43 x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          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 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->P_id_U61 x12 x14 <= P_id_U61 x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          
          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__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_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__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_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_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_a__and_bounded :
         forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__and x12 x11.
        Proof.
          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_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__U61_bounded :
         forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U61 x12 x11.
        Proof.
          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 x11, (0 <= x11) ->0 <= P_id_a__U32 x11.
        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 x13 x11, 
          (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U51 x13 x12 x11.
        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_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_U12_bounded : forall x11, (0 <= x11) ->0 <= P_id_U12 x11.
        Proof.
          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 x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U52 x12 x11.
        Proof.
          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_U43_bounded : forall x11, (0 <= x11) ->0 <= P_id_U43 x11.
        Proof.
          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_a__U12_bounded :
         forall x11, (0 <= x11) ->0 <= P_id_a__U12 x11.
        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__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__U42_bounded :
         forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U42 x12 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_U53_bounded : forall x11, (0 <= x11) ->0 <= P_id_U53 x11.
        Proof.
          intros .
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_and_bounded :
         forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_and x12 x11.
        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_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__U51_bounded :
         forall x12 x13 x11, 
          (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U51 x13 x12 x11.
        Proof.
          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 x11, (0 <= x11) ->0 <= P_id_a__U22 x11.
        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 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U42 x12 x11.
        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_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_U22_bounded : forall x11, (0 <= x11) ->0 <= P_id_U22 x11.
        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__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_U52_bounded :
         forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U52 x12 x11.
        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_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_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__U53_bounded :
         forall x11, (0 <= x11) ->0 <= P_id_a__U53 x11.
        Proof.
          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_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.
        
        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_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_U32_bounded : forall x11, (0 <= x11) ->0 <= P_id_U32 x11.
        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__U43_bounded :
         forall x11, (0 <= x11) ->0 <= P_id_a__U43 x11.
        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 x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U61 x12 x11.
        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.
        
        Definition measure  := 
          InterpZ.measure 0 P_id_a__zeros P_id_isNatKind P_id_a__isNatIList 
           P_id_a__U21 P_id_U41 P_id_a__and P_id_a__U11 P_id_U21 P_id_a__U61 
           P_id_a__U32 P_id_U51 P_id_isNatIListKind P_id_0 P_id_U12 
           P_id_a__U52 P_id_a__isNat P_id_U43 P_id_a__isNatIListKind 
           P_id_a__U12 P_id_U31 P_id_a__length P_id_a__U42 P_id_U53 P_id_and 
           P_id_cons P_id_U11 P_id_a__U51 P_id_a__U22 P_id_U42 P_id_length 
           P_id_tt P_id_U22 P_id_s P_id_a__U41 P_id_U52 P_id_nil P_id_zeros 
           P_id_isNatList P_id_a__U53 P_id_a__U31 P_id_isNatIList 
           P_id_a__isNatKind P_id_a__isNatList P_id_U32 P_id_mark 
           P_id_a__U43 P_id_U61 P_id_isNat.
        
        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_isNatKind 
                           (x11::nil)) =>
                         P_id_isNatKind (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__U21 (x12::
                           x11::nil)) =>
                         P_id_a__U21 (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_a__and (x12::
                           x11::nil)) =>
                         P_id_a__and (measure x12) (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_U21 (x12::x11::nil)) =>
                         P_id_U21 (measure x12) (measure x11)
                        | (algebra.Alg.Term algebra.F.id_a__U61 (x12::
                           x11::nil)) =>
                         P_id_a__U61 (measure x12) (measure x11)
                        | (algebra.Alg.Term algebra.F.id_a__U32 (x11::nil)) =>
                         P_id_a__U32 (measure x11)
                        | (algebra.Alg.Term algebra.F.id_U51 (x13::x12::
                           x11::nil)) =>
                         P_id_U51 (measure x13) (measure x12) (measure x11)
                        | (algebra.Alg.Term algebra.F.id_isNatIListKind 
                           (x11::nil)) =>
                         P_id_isNatIListKind (measure x11)
                        | (algebra.Alg.Term algebra.F.id_0 nil) => P_id_0 
                        | (algebra.Alg.Term algebra.F.id_U12 (x11::nil)) =>
                         P_id_U12 (measure x11)
                        | (algebra.Alg.Term algebra.F.id_a__U52 (x12::
                           x11::nil)) =>
                         P_id_a__U52 (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_U43 (x11::nil)) =>
                         P_id_U43 (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__U12 (x11::nil)) =>
                         P_id_a__U12 (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__length 
                           (x11::nil)) =>
                         P_id_a__length (measure x11)
                        | (algebra.Alg.Term algebra.F.id_a__U42 (x12::
                           x11::nil)) =>
                         P_id_a__U42 (measure x12) (measure x11)
                        | (algebra.Alg.Term algebra.F.id_U53 (x11::nil)) =>
                         P_id_U53 (measure x11)
                        | (algebra.Alg.Term algebra.F.id_and (x12::x11::nil)) =>
                         P_id_and (measure x12) (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_U11 (x12::x11::nil)) =>
                         P_id_U11 (measure x12) (measure x11)
                        | (algebra.Alg.Term algebra.F.id_a__U51 (x13::x12::
                           x11::nil)) =>
                         P_id_a__U51 (measure x13) (measure x12) 
                          (measure x11)
                        | (algebra.Alg.Term algebra.F.id_a__U22 (x11::nil)) =>
                         P_id_a__U22 (measure x11)
                        | (algebra.Alg.Term algebra.F.id_U42 (x12::x11::nil)) =>
                         P_id_U42 (measure x12) (measure x11)
                        | (algebra.Alg.Term algebra.F.id_length (x11::nil)) =>
                         P_id_length (measure x11)
                        | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt 
                        | (algebra.Alg.Term algebra.F.id_U22 (x11::nil)) =>
                         P_id_U22 (measure x11)
                        | (algebra.Alg.Term algebra.F.id_s (x11::nil)) =>
                         P_id_s (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_U52 (x12::x11::nil)) =>
                         P_id_U52 (measure x12) (measure x11)
                        | (algebra.Alg.Term algebra.F.id_nil nil) =>
                         P_id_nil 
                        | (algebra.Alg.Term algebra.F.id_zeros nil) =>
                         P_id_zeros 
                        | (algebra.Alg.Term algebra.F.id_isNatList 
                           (x11::nil)) =>
                         P_id_isNatList (measure x11)
                        | (algebra.Alg.Term algebra.F.id_a__U53 (x11::nil)) =>
                         P_id_a__U53 (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_isNatIList 
                           (x11::nil)) =>
                         P_id_isNatIList (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__isNatList 
                           (x11::nil)) =>
                         P_id_a__isNatList (measure x11)
                        | (algebra.Alg.Term algebra.F.id_U32 (x11::nil)) =>
                         P_id_U32 (measure x11)
                        | (algebra.Alg.Term algebra.F.id_mark (x11::nil)) =>
                         P_id_mark (measure x11)
                        | (algebra.Alg.Term algebra.F.id_a__U43 (x11::nil)) =>
                         P_id_a__U43 (measure x11)
                        | (algebra.Alg.Term algebra.F.id_U61 (x12::x11::nil)) =>
                         P_id_U61 (measure x12) (measure x11)
                        | (algebra.Alg.Term algebra.F.id_isNat (x11::nil)) =>
                         P_id_isNat (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_isNatKind_monotonic;assumption.
          intros ;apply P_id_a__isNatIList_monotonic;assumption.
          intros ;apply P_id_a__U21_monotonic;assumption.
          intros ;apply P_id_U41_monotonic;assumption.
          intros ;apply P_id_a__and_monotonic;assumption.
          intros ;apply P_id_a__U11_monotonic;assumption.
          intros ;apply P_id_U21_monotonic;assumption.
          intros ;apply P_id_a__U61_monotonic;assumption.
          intros ;apply P_id_a__U32_monotonic;assumption.
          intros ;apply P_id_U51_monotonic;assumption.
          intros ;apply P_id_isNatIListKind_monotonic;assumption.
          intros ;apply P_id_U12_monotonic;assumption.
          intros ;apply P_id_a__U52_monotonic;assumption.
          intros ;apply P_id_a__isNat_monotonic;assumption.
          intros ;apply P_id_U43_monotonic;assumption.
          intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
          intros ;apply P_id_a__U12_monotonic;assumption.
          intros ;apply P_id_U31_monotonic;assumption.
          intros ;apply P_id_a__length_monotonic;assumption.
          intros ;apply P_id_a__U42_monotonic;assumption.
          intros ;apply P_id_U53_monotonic;assumption.
          intros ;apply P_id_and_monotonic;assumption.
          intros ;apply P_id_cons_monotonic;assumption.
          intros ;apply P_id_U11_monotonic;assumption.
          intros ;apply P_id_a__U51_monotonic;assumption.
          intros ;apply P_id_a__U22_monotonic;assumption.
          intros ;apply P_id_U42_monotonic;assumption.
          intros ;apply P_id_length_monotonic;assumption.
          intros ;apply P_id_U22_monotonic;assumption.
          intros ;apply P_id_s_monotonic;assumption.
          intros ;apply P_id_a__U41_monotonic;assumption.
          intros ;apply P_id_U52_monotonic;assumption.
          intros ;apply P_id_isNatList_monotonic;assumption.
          intros ;apply P_id_a__U53_monotonic;assumption.
          intros ;apply P_id_a__U31_monotonic;assumption.
          intros ;apply P_id_isNatIList_monotonic;assumption.
          intros ;apply P_id_a__isNatKind_monotonic;assumption.
          intros ;apply P_id_a__isNatList_monotonic;assumption.
          intros ;apply P_id_U32_monotonic;assumption.
          intros ;apply P_id_mark_monotonic;assumption.
          intros ;apply P_id_a__U43_monotonic;assumption.
          intros ;apply P_id_U61_monotonic;assumption.
          intros ;apply P_id_isNat_monotonic;assumption.
          intros ;apply P_id_a__zeros_bounded;assumption.
          intros ;apply P_id_isNatKind_bounded;assumption.
          intros ;apply P_id_a__isNatIList_bounded;assumption.
          intros ;apply P_id_a__U21_bounded;assumption.
          intros ;apply P_id_U41_bounded;assumption.
          intros ;apply P_id_a__and_bounded;assumption.
          intros ;apply P_id_a__U11_bounded;assumption.
          intros ;apply P_id_U21_bounded;assumption.
          intros ;apply P_id_a__U61_bounded;assumption.
          intros ;apply P_id_a__U32_bounded;assumption.
          intros ;apply P_id_U51_bounded;assumption.
          intros ;apply P_id_isNatIListKind_bounded;assumption.
          intros ;apply P_id_0_bounded;assumption.
          intros ;apply P_id_U12_bounded;assumption.
          intros ;apply P_id_a__U52_bounded;assumption.
          intros ;apply P_id_a__isNat_bounded;assumption.
          intros ;apply P_id_U43_bounded;assumption.
          intros ;apply P_id_a__isNatIListKind_bounded;assumption.
          intros ;apply P_id_a__U12_bounded;assumption.
          intros ;apply P_id_U31_bounded;assumption.
          intros ;apply P_id_a__length_bounded;assumption.
          intros ;apply P_id_a__U42_bounded;assumption.
          intros ;apply P_id_U53_bounded;assumption.
          intros ;apply P_id_and_bounded;assumption.
          intros ;apply P_id_cons_bounded;assumption.
          intros ;apply P_id_U11_bounded;assumption.
          intros ;apply P_id_a__U51_bounded;assumption.
          intros ;apply P_id_a__U22_bounded;assumption.
          intros ;apply P_id_U42_bounded;assumption.
          intros ;apply P_id_length_bounded;assumption.
          intros ;apply P_id_tt_bounded;assumption.
          intros ;apply P_id_U22_bounded;assumption.
          intros ;apply P_id_s_bounded;assumption.
          intros ;apply P_id_a__U41_bounded;assumption.
          intros ;apply P_id_U52_bounded;assumption.
          intros ;apply P_id_nil_bounded;assumption.
          intros ;apply P_id_zeros_bounded;assumption.
          intros ;apply P_id_isNatList_bounded;assumption.
          intros ;apply P_id_a__U53_bounded;assumption.
          intros ;apply P_id_a__U31_bounded;assumption.
          intros ;apply P_id_isNatIList_bounded;assumption.
          intros ;apply P_id_a__isNatKind_bounded;assumption.
          intros ;apply P_id_a__isNatList_bounded;assumption.
          intros ;apply P_id_U32_bounded;assumption.
          intros ;apply P_id_mark_bounded;assumption.
          intros ;apply P_id_a__U43_bounded;assumption.
          intros ;apply P_id_U61_bounded;assumption.
          intros ;apply P_id_isNat_bounded;assumption.
          apply rules_monotonic.
        Qed.
        
        Definition P_id_A__LENGTH (x11:Z) := 0.
        
        Definition P_id_A__U11 (x11:Z) (x12:Z) := 0.
        
        Definition P_id_A__U41 (x11:Z) (x12:Z) (x13:Z) := 1* x12 + 2* x13.
        
        Definition P_id_A__ISNATKIND (x11:Z) := 0.
        
        Definition P_id_A__U22 (x11:Z) := 0.
        
        Definition P_id_A__U51 (x11:Z) (x12:Z) (x13:Z) := 0.
        
        Definition P_id_A__AND (x11:Z) (x12:Z) := 0.
        
        Definition P_id_A__ISNATLIST (x11:Z) := 0.
        
        Definition P_id_A__U43 (x11:Z) := 0.
        
        Definition P_id_A__U31 (x11:Z) (x12:Z) := 0.
        
        Definition P_id_A__U53 (x11:Z) := 0.
        
        Definition P_id_MARK (x11:Z) := 0.
        
        Definition P_id_A__U12 (x11:Z) := 0.
        
        Definition P_id_A__U42 (x11:Z) (x12:Z) := 1* x11 + 2* x12.
        
        Definition P_id_A__ZEROS  := 0.
        
        Definition P_id_A__ISNAT (x11:Z) := 0.
        
        Definition P_id_A__U52 (x11:Z) (x12:Z) := 0.
        
        Definition P_id_A__ISNATILISTKIND (x11:Z) := 0.
        
        Definition P_id_A__U21 (x11:Z) (x12:Z) := 0.
        
        Definition P_id_A__ISNATILIST (x11:Z) := 2* x11.
        
        Definition P_id_A__U32 (x11:Z) := 0.
        
        Definition P_id_A__U61 (x11:Z) (x12:Z) := 0.
        
        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__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__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__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__U22_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_A__U22 x12 <= P_id_A__U22 x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_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 x16 x12 x14 x13 x11 x15, 
          (0 <= x16)/\ (x16 <= x15) ->
           (0 <= x14)/\ (x14 <= x13) ->
            (0 <= x12)/\ (x12 <= x11) ->
             P_id_A__U51 x12 x14 x16 <= P_id_A__U51 x11 x13 x15.
        Proof.
          intros x16 x15 x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          intros [H_5 H_4].
          
          cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
           (auto with zarith)||(repeat (translate_vars );prove_ineq ).
        Qed.
        
        Lemma P_id_A__AND_monotonic :
         forall x12 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_A__AND x12 x14 <= P_id_A__AND x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_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__U43_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_A__U43 x12 <= P_id_A__U43 x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          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__U53_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_A__U53 x12 <= P_id_A__U53 x11.
        Proof.
          intros x12 x11.
          intros [H_1 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__U12_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_A__U12 x12 <= P_id_A__U12 x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_0].
          
          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 x12 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_A__U42 x12 x14 <= P_id_A__U42 x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_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_A__U52_monotonic :
         forall x12 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_A__U52 x12 x14 <= P_id_A__U52 x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          intros [H_3 H_2].
          
          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__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__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__U32_monotonic :
         forall x12 x11, 
          (0 <= x12)/\ (x12 <= x11) ->P_id_A__U32 x12 <= P_id_A__U32 x11.
        Proof.
          intros x12 x11.
          intros [H_1 H_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 x14 x13 x11, 
          (0 <= x14)/\ (x14 <= x13) ->
           (0 <= x12)/\ (x12 <= x11) ->
            P_id_A__U61 x12 x14 <= P_id_A__U61 x11 x13.
        Proof.
          intros x14 x13 x12 x11.
          intros [H_1 H_0].
          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_isNatKind 
           P_id_a__isNatIList P_id_a__U21 P_id_U41 P_id_a__and P_id_a__U11 
           P_id_U21 P_id_a__U61 P_id_a__U32 P_id_U51 P_id_isNatIListKind 
           P_id_0 P_id_U12 P_id_a__U52 P_id_a__isNat P_id_U43 
           P_id_a__isNatIListKind P_id_a__U12 P_id_U31 P_id_a__length 
           P_id_a__U42 P_id_U53 P_id_and P_id_cons P_id_U11 P_id_a__U51 
           P_id_a__U22 P_id_U42 P_id_length P_id_tt P_id_U22 P_id_s 
           P_id_a__U41 P_id_U52 P_id_nil P_id_zeros P_id_isNatList 
           P_id_a__U53 P_id_a__U31 P_id_isNatIList P_id_a__isNatKind 
           P_id_a__isNatList P_id_U32 P_id_mark P_id_a__U43 P_id_U61 
           P_id_isNat P_id_A__LENGTH P_id_A__U11 P_id_A__U41 
           P_id_A__ISNATKIND P_id_A__U22 P_id_A__U51 P_id_A__AND 
           P_id_A__ISNATLIST P_id_A__U43 P_id_A__U31 P_id_A__U53 P_id_MARK 
           P_id_A__U12 P_id_A__U42 P_id_A__ZEROS P_id_A__ISNAT P_id_A__U52 
           P_id_A__ISNATILISTKIND P_id_A__U21 P_id_A__ISNATILIST P_id_A__U32 
           P_id_A__U61.
        
        Lemma marked_measure_equation :
         forall t, 
          marked_measure t = match t with
                               | (algebra.Alg.Term algebra.F.id_a__length 
                                  (x11::nil)) =>
                                P_id_A__LENGTH (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__U41 (x13::
                                  x12::x11::nil)) =>
                                P_id_A__U41 (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__U22 
                                  (x11::nil)) =>
                                P_id_A__U22 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U51 (x13::
                                  x12::x11::nil)) =>
                                P_id_A__U51 (measure x13) (measure x12) 
                                 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__and (x12::
                                  x11::nil)) =>
                                P_id_A__AND (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__U43 
                                  (x11::nil)) =>
                                P_id_A__U43 (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__U53 
                                  (x11::nil)) =>
                                P_id_A__U53 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_mark 
                                  (x11::nil)) =>
                                P_id_MARK (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U12 
                                  (x11::nil)) =>
                                P_id_A__U12 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U42 (x12::
                                  x11::nil)) =>
                                P_id_A__U42 (measure x12) (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__zeros nil) =>
                                P_id_A__ZEROS 
                               | (algebra.Alg.Term algebra.F.id_a__isNat 
                                  (x11::nil)) =>
                                P_id_A__ISNAT (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U52 (x12::
                                  x11::nil)) =>
                                P_id_A__U52 (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_a__U21 (x12::
                                  x11::nil)) =>
                                P_id_A__U21 (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__U32 
                                  (x11::nil)) =>
                                P_id_A__U32 (measure x11)
                               | (algebra.Alg.Term algebra.F.id_a__U61 (x12::
                                  x11::nil)) =>
                                P_id_A__U61 (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_isNatKind_monotonic;assumption.
          intros ;apply P_id_a__isNatIList_monotonic;assumption.
          intros ;apply P_id_a__U21_monotonic;assumption.
          intros ;apply P_id_U41_monotonic;assumption.
          intros ;apply P_id_a__and_monotonic;assumption.
          intros ;apply P_id_a__U11_monotonic;assumption.
          intros ;apply P_id_U21_monotonic;assumption.
          intros ;apply P_id_a__U61_monotonic;assumption.
          intros ;apply P_id_a__U32_monotonic;assumption.
          intros ;apply P_id_U51_monotonic;assumption.
          intros ;apply P_id_isNatIListKind_monotonic;assumption.
          intros ;apply P_id_U12_monotonic;assumption.
          intros ;apply P_id_a__U52_monotonic;assumption.
          intros ;apply P_id_a__isNat_monotonic;assumption.
          intros ;apply P_id_U43_monotonic;assumption.
          intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
          intros ;apply P_id_a__U12_monotonic;assumption.
          intros ;apply P_id_U31_monotonic;assumption.
          intros ;apply P_id_a__length_monotonic;assumption.
          intros ;apply P_id_a__U42_monotonic;assumption.
          intros ;apply P_id_U53_monotonic;assumption.
          intros ;apply P_id_and_monotonic;assumption.
          intros ;apply P_id_cons_monotonic;assumption.
          intros ;apply P_id_U11_monotonic;assumption.
          intros ;apply P_id_a__U51_monotonic;assumption.
          intros ;apply P_id_a__U22_monotonic;assumption.
          intros ;apply P_id_U42_monotonic;assumption.
          intros ;apply P_id_length_monotonic;assumption.
          intros ;apply P_id_U22_monotonic;assumption.
          intros ;apply P_id_s_monotonic;assumption.
          intros ;apply P_id_a__U41_monotonic;assumption.
          intros ;apply P_id_U52_monotonic;assumption.
          intros ;apply P_id_isNatList_monotonic;assumption.
          intros ;apply P_id_a__U53_monotonic;assumption.
          intros ;apply P_id_a__U31_monotonic;assumption.
          intros ;apply P_id_isNatIList_monotonic;assumption.
          intros ;apply P_id_a__isNatKind_monotonic;assumption.
          intros ;apply P_id_a__isNatList_monotonic;assumption.
          intros ;apply P_id_U32_monotonic;assumption.
          intros ;apply P_id_mark_monotonic;assumption.
          intros ;apply P_id_a__U43_monotonic;assumption.
          intros ;apply P_id_U61_monotonic;assumption.
          intros ;apply P_id_isNat_monotonic;assumption.
          intros ;apply P_id_a__zeros_bounded;assumption.
          intros ;apply P_id_isNatKind_bounded;assumption.
          intros ;apply P_id_a__isNatIList_bounded;assumption.
          intros ;apply P_id_a__U21_bounded;assumption.
          intros ;apply P_id_U41_bounded;assumption.
          intros ;apply P_id_a__and_bounded;assumption.
          intros ;apply P_id_a__U11_bounded;assumption.
          intros ;apply P_id_U21_bounded;assumption.
          intros ;apply P_id_a__U61_bounded;assumption.
          intros ;apply P_id_a__U32_bounded;assumption.
          intros ;apply P_id_U51_bounded;assumption.
          intros ;apply P_id_isNatIListKind_bounded;assumption.
          intros ;apply P_id_0_bounded;assumption.
          intros ;apply P_id_U12_bounded;assumption.
          intros ;apply P_id_a__U52_bounded;assumption.
          intros ;apply P_id_a__isNat_bounded;assumption.
          intros ;apply P_id_U43_bounded;assumption.
          intros ;apply P_id_a__isNatIListKind_bounded;assumption.
          intros ;apply P_id_a__U12_bounded;assumption.
          intros ;apply P_id_U31_bounded;assumption.
          intros ;apply P_id_a__length_bounded;assumption.
          intros ;apply P_id_a__U42_bounded;assumption.
          intros ;apply P_id_U53_bounded;assumption.
          intros ;apply P_id_and_bounded;assumption.
          intros ;apply P_id_cons_bounded;assumption.
          intros ;apply P_id_U11_bounded;assumption.
          intros ;apply P_id_a__U51_bounded;assumption.
          intros ;apply P_id_a__U22_bounded;assumption.
          intros ;apply P_id_U42_bounded;assumption.
          intros ;apply P_id_length_bounded;assumption.
          intros ;apply P_id_tt_bounded;assumption.
          intros ;apply P_id_U22_bounded;assumption.
          intros ;apply P_id_s_bounded;assumption.
          intros ;apply P_id_a__U41_bounded;assumption.
          intros ;apply P_id_U52_bounded;assumption.
          intros ;apply P_id_nil_bounded;assumption.
          intros ;apply P_id_zeros_bounded;assumption.
          intros ;apply P_id_isNatList_bounded;assumption.
          intros ;apply P_id_a__U53_bounded;assumption.
          intros ;apply P_id_a__U31_bounded;assumption.
          intros ;apply P_id_isNatIList_bounded;assumption.
          intros ;apply P_id_a__isNatKind_bounded;assumption.
          intros ;apply P_id_a__isNatList_bounded;assumption.
          intros ;apply P_id_U32_bounded;assumption.
          intros ;apply P_id_mark_bounded;assumption.
          intros ;apply P_id_a__U43_bounded;assumption.
          intros ;apply P_id_U61_bounded;assumption.
          intros ;apply P_id_isNat_bounded;assumption.
          apply rules_monotonic.
          intros ;apply P_id_A__LENGTH_monotonic;assumption.
          intros ;apply P_id_A__U11_monotonic;assumption.
          intros ;apply P_id_A__U41_monotonic;assumption.
          intros ;apply P_id_A__ISNATKIND_monotonic;assumption.
          intros ;apply P_id_A__U22_monotonic;assumption.
          intros ;apply P_id_A__U51_monotonic;assumption.
          intros ;apply P_id_A__AND_monotonic;assumption.
          intros ;apply P_id_A__ISNATLIST_monotonic;assumption.
          intros ;apply P_id_A__U43_monotonic;assumption.
          intros ;apply P_id_A__U31_monotonic;assumption.
          intros ;apply P_id_A__U53_monotonic;assumption.
          intros ;apply P_id_MARK_monotonic;assumption.
          intros ;apply P_id_A__U12_monotonic;assumption.
          intros ;apply P_id_A__U42_monotonic;assumption.
          intros ;apply P_id_A__ISNAT_monotonic;assumption.
          intros ;apply P_id_A__U52_monotonic;assumption.
          intros ;apply P_id_A__ISNATILISTKIND_monotonic;assumption.
          intros ;apply P_id_A__U21_monotonic;assumption.
          intros ;apply P_id_A__ISNATILIST_monotonic;assumption.
          intros ;apply P_id_A__U32_monotonic;assumption.
          intros ;apply P_id_A__U61_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_12_large_scc_2_large_large_scc_8_strict_in_lt :
         Relation_Definitions.inclusion _ 
          DP_R_xml_0_scc_12_large_scc_2_large_large_scc_8_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_12_large_scc_2_large_large_scc_8_large_in_le :
         Relation_Definitions.inclusion _ 
          DP_R_xml_0_scc_12_large_scc_2_large_large_scc_8_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_12_large_scc_2_large_large_scc_8_large  := 
          WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_8_large.wf.
        
        
        Lemma wf :
         well_founded WF_DP_R_xml_0_scc_12_large_scc_2_large_large.DP_R_xml_0_scc_12_large_scc_2_large_large_scc_8
          .
        Proof.
          intros x.
          apply (well_founded_ind wf_lt).
          clear x.
          intros x.
          pattern x.
          
          apply (@Acc_ind _ 
                  DP_R_xml_0_scc_12_large_scc_2_large_large_scc_8_large).
          clear x.
          intros x _ IHx IHx'.
          constructor.
          intros y H.
          
          destruct H;
           (apply IHx';
             apply DP_R_xml_0_scc_12_large_scc_2_large_large_scc_8_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_12_large_scc_2_large_large_scc_8_large_in_le;
              econstructor eassumption])).
          apply wf_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_8_large.
        Qed.
       End WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_8.
       
       Definition wf_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_8  := 
         WF_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_8.wf.
       
       
       Lemma acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_8 :
        forall x y, 
         (DP_R_xml_0_scc_12_large_scc_2_large_large_scc_8 x y) ->
          Acc WF_DP_R_xml_0_scc_12_large_scc_2_large.DP_R_xml_0_scc_12_large_scc_2_large_large 
           x.
       Proof.
         intros x.
         pattern x.
         apply (@Acc_ind _ DP_R_xml_0_scc_12_large_scc_2_large_large_scc_8).
         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_12_large_scc_2_large_large_non_scc_7;
              econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
            ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_non_scc_6;
               econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
             ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_non_scc_4;
                econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
              ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_non_scc_3;
                 econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
               ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_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_12_large_scc_2_large_large_scc_8.
       Qed.
       
       
       Lemma wf :
        well_founded WF_DP_R_xml_0_scc_12_large_scc_2_large.DP_R_xml_0_scc_12_large_scc_2_large_large
         .
       Proof.
         constructor;intros _y _h;inversion _h;clear _h;subst;
          (eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_non_scc_7;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
          ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_non_scc_6;
             econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
           ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_non_scc_5;
              econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
            ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_non_scc_4;
               econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
             ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_non_scc_3;
                econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
              ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_non_scc_2;
                 econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
               ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_non_scc_1;
                  econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
                ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_non_scc_0;
                   econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
                 ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_8;
                    econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
                  ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_7;
                     
                     econstructor 
                     (eassumption)||(algebra.Alg_ext.star_refl' ))||
                   ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_6;
                      
                      econstructor 
                      (eassumption)||(algebra.Alg_ext.star_refl' ))||
                    ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_5;
                       
                       econstructor 
                       (eassumption)||(algebra.Alg_ext.star_refl' ))||
                     ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_4;
                        
                        econstructor 
                        (eassumption)||(algebra.Alg_ext.star_refl' ))||
                      ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_3;
                         
                         econstructor 
                         (eassumption)||(algebra.Alg_ext.star_refl' ))||
                       ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_2;
                          
                          econstructor 
                          (eassumption)||(algebra.Alg_ext.star_refl' ))||
                        ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_large_scc_1;
                           
                           econstructor 
                           (eassumption)||(algebra.Alg_ext.star_refl' ))||
                         ((eapply acc_DP_R_xml_0_scc_12_large_scc_2_large_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_12_large_scc_2_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_isNatKind (x11:Z) := 0.
      
      Definition P_id_a__isNatIList (x11:Z) := 1.
      
      Definition P_id_a__U21 (x11:Z) (x12:Z) := 1* x11.
      
      Definition P_id_U41 (x11:Z) (x12:Z) (x13:Z) := 1 + 1* x11.
      
      Definition P_id_a__and (x11:Z) (x12:Z) := 2* x11 + 2* x12.
      
      Definition P_id_a__U11 (x11:Z) (x12:Z) := 1* x11.
      
      Definition P_id_U21 (x11:Z) (x12:Z) := 1* x11.
      
      Definition P_id_a__U61 (x11:Z) (x12:Z) := 0.
      
      Definition P_id_a__U32 (x11:Z) := 1 + 2* x11.
      
      Definition P_id_U51 (x11:Z) (x12:Z) (x13:Z) := 2* x11.
      
      Definition P_id_isNatIListKind (x11:Z) := 0.
      
      Definition P_id_0  := 0.
      
      Definition P_id_U12 (x11:Z) := 2* x11.
      
      Definition P_id_a__U52 (x11:Z) (x12:Z) := 2* x11.
      
      Definition P_id_a__isNat (x11:Z) := 0.
      
      Definition P_id_U43 (x11:Z) := 1* x11.
      
      Definition P_id_a__isNatIListKind (x11:Z) := 0.
      
      Definition P_id_a__U12 (x11:Z) := 2* x11.
      
      Definition P_id_U31 (x11:Z) (x12:Z) := 1 + 2* x11.
      
      Definition P_id_a__length (x11:Z) := 0.
      
      Definition P_id_a__U42 (x11:Z) (x12:Z) := 1 + 1* x11.
      
      Definition P_id_U53 (x11:Z) := 2* x11.
      
      Definition P_id_and (x11:Z) (x12:Z) := 2* x11 + 2* x12.
      
      Definition P_id_cons (x11:Z) (x12:Z) := 0.
      
      Definition P_id_U11 (x11:Z) (x12:Z) := 1* x11.
      
      Definition P_id_a__U51 (x11:Z) (x12:Z) (x13:Z) := 2* x11.
      
      Definition P_id_a__U22 (x11:Z) := 1* x11.
      
      Definition P_id_U42 (x11:Z) (x12:Z) := 1 + 1* x11.
      
      Definition P_id_length (x11:Z) := 0.
      
      Definition P_id_tt  := 0.
      
      Definition P_id_U22 (x11:Z) := 1* x11.
      
      Definition P_id_s (x11:Z) := 1* x11.
      
      Definition P_id_a__U41 (x11:Z) (x12:Z) (x13:Z) := 1 + 1* x11.
      
      Definition P_id_U52 (x11:Z) (x12:Z) := 2* x11.
      
      Definition P_id_nil  := 1.
      
      Definition P_id_zeros  := 0.
      
      Definition P_id_isNatList (x11:Z) := 0.
      
      Definition P_id_a__U53 (x11:Z) := 2* x11.
      
      Definition P_id_a__U31 (x11:Z) (x12:Z) := 1 + 2* x11.
      
      Definition P_id_isNatIList (x11:Z) := 1.
      
      Definition P_id_a__isNatKind (x11:Z) := 0.
      
      Definition P_id_a__isNatList (x11:Z) := 0.
      
      Definition P_id_U32 (x11:Z) := 1 + 2* x11.
      
      Definition P_id_mark (x11:Z) := 2* x11.
      
      Definition P_id_a__U43 (x11:Z) := 1* x11.
      
      Definition P_id_U61 (x11:Z) (x12:Z) := 0.
      
      Definition P_id_isNat (x11:Z) := 0.
      
      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__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__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_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_a__and_monotonic :
       forall x12 x14 x13 x11, 
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__and x12 x14 <= P_id_a__and x11 x13.
      Proof.
        intros x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        
        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_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__U61_monotonic :
       forall x12 x14 x13 x11, 
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U61 x12 x14 <= P_id_a__U61 x11 x13.
      Proof.
        intros x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        
        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 x11, 
        (0 <= x12)/\ (x12 <= x11) ->P_id_a__U32 x12 <= P_id_a__U32 x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U51_monotonic :
       forall x16 x12 x14 x13 x11 x15, 
        (0 <= x16)/\ (x16 <= x15) ->
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_U51 x12 x14 x16 <= P_id_U51 x11 x13 x15.
      Proof.
        intros x16 x15 x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        intros [H_5 H_4].
        
        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_U12_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->P_id_U12 x12 <= P_id_U12 x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_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 x14 x13 x11, 
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U52 x12 x14 <= P_id_a__U52 x11 x13.
      Proof.
        intros x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_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_U43_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->P_id_U43 x12 <= P_id_U43 x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_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__U12_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->P_id_a__U12 x12 <= P_id_a__U12 x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_0].
        
        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__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__U42_monotonic :
       forall x12 x14 x13 x11, 
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U42 x12 x14 <= P_id_a__U42 x11 x13.
      Proof.
        intros x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U53_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->P_id_U53 x12 <= P_id_U53 x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_and_monotonic :
       forall x12 x14 x13 x11, 
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->P_id_and x12 x14 <= P_id_and x11 x13.
      Proof.
        intros x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        
        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_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__U51_monotonic :
       forall x16 x12 x14 x13 x11 x15, 
        (0 <= x16)/\ (x16 <= x15) ->
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_a__U51 x12 x14 x16 <= P_id_a__U51 x11 x13 x15.
      Proof.
        intros x16 x15 x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        intros [H_5 H_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 x11, 
        (0 <= x12)/\ (x12 <= x11) ->P_id_a__U22 x12 <= P_id_a__U22 x11.
      Proof.
        intros x12 x11.
        intros [H_1 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 x12 x14 x13 x11, 
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->P_id_U42 x12 x14 <= P_id_U42 x11 x13.
      Proof.
        intros x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        
        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_U22_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->P_id_U22 x12 <= P_id_U22 x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_0].
        
        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__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_U52_monotonic :
       forall x12 x14 x13 x11, 
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->P_id_U52 x12 x14 <= P_id_U52 x11 x13.
      Proof.
        intros x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 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__U53_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->P_id_a__U53 x12 <= P_id_a__U53 x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_0].
        
        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_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__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__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_U32_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->P_id_U32 x12 <= P_id_U32 x11.
      Proof.
        intros x12 x11.
        intros [H_1 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__U43_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->P_id_a__U43 x12 <= P_id_a__U43 x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_0].
        
        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 x14 x13 x11, 
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->P_id_U61 x12 x14 <= P_id_U61 x11 x13.
      Proof.
        intros x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        
        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__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_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__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_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_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_a__and_bounded :
       forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__and x12 x11.
      Proof.
        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_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__U61_bounded :
       forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U61 x12 x11.
      Proof.
        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 x11, (0 <= x11) ->0 <= P_id_a__U32 x11.
      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 x13 x11, 
        (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U51 x13 x12 x11.
      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_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_U12_bounded : forall x11, (0 <= x11) ->0 <= P_id_U12 x11.
      Proof.
        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 x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U52 x12 x11.
      Proof.
        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_U43_bounded : forall x11, (0 <= x11) ->0 <= P_id_U43 x11.
      Proof.
        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_a__U12_bounded :
       forall x11, (0 <= x11) ->0 <= P_id_a__U12 x11.
      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__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__U42_bounded :
       forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U42 x12 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U53_bounded : forall x11, (0 <= x11) ->0 <= P_id_U53 x11.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_and_bounded :
       forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_and x12 x11.
      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_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__U51_bounded :
       forall x12 x13 x11, 
        (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U51 x13 x12 x11.
      Proof.
        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 x11, (0 <= x11) ->0 <= P_id_a__U22 x11.
      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 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U42 x12 x11.
      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_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_U22_bounded : forall x11, (0 <= x11) ->0 <= P_id_U22 x11.
      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__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_U52_bounded :
       forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U52 x12 x11.
      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_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_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__U53_bounded :
       forall x11, (0 <= x11) ->0 <= P_id_a__U53 x11.
      Proof.
        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_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.
      
      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_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_U32_bounded : forall x11, (0 <= x11) ->0 <= P_id_U32 x11.
      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__U43_bounded :
       forall x11, (0 <= x11) ->0 <= P_id_a__U43 x11.
      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 x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U61 x12 x11.
      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.
      
      Definition measure  := 
        InterpZ.measure 0 P_id_a__zeros P_id_isNatKind P_id_a__isNatIList 
         P_id_a__U21 P_id_U41 P_id_a__and P_id_a__U11 P_id_U21 P_id_a__U61 
         P_id_a__U32 P_id_U51 P_id_isNatIListKind P_id_0 P_id_U12 
         P_id_a__U52 P_id_a__isNat P_id_U43 P_id_a__isNatIListKind 
         P_id_a__U12 P_id_U31 P_id_a__length P_id_a__U42 P_id_U53 P_id_and 
         P_id_cons P_id_U11 P_id_a__U51 P_id_a__U22 P_id_U42 P_id_length 
         P_id_tt P_id_U22 P_id_s P_id_a__U41 P_id_U52 P_id_nil P_id_zeros 
         P_id_isNatList P_id_a__U53 P_id_a__U31 P_id_isNatIList 
         P_id_a__isNatKind P_id_a__isNatList P_id_U32 P_id_mark P_id_a__U43 
         P_id_U61 P_id_isNat.
      
      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_isNatKind (x11::nil)) =>
                       P_id_isNatKind (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__U21 (x12::
                         x11::nil)) =>
                       P_id_a__U21 (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_a__and (x12::
                         x11::nil)) =>
                       P_id_a__and (measure x12) (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_U21 (x12::x11::nil)) =>
                       P_id_U21 (measure x12) (measure x11)
                      | (algebra.Alg.Term algebra.F.id_a__U61 (x12::
                         x11::nil)) =>
                       P_id_a__U61 (measure x12) (measure x11)
                      | (algebra.Alg.Term algebra.F.id_a__U32 (x11::nil)) =>
                       P_id_a__U32 (measure x11)
                      | (algebra.Alg.Term algebra.F.id_U51 (x13::x12::
                         x11::nil)) =>
                       P_id_U51 (measure x13) (measure x12) (measure x11)
                      | (algebra.Alg.Term algebra.F.id_isNatIListKind 
                         (x11::nil)) =>
                       P_id_isNatIListKind (measure x11)
                      | (algebra.Alg.Term algebra.F.id_0 nil) => P_id_0 
                      | (algebra.Alg.Term algebra.F.id_U12 (x11::nil)) =>
                       P_id_U12 (measure x11)
                      | (algebra.Alg.Term algebra.F.id_a__U52 (x12::
                         x11::nil)) =>
                       P_id_a__U52 (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_U43 (x11::nil)) =>
                       P_id_U43 (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__U12 (x11::nil)) =>
                       P_id_a__U12 (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__length (x11::nil)) =>
                       P_id_a__length (measure x11)
                      | (algebra.Alg.Term algebra.F.id_a__U42 (x12::
                         x11::nil)) =>
                       P_id_a__U42 (measure x12) (measure x11)
                      | (algebra.Alg.Term algebra.F.id_U53 (x11::nil)) =>
                       P_id_U53 (measure x11)
                      | (algebra.Alg.Term algebra.F.id_and (x12::x11::nil)) =>
                       P_id_and (measure x12) (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_U11 (x12::x11::nil)) =>
                       P_id_U11 (measure x12) (measure x11)
                      | (algebra.Alg.Term algebra.F.id_a__U51 (x13::x12::
                         x11::nil)) =>
                       P_id_a__U51 (measure x13) (measure x12) (measure x11)
                      | (algebra.Alg.Term algebra.F.id_a__U22 (x11::nil)) =>
                       P_id_a__U22 (measure x11)
                      | (algebra.Alg.Term algebra.F.id_U42 (x12::x11::nil)) =>
                       P_id_U42 (measure x12) (measure x11)
                      | (algebra.Alg.Term algebra.F.id_length (x11::nil)) =>
                       P_id_length (measure x11)
                      | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt 
                      | (algebra.Alg.Term algebra.F.id_U22 (x11::nil)) =>
                       P_id_U22 (measure x11)
                      | (algebra.Alg.Term algebra.F.id_s (x11::nil)) =>
                       P_id_s (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_U52 (x12::x11::nil)) =>
                       P_id_U52 (measure x12) (measure x11)
                      | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil 
                      | (algebra.Alg.Term algebra.F.id_zeros nil) =>
                       P_id_zeros 
                      | (algebra.Alg.Term algebra.F.id_isNatList (x11::nil)) =>
                       P_id_isNatList (measure x11)
                      | (algebra.Alg.Term algebra.F.id_a__U53 (x11::nil)) =>
                       P_id_a__U53 (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_isNatIList (x11::nil)) =>
                       P_id_isNatIList (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__isNatList 
                         (x11::nil)) =>
                       P_id_a__isNatList (measure x11)
                      | (algebra.Alg.Term algebra.F.id_U32 (x11::nil)) =>
                       P_id_U32 (measure x11)
                      | (algebra.Alg.Term algebra.F.id_mark (x11::nil)) =>
                       P_id_mark (measure x11)
                      | (algebra.Alg.Term algebra.F.id_a__U43 (x11::nil)) =>
                       P_id_a__U43 (measure x11)
                      | (algebra.Alg.Term algebra.F.id_U61 (x12::x11::nil)) =>
                       P_id_U61 (measure x12) (measure x11)
                      | (algebra.Alg.Term algebra.F.id_isNat (x11::nil)) =>
                       P_id_isNat (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_isNatKind_monotonic;assumption.
        intros ;apply P_id_a__isNatIList_monotonic;assumption.
        intros ;apply P_id_a__U21_monotonic;assumption.
        intros ;apply P_id_U41_monotonic;assumption.
        intros ;apply P_id_a__and_monotonic;assumption.
        intros ;apply P_id_a__U11_monotonic;assumption.
        intros ;apply P_id_U21_monotonic;assumption.
        intros ;apply P_id_a__U61_monotonic;assumption.
        intros ;apply P_id_a__U32_monotonic;assumption.
        intros ;apply P_id_U51_monotonic;assumption.
        intros ;apply P_id_isNatIListKind_monotonic;assumption.
        intros ;apply P_id_U12_monotonic;assumption.
        intros ;apply P_id_a__U52_monotonic;assumption.
        intros ;apply P_id_a__isNat_monotonic;assumption.
        intros ;apply P_id_U43_monotonic;assumption.
        intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
        intros ;apply P_id_a__U12_monotonic;assumption.
        intros ;apply P_id_U31_monotonic;assumption.
        intros ;apply P_id_a__length_monotonic;assumption.
        intros ;apply P_id_a__U42_monotonic;assumption.
        intros ;apply P_id_U53_monotonic;assumption.
        intros ;apply P_id_and_monotonic;assumption.
        intros ;apply P_id_cons_monotonic;assumption.
        intros ;apply P_id_U11_monotonic;assumption.
        intros ;apply P_id_a__U51_monotonic;assumption.
        intros ;apply P_id_a__U22_monotonic;assumption.
        intros ;apply P_id_U42_monotonic;assumption.
        intros ;apply P_id_length_monotonic;assumption.
        intros ;apply P_id_U22_monotonic;assumption.
        intros ;apply P_id_s_monotonic;assumption.
        intros ;apply P_id_a__U41_monotonic;assumption.
        intros ;apply P_id_U52_monotonic;assumption.
        intros ;apply P_id_isNatList_monotonic;assumption.
        intros ;apply P_id_a__U53_monotonic;assumption.
        intros ;apply P_id_a__U31_monotonic;assumption.
        intros ;apply P_id_isNatIList_monotonic;assumption.
        intros ;apply P_id_a__isNatKind_monotonic;assumption.
        intros ;apply P_id_a__isNatList_monotonic;assumption.
        intros ;apply P_id_U32_monotonic;assumption.
        intros ;apply P_id_mark_monotonic;assumption.
        intros ;apply P_id_a__U43_monotonic;assumption.
        intros ;apply P_id_U61_monotonic;assumption.
        intros ;apply P_id_isNat_monotonic;assumption.
        intros ;apply P_id_a__zeros_bounded;assumption.
        intros ;apply P_id_isNatKind_bounded;assumption.
        intros ;apply P_id_a__isNatIList_bounded;assumption.
        intros ;apply P_id_a__U21_bounded;assumption.
        intros ;apply P_id_U41_bounded;assumption.
        intros ;apply P_id_a__and_bounded;assumption.
        intros ;apply P_id_a__U11_bounded;assumption.
        intros ;apply P_id_U21_bounded;assumption.
        intros ;apply P_id_a__U61_bounded;assumption.
        intros ;apply P_id_a__U32_bounded;assumption.
        intros ;apply P_id_U51_bounded;assumption.
        intros ;apply P_id_isNatIListKind_bounded;assumption.
        intros ;apply P_id_0_bounded;assumption.
        intros ;apply P_id_U12_bounded;assumption.
        intros ;apply P_id_a__U52_bounded;assumption.
        intros ;apply P_id_a__isNat_bounded;assumption.
        intros ;apply P_id_U43_bounded;assumption.
        intros ;apply P_id_a__isNatIListKind_bounded;assumption.
        intros ;apply P_id_a__U12_bounded;assumption.
        intros ;apply P_id_U31_bounded;assumption.
        intros ;apply P_id_a__length_bounded;assumption.
        intros ;apply P_id_a__U42_bounded;assumption.
        intros ;apply P_id_U53_bounded;assumption.
        intros ;apply P_id_and_bounded;assumption.
        intros ;apply P_id_cons_bounded;assumption.
        intros ;apply P_id_U11_bounded;assumption.
        intros ;apply P_id_a__U51_bounded;assumption.
        intros ;apply P_id_a__U22_bounded;assumption.
        intros ;apply P_id_U42_bounded;assumption.
        intros ;apply P_id_length_bounded;assumption.
        intros ;apply P_id_tt_bounded;assumption.
        intros ;apply P_id_U22_bounded;assumption.
        intros ;apply P_id_s_bounded;assumption.
        intros ;apply P_id_a__U41_bounded;assumption.
        intros ;apply P_id_U52_bounded;assumption.
        intros ;apply P_id_nil_bounded;assumption.
        intros ;apply P_id_zeros_bounded;assumption.
        intros ;apply P_id_isNatList_bounded;assumption.
        intros ;apply P_id_a__U53_bounded;assumption.
        intros ;apply P_id_a__U31_bounded;assumption.
        intros ;apply P_id_isNatIList_bounded;assumption.
        intros ;apply P_id_a__isNatKind_bounded;assumption.
        intros ;apply P_id_a__isNatList_bounded;assumption.
        intros ;apply P_id_U32_bounded;assumption.
        intros ;apply P_id_mark_bounded;assumption.
        intros ;apply P_id_a__U43_bounded;assumption.
        intros ;apply P_id_U61_bounded;assumption.
        intros ;apply P_id_isNat_bounded;assumption.
        apply rules_monotonic.
      Qed.
      
      Definition P_id_A__LENGTH (x11:Z) := 0.
      
      Definition P_id_A__U11 (x11:Z) (x12:Z) := 0.
      
      Definition P_id_A__U41 (x11:Z) (x12:Z) (x13:Z) := 0.
      
      Definition P_id_A__ISNATKIND (x11:Z) := 0.
      
      Definition P_id_A__U22 (x11:Z) := 0.
      
      Definition P_id_A__U51 (x11:Z) (x12:Z) (x13:Z) := 0.
      
      Definition P_id_A__AND (x11:Z) (x12:Z) := 2* x12.
      
      Definition P_id_A__ISNATLIST (x11:Z) := 0.
      
      Definition P_id_A__U43 (x11:Z) := 0.
      
      Definition P_id_A__U31 (x11:Z) (x12:Z) := 0.
      
      Definition P_id_A__U53 (x11:Z) := 0.
      
      Definition P_id_MARK (x11:Z) := 2* x11.
      
      Definition P_id_A__U12 (x11:Z) := 0.
      
      Definition P_id_A__U42 (x11:Z) (x12:Z) := 0.
      
      Definition P_id_A__ZEROS  := 0.
      
      Definition P_id_A__ISNAT (x11:Z) := 0.
      
      Definition P_id_A__U52 (x11:Z) (x12:Z) := 0.
      
      Definition P_id_A__ISNATILISTKIND (x11:Z) := 0.
      
      Definition P_id_A__U21 (x11:Z) (x12:Z) := 0.
      
      Definition P_id_A__ISNATILIST (x11:Z) := 0.
      
      Definition P_id_A__U32 (x11:Z) := 0.
      
      Definition P_id_A__U61 (x11:Z) (x12:Z) := 0.
      
      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__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__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__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__U22_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->P_id_A__U22 x12 <= P_id_A__U22 x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_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 x16 x12 x14 x13 x11 x15, 
        (0 <= x16)/\ (x16 <= x15) ->
         (0 <= x14)/\ (x14 <= x13) ->
          (0 <= x12)/\ (x12 <= x11) ->
           P_id_A__U51 x12 x14 x16 <= P_id_A__U51 x11 x13 x15.
      Proof.
        intros x16 x15 x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        intros [H_5 H_4].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_A__AND_monotonic :
       forall x12 x14 x13 x11, 
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__AND x12 x14 <= P_id_A__AND x11 x13.
      Proof.
        intros x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_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__U43_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->P_id_A__U43 x12 <= P_id_A__U43 x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_0].
        
        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__U53_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->P_id_A__U53 x12 <= P_id_A__U53 x11.
      Proof.
        intros x12 x11.
        intros [H_1 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__U12_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->P_id_A__U12 x12 <= P_id_A__U12 x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_0].
        
        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 x12 x14 x13 x11, 
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U42 x12 x14 <= P_id_A__U42 x11 x13.
      Proof.
        intros x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_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_A__U52_monotonic :
       forall x12 x14 x13 x11, 
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U52 x12 x14 <= P_id_A__U52 x11 x13.
      Proof.
        intros x14 x13 x12 x11.
        intros [H_1 H_0].
        intros [H_3 H_2].
        
        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__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__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__U32_monotonic :
       forall x12 x11, 
        (0 <= x12)/\ (x12 <= x11) ->P_id_A__U32 x12 <= P_id_A__U32 x11.
      Proof.
        intros x12 x11.
        intros [H_1 H_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 x14 x13 x11, 
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U61 x12 x14 <= P_id_A__U61 x11 x13.
      Proof.
        intros x14 x13 x12 x11.
        intros [H_1 H_0].
        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_isNatKind 
         P_id_a__isNatIList P_id_a__U21 P_id_U41 P_id_a__and P_id_a__U11 
         P_id_U21 P_id_a__U61 P_id_a__U32 P_id_U51 P_id_isNatIListKind 
         P_id_0 P_id_U12 P_id_a__U52 P_id_a__isNat P_id_U43 
         P_id_a__isNatIListKind P_id_a__U12 P_id_U31 P_id_a__length 
         P_id_a__U42 P_id_U53 P_id_and P_id_cons P_id_U11 P_id_a__U51 
         P_id_a__U22 P_id_U42 P_id_length P_id_tt P_id_U22 P_id_s 
         P_id_a__U41 P_id_U52 P_id_nil P_id_zeros P_id_isNatList P_id_a__U53 
         P_id_a__U31 P_id_isNatIList P_id_a__isNatKind P_id_a__isNatList 
         P_id_U32 P_id_mark P_id_a__U43 P_id_U61 P_id_isNat P_id_A__LENGTH 
         P_id_A__U11 P_id_A__U41 P_id_A__ISNATKIND P_id_A__U22 P_id_A__U51 
         P_id_A__AND P_id_A__ISNATLIST P_id_A__U43 P_id_A__U31 P_id_A__U53 
         P_id_MARK P_id_A__U12 P_id_A__U42 P_id_A__ZEROS P_id_A__ISNAT 
         P_id_A__U52 P_id_A__ISNATILISTKIND P_id_A__U21 P_id_A__ISNATILIST 
         P_id_A__U32 P_id_A__U61.
      
      Lemma marked_measure_equation :
       forall t, 
        marked_measure t = match t with
                             | (algebra.Alg.Term algebra.F.id_a__length 
                                (x11::nil)) =>
                              P_id_A__LENGTH (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__U41 (x13::
                                x12::x11::nil)) =>
                              P_id_A__U41 (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__U22 
                                (x11::nil)) =>
                              P_id_A__U22 (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U51 (x13::
                                x12::x11::nil)) =>
                              P_id_A__U51 (measure x13) (measure x12) 
                               (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__and (x12::
                                x11::nil)) =>
                              P_id_A__AND (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__U43 
                                (x11::nil)) =>
                              P_id_A__U43 (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__U53 
                                (x11::nil)) =>
                              P_id_A__U53 (measure x11)
                             | (algebra.Alg.Term algebra.F.id_mark 
                                (x11::nil)) =>
                              P_id_MARK (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U12 
                                (x11::nil)) =>
                              P_id_A__U12 (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U42 (x12::
                                x11::nil)) =>
                              P_id_A__U42 (measure x12) (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__zeros nil) =>
                              P_id_A__ZEROS 
                             | (algebra.Alg.Term algebra.F.id_a__isNat 
                                (x11::nil)) =>
                              P_id_A__ISNAT (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U52 (x12::
                                x11::nil)) =>
                              P_id_A__U52 (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_a__U21 (x12::
                                x11::nil)) =>
                              P_id_A__U21 (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__U32 
                                (x11::nil)) =>
                              P_id_A__U32 (measure x11)
                             | (algebra.Alg.Term algebra.F.id_a__U61 (x12::
                                x11::nil)) =>
                              P_id_A__U61 (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_isNatKind_monotonic;assumption.
        intros ;apply P_id_a__isNatIList_monotonic;assumption.
        intros ;apply P_id_a__U21_monotonic;assumption.
        intros ;apply P_id_U41_monotonic;assumption.
        intros ;apply P_id_a__and_monotonic;assumption.
        intros ;apply P_id_a__U11_monotonic;assumption.
        intros ;apply P_id_U21_monotonic;assumption.
        intros ;apply P_id_a__U61_monotonic;assumption.
        intros ;apply P_id_a__U32_monotonic;assumption.
        intros ;apply P_id_U51_monotonic;assumption.
        intros ;apply P_id_isNatIListKind_monotonic;assumption.
        intros ;apply P_id_U12_monotonic;assumption.
        intros ;apply P_id_a__U52_monotonic;assumption.
        intros ;apply P_id_a__isNat_monotonic;assumption.
        intros ;apply P_id_U43_monotonic;assumption.
        intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
        intros ;apply P_id_a__U12_monotonic;assumption.
        intros ;apply P_id_U31_monotonic;assumption.
        intros ;apply P_id_a__length_monotonic;assumption.
        intros ;apply P_id_a__U42_monotonic;assumption.
        intros ;apply P_id_U53_monotonic;assumption.
        intros ;apply P_id_and_monotonic;assumption.
        intros ;apply P_id_cons_monotonic;assumption.
        intros ;apply P_id_U11_monotonic;assumption.
        intros ;apply P_id_a__U51_monotonic;assumption.
        intros ;apply P_id_a__U22_monotonic;assumption.
        intros ;apply P_id_U42_monotonic;assumption.
        intros ;apply P_id_length_monotonic;assumption.
        intros ;apply P_id_U22_monotonic;assumption.
        intros ;apply P_id_s_monotonic;assumption.
        intros ;apply P_id_a__U41_monotonic;assumption.
        intros ;apply P_id_U52_monotonic;assumption.
        intros ;apply P_id_isNatList_monotonic;assumption.
        intros ;apply P_id_a__U53_monotonic;assumption.
        intros ;apply P_id_a__U31_monotonic;assumption.
        intros ;apply P_id_isNatIList_monotonic;assumption.
        intros ;apply P_id_a__isNatKind_monotonic;assumption.
        intros ;apply P_id_a__isNatList_monotonic;assumption.
        intros ;apply P_id_U32_monotonic;assumption.
        intros ;apply P_id_mark_monotonic;assumption.
        intros ;apply P_id_a__U43_monotonic;assumption.
        intros ;apply P_id_U61_monotonic;assumption.
        intros ;apply P_id_isNat_monotonic;assumption.
        intros ;apply P_id_a__zeros_bounded;assumption.
        intros ;apply P_id_isNatKind_bounded;assumption.
        intros ;apply P_id_a__isNatIList_bounded;assumption.
        intros ;apply P_id_a__U21_bounded;assumption.
        intros ;apply P_id_U41_bounded;assumption.
        intros ;apply P_id_a__and_bounded;assumption.
        intros ;apply P_id_a__U11_bounded;assumption.
        intros ;apply P_id_U21_bounded;assumption.
        intros ;apply P_id_a__U61_bounded;assumption.
        intros ;apply P_id_a__U32_bounded;assumption.
        intros ;apply P_id_U51_bounded;assumption.
        intros ;apply P_id_isNatIListKind_bounded;assumption.
        intros ;apply P_id_0_bounded;assumption.
        intros ;apply P_id_U12_bounded;assumption.
        intros ;apply P_id_a__U52_bounded;assumption.
        intros ;apply P_id_a__isNat_bounded;assumption.
        intros ;apply P_id_U43_bounded;assumption.
        intros ;apply P_id_a__isNatIListKind_bounded;assumption.
        intros ;apply P_id_a__U12_bounded;assumption.
        intros ;apply P_id_U31_bounded;assumption.
        intros ;apply P_id_a__length_bounded;assumption.
        intros ;apply P_id_a__U42_bounded;assumption.
        intros ;apply P_id_U53_bounded;assumption.
        intros ;apply P_id_and_bounded;assumption.
        intros ;apply P_id_cons_bounded;assumption.
        intros ;apply P_id_U11_bounded;assumption.
        intros ;apply P_id_a__U51_bounded;assumption.
        intros ;apply P_id_a__U22_bounded;assumption.
        intros ;apply P_id_U42_bounded;assumption.
        intros ;apply P_id_length_bounded;assumption.
        intros ;apply P_id_tt_bounded;assumption.
        intros ;apply P_id_U22_bounded;assumption.
        intros ;apply P_id_s_bounded;assumption.
        intros ;apply P_id_a__U41_bounded;assumption.
        intros ;apply P_id_U52_bounded;assumption.
        intros ;apply P_id_nil_bounded;assumption.
        intros ;apply P_id_zeros_bounded;assumption.
        intros ;apply P_id_isNatList_bounded;assumption.
        intros ;apply P_id_a__U53_bounded;assumption.
        intros ;apply P_id_a__U31_bounded;assumption.
        intros ;apply P_id_isNatIList_bounded;assumption.
        intros ;apply P_id_a__isNatKind_bounded;assumption.
        intros ;apply P_id_a__isNatList_bounded;assumption.
        intros ;apply P_id_U32_bounded;assumption.
        intros ;apply P_id_mark_bounded;assumption.
        intros ;apply P_id_a__U43_bounded;assumption.
        intros ;apply P_id_U61_bounded;assumption.
        intros ;apply P_id_isNat_bounded;assumption.
        apply rules_monotonic.
        intros ;apply P_id_A__LENGTH_monotonic;assumption.
        intros ;apply P_id_A__U11_monotonic;assumption.
        intros ;apply P_id_A__U41_monotonic;assumption.
        intros ;apply P_id_A__ISNATKIND_monotonic;assumption.
        intros ;apply P_id_A__U22_monotonic;assumption.
        intros ;apply P_id_A__U51_monotonic;assumption.
        intros ;apply P_id_A__AND_monotonic;assumption.
        intros ;apply P_id_A__ISNATLIST_monotonic;assumption.
        intros ;apply P_id_A__U43_monotonic;assumption.
        intros ;apply P_id_A__U31_monotonic;assumption.
        intros ;apply P_id_A__U53_monotonic;assumption.
        intros ;apply P_id_MARK_monotonic;assumption.
        intros ;apply P_id_A__U12_monotonic;assumption.
        intros ;apply P_id_A__U42_monotonic;assumption.
        intros ;apply P_id_A__ISNAT_monotonic;assumption.
        intros ;apply P_id_A__U52_monotonic;assumption.
        intros ;apply P_id_A__ISNATILISTKIND_monotonic;assumption.
        intros ;apply P_id_A__U21_monotonic;assumption.
        intros ;apply P_id_A__ISNATILIST_monotonic;assumption.
        intros ;apply P_id_A__U32_monotonic;assumption.
        intros ;apply P_id_A__U61_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_12_large_scc_2_large_strict_in_lt :
       Relation_Definitions.inclusion _ 
        DP_R_xml_0_scc_12_large_scc_2_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_12_large_scc_2_large_large_in_le :
       Relation_Definitions.inclusion _ 
        DP_R_xml_0_scc_12_large_scc_2_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_12_large_scc_2_large_large  := 
        WF_DP_R_xml_0_scc_12_large_scc_2_large_large.wf.
      
      
      Lemma wf :
       well_founded WF_DP_R_xml_0_scc_12_large_scc_2.DP_R_xml_0_scc_12_large_scc_2_large
        .
      Proof.
        intros x.
        apply (well_founded_ind wf_lt).
        clear x.
        intros x.
        pattern x.
        apply (@Acc_ind _ DP_R_xml_0_scc_12_large_scc_2_large_large).
        clear x.
        intros x _ IHx IHx'.
        constructor.
        intros y H.
        
        destruct H;
         (apply IHx';apply DP_R_xml_0_scc_12_large_scc_2_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_12_large_scc_2_large_large_in_le;
            econstructor eassumption])).
        apply wf_DP_R_xml_0_scc_12_large_scc_2_large_large.
      Qed.
     End WF_DP_R_xml_0_scc_12_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_isNatKind (x11:Z) := 0.
     
     Definition P_id_a__isNatIList (x11:Z) := 0.
     
     Definition P_id_a__U21 (x11:Z) (x12:Z) := 2* x11.
     
     Definition P_id_U41 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
     
     Definition P_id_a__and (x11:Z) (x12:Z) := 2* x11 + 2* x12.
     
     Definition P_id_a__U11 (x11:Z) (x12:Z) := 2* x11.
     
     Definition P_id_U21 (x11:Z) (x12:Z) := 2* x11.
     
     Definition P_id_a__U61 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_a__U32 (x11:Z) := 2* x11.
     
     Definition P_id_U51 (x11:Z) (x12:Z) (x13:Z) := 2* x11.
     
     Definition P_id_isNatIListKind (x11:Z) := 0.
     
     Definition P_id_0  := 0.
     
     Definition P_id_U12 (x11:Z) := 1* x11.
     
     Definition P_id_a__U52 (x11:Z) (x12:Z) := 2* x11.
     
     Definition P_id_a__isNat (x11:Z) := 0.
     
     Definition P_id_U43 (x11:Z) := 1* x11.
     
     Definition P_id_a__isNatIListKind (x11:Z) := 0.
     
     Definition P_id_a__U12 (x11:Z) := 1* x11.
     
     Definition P_id_U31 (x11:Z) (x12:Z) := 2* x11.
     
     Definition P_id_a__length (x11:Z) := 0.
     
     Definition P_id_a__U42 (x11:Z) (x12:Z) := 2* x11.
     
     Definition P_id_U53 (x11:Z) := 1* x11.
     
     Definition P_id_and (x11:Z) (x12:Z) := 2* x11 + 2* x12.
     
     Definition P_id_cons (x11:Z) (x12:Z) := 3 + 1* x11.
     
     Definition P_id_U11 (x11:Z) (x12:Z) := 2* x11.
     
     Definition P_id_a__U51 (x11:Z) (x12:Z) (x13:Z) := 2* x11.
     
     Definition P_id_a__U22 (x11:Z) := 1* x11.
     
     Definition P_id_U42 (x11:Z) (x12:Z) := 2* x11.
     
     Definition P_id_length (x11:Z) := 0.
     
     Definition P_id_tt  := 0.
     
     Definition P_id_U22 (x11:Z) := 1* x11.
     
     Definition P_id_s (x11:Z) := 1* x11.
     
     Definition P_id_a__U41 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
     
     Definition P_id_U52 (x11:Z) (x12:Z) := 2* x11.
     
     Definition P_id_nil  := 0.
     
     Definition P_id_zeros  := 3.
     
     Definition P_id_isNatList (x11:Z) := 0.
     
     Definition P_id_a__U53 (x11:Z) := 1* x11.
     
     Definition P_id_a__U31 (x11:Z) (x12:Z) := 2* x11.
     
     Definition P_id_isNatIList (x11:Z) := 0.
     
     Definition P_id_a__isNatKind (x11:Z) := 0.
     
     Definition P_id_a__isNatList (x11:Z) := 0.
     
     Definition P_id_U32 (x11:Z) := 2* x11.
     
     Definition P_id_mark (x11:Z) := 1* x11.
     
     Definition P_id_a__U43 (x11:Z) := 1* x11.
     
     Definition P_id_U61 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_isNat (x11:Z) := 0.
     
     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__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__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_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_a__and_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__and x12 x14 <= P_id_a__and x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       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_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__U61_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U61 x12 x14 <= P_id_a__U61 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       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 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U32 x12 <= P_id_a__U32 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U51_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_U51 x12 x14 x16 <= P_id_U51 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       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_U12_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U12 x12 <= P_id_U12 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_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 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U52 x12 x14 <= P_id_a__U52 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_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_U43_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U43 x12 <= P_id_U43 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_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__U12_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U12 x12 <= P_id_a__U12 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       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__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__U42_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_a__U42 x12 x14 <= P_id_a__U42 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U53_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U53 x12 <= P_id_U53 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_and_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_and x12 x14 <= P_id_and x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       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_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__U51_monotonic :
      forall x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_a__U51 x12 x14 x16 <= P_id_a__U51 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_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 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U22 x12 <= P_id_a__U22 x11.
     Proof.
       intros x12 x11.
       intros [H_1 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 x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U42 x12 x14 <= P_id_U42 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       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_U22_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U22 x12 <= P_id_U22 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       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__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_U52_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U52 x12 x14 <= P_id_U52 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 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__U53_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U53 x12 <= P_id_a__U53 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       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_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__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__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_U32_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_U32 x12 <= P_id_U32 x11.
     Proof.
       intros x12 x11.
       intros [H_1 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__U43_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_a__U43 x12 <= P_id_a__U43 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       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 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->P_id_U61 x12 x14 <= P_id_U61 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       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__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_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__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_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_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_a__and_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__and x12 x11.
     Proof.
       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_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__U61_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U61 x12 x11.
     Proof.
       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 x11, (0 <= x11) ->0 <= P_id_a__U32 x11.
     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 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U51 x13 x12 x11.
     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_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_U12_bounded : forall x11, (0 <= x11) ->0 <= P_id_U12 x11.
     Proof.
       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 x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U52 x12 x11.
     Proof.
       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_U43_bounded : forall x11, (0 <= x11) ->0 <= P_id_U43 x11.
     Proof.
       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_a__U12_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U12 x11.
     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__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__U42_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U42 x12 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_U53_bounded : forall x11, (0 <= x11) ->0 <= P_id_U53 x11.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_and_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_and x12 x11.
     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_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__U51_bounded :
      forall x12 x13 x11, 
       (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U51 x13 x12 x11.
     Proof.
       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 x11, (0 <= x11) ->0 <= P_id_a__U22 x11.
     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 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U42 x12 x11.
     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_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_U22_bounded : forall x11, (0 <= x11) ->0 <= P_id_U22 x11.
     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__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_U52_bounded :
      forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U52 x12 x11.
     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_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_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__U53_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U53 x11.
     Proof.
       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_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.
     
     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_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_U32_bounded : forall x11, (0 <= x11) ->0 <= P_id_U32 x11.
     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__U43_bounded :
      forall x11, (0 <= x11) ->0 <= P_id_a__U43 x11.
     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 x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U61 x12 x11.
     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.
     
     Definition measure  := 
       InterpZ.measure 0 P_id_a__zeros P_id_isNatKind P_id_a__isNatIList 
        P_id_a__U21 P_id_U41 P_id_a__and P_id_a__U11 P_id_U21 P_id_a__U61 
        P_id_a__U32 P_id_U51 P_id_isNatIListKind P_id_0 P_id_U12 P_id_a__U52 
        P_id_a__isNat P_id_U43 P_id_a__isNatIListKind P_id_a__U12 P_id_U31 
        P_id_a__length P_id_a__U42 P_id_U53 P_id_and P_id_cons P_id_U11 
        P_id_a__U51 P_id_a__U22 P_id_U42 P_id_length P_id_tt P_id_U22 
        P_id_s P_id_a__U41 P_id_U52 P_id_nil P_id_zeros P_id_isNatList 
        P_id_a__U53 P_id_a__U31 P_id_isNatIList P_id_a__isNatKind 
        P_id_a__isNatList P_id_U32 P_id_mark P_id_a__U43 P_id_U61 P_id_isNat.
     
     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_isNatKind (x11::nil)) =>
                      P_id_isNatKind (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__U21 (x12::x11::nil)) =>
                      P_id_a__U21 (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_a__and (x12::x11::nil)) =>
                      P_id_a__and (measure x12) (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_U21 (x12::x11::nil)) =>
                      P_id_U21 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U61 (x12::x11::nil)) =>
                      P_id_a__U61 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U32 (x11::nil)) =>
                      P_id_a__U32 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U51 (x13::x12::
                        x11::nil)) =>
                      P_id_U51 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_isNatIListKind 
                        (x11::nil)) =>
                      P_id_isNatIListKind (measure x11)
                     | (algebra.Alg.Term algebra.F.id_0 nil) => P_id_0 
                     | (algebra.Alg.Term algebra.F.id_U12 (x11::nil)) =>
                      P_id_U12 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U52 (x12::x11::nil)) =>
                      P_id_a__U52 (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_U43 (x11::nil)) =>
                      P_id_U43 (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__U12 (x11::nil)) =>
                      P_id_a__U12 (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__length (x11::nil)) =>
                      P_id_a__length (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U42 (x12::x11::nil)) =>
                      P_id_a__U42 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U53 (x11::nil)) =>
                      P_id_U53 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_and (x12::x11::nil)) =>
                      P_id_and (measure x12) (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_U11 (x12::x11::nil)) =>
                      P_id_U11 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U51 (x13::x12::
                        x11::nil)) =>
                      P_id_a__U51 (measure x13) (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U22 (x11::nil)) =>
                      P_id_a__U22 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U42 (x12::x11::nil)) =>
                      P_id_U42 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_length (x11::nil)) =>
                      P_id_length (measure x11)
                     | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt 
                     | (algebra.Alg.Term algebra.F.id_U22 (x11::nil)) =>
                      P_id_U22 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_s (x11::nil)) =>
                      P_id_s (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_U52 (x12::x11::nil)) =>
                      P_id_U52 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil 
                     | (algebra.Alg.Term algebra.F.id_zeros nil) =>
                      P_id_zeros 
                     | (algebra.Alg.Term algebra.F.id_isNatList (x11::nil)) =>
                      P_id_isNatList (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U53 (x11::nil)) =>
                      P_id_a__U53 (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_isNatIList (x11::nil)) =>
                      P_id_isNatIList (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__isNatList 
                        (x11::nil)) =>
                      P_id_a__isNatList (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U32 (x11::nil)) =>
                      P_id_U32 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_mark (x11::nil)) =>
                      P_id_mark (measure x11)
                     | (algebra.Alg.Term algebra.F.id_a__U43 (x11::nil)) =>
                      P_id_a__U43 (measure x11)
                     | (algebra.Alg.Term algebra.F.id_U61 (x12::x11::nil)) =>
                      P_id_U61 (measure x12) (measure x11)
                     | (algebra.Alg.Term algebra.F.id_isNat (x11::nil)) =>
                      P_id_isNat (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_isNatKind_monotonic;assumption.
       intros ;apply P_id_a__isNatIList_monotonic;assumption.
       intros ;apply P_id_a__U21_monotonic;assumption.
       intros ;apply P_id_U41_monotonic;assumption.
       intros ;apply P_id_a__and_monotonic;assumption.
       intros ;apply P_id_a__U11_monotonic;assumption.
       intros ;apply P_id_U21_monotonic;assumption.
       intros ;apply P_id_a__U61_monotonic;assumption.
       intros ;apply P_id_a__U32_monotonic;assumption.
       intros ;apply P_id_U51_monotonic;assumption.
       intros ;apply P_id_isNatIListKind_monotonic;assumption.
       intros ;apply P_id_U12_monotonic;assumption.
       intros ;apply P_id_a__U52_monotonic;assumption.
       intros ;apply P_id_a__isNat_monotonic;assumption.
       intros ;apply P_id_U43_monotonic;assumption.
       intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
       intros ;apply P_id_a__U12_monotonic;assumption.
       intros ;apply P_id_U31_monotonic;assumption.
       intros ;apply P_id_a__length_monotonic;assumption.
       intros ;apply P_id_a__U42_monotonic;assumption.
       intros ;apply P_id_U53_monotonic;assumption.
       intros ;apply P_id_and_monotonic;assumption.
       intros ;apply P_id_cons_monotonic;assumption.
       intros ;apply P_id_U11_monotonic;assumption.
       intros ;apply P_id_a__U51_monotonic;assumption.
       intros ;apply P_id_a__U22_monotonic;assumption.
       intros ;apply P_id_U42_monotonic;assumption.
       intros ;apply P_id_length_monotonic;assumption.
       intros ;apply P_id_U22_monotonic;assumption.
       intros ;apply P_id_s_monotonic;assumption.
       intros ;apply P_id_a__U41_monotonic;assumption.
       intros ;apply P_id_U52_monotonic;assumption.
       intros ;apply P_id_isNatList_monotonic;assumption.
       intros ;apply P_id_a__U53_monotonic;assumption.
       intros ;apply P_id_a__U31_monotonic;assumption.
       intros ;apply P_id_isNatIList_monotonic;assumption.
       intros ;apply P_id_a__isNatKind_monotonic;assumption.
       intros ;apply P_id_a__isNatList_monotonic;assumption.
       intros ;apply P_id_U32_monotonic;assumption.
       intros ;apply P_id_mark_monotonic;assumption.
       intros ;apply P_id_a__U43_monotonic;assumption.
       intros ;apply P_id_U61_monotonic;assumption.
       intros ;apply P_id_isNat_monotonic;assumption.
       intros ;apply P_id_a__zeros_bounded;assumption.
       intros ;apply P_id_isNatKind_bounded;assumption.
       intros ;apply P_id_a__isNatIList_bounded;assumption.
       intros ;apply P_id_a__U21_bounded;assumption.
       intros ;apply P_id_U41_bounded;assumption.
       intros ;apply P_id_a__and_bounded;assumption.
       intros ;apply P_id_a__U11_bounded;assumption.
       intros ;apply P_id_U21_bounded;assumption.
       intros ;apply P_id_a__U61_bounded;assumption.
       intros ;apply P_id_a__U32_bounded;assumption.
       intros ;apply P_id_U51_bounded;assumption.
       intros ;apply P_id_isNatIListKind_bounded;assumption.
       intros ;apply P_id_0_bounded;assumption.
       intros ;apply P_id_U12_bounded;assumption.
       intros ;apply P_id_a__U52_bounded;assumption.
       intros ;apply P_id_a__isNat_bounded;assumption.
       intros ;apply P_id_U43_bounded;assumption.
       intros ;apply P_id_a__isNatIListKind_bounded;assumption.
       intros ;apply P_id_a__U12_bounded;assumption.
       intros ;apply P_id_U31_bounded;assumption.
       intros ;apply P_id_a__length_bounded;assumption.
       intros ;apply P_id_a__U42_bounded;assumption.
       intros ;apply P_id_U53_bounded;assumption.
       intros ;apply P_id_and_bounded;assumption.
       intros ;apply P_id_cons_bounded;assumption.
       intros ;apply P_id_U11_bounded;assumption.
       intros ;apply P_id_a__U51_bounded;assumption.
       intros ;apply P_id_a__U22_bounded;assumption.
       intros ;apply P_id_U42_bounded;assumption.
       intros ;apply P_id_length_bounded;assumption.
       intros ;apply P_id_tt_bounded;assumption.
       intros ;apply P_id_U22_bounded;assumption.
       intros ;apply P_id_s_bounded;assumption.
       intros ;apply P_id_a__U41_bounded;assumption.
       intros ;apply P_id_U52_bounded;assumption.
       intros ;apply P_id_nil_bounded;assumption.
       intros ;apply P_id_zeros_bounded;assumption.
       intros ;apply P_id_isNatList_bounded;assumption.
       intros ;apply P_id_a__U53_bounded;assumption.
       intros ;apply P_id_a__U31_bounded;assumption.
       intros ;apply P_id_isNatIList_bounded;assumption.
       intros ;apply P_id_a__isNatKind_bounded;assumption.
       intros ;apply P_id_a__isNatList_bounded;assumption.
       intros ;apply P_id_U32_bounded;assumption.
       intros ;apply P_id_mark_bounded;assumption.
       intros ;apply P_id_a__U43_bounded;assumption.
       intros ;apply P_id_U61_bounded;assumption.
       intros ;apply P_id_isNat_bounded;assumption.
       apply rules_monotonic.
     Qed.
     
     Definition P_id_A__LENGTH (x11:Z) := 0.
     
     Definition P_id_A__U11 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__U41 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__ISNATKIND (x11:Z) := 0.
     
     Definition P_id_A__U22 (x11:Z) := 0.
     
     Definition P_id_A__U51 (x11:Z) (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__AND (x11:Z) (x12:Z) := 1* x12.
     
     Definition P_id_A__ISNATLIST (x11:Z) := 0.
     
     Definition P_id_A__U43 (x11:Z) := 0.
     
     Definition P_id_A__U31 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__U53 (x11:Z) := 0.
     
     Definition P_id_MARK (x11:Z) := 1* x11.
     
     Definition P_id_A__U12 (x11:Z) := 0.
     
     Definition P_id_A__U42 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__ZEROS  := 0.
     
     Definition P_id_A__ISNAT (x11:Z) := 0.
     
     Definition P_id_A__U52 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__ISNATILISTKIND (x11:Z) := 0.
     
     Definition P_id_A__U21 (x11:Z) (x12:Z) := 0.
     
     Definition P_id_A__ISNATILIST (x11:Z) := 0.
     
     Definition P_id_A__U32 (x11:Z) := 0.
     
     Definition P_id_A__U61 (x11:Z) (x12:Z) := 0.
     
     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__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__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__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__U22_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U22 x12 <= P_id_A__U22 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_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 x16 x12 x14 x13 x11 x15, 
       (0 <= x16)/\ (x16 <= x15) ->
        (0 <= x14)/\ (x14 <= x13) ->
         (0 <= x12)/\ (x12 <= x11) ->
          P_id_A__U51 x12 x14 x16 <= P_id_A__U51 x11 x13 x15.
     Proof.
       intros x16 x15 x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__AND_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__AND x12 x14 <= P_id_A__AND x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_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__U43_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U43 x12 <= P_id_A__U43 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       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__U53_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U53 x12 <= P_id_A__U53 x11.
     Proof.
       intros x12 x11.
       intros [H_1 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__U12_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U12 x12 <= P_id_A__U12 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_0].
       
       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 x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U42 x12 x14 <= P_id_A__U42 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_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_A__U52_monotonic :
      forall x12 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U52 x12 x14 <= P_id_A__U52 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       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__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__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__U32_monotonic :
      forall x12 x11, 
       (0 <= x12)/\ (x12 <= x11) ->P_id_A__U32 x12 <= P_id_A__U32 x11.
     Proof.
       intros x12 x11.
       intros [H_1 H_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 x14 x13 x11, 
       (0 <= x14)/\ (x14 <= x13) ->
        (0 <= x12)/\ (x12 <= x11) ->
         P_id_A__U61 x12 x14 <= P_id_A__U61 x11 x13.
     Proof.
       intros x14 x13 x12 x11.
       intros [H_1 H_0].
       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_isNatKind 
        P_id_a__isNatIList P_id_a__U21 P_id_U41 P_id_a__and P_id_a__U11 
        P_id_U21 P_id_a__U61 P_id_a__U32 P_id_U51 P_id_isNatIListKind 
        P_id_0 P_id_U12 P_id_a__U52 P_id_a__isNat P_id_U43 
        P_id_a__isNatIListKind P_id_a__U12 P_id_U31 P_id_a__length 
        P_id_a__U42 P_id_U53 P_id_and P_id_cons P_id_U11 P_id_a__U51 
        P_id_a__U22 P_id_U42 P_id_length P_id_tt P_id_U22 P_id_s P_id_a__U41 
        P_id_U52 P_id_nil P_id_zeros P_id_isNatList P_id_a__U53 P_id_a__U31 
        P_id_isNatIList P_id_a__isNatKind P_id_a__isNatList P_id_U32 
        P_id_mark P_id_a__U43 P_id_U61 P_id_isNat P_id_A__LENGTH P_id_A__U11 
        P_id_A__U41 P_id_A__ISNATKIND P_id_A__U22 P_id_A__U51 P_id_A__AND 
        P_id_A__ISNATLIST P_id_A__U43 P_id_A__U31 P_id_A__U53 P_id_MARK 
        P_id_A__U12 P_id_A__U42 P_id_A__ZEROS P_id_A__ISNAT P_id_A__U52 
        P_id_A__ISNATILISTKIND P_id_A__U21 P_id_A__ISNATILIST P_id_A__U32 
        P_id_A__U61.
     
     Lemma marked_measure_equation :
      forall t, 
       marked_measure t = match t with
                            | (algebra.Alg.Term algebra.F.id_a__length 
                               (x11::nil)) =>
                             P_id_A__LENGTH (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__U41 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U41 (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__U22 
                               (x11::nil)) =>
                             P_id_A__U22 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U51 (x13::
                               x12::x11::nil)) =>
                             P_id_A__U51 (measure x13) (measure x12) 
                              (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__and (x12::
                               x11::nil)) =>
                             P_id_A__AND (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__U43 
                               (x11::nil)) =>
                             P_id_A__U43 (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__U53 
                               (x11::nil)) =>
                             P_id_A__U53 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_mark (x11::nil)) =>
                             P_id_MARK (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U12 
                               (x11::nil)) =>
                             P_id_A__U12 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U42 (x12::
                               x11::nil)) =>
                             P_id_A__U42 (measure x12) (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__zeros nil) =>
                             P_id_A__ZEROS 
                            | (algebra.Alg.Term algebra.F.id_a__isNat 
                               (x11::nil)) =>
                             P_id_A__ISNAT (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U52 (x12::
                               x11::nil)) =>
                             P_id_A__U52 (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_a__U21 (x12::
                               x11::nil)) =>
                             P_id_A__U21 (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__U32 
                               (x11::nil)) =>
                             P_id_A__U32 (measure x11)
                            | (algebra.Alg.Term algebra.F.id_a__U61 (x12::
                               x11::nil)) =>
                             P_id_A__U61 (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_isNatKind_monotonic;assumption.
       intros ;apply P_id_a__isNatIList_monotonic;assumption.
       intros ;apply P_id_a__U21_monotonic;assumption.
       intros ;apply P_id_U41_monotonic;assumption.
       intros ;apply P_id_a__and_monotonic;assumption.
       intros ;apply P_id_a__U11_monotonic;assumption.
       intros ;apply P_id_U21_monotonic;assumption.
       intros ;apply P_id_a__U61_monotonic;assumption.
       intros ;apply P_id_a__U32_monotonic;assumption.
       intros ;apply P_id_U51_monotonic;assumption.
       intros ;apply P_id_isNatIListKind_monotonic;assumption.
       intros ;apply P_id_U12_monotonic;assumption.
       intros ;apply P_id_a__U52_monotonic;assumption.
       intros ;apply P_id_a__isNat_monotonic;assumption.
       intros ;apply P_id_U43_monotonic;assumption.
       intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
       intros ;apply P_id_a__U12_monotonic;assumption.
       intros ;apply P_id_U31_monotonic;assumption.
       intros ;apply P_id_a__length_monotonic;assumption.
       intros ;apply P_id_a__U42_monotonic;assumption.
       intros ;apply P_id_U53_monotonic;assumption.
       intros ;apply P_id_and_monotonic;assumption.
       intros ;apply P_id_cons_monotonic;assumption.
       intros ;apply P_id_U11_monotonic;assumption.
       intros ;apply P_id_a__U51_monotonic;assumption.
       intros ;apply P_id_a__U22_monotonic;assumption.
       intros ;apply P_id_U42_monotonic;assumption.
       intros ;apply P_id_length_monotonic;assumption.
       intros ;apply P_id_U22_monotonic;assumption.
       intros ;apply P_id_s_monotonic;assumption.
       intros ;apply P_id_a__U41_monotonic;assumption.
       intros ;apply P_id_U52_monotonic;assumption.
       intros ;apply P_id_isNatList_monotonic;assumption.
       intros ;apply P_id_a__U53_monotonic;assumption.
       intros ;apply P_id_a__U31_monotonic;assumption.
       intros ;apply P_id_isNatIList_monotonic;assumption.
       intros ;apply P_id_a__isNatKind_monotonic;assumption.
       intros ;apply P_id_a__isNatList_monotonic;assumption.
       intros ;apply P_id_U32_monotonic;assumption.
       intros ;apply P_id_mark_monotonic;assumption.
       intros ;apply P_id_a__U43_monotonic;assumption.
       intros ;apply P_id_U61_monotonic;assumption.
       intros ;apply P_id_isNat_monotonic;assumption.
       intros ;apply P_id_a__zeros_bounded;assumption.
       intros ;apply P_id_isNatKind_bounded;assumption.
       intros ;apply P_id_a__isNatIList_bounded;assumption.
       intros ;apply P_id_a__U21_bounded;assumption.
       intros ;apply P_id_U41_bounded;assumption.
       intros ;apply P_id_a__and_bounded;assumption.
       intros ;apply P_id_a__U11_bounded;assumption.
       intros ;apply P_id_U21_bounded;assumption.
       intros ;apply P_id_a__U61_bounded;assumption.
       intros ;apply P_id_a__U32_bounded;assumption.
       intros ;apply P_id_U51_bounded;assumption.
       intros ;apply P_id_isNatIListKind_bounded;assumption.
       intros ;apply P_id_0_bounded;assumption.
       intros ;apply P_id_U12_bounded;assumption.
       intros ;apply P_id_a__U52_bounded;assumption.
       intros ;apply P_id_a__isNat_bounded;assumption.
       intros ;apply P_id_U43_bounded;assumption.
       intros ;apply P_id_a__isNatIListKind_bounded;assumption.
       intros ;apply P_id_a__U12_bounded;assumption.
       intros ;apply P_id_U31_bounded;assumption.
       intros ;apply P_id_a__length_bounded;assumption.
       intros ;apply P_id_a__U42_bounded;assumption.
       intros ;apply P_id_U53_bounded;assumption.
       intros ;apply P_id_and_bounded;assumption.
       intros ;apply P_id_cons_bounded;assumption.
       intros ;apply P_id_U11_bounded;assumption.
       intros ;apply P_id_a__U51_bounded;assumption.
       intros ;apply P_id_a__U22_bounded;assumption.
       intros ;apply P_id_U42_bounded;assumption.
       intros ;apply P_id_length_bounded;assumption.
       intros ;apply P_id_tt_bounded;assumption.
       intros ;apply P_id_U22_bounded;assumption.
       intros ;apply P_id_s_bounded;assumption.
       intros ;apply P_id_a__U41_bounded;assumption.
       intros ;apply P_id_U52_bounded;assumption.
       intros ;apply P_id_nil_bounded;assumption.
       intros ;apply P_id_zeros_bounded;assumption.
       intros ;apply P_id_isNatList_bounded;assumption.
       intros ;apply P_id_a__U53_bounded;assumption.
       intros ;apply P_id_a__U31_bounded;assumption.
       intros ;apply P_id_isNatIList_bounded;assumption.
       intros ;apply P_id_a__isNatKind_bounded;assumption.
       intros ;apply P_id_a__isNatList_bounded;assumption.
       intros ;apply P_id_U32_bounded;assumption.
       intros ;apply P_id_mark_bounded;assumption.
       intros ;apply P_id_a__U43_bounded;assumption.
       intros ;apply P_id_U61_bounded;assumption.
       intros ;apply P_id_isNat_bounded;assumption.
       apply rules_monotonic.
       intros ;apply P_id_A__LENGTH_monotonic;assumption.
       intros ;apply P_id_A__U11_monotonic;assumption.
       intros ;apply P_id_A__U41_monotonic;assumption.
       intros ;apply P_id_A__ISNATKIND_monotonic;assumption.
       intros ;apply P_id_A__U22_monotonic;assumption.
       intros ;apply P_id_A__U51_monotonic;assumption.
       intros ;apply P_id_A__AND_monotonic;assumption.
       intros ;apply P_id_A__ISNATLIST_monotonic;assumption.
       intros ;apply P_id_A__U43_monotonic;assumption.
       intros ;apply P_id_A__U31_monotonic;assumption.
       intros ;apply P_id_A__U53_monotonic;assumption.
       intros ;apply P_id_MARK_monotonic;assumption.
       intros ;apply P_id_A__U12_monotonic;assumption.
       intros ;apply P_id_A__U42_monotonic;assumption.
       intros ;apply P_id_A__ISNAT_monotonic;assumption.
       intros ;apply P_id_A__U52_monotonic;assumption.
       intros ;apply P_id_A__ISNATILISTKIND_monotonic;assumption.
       intros ;apply P_id_A__U21_monotonic;assumption.
       intros ;apply P_id_A__ISNATILIST_monotonic;assumption.
       intros ;apply P_id_A__U32_monotonic;assumption.
       intros ;apply P_id_A__U61_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_12_large_scc_2_strict_in_lt :
      Relation_Definitions.inclusion _ DP_R_xml_0_scc_12_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_12_large_scc_2_large_in_le :
      Relation_Definitions.inclusion _ DP_R_xml_0_scc_12_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_12_large_scc_2_large  := 
       WF_DP_R_xml_0_scc_12_large_scc_2_large.wf.
     
     
     Lemma wf :
      well_founded WF_DP_R_xml_0_scc_12_large.DP_R_xml_0_scc_12_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_12_large_scc_2_large).
       clear x.
       intros x _ IHx IHx'.
       constructor.
       intros y H.
       
       destruct H;
        (apply IHx';apply DP_R_xml_0_scc_12_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_12_large_scc_2_large_in_le;
           econstructor eassumption])).
       apply wf_DP_R_xml_0_scc_12_large_scc_2_large.
     Qed.
    End WF_DP_R_xml_0_scc_12_large_scc_2.
    
    Definition wf_DP_R_xml_0_scc_12_large_scc_2  := 
      WF_DP_R_xml_0_scc_12_large_scc_2.wf.
    
    
    Lemma acc_DP_R_xml_0_scc_12_large_scc_2 :
     forall x y, 
      (DP_R_xml_0_scc_12_large_scc_2 x y) ->
       Acc WF_DP_R_xml_0_scc_12.DP_R_xml_0_scc_12_large x.
    Proof.
      intros x.
      pattern x.
      apply (@Acc_ind _ DP_R_xml_0_scc_12_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_12_large_scc_2.
    Qed.
    
    
    Lemma wf : well_founded WF_DP_R_xml_0_scc_12.DP_R_xml_0_scc_12_large.
    Proof.
      constructor;intros _y _h;inversion _h;clear _h;subst;
       (eapply acc_DP_R_xml_0_scc_12_large_scc_2;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_scc_12_large_scc_1;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((eapply acc_DP_R_xml_0_scc_12_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_12_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_isNatKind (x11:Z) := 0.
   
   Definition P_id_a__isNatIList (x11:Z) := 0.
   
   Definition P_id_a__U21 (x11:Z) (x12:Z) := 2* x11.
   
   Definition P_id_U41 (x11:Z) (x12:Z) (x13:Z) := 2* x11.
   
   Definition P_id_a__and (x11:Z) (x12:Z) := 2* x11 + 1* x12.
   
   Definition P_id_a__U11 (x11:Z) (x12:Z) := 1* x11.
   
   Definition P_id_U21 (x11:Z) (x12:Z) := 2* x11.
   
   Definition P_id_a__U61 (x11:Z) (x12:Z) := 1 + 2* x11 + 1* x12.
   
   Definition P_id_a__U32 (x11:Z) := 2* x11.
   
   Definition P_id_U51 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
   
   Definition P_id_isNatIListKind (x11:Z) := 0.
   
   Definition P_id_0  := 0.
   
   Definition P_id_U12 (x11:Z) := 1* x11.
   
   Definition P_id_a__U52 (x11:Z) (x12:Z) := 2* x11.
   
   Definition P_id_a__isNat (x11:Z) := 0.
   
   Definition P_id_U43 (x11:Z) := 2* x11.
   
   Definition P_id_a__isNatIListKind (x11:Z) := 0.
   
   Definition P_id_a__U12 (x11:Z) := 1* x11.
   
   Definition P_id_U31 (x11:Z) (x12:Z) := 2* x11.
   
   Definition P_id_a__length (x11:Z) := 1 + 1* x11.
   
   Definition P_id_a__U42 (x11:Z) (x12:Z) := 1* x11.
   
   Definition P_id_U53 (x11:Z) := 1* x11.
   
   Definition P_id_and (x11:Z) (x12:Z) := 2* x11 + 1* x12.
   
   Definition P_id_cons (x11:Z) (x12:Z) := 2* x11 + 2* x12.
   
   Definition P_id_U11 (x11:Z) (x12:Z) := 1* x11.
   
   Definition P_id_a__U51 (x11:Z) (x12:Z) (x13:Z) := 1* x11.
   
   Definition P_id_a__U22 (x11:Z) := 2* x11.
   
   Definition P_id_U42 (x11:Z) (x12:Z) := 1* x11.
   
   Definition P_id_length (x11:Z) := 1 + 1* x11.
   
   Definition P_id_tt  := 0.
   
   Definition P_id_U22 (x11:Z) := 2* x11.
   
   Definition P_id_s (x11:Z) := 1* x11.
   
   Definition P_id_a__U41 (x11:Z) (x12:Z) (x13:Z) := 2* x11.
   
   Definition P_id_U52 (x11:Z) (x12:Z) := 2* x11.
   
   Definition P_id_nil  := 3.
   
   Definition P_id_zeros  := 0.
   
   Definition P_id_isNatList (x11:Z) := 0.
   
   Definition P_id_a__U53 (x11:Z) := 1* x11.
   
   Definition P_id_a__U31 (x11:Z) (x12:Z) := 2* x11.
   
   Definition P_id_isNatIList (x11:Z) := 0.
   
   Definition P_id_a__isNatKind (x11:Z) := 0.
   
   Definition P_id_a__isNatList (x11:Z) := 0.
   
   Definition P_id_U32 (x11:Z) := 2* x11.
   
   Definition P_id_mark (x11:Z) := 1* x11.
   
   Definition P_id_a__U43 (x11:Z) := 2* x11.
   
   Definition P_id_U61 (x11:Z) (x12:Z) := 1 + 2* x11 + 1* x12.
   
   Definition P_id_isNat (x11:Z) := 0.
   
   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__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__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_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_a__and_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_a__and x12 x14 <= P_id_a__and x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     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_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__U61_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_a__U61 x12 x14 <= P_id_a__U61 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     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 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_a__U32 x12 <= P_id_a__U32 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U51_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_U51 x12 x14 x16 <= P_id_U51 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     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_U12_monotonic :
    forall x12 x11, (0 <= x12)/\ (x12 <= x11) ->P_id_U12 x12 <= P_id_U12 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_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 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_a__U52 x12 x14 <= P_id_a__U52 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_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_U43_monotonic :
    forall x12 x11, (0 <= x12)/\ (x12 <= x11) ->P_id_U43 x12 <= P_id_U43 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_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__U12_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_a__U12 x12 <= P_id_a__U12 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     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__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__U42_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_a__U42 x12 x14 <= P_id_a__U42 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U53_monotonic :
    forall x12 x11, (0 <= x12)/\ (x12 <= x11) ->P_id_U53 x12 <= P_id_U53 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_and_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_and x12 x14 <= P_id_and x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     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_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__U51_monotonic :
    forall x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_a__U51 x12 x14 x16 <= P_id_a__U51 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_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 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_a__U22 x12 <= P_id_a__U22 x11.
   Proof.
     intros x12 x11.
     intros [H_1 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 x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_U42 x12 x14 <= P_id_U42 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     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_U22_monotonic :
    forall x12 x11, (0 <= x12)/\ (x12 <= x11) ->P_id_U22 x12 <= P_id_U22 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     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__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_U52_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_U52 x12 x14 <= P_id_U52 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 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__U53_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_a__U53 x12 <= P_id_a__U53 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     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_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__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__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_U32_monotonic :
    forall x12 x11, (0 <= x12)/\ (x12 <= x11) ->P_id_U32 x12 <= P_id_U32 x11.
   Proof.
     intros x12 x11.
     intros [H_1 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__U43_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_a__U43 x12 <= P_id_a__U43 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     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 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_U61 x12 x14 <= P_id_U61 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     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__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_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__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_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_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_a__and_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__and x12 x11.
   Proof.
     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_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__U61_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U61 x12 x11.
   Proof.
     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 x11, (0 <= x11) ->0 <= P_id_a__U32 x11.
   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 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_U51 x13 x12 x11.
   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_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_U12_bounded : forall x11, (0 <= x11) ->0 <= P_id_U12 x11.
   Proof.
     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 x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U52 x12 x11.
   Proof.
     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_U43_bounded : forall x11, (0 <= x11) ->0 <= P_id_U43 x11.
   Proof.
     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_a__U12_bounded : forall x11, (0 <= x11) ->0 <= P_id_a__U12 x11.
   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__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__U42_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_a__U42 x12 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U53_bounded : forall x11, (0 <= x11) ->0 <= P_id_U53 x11.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_and_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_and x12 x11.
   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_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__U51_bounded :
    forall x12 x13 x11, 
     (0 <= x11) ->(0 <= x12) ->(0 <= x13) ->0 <= P_id_a__U51 x13 x12 x11.
   Proof.
     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 x11, (0 <= x11) ->0 <= P_id_a__U22 x11.
   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 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U42 x12 x11.
   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_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_U22_bounded : forall x11, (0 <= x11) ->0 <= P_id_U22 x11.
   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__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_U52_bounded :
    forall x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U52 x12 x11.
   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_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_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__U53_bounded : forall x11, (0 <= x11) ->0 <= P_id_a__U53 x11.
   Proof.
     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_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.
   
   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_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_U32_bounded : forall x11, (0 <= x11) ->0 <= P_id_U32 x11.
   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__U43_bounded : forall x11, (0 <= x11) ->0 <= P_id_a__U43 x11.
   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 x12 x11, (0 <= x11) ->(0 <= x12) ->0 <= P_id_U61 x12 x11.
   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.
   
   Definition measure  := 
     InterpZ.measure 0 P_id_a__zeros P_id_isNatKind P_id_a__isNatIList 
      P_id_a__U21 P_id_U41 P_id_a__and P_id_a__U11 P_id_U21 P_id_a__U61 
      P_id_a__U32 P_id_U51 P_id_isNatIListKind P_id_0 P_id_U12 P_id_a__U52 
      P_id_a__isNat P_id_U43 P_id_a__isNatIListKind P_id_a__U12 P_id_U31 
      P_id_a__length P_id_a__U42 P_id_U53 P_id_and P_id_cons P_id_U11 
      P_id_a__U51 P_id_a__U22 P_id_U42 P_id_length P_id_tt P_id_U22 P_id_s 
      P_id_a__U41 P_id_U52 P_id_nil P_id_zeros P_id_isNatList P_id_a__U53 
      P_id_a__U31 P_id_isNatIList P_id_a__isNatKind P_id_a__isNatList 
      P_id_U32 P_id_mark P_id_a__U43 P_id_U61 P_id_isNat.
   
   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_isNatKind (x11::nil)) =>
                    P_id_isNatKind (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__U21 (x12::x11::nil)) =>
                    P_id_a__U21 (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_a__and (x12::x11::nil)) =>
                    P_id_a__and (measure x12) (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_U21 (x12::x11::nil)) =>
                    P_id_U21 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U61 (x12::x11::nil)) =>
                    P_id_a__U61 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U32 (x11::nil)) =>
                    P_id_a__U32 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U51 (x13::x12::x11::nil)) =>
                    P_id_U51 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_isNatIListKind 
                      (x11::nil)) =>
                    P_id_isNatIListKind (measure x11)
                   | (algebra.Alg.Term algebra.F.id_0 nil) => P_id_0 
                   | (algebra.Alg.Term algebra.F.id_U12 (x11::nil)) =>
                    P_id_U12 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U52 (x12::x11::nil)) =>
                    P_id_a__U52 (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_U43 (x11::nil)) =>
                    P_id_U43 (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__U12 (x11::nil)) =>
                    P_id_a__U12 (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__length (x11::nil)) =>
                    P_id_a__length (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U42 (x12::x11::nil)) =>
                    P_id_a__U42 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U53 (x11::nil)) =>
                    P_id_U53 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_and (x12::x11::nil)) =>
                    P_id_and (measure x12) (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_U11 (x12::x11::nil)) =>
                    P_id_U11 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U51 (x13::x12::
                      x11::nil)) =>
                    P_id_a__U51 (measure x13) (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U22 (x11::nil)) =>
                    P_id_a__U22 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U42 (x12::x11::nil)) =>
                    P_id_U42 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_length (x11::nil)) =>
                    P_id_length (measure x11)
                   | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt 
                   | (algebra.Alg.Term algebra.F.id_U22 (x11::nil)) =>
                    P_id_U22 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_s (x11::nil)) =>
                    P_id_s (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_U52 (x12::x11::nil)) =>
                    P_id_U52 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil 
                   | (algebra.Alg.Term algebra.F.id_zeros nil) => P_id_zeros 
                   | (algebra.Alg.Term algebra.F.id_isNatList (x11::nil)) =>
                    P_id_isNatList (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U53 (x11::nil)) =>
                    P_id_a__U53 (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_isNatIList (x11::nil)) =>
                    P_id_isNatIList (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__isNatList (x11::nil)) =>
                    P_id_a__isNatList (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U32 (x11::nil)) =>
                    P_id_U32 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_mark (x11::nil)) =>
                    P_id_mark (measure x11)
                   | (algebra.Alg.Term algebra.F.id_a__U43 (x11::nil)) =>
                    P_id_a__U43 (measure x11)
                   | (algebra.Alg.Term algebra.F.id_U61 (x12::x11::nil)) =>
                    P_id_U61 (measure x12) (measure x11)
                   | (algebra.Alg.Term algebra.F.id_isNat (x11::nil)) =>
                    P_id_isNat (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_isNatKind_monotonic;assumption.
     intros ;apply P_id_a__isNatIList_monotonic;assumption.
     intros ;apply P_id_a__U21_monotonic;assumption.
     intros ;apply P_id_U41_monotonic;assumption.
     intros ;apply P_id_a__and_monotonic;assumption.
     intros ;apply P_id_a__U11_monotonic;assumption.
     intros ;apply P_id_U21_monotonic;assumption.
     intros ;apply P_id_a__U61_monotonic;assumption.
     intros ;apply P_id_a__U32_monotonic;assumption.
     intros ;apply P_id_U51_monotonic;assumption.
     intros ;apply P_id_isNatIListKind_monotonic;assumption.
     intros ;apply P_id_U12_monotonic;assumption.
     intros ;apply P_id_a__U52_monotonic;assumption.
     intros ;apply P_id_a__isNat_monotonic;assumption.
     intros ;apply P_id_U43_monotonic;assumption.
     intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
     intros ;apply P_id_a__U12_monotonic;assumption.
     intros ;apply P_id_U31_monotonic;assumption.
     intros ;apply P_id_a__length_monotonic;assumption.
     intros ;apply P_id_a__U42_monotonic;assumption.
     intros ;apply P_id_U53_monotonic;assumption.
     intros ;apply P_id_and_monotonic;assumption.
     intros ;apply P_id_cons_monotonic;assumption.
     intros ;apply P_id_U11_monotonic;assumption.
     intros ;apply P_id_a__U51_monotonic;assumption.
     intros ;apply P_id_a__U22_monotonic;assumption.
     intros ;apply P_id_U42_monotonic;assumption.
     intros ;apply P_id_length_monotonic;assumption.
     intros ;apply P_id_U22_monotonic;assumption.
     intros ;apply P_id_s_monotonic;assumption.
     intros ;apply P_id_a__U41_monotonic;assumption.
     intros ;apply P_id_U52_monotonic;assumption.
     intros ;apply P_id_isNatList_monotonic;assumption.
     intros ;apply P_id_a__U53_monotonic;assumption.
     intros ;apply P_id_a__U31_monotonic;assumption.
     intros ;apply P_id_isNatIList_monotonic;assumption.
     intros ;apply P_id_a__isNatKind_monotonic;assumption.
     intros ;apply P_id_a__isNatList_monotonic;assumption.
     intros ;apply P_id_U32_monotonic;assumption.
     intros ;apply P_id_mark_monotonic;assumption.
     intros ;apply P_id_a__U43_monotonic;assumption.
     intros ;apply P_id_U61_monotonic;assumption.
     intros ;apply P_id_isNat_monotonic;assumption.
     intros ;apply P_id_a__zeros_bounded;assumption.
     intros ;apply P_id_isNatKind_bounded;assumption.
     intros ;apply P_id_a__isNatIList_bounded;assumption.
     intros ;apply P_id_a__U21_bounded;assumption.
     intros ;apply P_id_U41_bounded;assumption.
     intros ;apply P_id_a__and_bounded;assumption.
     intros ;apply P_id_a__U11_bounded;assumption.
     intros ;apply P_id_U21_bounded;assumption.
     intros ;apply P_id_a__U61_bounded;assumption.
     intros ;apply P_id_a__U32_bounded;assumption.
     intros ;apply P_id_U51_bounded;assumption.
     intros ;apply P_id_isNatIListKind_bounded;assumption.
     intros ;apply P_id_0_bounded;assumption.
     intros ;apply P_id_U12_bounded;assumption.
     intros ;apply P_id_a__U52_bounded;assumption.
     intros ;apply P_id_a__isNat_bounded;assumption.
     intros ;apply P_id_U43_bounded;assumption.
     intros ;apply P_id_a__isNatIListKind_bounded;assumption.
     intros ;apply P_id_a__U12_bounded;assumption.
     intros ;apply P_id_U31_bounded;assumption.
     intros ;apply P_id_a__length_bounded;assumption.
     intros ;apply P_id_a__U42_bounded;assumption.
     intros ;apply P_id_U53_bounded;assumption.
     intros ;apply P_id_and_bounded;assumption.
     intros ;apply P_id_cons_bounded;assumption.
     intros ;apply P_id_U11_bounded;assumption.
     intros ;apply P_id_a__U51_bounded;assumption.
     intros ;apply P_id_a__U22_bounded;assumption.
     intros ;apply P_id_U42_bounded;assumption.
     intros ;apply P_id_length_bounded;assumption.
     intros ;apply P_id_tt_bounded;assumption.
     intros ;apply P_id_U22_bounded;assumption.
     intros ;apply P_id_s_bounded;assumption.
     intros ;apply P_id_a__U41_bounded;assumption.
     intros ;apply P_id_U52_bounded;assumption.
     intros ;apply P_id_nil_bounded;assumption.
     intros ;apply P_id_zeros_bounded;assumption.
     intros ;apply P_id_isNatList_bounded;assumption.
     intros ;apply P_id_a__U53_bounded;assumption.
     intros ;apply P_id_a__U31_bounded;assumption.
     intros ;apply P_id_isNatIList_bounded;assumption.
     intros ;apply P_id_a__isNatKind_bounded;assumption.
     intros ;apply P_id_a__isNatList_bounded;assumption.
     intros ;apply P_id_U32_bounded;assumption.
     intros ;apply P_id_mark_bounded;assumption.
     intros ;apply P_id_a__U43_bounded;assumption.
     intros ;apply P_id_U61_bounded;assumption.
     intros ;apply P_id_isNat_bounded;assumption.
     apply rules_monotonic.
   Qed.
   
   Definition P_id_A__LENGTH (x11:Z) := 1 + 1* x11.
   
   Definition P_id_A__U11 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_A__U41 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_A__ISNATKIND (x11:Z) := 0.
   
   Definition P_id_A__U22 (x11:Z) := 0.
   
   Definition P_id_A__U51 (x11:Z) (x12:Z) (x13:Z) := 0.
   
   Definition P_id_A__AND (x11:Z) (x12:Z) := 2* x12.
   
   Definition P_id_A__ISNATLIST (x11:Z) := 0.
   
   Definition P_id_A__U43 (x11:Z) := 0.
   
   Definition P_id_A__U31 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_A__U53 (x11:Z) := 0.
   
   Definition P_id_MARK (x11:Z) := 2* x11.
   
   Definition P_id_A__U12 (x11:Z) := 0.
   
   Definition P_id_A__U42 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_A__ZEROS  := 0.
   
   Definition P_id_A__ISNAT (x11:Z) := 0.
   
   Definition P_id_A__U52 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_A__ISNATILISTKIND (x11:Z) := 0.
   
   Definition P_id_A__U21 (x11:Z) (x12:Z) := 0.
   
   Definition P_id_A__ISNATILIST (x11:Z) := 0.
   
   Definition P_id_A__U32 (x11:Z) := 0.
   
   Definition P_id_A__U61 (x11:Z) (x12:Z) := 1 + 2* x12.
   
   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__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__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__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__U22_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_A__U22 x12 <= P_id_A__U22 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_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 x16 x12 x14 x13 x11 x15, 
     (0 <= x16)/\ (x16 <= x15) ->
      (0 <= x14)/\ (x14 <= x13) ->
       (0 <= x12)/\ (x12 <= x11) ->
        P_id_A__U51 x12 x14 x16 <= P_id_A__U51 x11 x13 x15.
   Proof.
     intros x16 x15 x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__AND_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_A__AND x12 x14 <= P_id_A__AND x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_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__U43_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_A__U43 x12 <= P_id_A__U43 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     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__U53_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_A__U53 x12 <= P_id_A__U53 x11.
   Proof.
     intros x12 x11.
     intros [H_1 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__U12_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_A__U12 x12 <= P_id_A__U12 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_0].
     
     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 x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_A__U42 x12 x14 <= P_id_A__U42 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_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_A__U52_monotonic :
    forall x12 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_A__U52 x12 x14 <= P_id_A__U52 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     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__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__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__U32_monotonic :
    forall x12 x11, 
     (0 <= x12)/\ (x12 <= x11) ->P_id_A__U32 x12 <= P_id_A__U32 x11.
   Proof.
     intros x12 x11.
     intros [H_1 H_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 x14 x13 x11, 
     (0 <= x14)/\ (x14 <= x13) ->
      (0 <= x12)/\ (x12 <= x11) ->P_id_A__U61 x12 x14 <= P_id_A__U61 x11 x13.
   Proof.
     intros x14 x13 x12 x11.
     intros [H_1 H_0].
     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_isNatKind 
      P_id_a__isNatIList P_id_a__U21 P_id_U41 P_id_a__and P_id_a__U11 
      P_id_U21 P_id_a__U61 P_id_a__U32 P_id_U51 P_id_isNatIListKind P_id_0 
      P_id_U12 P_id_a__U52 P_id_a__isNat P_id_U43 P_id_a__isNatIListKind 
      P_id_a__U12 P_id_U31 P_id_a__length P_id_a__U42 P_id_U53 P_id_and 
      P_id_cons P_id_U11 P_id_a__U51 P_id_a__U22 P_id_U42 P_id_length 
      P_id_tt P_id_U22 P_id_s P_id_a__U41 P_id_U52 P_id_nil P_id_zeros 
      P_id_isNatList P_id_a__U53 P_id_a__U31 P_id_isNatIList 
      P_id_a__isNatKind P_id_a__isNatList P_id_U32 P_id_mark P_id_a__U43 
      P_id_U61 P_id_isNat P_id_A__LENGTH P_id_A__U11 P_id_A__U41 
      P_id_A__ISNATKIND P_id_A__U22 P_id_A__U51 P_id_A__AND 
      P_id_A__ISNATLIST P_id_A__U43 P_id_A__U31 P_id_A__U53 P_id_MARK 
      P_id_A__U12 P_id_A__U42 P_id_A__ZEROS P_id_A__ISNAT P_id_A__U52 
      P_id_A__ISNATILISTKIND P_id_A__U21 P_id_A__ISNATILIST P_id_A__U32 
      P_id_A__U61.
   
   Lemma marked_measure_equation :
    forall t, 
     marked_measure t = match t with
                          | (algebra.Alg.Term algebra.F.id_a__length 
                             (x11::nil)) =>
                           P_id_A__LENGTH (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__U41 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U41 (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__U22 (x11::nil)) =>
                           P_id_A__U22 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U51 (x13::x12::
                             x11::nil)) =>
                           P_id_A__U51 (measure x13) (measure x12) 
                            (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__and (x12::
                             x11::nil)) =>
                           P_id_A__AND (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__U43 (x11::nil)) =>
                           P_id_A__U43 (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__U53 (x11::nil)) =>
                           P_id_A__U53 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_mark (x11::nil)) =>
                           P_id_MARK (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U12 (x11::nil)) =>
                           P_id_A__U12 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U42 (x12::
                             x11::nil)) =>
                           P_id_A__U42 (measure x12) (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__zeros nil) =>
                           P_id_A__ZEROS 
                          | (algebra.Alg.Term algebra.F.id_a__isNat 
                             (x11::nil)) =>
                           P_id_A__ISNAT (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U52 (x12::
                             x11::nil)) =>
                           P_id_A__U52 (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_a__U21 (x12::
                             x11::nil)) =>
                           P_id_A__U21 (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__U32 (x11::nil)) =>
                           P_id_A__U32 (measure x11)
                          | (algebra.Alg.Term algebra.F.id_a__U61 (x12::
                             x11::nil)) =>
                           P_id_A__U61 (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_isNatKind_monotonic;assumption.
     intros ;apply P_id_a__isNatIList_monotonic;assumption.
     intros ;apply P_id_a__U21_monotonic;assumption.
     intros ;apply P_id_U41_monotonic;assumption.
     intros ;apply P_id_a__and_monotonic;assumption.
     intros ;apply P_id_a__U11_monotonic;assumption.
     intros ;apply P_id_U21_monotonic;assumption.
     intros ;apply P_id_a__U61_monotonic;assumption.
     intros ;apply P_id_a__U32_monotonic;assumption.
     intros ;apply P_id_U51_monotonic;assumption.
     intros ;apply P_id_isNatIListKind_monotonic;assumption.
     intros ;apply P_id_U12_monotonic;assumption.
     intros ;apply P_id_a__U52_monotonic;assumption.
     intros ;apply P_id_a__isNat_monotonic;assumption.
     intros ;apply P_id_U43_monotonic;assumption.
     intros ;apply P_id_a__isNatIListKind_monotonic;assumption.
     intros ;apply P_id_a__U12_monotonic;assumption.
     intros ;apply P_id_U31_monotonic;assumption.
     intros ;apply P_id_a__length_monotonic;assumption.
     intros ;apply P_id_a__U42_monotonic;assumption.
     intros ;apply P_id_U53_monotonic;assumption.
     intros ;apply P_id_and_monotonic;assumption.
     intros ;apply P_id_cons_monotonic;assumption.
     intros ;apply P_id_U11_monotonic;assumption.
     intros ;apply P_id_a__U51_monotonic;assumption.
     intros ;apply P_id_a__U22_monotonic;assumption.
     intros ;apply P_id_U42_monotonic;assumption.
     intros ;apply P_id_length_monotonic;assumption.
     intros ;apply P_id_U22_monotonic;assumption.
     intros ;apply P_id_s_monotonic;assumption.
     intros ;apply P_id_a__U41_monotonic;assumption.
     intros ;apply P_id_U52_monotonic;assumption.
     intros ;apply P_id_isNatList_monotonic;assumption.
     intros ;apply P_id_a__U53_monotonic;assumption.
     intros ;apply P_id_a__U31_monotonic;assumption.
     intros ;apply P_id_isNatIList_monotonic;assumption.
     intros ;apply P_id_a__isNatKind_monotonic;assumption.
     intros ;apply P_id_a__isNatList_monotonic;assumption.
     intros ;apply P_id_U32_monotonic;assumption.
     intros ;apply P_id_mark_monotonic;assumption.
     intros ;apply P_id_a__U43_monotonic;assumption.
     intros ;apply P_id_U61_monotonic;assumption.
     intros ;apply P_id_isNat_monotonic;assumption.
     intros ;apply P_id_a__zeros_bounded;assumption.
     intros ;apply P_id_isNatKind_bounded;assumption.
     intros ;apply P_id_a__isNatIList_bounded;assumption.
     intros ;apply P_id_a__U21_bounded;assumption.
     intros ;apply P_id_U41_bounded;assumption.
     intros ;apply P_id_a__and_bounded;assumption.
     intros ;apply P_id_a__U11_bounded;assumption.
     intros ;apply P_id_U21_bounded;assumption.
     intros ;apply P_id_a__U61_bounded;assumption.
     intros ;apply P_id_a__U32_bounded;assumption.
     intros ;apply P_id_U51_bounded;assumption.
     intros ;apply P_id_isNatIListKind_bounded;assumption.
     intros ;apply P_id_0_bounded;assumption.
     intros ;apply P_id_U12_bounded;assumption.
     intros ;apply P_id_a__U52_bounded;assumption.
     intros ;apply P_id_a__isNat_bounded;assumption.
     intros ;apply P_id_U43_bounded;assumption.
     intros ;apply P_id_a__isNatIListKind_bounded;assumption.
     intros ;apply P_id_a__U12_bounded;assumption.
     intros ;apply P_id_U31_bounded;assumption.
     intros ;apply P_id_a__length_bounded;assumption.
     intros ;apply P_id_a__U42_bounded;assumption.
     intros ;apply P_id_U53_bounded;assumption.
     intros ;apply P_id_and_bounded;assumption.
     intros ;apply P_id_cons_bounded;assumption.
     intros ;apply P_id_U11_bounded;assumption.
     intros ;apply P_id_a__U51_bounded;assumption.
     intros ;apply P_id_a__U22_bounded;assumption.
     intros ;apply P_id_U42_bounded;assumption.
     intros ;apply P_id_length_bounded;assumption.
     intros ;apply P_id_tt_bounded;assumption.
     intros ;apply P_id_U22_bounded;assumption.
     intros ;apply P_id_s_bounded;assumption.
     intros ;apply P_id_a__U41_bounded;assumption.
     intros ;apply P_id_U52_bounded;assumption.
     intros ;apply P_id_nil_bounded;assumption.
     intros ;apply P_id_zeros_bounded;assumption.
     intros ;apply P_id_isNatList_bounded;assumption.
     intros ;apply P_id_a__U53_bounded;assumption.
     intros ;apply P_id_a__U31_bounded;assumption.
     intros ;apply P_id_isNatIList_bounded;assumption.
     intros ;apply P_id_a__isNatKind_bounded;assumption.
     intros ;apply P_id_a__isNatList_bounded;assumption.
     intros ;apply P_id_U32_bounded;assumption.
     intros ;apply P_id_mark_bounded;assumption.
     intros ;apply P_id_a__U43_bounded;assumption.
     intros ;apply P_id_U61_bounded;assumption.
     intros ;apply P_id_isNat_bounded;assumption.
     apply rules_monotonic.
     intros ;apply P_id_A__LENGTH_monotonic;assumption.
     intros ;apply P_id_A__U11_monotonic;assumption.
     intros ;apply P_id_A__U41_monotonic;assumption.
     intros ;apply P_id_A__ISNATKIND_monotonic;assumption.
     intros ;apply P_id_A__U22_monotonic;assumption.
     intros ;apply P_id_A__U51_monotonic;assumption.
     intros ;apply P_id_A__AND_monotonic;assumption.
     intros ;apply P_id_A__ISNATLIST_monotonic;assumption.
     intros ;apply P_id_A__U43_monotonic;assumption.
     intros ;apply P_id_A__U31_monotonic;assumption.
     intros ;apply P_id_A__U53_monotonic;assumption.
     intros ;apply P_id_MARK_monotonic;assumption.
     intros ;apply P_id_A__U12_monotonic;assumption.
     intros ;apply P_id_A__U42_monotonic;assumption.
     intros ;apply P_id_A__ISNAT_monotonic;assumption.
     intros ;apply P_id_A__U52_monotonic;assumption.
     intros ;apply P_id_A__ISNATILISTKIND_monotonic;assumption.
     intros ;apply P_id_A__U21_monotonic;assumption.
     intros ;apply P_id_A__ISNATILIST_monotonic;assumption.
     intros ;apply P_id_A__U32_monotonic;assumption.
     intros ;apply P_id_A__U61_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_12_strict_in_lt :
    Relation_Definitions.inclusion _ DP_R_xml_0_scc_12_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_12_large_in_le :
    Relation_Definitions.inclusion _ DP_R_xml_0_scc_12_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_12_large  := WF_DP_R_xml_0_scc_12_large.wf.
   
   
   Lemma wf : well_founded WF_DP_R_xml_0.DP_R_xml_0_scc_12.
   Proof.
     intros x.
     apply (well_founded_ind wf_lt).
     clear x.
     intros x.
     pattern x.
     apply (@Acc_ind _ DP_R_xml_0_scc_12_large).
     clear x.
     intros x _ IHx IHx'.
     constructor.
     intros y H.
     
     destruct H;
      (apply IHx';apply DP_R_xml_0_scc_12_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_12_large_in_le;econstructor eassumption])).
     apply wf_DP_R_xml_0_scc_12_large.
   Qed.
  End WF_DP_R_xml_0_scc_12.
  
  Definition wf_DP_R_xml_0_scc_12  := WF_DP_R_xml_0_scc_12.wf.
  
  
  Lemma acc_DP_R_xml_0_scc_12 :
   forall x y, (DP_R_xml_0_scc_12 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_12).
    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_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' ))||
                 ((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_12.
  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_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_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_GM.trs/a3pat.v" ***
*** End: ***
 *)