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_intersect'ii'in *)
    | id_intersect'ii'in : symb
     (* id_tautology'i'out *)
    | id_tautology'i'out : symb
     (* id_u'6'1 *)
    | id_u'6'1 : symb
     (* id_u'3'1 *)
    | id_u'3'1 : symb
     (* id_u'12'1 *)
    | id_u'12'1 : symb
     (* id_u'2'1 *)
    | id_u'2'1 : symb
     (* id_u'9'1 *)
    | id_u'9'1 : symb
     (* id_iff *)
    | id_iff : symb
     (* id_u'14'1 *)
    | id_u'14'1 : symb
     (* id_intersect'ii'out *)
    | id_intersect'ii'out : symb
     (* id_u'7'1 *)
    | id_u'7'1 : symb
     (* id_x'2d *)
    | id_x'2d : symb
     (* id_u'13'1 *)
    | id_u'13'1 : symb
     (* id_sequent *)
    | id_sequent : symb
     (* id_u'10'1 *)
    | id_u'10'1 : symb
     (* id_x'2a *)
    | id_x'2a : symb
     (* id_tautology'i'in *)
    | id_tautology'i'in : symb
     (* id_cons *)
    | id_cons : symb
     (* id_u'6'2 *)
    | id_u'6'2 : symb
     (* id_x'2b *)
    | id_x'2b : symb
     (* id_u'12'2 *)
    | id_u'12'2 : symb
     (* id_reduce'ii'in *)
    | id_reduce'ii'in : symb
     (* id_p *)
    | id_p : symb
     (* id_u'4'1 *)
    | id_u'4'1 : symb
     (* id_u'15'1 *)
    | id_u'15'1 : symb
     (* id_u'1'1 *)
    | id_u'1'1 : symb
     (* id_u'8'1 *)
    | id_u'8'1 : symb
     (* id_reduce'ii'out *)
    | id_reduce'ii'out : symb
     (* id_nil *)
    | id_nil : symb
     (* id_if *)
    | id_if : symb
     (* id_u'11'1 *)
    | id_u'11'1 : symb
     (* id_u'5'1 *)
    | id_u'5'1 : symb
     (* id_u'16'1 *)
    | id_u'16'1 : symb
  .
  
  
  Definition symb_eq_bool (f1 f2:symb) : bool := 
    match f1,f2 with
      | id_intersect'ii'in,id_intersect'ii'in => true
      | id_tautology'i'out,id_tautology'i'out => true
      | id_u'6'1,id_u'6'1 => true
      | id_u'3'1,id_u'3'1 => true
      | id_u'12'1,id_u'12'1 => true
      | id_u'2'1,id_u'2'1 => true
      | id_u'9'1,id_u'9'1 => true
      | id_iff,id_iff => true
      | id_u'14'1,id_u'14'1 => true
      | id_intersect'ii'out,id_intersect'ii'out => true
      | id_u'7'1,id_u'7'1 => true
      | id_x'2d,id_x'2d => true
      | id_u'13'1,id_u'13'1 => true
      | id_sequent,id_sequent => true
      | id_u'10'1,id_u'10'1 => true
      | id_x'2a,id_x'2a => true
      | id_tautology'i'in,id_tautology'i'in => true
      | id_cons,id_cons => true
      | id_u'6'2,id_u'6'2 => true
      | id_x'2b,id_x'2b => true
      | id_u'12'2,id_u'12'2 => true
      | id_reduce'ii'in,id_reduce'ii'in => true
      | id_p,id_p => true
      | id_u'4'1,id_u'4'1 => true
      | id_u'15'1,id_u'15'1 => true
      | id_u'1'1,id_u'1'1 => true
      | id_u'8'1,id_u'8'1 => true
      | id_reduce'ii'out,id_reduce'ii'out => true
      | id_nil,id_nil => true
      | id_if,id_if => true
      | id_u'11'1,id_u'11'1 => true
      | id_u'5'1,id_u'5'1 => true
      | id_u'16'1,id_u'16'1 => 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_intersect'ii'in,id_intersect'ii'in => refl_equal _
             | id_tautology'i'out,id_tautology'i'out => refl_equal _
             | id_u'6'1,id_u'6'1 => refl_equal _
             | id_u'3'1,id_u'3'1 => refl_equal _
             | id_u'12'1,id_u'12'1 => refl_equal _
             | id_u'2'1,id_u'2'1 => refl_equal _
             | id_u'9'1,id_u'9'1 => refl_equal _
             | id_iff,id_iff => refl_equal _
             | id_u'14'1,id_u'14'1 => refl_equal _
             | id_intersect'ii'out,id_intersect'ii'out => refl_equal _
             | id_u'7'1,id_u'7'1 => refl_equal _
             | id_x'2d,id_x'2d => refl_equal _
             | id_u'13'1,id_u'13'1 => refl_equal _
             | id_sequent,id_sequent => refl_equal _
             | id_u'10'1,id_u'10'1 => refl_equal _
             | id_x'2a,id_x'2a => refl_equal _
             | id_tautology'i'in,id_tautology'i'in => refl_equal _
             | id_cons,id_cons => refl_equal _
             | id_u'6'2,id_u'6'2 => refl_equal _
             | id_x'2b,id_x'2b => refl_equal _
             | id_u'12'2,id_u'12'2 => refl_equal _
             | id_reduce'ii'in,id_reduce'ii'in => refl_equal _
             | id_p,id_p => refl_equal _
             | id_u'4'1,id_u'4'1 => refl_equal _
             | id_u'15'1,id_u'15'1 => refl_equal _
             | id_u'1'1,id_u'1'1 => refl_equal _
             | id_u'8'1,id_u'8'1 => refl_equal _
             | id_reduce'ii'out,id_reduce'ii'out => refl_equal _
             | id_nil,id_nil => refl_equal _
             | id_if,id_if => refl_equal _
             | id_u'11'1,id_u'11'1 => refl_equal _
             | id_u'5'1,id_u'5'1 => refl_equal _
             | id_u'16'1,id_u'16'1 => refl_equal _
             | _,_ => _
             end;intros abs;discriminate.
  Defined.
  
  
  Definition arity (f:symb) := 
    match f with
      | id_intersect'ii'in => term.Free 2
      | id_tautology'i'out => term.Free 0
      | id_u'6'1 => term.Free 5
      | id_u'3'1 => term.Free 1
      | id_u'12'1 => term.Free 5
      | id_u'2'1 => term.Free 1
      | id_u'9'1 => term.Free 1
      | id_iff => term.Free 2
      | id_u'14'1 => term.Free 1
      | id_intersect'ii'out => term.Free 0
      | id_u'7'1 => term.Free 1
      | id_x'2d => term.Free 1
      | id_u'13'1 => term.Free 1
      | id_sequent => term.Free 2
      | id_u'10'1 => term.Free 1
      | id_x'2a => term.Free 2
      | id_tautology'i'in => term.Free 1
      | id_cons => term.Free 2
      | id_u'6'2 => term.Free 1
      | id_x'2b => term.Free 2
      | id_u'12'2 => term.Free 1
      | id_reduce'ii'in => term.Free 2
      | id_p => term.Free 1
      | id_u'4'1 => term.Free 1
      | id_u'15'1 => term.Free 1
      | id_u'1'1 => term.Free 1
      | id_u'8'1 => term.Free 1
      | id_reduce'ii'out => term.Free 0
      | id_nil => term.Free 0
      | id_if => term.Free 2
      | id_u'11'1 => term.Free 1
      | id_u'5'1 => term.Free 1
      | id_u'16'1 => term.Free 1
      end.
  
  
  Definition symb_order (f1 f2:symb) : bool := 
    match f1,f2 with
      | id_intersect'ii'in,id_intersect'ii'in => true
      | id_intersect'ii'in,id_tautology'i'out => false
      | id_intersect'ii'in,id_u'6'1 => false
      | id_intersect'ii'in,id_u'3'1 => false
      | id_intersect'ii'in,id_u'12'1 => false
      | id_intersect'ii'in,id_u'2'1 => false
      | id_intersect'ii'in,id_u'9'1 => false
      | id_intersect'ii'in,id_iff => false
      | id_intersect'ii'in,id_u'14'1 => false
      | id_intersect'ii'in,id_intersect'ii'out => false
      | id_intersect'ii'in,id_u'7'1 => false
      | id_intersect'ii'in,id_x'2d => false
      | id_intersect'ii'in,id_u'13'1 => false
      | id_intersect'ii'in,id_sequent => false
      | id_intersect'ii'in,id_u'10'1 => false
      | id_intersect'ii'in,id_x'2a => false
      | id_intersect'ii'in,id_tautology'i'in => false
      | id_intersect'ii'in,id_cons => false
      | id_intersect'ii'in,id_u'6'2 => false
      | id_intersect'ii'in,id_x'2b => false
      | id_intersect'ii'in,id_u'12'2 => false
      | id_intersect'ii'in,id_reduce'ii'in => false
      | id_intersect'ii'in,id_p => false
      | id_intersect'ii'in,id_u'4'1 => false
      | id_intersect'ii'in,id_u'15'1 => false
      | id_intersect'ii'in,id_u'1'1 => false
      | id_intersect'ii'in,id_u'8'1 => false
      | id_intersect'ii'in,id_reduce'ii'out => false
      | id_intersect'ii'in,id_nil => false
      | id_intersect'ii'in,id_if => false
      | id_intersect'ii'in,id_u'11'1 => false
      | id_intersect'ii'in,id_u'5'1 => false
      | id_intersect'ii'in,id_u'16'1 => false
      | id_tautology'i'out,id_intersect'ii'in => true
      | id_tautology'i'out,id_tautology'i'out => true
      | id_tautology'i'out,id_u'6'1 => false
      | id_tautology'i'out,id_u'3'1 => false
      | id_tautology'i'out,id_u'12'1 => false
      | id_tautology'i'out,id_u'2'1 => false
      | id_tautology'i'out,id_u'9'1 => false
      | id_tautology'i'out,id_iff => false
      | id_tautology'i'out,id_u'14'1 => false
      | id_tautology'i'out,id_intersect'ii'out => false
      | id_tautology'i'out,id_u'7'1 => false
      | id_tautology'i'out,id_x'2d => false
      | id_tautology'i'out,id_u'13'1 => false
      | id_tautology'i'out,id_sequent => false
      | id_tautology'i'out,id_u'10'1 => false
      | id_tautology'i'out,id_x'2a => false
      | id_tautology'i'out,id_tautology'i'in => false
      | id_tautology'i'out,id_cons => false
      | id_tautology'i'out,id_u'6'2 => false
      | id_tautology'i'out,id_x'2b => false
      | id_tautology'i'out,id_u'12'2 => false
      | id_tautology'i'out,id_reduce'ii'in => false
      | id_tautology'i'out,id_p => false
      | id_tautology'i'out,id_u'4'1 => false
      | id_tautology'i'out,id_u'15'1 => false
      | id_tautology'i'out,id_u'1'1 => false
      | id_tautology'i'out,id_u'8'1 => false
      | id_tautology'i'out,id_reduce'ii'out => false
      | id_tautology'i'out,id_nil => false
      | id_tautology'i'out,id_if => false
      | id_tautology'i'out,id_u'11'1 => false
      | id_tautology'i'out,id_u'5'1 => false
      | id_tautology'i'out,id_u'16'1 => false
      | id_u'6'1,id_intersect'ii'in => true
      | id_u'6'1,id_tautology'i'out => true
      | id_u'6'1,id_u'6'1 => true
      | id_u'6'1,id_u'3'1 => false
      | id_u'6'1,id_u'12'1 => false
      | id_u'6'1,id_u'2'1 => false
      | id_u'6'1,id_u'9'1 => false
      | id_u'6'1,id_iff => false
      | id_u'6'1,id_u'14'1 => false
      | id_u'6'1,id_intersect'ii'out => false
      | id_u'6'1,id_u'7'1 => false
      | id_u'6'1,id_x'2d => false
      | id_u'6'1,id_u'13'1 => false
      | id_u'6'1,id_sequent => false
      | id_u'6'1,id_u'10'1 => false
      | id_u'6'1,id_x'2a => false
      | id_u'6'1,id_tautology'i'in => false
      | id_u'6'1,id_cons => false
      | id_u'6'1,id_u'6'2 => false
      | id_u'6'1,id_x'2b => false
      | id_u'6'1,id_u'12'2 => false
      | id_u'6'1,id_reduce'ii'in => false
      | id_u'6'1,id_p => false
      | id_u'6'1,id_u'4'1 => false
      | id_u'6'1,id_u'15'1 => false
      | id_u'6'1,id_u'1'1 => false
      | id_u'6'1,id_u'8'1 => false
      | id_u'6'1,id_reduce'ii'out => false
      | id_u'6'1,id_nil => false
      | id_u'6'1,id_if => false
      | id_u'6'1,id_u'11'1 => false
      | id_u'6'1,id_u'5'1 => false
      | id_u'6'1,id_u'16'1 => false
      | id_u'3'1,id_intersect'ii'in => true
      | id_u'3'1,id_tautology'i'out => true
      | id_u'3'1,id_u'6'1 => true
      | id_u'3'1,id_u'3'1 => true
      | id_u'3'1,id_u'12'1 => false
      | id_u'3'1,id_u'2'1 => false
      | id_u'3'1,id_u'9'1 => false
      | id_u'3'1,id_iff => false
      | id_u'3'1,id_u'14'1 => false
      | id_u'3'1,id_intersect'ii'out => false
      | id_u'3'1,id_u'7'1 => false
      | id_u'3'1,id_x'2d => false
      | id_u'3'1,id_u'13'1 => false
      | id_u'3'1,id_sequent => false
      | id_u'3'1,id_u'10'1 => false
      | id_u'3'1,id_x'2a => false
      | id_u'3'1,id_tautology'i'in => false
      | id_u'3'1,id_cons => false
      | id_u'3'1,id_u'6'2 => false
      | id_u'3'1,id_x'2b => false
      | id_u'3'1,id_u'12'2 => false
      | id_u'3'1,id_reduce'ii'in => false
      | id_u'3'1,id_p => false
      | id_u'3'1,id_u'4'1 => false
      | id_u'3'1,id_u'15'1 => false
      | id_u'3'1,id_u'1'1 => false
      | id_u'3'1,id_u'8'1 => false
      | id_u'3'1,id_reduce'ii'out => false
      | id_u'3'1,id_nil => false
      | id_u'3'1,id_if => false
      | id_u'3'1,id_u'11'1 => false
      | id_u'3'1,id_u'5'1 => false
      | id_u'3'1,id_u'16'1 => false
      | id_u'12'1,id_intersect'ii'in => true
      | id_u'12'1,id_tautology'i'out => true
      | id_u'12'1,id_u'6'1 => true
      | id_u'12'1,id_u'3'1 => true
      | id_u'12'1,id_u'12'1 => true
      | id_u'12'1,id_u'2'1 => false
      | id_u'12'1,id_u'9'1 => false
      | id_u'12'1,id_iff => false
      | id_u'12'1,id_u'14'1 => false
      | id_u'12'1,id_intersect'ii'out => false
      | id_u'12'1,id_u'7'1 => false
      | id_u'12'1,id_x'2d => false
      | id_u'12'1,id_u'13'1 => false
      | id_u'12'1,id_sequent => false
      | id_u'12'1,id_u'10'1 => false
      | id_u'12'1,id_x'2a => false
      | id_u'12'1,id_tautology'i'in => false
      | id_u'12'1,id_cons => false
      | id_u'12'1,id_u'6'2 => false
      | id_u'12'1,id_x'2b => false
      | id_u'12'1,id_u'12'2 => false
      | id_u'12'1,id_reduce'ii'in => false
      | id_u'12'1,id_p => false
      | id_u'12'1,id_u'4'1 => false
      | id_u'12'1,id_u'15'1 => false
      | id_u'12'1,id_u'1'1 => false
      | id_u'12'1,id_u'8'1 => false
      | id_u'12'1,id_reduce'ii'out => false
      | id_u'12'1,id_nil => false
      | id_u'12'1,id_if => false
      | id_u'12'1,id_u'11'1 => false
      | id_u'12'1,id_u'5'1 => false
      | id_u'12'1,id_u'16'1 => false
      | id_u'2'1,id_intersect'ii'in => true
      | id_u'2'1,id_tautology'i'out => true
      | id_u'2'1,id_u'6'1 => true
      | id_u'2'1,id_u'3'1 => true
      | id_u'2'1,id_u'12'1 => true
      | id_u'2'1,id_u'2'1 => true
      | id_u'2'1,id_u'9'1 => false
      | id_u'2'1,id_iff => false
      | id_u'2'1,id_u'14'1 => false
      | id_u'2'1,id_intersect'ii'out => false
      | id_u'2'1,id_u'7'1 => false
      | id_u'2'1,id_x'2d => false
      | id_u'2'1,id_u'13'1 => false
      | id_u'2'1,id_sequent => false
      | id_u'2'1,id_u'10'1 => false
      | id_u'2'1,id_x'2a => false
      | id_u'2'1,id_tautology'i'in => false
      | id_u'2'1,id_cons => false
      | id_u'2'1,id_u'6'2 => false
      | id_u'2'1,id_x'2b => false
      | id_u'2'1,id_u'12'2 => false
      | id_u'2'1,id_reduce'ii'in => false
      | id_u'2'1,id_p => false
      | id_u'2'1,id_u'4'1 => false
      | id_u'2'1,id_u'15'1 => false
      | id_u'2'1,id_u'1'1 => false
      | id_u'2'1,id_u'8'1 => false
      | id_u'2'1,id_reduce'ii'out => false
      | id_u'2'1,id_nil => false
      | id_u'2'1,id_if => false
      | id_u'2'1,id_u'11'1 => false
      | id_u'2'1,id_u'5'1 => false
      | id_u'2'1,id_u'16'1 => false
      | id_u'9'1,id_intersect'ii'in => true
      | id_u'9'1,id_tautology'i'out => true
      | id_u'9'1,id_u'6'1 => true
      | id_u'9'1,id_u'3'1 => true
      | id_u'9'1,id_u'12'1 => true
      | id_u'9'1,id_u'2'1 => true
      | id_u'9'1,id_u'9'1 => true
      | id_u'9'1,id_iff => false
      | id_u'9'1,id_u'14'1 => false
      | id_u'9'1,id_intersect'ii'out => false
      | id_u'9'1,id_u'7'1 => false
      | id_u'9'1,id_x'2d => false
      | id_u'9'1,id_u'13'1 => false
      | id_u'9'1,id_sequent => false
      | id_u'9'1,id_u'10'1 => false
      | id_u'9'1,id_x'2a => false
      | id_u'9'1,id_tautology'i'in => false
      | id_u'9'1,id_cons => false
      | id_u'9'1,id_u'6'2 => false
      | id_u'9'1,id_x'2b => false
      | id_u'9'1,id_u'12'2 => false
      | id_u'9'1,id_reduce'ii'in => false
      | id_u'9'1,id_p => false
      | id_u'9'1,id_u'4'1 => false
      | id_u'9'1,id_u'15'1 => false
      | id_u'9'1,id_u'1'1 => false
      | id_u'9'1,id_u'8'1 => false
      | id_u'9'1,id_reduce'ii'out => false
      | id_u'9'1,id_nil => false
      | id_u'9'1,id_if => false
      | id_u'9'1,id_u'11'1 => false
      | id_u'9'1,id_u'5'1 => false
      | id_u'9'1,id_u'16'1 => false
      | id_iff,id_intersect'ii'in => true
      | id_iff,id_tautology'i'out => true
      | id_iff,id_u'6'1 => true
      | id_iff,id_u'3'1 => true
      | id_iff,id_u'12'1 => true
      | id_iff,id_u'2'1 => true
      | id_iff,id_u'9'1 => true
      | id_iff,id_iff => true
      | id_iff,id_u'14'1 => false
      | id_iff,id_intersect'ii'out => false
      | id_iff,id_u'7'1 => false
      | id_iff,id_x'2d => false
      | id_iff,id_u'13'1 => false
      | id_iff,id_sequent => false
      | id_iff,id_u'10'1 => false
      | id_iff,id_x'2a => false
      | id_iff,id_tautology'i'in => false
      | id_iff,id_cons => false
      | id_iff,id_u'6'2 => false
      | id_iff,id_x'2b => false
      | id_iff,id_u'12'2 => false
      | id_iff,id_reduce'ii'in => false
      | id_iff,id_p => false
      | id_iff,id_u'4'1 => false
      | id_iff,id_u'15'1 => false
      | id_iff,id_u'1'1 => false
      | id_iff,id_u'8'1 => false
      | id_iff,id_reduce'ii'out => false
      | id_iff,id_nil => false
      | id_iff,id_if => false
      | id_iff,id_u'11'1 => false
      | id_iff,id_u'5'1 => false
      | id_iff,id_u'16'1 => false
      | id_u'14'1,id_intersect'ii'in => true
      | id_u'14'1,id_tautology'i'out => true
      | id_u'14'1,id_u'6'1 => true
      | id_u'14'1,id_u'3'1 => true
      | id_u'14'1,id_u'12'1 => true
      | id_u'14'1,id_u'2'1 => true
      | id_u'14'1,id_u'9'1 => true
      | id_u'14'1,id_iff => true
      | id_u'14'1,id_u'14'1 => true
      | id_u'14'1,id_intersect'ii'out => false
      | id_u'14'1,id_u'7'1 => false
      | id_u'14'1,id_x'2d => false
      | id_u'14'1,id_u'13'1 => false
      | id_u'14'1,id_sequent => false
      | id_u'14'1,id_u'10'1 => false
      | id_u'14'1,id_x'2a => false
      | id_u'14'1,id_tautology'i'in => false
      | id_u'14'1,id_cons => false
      | id_u'14'1,id_u'6'2 => false
      | id_u'14'1,id_x'2b => false
      | id_u'14'1,id_u'12'2 => false
      | id_u'14'1,id_reduce'ii'in => false
      | id_u'14'1,id_p => false
      | id_u'14'1,id_u'4'1 => false
      | id_u'14'1,id_u'15'1 => false
      | id_u'14'1,id_u'1'1 => false
      | id_u'14'1,id_u'8'1 => false
      | id_u'14'1,id_reduce'ii'out => false
      | id_u'14'1,id_nil => false
      | id_u'14'1,id_if => false
      | id_u'14'1,id_u'11'1 => false
      | id_u'14'1,id_u'5'1 => false
      | id_u'14'1,id_u'16'1 => false
      | id_intersect'ii'out,id_intersect'ii'in => true
      | id_intersect'ii'out,id_tautology'i'out => true
      | id_intersect'ii'out,id_u'6'1 => true
      | id_intersect'ii'out,id_u'3'1 => true
      | id_intersect'ii'out,id_u'12'1 => true
      | id_intersect'ii'out,id_u'2'1 => true
      | id_intersect'ii'out,id_u'9'1 => true
      | id_intersect'ii'out,id_iff => true
      | id_intersect'ii'out,id_u'14'1 => true
      | id_intersect'ii'out,id_intersect'ii'out => true
      | id_intersect'ii'out,id_u'7'1 => false
      | id_intersect'ii'out,id_x'2d => false
      | id_intersect'ii'out,id_u'13'1 => false
      | id_intersect'ii'out,id_sequent => false
      | id_intersect'ii'out,id_u'10'1 => false
      | id_intersect'ii'out,id_x'2a => false
      | id_intersect'ii'out,id_tautology'i'in => false
      | id_intersect'ii'out,id_cons => false
      | id_intersect'ii'out,id_u'6'2 => false
      | id_intersect'ii'out,id_x'2b => false
      | id_intersect'ii'out,id_u'12'2 => false
      | id_intersect'ii'out,id_reduce'ii'in => false
      | id_intersect'ii'out,id_p => false
      | id_intersect'ii'out,id_u'4'1 => false
      | id_intersect'ii'out,id_u'15'1 => false
      | id_intersect'ii'out,id_u'1'1 => false
      | id_intersect'ii'out,id_u'8'1 => false
      | id_intersect'ii'out,id_reduce'ii'out => false
      | id_intersect'ii'out,id_nil => false
      | id_intersect'ii'out,id_if => false
      | id_intersect'ii'out,id_u'11'1 => false
      | id_intersect'ii'out,id_u'5'1 => false
      | id_intersect'ii'out,id_u'16'1 => false
      | id_u'7'1,id_intersect'ii'in => true
      | id_u'7'1,id_tautology'i'out => true
      | id_u'7'1,id_u'6'1 => true
      | id_u'7'1,id_u'3'1 => true
      | id_u'7'1,id_u'12'1 => true
      | id_u'7'1,id_u'2'1 => true
      | id_u'7'1,id_u'9'1 => true
      | id_u'7'1,id_iff => true
      | id_u'7'1,id_u'14'1 => true
      | id_u'7'1,id_intersect'ii'out => true
      | id_u'7'1,id_u'7'1 => true
      | id_u'7'1,id_x'2d => false
      | id_u'7'1,id_u'13'1 => false
      | id_u'7'1,id_sequent => false
      | id_u'7'1,id_u'10'1 => false
      | id_u'7'1,id_x'2a => false
      | id_u'7'1,id_tautology'i'in => false
      | id_u'7'1,id_cons => false
      | id_u'7'1,id_u'6'2 => false
      | id_u'7'1,id_x'2b => false
      | id_u'7'1,id_u'12'2 => false
      | id_u'7'1,id_reduce'ii'in => false
      | id_u'7'1,id_p => false
      | id_u'7'1,id_u'4'1 => false
      | id_u'7'1,id_u'15'1 => false
      | id_u'7'1,id_u'1'1 => false
      | id_u'7'1,id_u'8'1 => false
      | id_u'7'1,id_reduce'ii'out => false
      | id_u'7'1,id_nil => false
      | id_u'7'1,id_if => false
      | id_u'7'1,id_u'11'1 => false
      | id_u'7'1,id_u'5'1 => false
      | id_u'7'1,id_u'16'1 => false
      | id_x'2d,id_intersect'ii'in => true
      | id_x'2d,id_tautology'i'out => true
      | id_x'2d,id_u'6'1 => true
      | id_x'2d,id_u'3'1 => true
      | id_x'2d,id_u'12'1 => true
      | id_x'2d,id_u'2'1 => true
      | id_x'2d,id_u'9'1 => true
      | id_x'2d,id_iff => true
      | id_x'2d,id_u'14'1 => true
      | id_x'2d,id_intersect'ii'out => true
      | id_x'2d,id_u'7'1 => true
      | id_x'2d,id_x'2d => true
      | id_x'2d,id_u'13'1 => false
      | id_x'2d,id_sequent => false
      | id_x'2d,id_u'10'1 => false
      | id_x'2d,id_x'2a => false
      | id_x'2d,id_tautology'i'in => false
      | id_x'2d,id_cons => false
      | id_x'2d,id_u'6'2 => false
      | id_x'2d,id_x'2b => false
      | id_x'2d,id_u'12'2 => false
      | id_x'2d,id_reduce'ii'in => false
      | id_x'2d,id_p => false
      | id_x'2d,id_u'4'1 => false
      | id_x'2d,id_u'15'1 => false
      | id_x'2d,id_u'1'1 => false
      | id_x'2d,id_u'8'1 => false
      | id_x'2d,id_reduce'ii'out => false
      | id_x'2d,id_nil => false
      | id_x'2d,id_if => false
      | id_x'2d,id_u'11'1 => false
      | id_x'2d,id_u'5'1 => false
      | id_x'2d,id_u'16'1 => false
      | id_u'13'1,id_intersect'ii'in => true
      | id_u'13'1,id_tautology'i'out => true
      | id_u'13'1,id_u'6'1 => true
      | id_u'13'1,id_u'3'1 => true
      | id_u'13'1,id_u'12'1 => true
      | id_u'13'1,id_u'2'1 => true
      | id_u'13'1,id_u'9'1 => true
      | id_u'13'1,id_iff => true
      | id_u'13'1,id_u'14'1 => true
      | id_u'13'1,id_intersect'ii'out => true
      | id_u'13'1,id_u'7'1 => true
      | id_u'13'1,id_x'2d => true
      | id_u'13'1,id_u'13'1 => true
      | id_u'13'1,id_sequent => false
      | id_u'13'1,id_u'10'1 => false
      | id_u'13'1,id_x'2a => false
      | id_u'13'1,id_tautology'i'in => false
      | id_u'13'1,id_cons => false
      | id_u'13'1,id_u'6'2 => false
      | id_u'13'1,id_x'2b => false
      | id_u'13'1,id_u'12'2 => false
      | id_u'13'1,id_reduce'ii'in => false
      | id_u'13'1,id_p => false
      | id_u'13'1,id_u'4'1 => false
      | id_u'13'1,id_u'15'1 => false
      | id_u'13'1,id_u'1'1 => false
      | id_u'13'1,id_u'8'1 => false
      | id_u'13'1,id_reduce'ii'out => false
      | id_u'13'1,id_nil => false
      | id_u'13'1,id_if => false
      | id_u'13'1,id_u'11'1 => false
      | id_u'13'1,id_u'5'1 => false
      | id_u'13'1,id_u'16'1 => false
      | id_sequent,id_intersect'ii'in => true
      | id_sequent,id_tautology'i'out => true
      | id_sequent,id_u'6'1 => true
      | id_sequent,id_u'3'1 => true
      | id_sequent,id_u'12'1 => true
      | id_sequent,id_u'2'1 => true
      | id_sequent,id_u'9'1 => true
      | id_sequent,id_iff => true
      | id_sequent,id_u'14'1 => true
      | id_sequent,id_intersect'ii'out => true
      | id_sequent,id_u'7'1 => true
      | id_sequent,id_x'2d => true
      | id_sequent,id_u'13'1 => true
      | id_sequent,id_sequent => true
      | id_sequent,id_u'10'1 => false
      | id_sequent,id_x'2a => false
      | id_sequent,id_tautology'i'in => false
      | id_sequent,id_cons => false
      | id_sequent,id_u'6'2 => false
      | id_sequent,id_x'2b => false
      | id_sequent,id_u'12'2 => false
      | id_sequent,id_reduce'ii'in => false
      | id_sequent,id_p => false
      | id_sequent,id_u'4'1 => false
      | id_sequent,id_u'15'1 => false
      | id_sequent,id_u'1'1 => false
      | id_sequent,id_u'8'1 => false
      | id_sequent,id_reduce'ii'out => false
      | id_sequent,id_nil => false
      | id_sequent,id_if => false
      | id_sequent,id_u'11'1 => false
      | id_sequent,id_u'5'1 => false
      | id_sequent,id_u'16'1 => false
      | id_u'10'1,id_intersect'ii'in => true
      | id_u'10'1,id_tautology'i'out => true
      | id_u'10'1,id_u'6'1 => true
      | id_u'10'1,id_u'3'1 => true
      | id_u'10'1,id_u'12'1 => true
      | id_u'10'1,id_u'2'1 => true
      | id_u'10'1,id_u'9'1 => true
      | id_u'10'1,id_iff => true
      | id_u'10'1,id_u'14'1 => true
      | id_u'10'1,id_intersect'ii'out => true
      | id_u'10'1,id_u'7'1 => true
      | id_u'10'1,id_x'2d => true
      | id_u'10'1,id_u'13'1 => true
      | id_u'10'1,id_sequent => true
      | id_u'10'1,id_u'10'1 => true
      | id_u'10'1,id_x'2a => false
      | id_u'10'1,id_tautology'i'in => false
      | id_u'10'1,id_cons => false
      | id_u'10'1,id_u'6'2 => false
      | id_u'10'1,id_x'2b => false
      | id_u'10'1,id_u'12'2 => false
      | id_u'10'1,id_reduce'ii'in => false
      | id_u'10'1,id_p => false
      | id_u'10'1,id_u'4'1 => false
      | id_u'10'1,id_u'15'1 => false
      | id_u'10'1,id_u'1'1 => false
      | id_u'10'1,id_u'8'1 => false
      | id_u'10'1,id_reduce'ii'out => false
      | id_u'10'1,id_nil => false
      | id_u'10'1,id_if => false
      | id_u'10'1,id_u'11'1 => false
      | id_u'10'1,id_u'5'1 => false
      | id_u'10'1,id_u'16'1 => false
      | id_x'2a,id_intersect'ii'in => true
      | id_x'2a,id_tautology'i'out => true
      | id_x'2a,id_u'6'1 => true
      | id_x'2a,id_u'3'1 => true
      | id_x'2a,id_u'12'1 => true
      | id_x'2a,id_u'2'1 => true
      | id_x'2a,id_u'9'1 => true
      | id_x'2a,id_iff => true
      | id_x'2a,id_u'14'1 => true
      | id_x'2a,id_intersect'ii'out => true
      | id_x'2a,id_u'7'1 => true
      | id_x'2a,id_x'2d => true
      | id_x'2a,id_u'13'1 => true
      | id_x'2a,id_sequent => true
      | id_x'2a,id_u'10'1 => true
      | id_x'2a,id_x'2a => true
      | id_x'2a,id_tautology'i'in => false
      | id_x'2a,id_cons => false
      | id_x'2a,id_u'6'2 => false
      | id_x'2a,id_x'2b => false
      | id_x'2a,id_u'12'2 => false
      | id_x'2a,id_reduce'ii'in => false
      | id_x'2a,id_p => false
      | id_x'2a,id_u'4'1 => false
      | id_x'2a,id_u'15'1 => false
      | id_x'2a,id_u'1'1 => false
      | id_x'2a,id_u'8'1 => false
      | id_x'2a,id_reduce'ii'out => false
      | id_x'2a,id_nil => false
      | id_x'2a,id_if => false
      | id_x'2a,id_u'11'1 => false
      | id_x'2a,id_u'5'1 => false
      | id_x'2a,id_u'16'1 => false
      | id_tautology'i'in,id_intersect'ii'in => true
      | id_tautology'i'in,id_tautology'i'out => true
      | id_tautology'i'in,id_u'6'1 => true
      | id_tautology'i'in,id_u'3'1 => true
      | id_tautology'i'in,id_u'12'1 => true
      | id_tautology'i'in,id_u'2'1 => true
      | id_tautology'i'in,id_u'9'1 => true
      | id_tautology'i'in,id_iff => true
      | id_tautology'i'in,id_u'14'1 => true
      | id_tautology'i'in,id_intersect'ii'out => true
      | id_tautology'i'in,id_u'7'1 => true
      | id_tautology'i'in,id_x'2d => true
      | id_tautology'i'in,id_u'13'1 => true
      | id_tautology'i'in,id_sequent => true
      | id_tautology'i'in,id_u'10'1 => true
      | id_tautology'i'in,id_x'2a => true
      | id_tautology'i'in,id_tautology'i'in => true
      | id_tautology'i'in,id_cons => false
      | id_tautology'i'in,id_u'6'2 => false
      | id_tautology'i'in,id_x'2b => false
      | id_tautology'i'in,id_u'12'2 => false
      | id_tautology'i'in,id_reduce'ii'in => false
      | id_tautology'i'in,id_p => false
      | id_tautology'i'in,id_u'4'1 => false
      | id_tautology'i'in,id_u'15'1 => false
      | id_tautology'i'in,id_u'1'1 => false
      | id_tautology'i'in,id_u'8'1 => false
      | id_tautology'i'in,id_reduce'ii'out => false
      | id_tautology'i'in,id_nil => false
      | id_tautology'i'in,id_if => false
      | id_tautology'i'in,id_u'11'1 => false
      | id_tautology'i'in,id_u'5'1 => false
      | id_tautology'i'in,id_u'16'1 => false
      | id_cons,id_intersect'ii'in => true
      | id_cons,id_tautology'i'out => true
      | id_cons,id_u'6'1 => true
      | id_cons,id_u'3'1 => true
      | id_cons,id_u'12'1 => true
      | id_cons,id_u'2'1 => true
      | id_cons,id_u'9'1 => true
      | id_cons,id_iff => true
      | id_cons,id_u'14'1 => true
      | id_cons,id_intersect'ii'out => true
      | id_cons,id_u'7'1 => true
      | id_cons,id_x'2d => true
      | id_cons,id_u'13'1 => true
      | id_cons,id_sequent => true
      | id_cons,id_u'10'1 => true
      | id_cons,id_x'2a => true
      | id_cons,id_tautology'i'in => true
      | id_cons,id_cons => true
      | id_cons,id_u'6'2 => false
      | id_cons,id_x'2b => false
      | id_cons,id_u'12'2 => false
      | id_cons,id_reduce'ii'in => false
      | id_cons,id_p => false
      | id_cons,id_u'4'1 => false
      | id_cons,id_u'15'1 => false
      | id_cons,id_u'1'1 => false
      | id_cons,id_u'8'1 => false
      | id_cons,id_reduce'ii'out => false
      | id_cons,id_nil => false
      | id_cons,id_if => false
      | id_cons,id_u'11'1 => false
      | id_cons,id_u'5'1 => false
      | id_cons,id_u'16'1 => false
      | id_u'6'2,id_intersect'ii'in => true
      | id_u'6'2,id_tautology'i'out => true
      | id_u'6'2,id_u'6'1 => true
      | id_u'6'2,id_u'3'1 => true
      | id_u'6'2,id_u'12'1 => true
      | id_u'6'2,id_u'2'1 => true
      | id_u'6'2,id_u'9'1 => true
      | id_u'6'2,id_iff => true
      | id_u'6'2,id_u'14'1 => true
      | id_u'6'2,id_intersect'ii'out => true
      | id_u'6'2,id_u'7'1 => true
      | id_u'6'2,id_x'2d => true
      | id_u'6'2,id_u'13'1 => true
      | id_u'6'2,id_sequent => true
      | id_u'6'2,id_u'10'1 => true
      | id_u'6'2,id_x'2a => true
      | id_u'6'2,id_tautology'i'in => true
      | id_u'6'2,id_cons => true
      | id_u'6'2,id_u'6'2 => true
      | id_u'6'2,id_x'2b => false
      | id_u'6'2,id_u'12'2 => false
      | id_u'6'2,id_reduce'ii'in => false
      | id_u'6'2,id_p => false
      | id_u'6'2,id_u'4'1 => false
      | id_u'6'2,id_u'15'1 => false
      | id_u'6'2,id_u'1'1 => false
      | id_u'6'2,id_u'8'1 => false
      | id_u'6'2,id_reduce'ii'out => false
      | id_u'6'2,id_nil => false
      | id_u'6'2,id_if => false
      | id_u'6'2,id_u'11'1 => false
      | id_u'6'2,id_u'5'1 => false
      | id_u'6'2,id_u'16'1 => false
      | id_x'2b,id_intersect'ii'in => true
      | id_x'2b,id_tautology'i'out => true
      | id_x'2b,id_u'6'1 => true
      | id_x'2b,id_u'3'1 => true
      | id_x'2b,id_u'12'1 => true
      | id_x'2b,id_u'2'1 => true
      | id_x'2b,id_u'9'1 => true
      | id_x'2b,id_iff => true
      | id_x'2b,id_u'14'1 => true
      | id_x'2b,id_intersect'ii'out => true
      | id_x'2b,id_u'7'1 => true
      | id_x'2b,id_x'2d => true
      | id_x'2b,id_u'13'1 => true
      | id_x'2b,id_sequent => true
      | id_x'2b,id_u'10'1 => true
      | id_x'2b,id_x'2a => true
      | id_x'2b,id_tautology'i'in => true
      | id_x'2b,id_cons => true
      | id_x'2b,id_u'6'2 => true
      | id_x'2b,id_x'2b => true
      | id_x'2b,id_u'12'2 => false
      | id_x'2b,id_reduce'ii'in => false
      | id_x'2b,id_p => false
      | id_x'2b,id_u'4'1 => false
      | id_x'2b,id_u'15'1 => false
      | id_x'2b,id_u'1'1 => false
      | id_x'2b,id_u'8'1 => false
      | id_x'2b,id_reduce'ii'out => false
      | id_x'2b,id_nil => false
      | id_x'2b,id_if => false
      | id_x'2b,id_u'11'1 => false
      | id_x'2b,id_u'5'1 => false
      | id_x'2b,id_u'16'1 => false
      | id_u'12'2,id_intersect'ii'in => true
      | id_u'12'2,id_tautology'i'out => true
      | id_u'12'2,id_u'6'1 => true
      | id_u'12'2,id_u'3'1 => true
      | id_u'12'2,id_u'12'1 => true
      | id_u'12'2,id_u'2'1 => true
      | id_u'12'2,id_u'9'1 => true
      | id_u'12'2,id_iff => true
      | id_u'12'2,id_u'14'1 => true
      | id_u'12'2,id_intersect'ii'out => true
      | id_u'12'2,id_u'7'1 => true
      | id_u'12'2,id_x'2d => true
      | id_u'12'2,id_u'13'1 => true
      | id_u'12'2,id_sequent => true
      | id_u'12'2,id_u'10'1 => true
      | id_u'12'2,id_x'2a => true
      | id_u'12'2,id_tautology'i'in => true
      | id_u'12'2,id_cons => true
      | id_u'12'2,id_u'6'2 => true
      | id_u'12'2,id_x'2b => true
      | id_u'12'2,id_u'12'2 => true
      | id_u'12'2,id_reduce'ii'in => false
      | id_u'12'2,id_p => false
      | id_u'12'2,id_u'4'1 => false
      | id_u'12'2,id_u'15'1 => false
      | id_u'12'2,id_u'1'1 => false
      | id_u'12'2,id_u'8'1 => false
      | id_u'12'2,id_reduce'ii'out => false
      | id_u'12'2,id_nil => false
      | id_u'12'2,id_if => false
      | id_u'12'2,id_u'11'1 => false
      | id_u'12'2,id_u'5'1 => false
      | id_u'12'2,id_u'16'1 => false
      | id_reduce'ii'in,id_intersect'ii'in => true
      | id_reduce'ii'in,id_tautology'i'out => true
      | id_reduce'ii'in,id_u'6'1 => true
      | id_reduce'ii'in,id_u'3'1 => true
      | id_reduce'ii'in,id_u'12'1 => true
      | id_reduce'ii'in,id_u'2'1 => true
      | id_reduce'ii'in,id_u'9'1 => true
      | id_reduce'ii'in,id_iff => true
      | id_reduce'ii'in,id_u'14'1 => true
      | id_reduce'ii'in,id_intersect'ii'out => true
      | id_reduce'ii'in,id_u'7'1 => true
      | id_reduce'ii'in,id_x'2d => true
      | id_reduce'ii'in,id_u'13'1 => true
      | id_reduce'ii'in,id_sequent => true
      | id_reduce'ii'in,id_u'10'1 => true
      | id_reduce'ii'in,id_x'2a => true
      | id_reduce'ii'in,id_tautology'i'in => true
      | id_reduce'ii'in,id_cons => true
      | id_reduce'ii'in,id_u'6'2 => true
      | id_reduce'ii'in,id_x'2b => true
      | id_reduce'ii'in,id_u'12'2 => true
      | id_reduce'ii'in,id_reduce'ii'in => true
      | id_reduce'ii'in,id_p => false
      | id_reduce'ii'in,id_u'4'1 => false
      | id_reduce'ii'in,id_u'15'1 => false
      | id_reduce'ii'in,id_u'1'1 => false
      | id_reduce'ii'in,id_u'8'1 => false
      | id_reduce'ii'in,id_reduce'ii'out => false
      | id_reduce'ii'in,id_nil => false
      | id_reduce'ii'in,id_if => false
      | id_reduce'ii'in,id_u'11'1 => false
      | id_reduce'ii'in,id_u'5'1 => false
      | id_reduce'ii'in,id_u'16'1 => false
      | id_p,id_intersect'ii'in => true
      | id_p,id_tautology'i'out => true
      | id_p,id_u'6'1 => true
      | id_p,id_u'3'1 => true
      | id_p,id_u'12'1 => true
      | id_p,id_u'2'1 => true
      | id_p,id_u'9'1 => true
      | id_p,id_iff => true
      | id_p,id_u'14'1 => true
      | id_p,id_intersect'ii'out => true
      | id_p,id_u'7'1 => true
      | id_p,id_x'2d => true
      | id_p,id_u'13'1 => true
      | id_p,id_sequent => true
      | id_p,id_u'10'1 => true
      | id_p,id_x'2a => true
      | id_p,id_tautology'i'in => true
      | id_p,id_cons => true
      | id_p,id_u'6'2 => true
      | id_p,id_x'2b => true
      | id_p,id_u'12'2 => true
      | id_p,id_reduce'ii'in => true
      | id_p,id_p => true
      | id_p,id_u'4'1 => false
      | id_p,id_u'15'1 => false
      | id_p,id_u'1'1 => false
      | id_p,id_u'8'1 => false
      | id_p,id_reduce'ii'out => false
      | id_p,id_nil => false
      | id_p,id_if => false
      | id_p,id_u'11'1 => false
      | id_p,id_u'5'1 => false
      | id_p,id_u'16'1 => false
      | id_u'4'1,id_intersect'ii'in => true
      | id_u'4'1,id_tautology'i'out => true
      | id_u'4'1,id_u'6'1 => true
      | id_u'4'1,id_u'3'1 => true
      | id_u'4'1,id_u'12'1 => true
      | id_u'4'1,id_u'2'1 => true
      | id_u'4'1,id_u'9'1 => true
      | id_u'4'1,id_iff => true
      | id_u'4'1,id_u'14'1 => true
      | id_u'4'1,id_intersect'ii'out => true
      | id_u'4'1,id_u'7'1 => true
      | id_u'4'1,id_x'2d => true
      | id_u'4'1,id_u'13'1 => true
      | id_u'4'1,id_sequent => true
      | id_u'4'1,id_u'10'1 => true
      | id_u'4'1,id_x'2a => true
      | id_u'4'1,id_tautology'i'in => true
      | id_u'4'1,id_cons => true
      | id_u'4'1,id_u'6'2 => true
      | id_u'4'1,id_x'2b => true
      | id_u'4'1,id_u'12'2 => true
      | id_u'4'1,id_reduce'ii'in => true
      | id_u'4'1,id_p => true
      | id_u'4'1,id_u'4'1 => true
      | id_u'4'1,id_u'15'1 => false
      | id_u'4'1,id_u'1'1 => false
      | id_u'4'1,id_u'8'1 => false
      | id_u'4'1,id_reduce'ii'out => false
      | id_u'4'1,id_nil => false
      | id_u'4'1,id_if => false
      | id_u'4'1,id_u'11'1 => false
      | id_u'4'1,id_u'5'1 => false
      | id_u'4'1,id_u'16'1 => false
      | id_u'15'1,id_intersect'ii'in => true
      | id_u'15'1,id_tautology'i'out => true
      | id_u'15'1,id_u'6'1 => true
      | id_u'15'1,id_u'3'1 => true
      | id_u'15'1,id_u'12'1 => true
      | id_u'15'1,id_u'2'1 => true
      | id_u'15'1,id_u'9'1 => true
      | id_u'15'1,id_iff => true
      | id_u'15'1,id_u'14'1 => true
      | id_u'15'1,id_intersect'ii'out => true
      | id_u'15'1,id_u'7'1 => true
      | id_u'15'1,id_x'2d => true
      | id_u'15'1,id_u'13'1 => true
      | id_u'15'1,id_sequent => true
      | id_u'15'1,id_u'10'1 => true
      | id_u'15'1,id_x'2a => true
      | id_u'15'1,id_tautology'i'in => true
      | id_u'15'1,id_cons => true
      | id_u'15'1,id_u'6'2 => true
      | id_u'15'1,id_x'2b => true
      | id_u'15'1,id_u'12'2 => true
      | id_u'15'1,id_reduce'ii'in => true
      | id_u'15'1,id_p => true
      | id_u'15'1,id_u'4'1 => true
      | id_u'15'1,id_u'15'1 => true
      | id_u'15'1,id_u'1'1 => false
      | id_u'15'1,id_u'8'1 => false
      | id_u'15'1,id_reduce'ii'out => false
      | id_u'15'1,id_nil => false
      | id_u'15'1,id_if => false
      | id_u'15'1,id_u'11'1 => false
      | id_u'15'1,id_u'5'1 => false
      | id_u'15'1,id_u'16'1 => false
      | id_u'1'1,id_intersect'ii'in => true
      | id_u'1'1,id_tautology'i'out => true
      | id_u'1'1,id_u'6'1 => true
      | id_u'1'1,id_u'3'1 => true
      | id_u'1'1,id_u'12'1 => true
      | id_u'1'1,id_u'2'1 => true
      | id_u'1'1,id_u'9'1 => true
      | id_u'1'1,id_iff => true
      | id_u'1'1,id_u'14'1 => true
      | id_u'1'1,id_intersect'ii'out => true
      | id_u'1'1,id_u'7'1 => true
      | id_u'1'1,id_x'2d => true
      | id_u'1'1,id_u'13'1 => true
      | id_u'1'1,id_sequent => true
      | id_u'1'1,id_u'10'1 => true
      | id_u'1'1,id_x'2a => true
      | id_u'1'1,id_tautology'i'in => true
      | id_u'1'1,id_cons => true
      | id_u'1'1,id_u'6'2 => true
      | id_u'1'1,id_x'2b => true
      | id_u'1'1,id_u'12'2 => true
      | id_u'1'1,id_reduce'ii'in => true
      | id_u'1'1,id_p => true
      | id_u'1'1,id_u'4'1 => true
      | id_u'1'1,id_u'15'1 => true
      | id_u'1'1,id_u'1'1 => true
      | id_u'1'1,id_u'8'1 => false
      | id_u'1'1,id_reduce'ii'out => false
      | id_u'1'1,id_nil => false
      | id_u'1'1,id_if => false
      | id_u'1'1,id_u'11'1 => false
      | id_u'1'1,id_u'5'1 => false
      | id_u'1'1,id_u'16'1 => false
      | id_u'8'1,id_intersect'ii'in => true
      | id_u'8'1,id_tautology'i'out => true
      | id_u'8'1,id_u'6'1 => true
      | id_u'8'1,id_u'3'1 => true
      | id_u'8'1,id_u'12'1 => true
      | id_u'8'1,id_u'2'1 => true
      | id_u'8'1,id_u'9'1 => true
      | id_u'8'1,id_iff => true
      | id_u'8'1,id_u'14'1 => true
      | id_u'8'1,id_intersect'ii'out => true
      | id_u'8'1,id_u'7'1 => true
      | id_u'8'1,id_x'2d => true
      | id_u'8'1,id_u'13'1 => true
      | id_u'8'1,id_sequent => true
      | id_u'8'1,id_u'10'1 => true
      | id_u'8'1,id_x'2a => true
      | id_u'8'1,id_tautology'i'in => true
      | id_u'8'1,id_cons => true
      | id_u'8'1,id_u'6'2 => true
      | id_u'8'1,id_x'2b => true
      | id_u'8'1,id_u'12'2 => true
      | id_u'8'1,id_reduce'ii'in => true
      | id_u'8'1,id_p => true
      | id_u'8'1,id_u'4'1 => true
      | id_u'8'1,id_u'15'1 => true
      | id_u'8'1,id_u'1'1 => true
      | id_u'8'1,id_u'8'1 => true
      | id_u'8'1,id_reduce'ii'out => false
      | id_u'8'1,id_nil => false
      | id_u'8'1,id_if => false
      | id_u'8'1,id_u'11'1 => false
      | id_u'8'1,id_u'5'1 => false
      | id_u'8'1,id_u'16'1 => false
      | id_reduce'ii'out,id_intersect'ii'in => true
      | id_reduce'ii'out,id_tautology'i'out => true
      | id_reduce'ii'out,id_u'6'1 => true
      | id_reduce'ii'out,id_u'3'1 => true
      | id_reduce'ii'out,id_u'12'1 => true
      | id_reduce'ii'out,id_u'2'1 => true
      | id_reduce'ii'out,id_u'9'1 => true
      | id_reduce'ii'out,id_iff => true
      | id_reduce'ii'out,id_u'14'1 => true
      | id_reduce'ii'out,id_intersect'ii'out => true
      | id_reduce'ii'out,id_u'7'1 => true
      | id_reduce'ii'out,id_x'2d => true
      | id_reduce'ii'out,id_u'13'1 => true
      | id_reduce'ii'out,id_sequent => true
      | id_reduce'ii'out,id_u'10'1 => true
      | id_reduce'ii'out,id_x'2a => true
      | id_reduce'ii'out,id_tautology'i'in => true
      | id_reduce'ii'out,id_cons => true
      | id_reduce'ii'out,id_u'6'2 => true
      | id_reduce'ii'out,id_x'2b => true
      | id_reduce'ii'out,id_u'12'2 => true
      | id_reduce'ii'out,id_reduce'ii'in => true
      | id_reduce'ii'out,id_p => true
      | id_reduce'ii'out,id_u'4'1 => true
      | id_reduce'ii'out,id_u'15'1 => true
      | id_reduce'ii'out,id_u'1'1 => true
      | id_reduce'ii'out,id_u'8'1 => true
      | id_reduce'ii'out,id_reduce'ii'out => true
      | id_reduce'ii'out,id_nil => false
      | id_reduce'ii'out,id_if => false
      | id_reduce'ii'out,id_u'11'1 => false
      | id_reduce'ii'out,id_u'5'1 => false
      | id_reduce'ii'out,id_u'16'1 => false
      | id_nil,id_intersect'ii'in => true
      | id_nil,id_tautology'i'out => true
      | id_nil,id_u'6'1 => true
      | id_nil,id_u'3'1 => true
      | id_nil,id_u'12'1 => true
      | id_nil,id_u'2'1 => true
      | id_nil,id_u'9'1 => true
      | id_nil,id_iff => true
      | id_nil,id_u'14'1 => true
      | id_nil,id_intersect'ii'out => true
      | id_nil,id_u'7'1 => true
      | id_nil,id_x'2d => true
      | id_nil,id_u'13'1 => true
      | id_nil,id_sequent => true
      | id_nil,id_u'10'1 => true
      | id_nil,id_x'2a => true
      | id_nil,id_tautology'i'in => true
      | id_nil,id_cons => true
      | id_nil,id_u'6'2 => true
      | id_nil,id_x'2b => true
      | id_nil,id_u'12'2 => true
      | id_nil,id_reduce'ii'in => true
      | id_nil,id_p => true
      | id_nil,id_u'4'1 => true
      | id_nil,id_u'15'1 => true
      | id_nil,id_u'1'1 => true
      | id_nil,id_u'8'1 => true
      | id_nil,id_reduce'ii'out => true
      | id_nil,id_nil => true
      | id_nil,id_if => false
      | id_nil,id_u'11'1 => false
      | id_nil,id_u'5'1 => false
      | id_nil,id_u'16'1 => false
      | id_if,id_intersect'ii'in => true
      | id_if,id_tautology'i'out => true
      | id_if,id_u'6'1 => true
      | id_if,id_u'3'1 => true
      | id_if,id_u'12'1 => true
      | id_if,id_u'2'1 => true
      | id_if,id_u'9'1 => true
      | id_if,id_iff => true
      | id_if,id_u'14'1 => true
      | id_if,id_intersect'ii'out => true
      | id_if,id_u'7'1 => true
      | id_if,id_x'2d => true
      | id_if,id_u'13'1 => true
      | id_if,id_sequent => true
      | id_if,id_u'10'1 => true
      | id_if,id_x'2a => true
      | id_if,id_tautology'i'in => true
      | id_if,id_cons => true
      | id_if,id_u'6'2 => true
      | id_if,id_x'2b => true
      | id_if,id_u'12'2 => true
      | id_if,id_reduce'ii'in => true
      | id_if,id_p => true
      | id_if,id_u'4'1 => true
      | id_if,id_u'15'1 => true
      | id_if,id_u'1'1 => true
      | id_if,id_u'8'1 => true
      | id_if,id_reduce'ii'out => true
      | id_if,id_nil => true
      | id_if,id_if => true
      | id_if,id_u'11'1 => false
      | id_if,id_u'5'1 => false
      | id_if,id_u'16'1 => false
      | id_u'11'1,id_intersect'ii'in => true
      | id_u'11'1,id_tautology'i'out => true
      | id_u'11'1,id_u'6'1 => true
      | id_u'11'1,id_u'3'1 => true
      | id_u'11'1,id_u'12'1 => true
      | id_u'11'1,id_u'2'1 => true
      | id_u'11'1,id_u'9'1 => true
      | id_u'11'1,id_iff => true
      | id_u'11'1,id_u'14'1 => true
      | id_u'11'1,id_intersect'ii'out => true
      | id_u'11'1,id_u'7'1 => true
      | id_u'11'1,id_x'2d => true
      | id_u'11'1,id_u'13'1 => true
      | id_u'11'1,id_sequent => true
      | id_u'11'1,id_u'10'1 => true
      | id_u'11'1,id_x'2a => true
      | id_u'11'1,id_tautology'i'in => true
      | id_u'11'1,id_cons => true
      | id_u'11'1,id_u'6'2 => true
      | id_u'11'1,id_x'2b => true
      | id_u'11'1,id_u'12'2 => true
      | id_u'11'1,id_reduce'ii'in => true
      | id_u'11'1,id_p => true
      | id_u'11'1,id_u'4'1 => true
      | id_u'11'1,id_u'15'1 => true
      | id_u'11'1,id_u'1'1 => true
      | id_u'11'1,id_u'8'1 => true
      | id_u'11'1,id_reduce'ii'out => true
      | id_u'11'1,id_nil => true
      | id_u'11'1,id_if => true
      | id_u'11'1,id_u'11'1 => true
      | id_u'11'1,id_u'5'1 => false
      | id_u'11'1,id_u'16'1 => false
      | id_u'5'1,id_intersect'ii'in => true
      | id_u'5'1,id_tautology'i'out => true
      | id_u'5'1,id_u'6'1 => true
      | id_u'5'1,id_u'3'1 => true
      | id_u'5'1,id_u'12'1 => true
      | id_u'5'1,id_u'2'1 => true
      | id_u'5'1,id_u'9'1 => true
      | id_u'5'1,id_iff => true
      | id_u'5'1,id_u'14'1 => true
      | id_u'5'1,id_intersect'ii'out => true
      | id_u'5'1,id_u'7'1 => true
      | id_u'5'1,id_x'2d => true
      | id_u'5'1,id_u'13'1 => true
      | id_u'5'1,id_sequent => true
      | id_u'5'1,id_u'10'1 => true
      | id_u'5'1,id_x'2a => true
      | id_u'5'1,id_tautology'i'in => true
      | id_u'5'1,id_cons => true
      | id_u'5'1,id_u'6'2 => true
      | id_u'5'1,id_x'2b => true
      | id_u'5'1,id_u'12'2 => true
      | id_u'5'1,id_reduce'ii'in => true
      | id_u'5'1,id_p => true
      | id_u'5'1,id_u'4'1 => true
      | id_u'5'1,id_u'15'1 => true
      | id_u'5'1,id_u'1'1 => true
      | id_u'5'1,id_u'8'1 => true
      | id_u'5'1,id_reduce'ii'out => true
      | id_u'5'1,id_nil => true
      | id_u'5'1,id_if => true
      | id_u'5'1,id_u'11'1 => true
      | id_u'5'1,id_u'5'1 => true
      | id_u'5'1,id_u'16'1 => false
      | id_u'16'1,id_intersect'ii'in => true
      | id_u'16'1,id_tautology'i'out => true
      | id_u'16'1,id_u'6'1 => true
      | id_u'16'1,id_u'3'1 => true
      | id_u'16'1,id_u'12'1 => true
      | id_u'16'1,id_u'2'1 => true
      | id_u'16'1,id_u'9'1 => true
      | id_u'16'1,id_iff => true
      | id_u'16'1,id_u'14'1 => true
      | id_u'16'1,id_intersect'ii'out => true
      | id_u'16'1,id_u'7'1 => true
      | id_u'16'1,id_x'2d => true
      | id_u'16'1,id_u'13'1 => true
      | id_u'16'1,id_sequent => true
      | id_u'16'1,id_u'10'1 => true
      | id_u'16'1,id_x'2a => true
      | id_u'16'1,id_tautology'i'in => true
      | id_u'16'1,id_cons => true
      | id_u'16'1,id_u'6'2 => true
      | id_u'16'1,id_x'2b => true
      | id_u'16'1,id_u'12'2 => true
      | id_u'16'1,id_reduce'ii'in => true
      | id_u'16'1,id_p => true
      | id_u'16'1,id_u'4'1 => true
      | id_u'16'1,id_u'15'1 => true
      | id_u'16'1,id_u'1'1 => true
      | id_u'16'1,id_u'8'1 => true
      | id_u'16'1,id_reduce'ii'out => true
      | id_u'16'1,id_nil => true
      | id_u'16'1,id_if => true
      | id_u'16'1,id_u'11'1 => true
      | id_u'16'1,id_u'5'1 => true
      | id_u'16'1,id_u'16'1 => 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 := 
    (* intersect'ii'in(cons(X_,X0_),cons(X_,X1_)) -> intersect'ii'out *)
   | R_xml_0_rule_0 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_intersect'ii'out nil) 
     (algebra.Alg.Term algebra.F.id_intersect'ii'in ((algebra.Alg.Term 
      algebra.F.id_cons ((algebra.Alg.Var 1)::(algebra.Alg.Var 2)::nil))::
      (algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Var 1)::
      (algebra.Alg.Var 3)::nil))::nil))
   
    (* intersect'ii'in(Xs_,cons(X0_,Ys_)) -> u'1'1(intersect'ii'in(Xs_,Ys_)) *)
   | R_xml_0_rule_1 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_u'1'1 ((algebra.Alg.Term 
                   algebra.F.id_intersect'ii'in ((algebra.Alg.Var 4)::
                   (algebra.Alg.Var 5)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_intersect'ii'in ((algebra.Alg.Var 4)::
      (algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Var 2)::
      (algebra.Alg.Var 5)::nil))::nil))
    (* u'1'1(intersect'ii'out) -> intersect'ii'out *)
   | R_xml_0_rule_2 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_intersect'ii'out nil) 
     (algebra.Alg.Term algebra.F.id_u'1'1 ((algebra.Alg.Term 
      algebra.F.id_intersect'ii'out nil)::nil))
   
    (* intersect'ii'in(cons(X0_,Xs_),Ys_) -> u'2'1(intersect'ii'in(Xs_,Ys_)) *)
   | R_xml_0_rule_3 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_u'2'1 ((algebra.Alg.Term 
                   algebra.F.id_intersect'ii'in ((algebra.Alg.Var 4)::
                   (algebra.Alg.Var 5)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_intersect'ii'in ((algebra.Alg.Term 
      algebra.F.id_cons ((algebra.Alg.Var 2)::(algebra.Alg.Var 4)::nil))::
      (algebra.Alg.Var 5)::nil))
    (* u'2'1(intersect'ii'out) -> intersect'ii'out *)
   | R_xml_0_rule_4 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_intersect'ii'out nil) 
     (algebra.Alg.Term algebra.F.id_u'2'1 ((algebra.Alg.Term 
      algebra.F.id_intersect'ii'out nil)::nil))
   
    (* reduce'ii'in(sequent(cons(if(A_,B_),Fs_),Gs_),NF_) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B_),A_),Fs_),Gs_),NF_)) *)
   | R_xml_0_rule_5 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_u'3'1 ((algebra.Alg.Term 
                   algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
                   algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons 
                   ((algebra.Alg.Term algebra.F.id_x'2b ((algebra.Alg.Term 
                   algebra.F.id_x'2d ((algebra.Alg.Var 7)::nil))::
                   (algebra.Alg.Var 6)::nil))::(algebra.Alg.Var 8)::nil))::
                   (algebra.Alg.Var 9)::nil))::
                   (algebra.Alg.Var 10)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
      algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons 
      ((algebra.Alg.Term algebra.F.id_if ((algebra.Alg.Var 6)::
      (algebra.Alg.Var 7)::nil))::(algebra.Alg.Var 8)::nil))::
      (algebra.Alg.Var 9)::nil))::(algebra.Alg.Var 10)::nil))
    (* u'3'1(reduce'ii'out) -> reduce'ii'out *)
   | R_xml_0_rule_6 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) 
     (algebra.Alg.Term algebra.F.id_u'3'1 ((algebra.Alg.Term 
      algebra.F.id_reduce'ii'out nil)::nil))
   
    (* reduce'ii'in(sequent(cons(iff(A_,B_),Fs_),Gs_),NF_) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A_,B_),if(B_,A_)),Fs_),Gs_),NF_)) *)
   | R_xml_0_rule_7 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_u'4'1 ((algebra.Alg.Term 
                   algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
                   algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons 
                   ((algebra.Alg.Term algebra.F.id_x'2a ((algebra.Alg.Term 
                   algebra.F.id_if ((algebra.Alg.Var 6)::
                   (algebra.Alg.Var 7)::nil))::(algebra.Alg.Term 
                   algebra.F.id_if ((algebra.Alg.Var 7)::
                   (algebra.Alg.Var 6)::nil))::nil))::
                   (algebra.Alg.Var 8)::nil))::(algebra.Alg.Var 9)::nil))::
                   (algebra.Alg.Var 10)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
      algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons 
      ((algebra.Alg.Term algebra.F.id_iff ((algebra.Alg.Var 6)::
      (algebra.Alg.Var 7)::nil))::(algebra.Alg.Var 8)::nil))::
      (algebra.Alg.Var 9)::nil))::(algebra.Alg.Var 10)::nil))
    (* u'4'1(reduce'ii'out) -> reduce'ii'out *)
   | R_xml_0_rule_8 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) 
     (algebra.Alg.Term algebra.F.id_u'4'1 ((algebra.Alg.Term 
      algebra.F.id_reduce'ii'out nil)::nil))
   
    (* reduce'ii'in(sequent(cons(x'2a(F1_,F2_),Fs_),Gs_),NF_) -> u'5'1(reduce'ii'in(sequent(cons(F1_,cons(F2_,Fs_)),Gs_),NF_)) *)
   | R_xml_0_rule_9 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_u'5'1 ((algebra.Alg.Term 
                   algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
                   algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons 
                   ((algebra.Alg.Var 11)::(algebra.Alg.Term 
                   algebra.F.id_cons ((algebra.Alg.Var 12)::
                   (algebra.Alg.Var 8)::nil))::nil))::
                   (algebra.Alg.Var 9)::nil))::
                   (algebra.Alg.Var 10)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
      algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons 
      ((algebra.Alg.Term algebra.F.id_x'2a ((algebra.Alg.Var 11)::
      (algebra.Alg.Var 12)::nil))::(algebra.Alg.Var 8)::nil))::
      (algebra.Alg.Var 9)::nil))::(algebra.Alg.Var 10)::nil))
    (* u'5'1(reduce'ii'out) -> reduce'ii'out *)
   | R_xml_0_rule_10 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) 
     (algebra.Alg.Term algebra.F.id_u'5'1 ((algebra.Alg.Term 
      algebra.F.id_reduce'ii'out nil)::nil))
   
    (* reduce'ii'in(sequent(cons(x'2b(F1_,F2_),Fs_),Gs_),NF_) -> u'6'1(reduce'ii'in(sequent(cons(F1_,Fs_),Gs_),NF_),F2_,Fs_,Gs_,NF_) *)
   | R_xml_0_rule_11 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_u'6'1 ((algebra.Alg.Term 
                   algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
                   algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons 
                   ((algebra.Alg.Var 11)::(algebra.Alg.Var 8)::nil))::
                   (algebra.Alg.Var 9)::nil))::(algebra.Alg.Var 10)::nil))::
                   (algebra.Alg.Var 12)::(algebra.Alg.Var 8)::
                   (algebra.Alg.Var 9)::(algebra.Alg.Var 10)::nil)) 
     (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
      algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons 
      ((algebra.Alg.Term algebra.F.id_x'2b ((algebra.Alg.Var 11)::
      (algebra.Alg.Var 12)::nil))::(algebra.Alg.Var 8)::nil))::
      (algebra.Alg.Var 9)::nil))::(algebra.Alg.Var 10)::nil))
   
    (* u'6'1(reduce'ii'out,F2_,Fs_,Gs_,NF_) -> u'6'2(reduce'ii'in(sequent(cons(F2_,Fs_),Gs_),NF_)) *)
   | R_xml_0_rule_12 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_u'6'2 ((algebra.Alg.Term 
                   algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
                   algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons 
                   ((algebra.Alg.Var 12)::(algebra.Alg.Var 8)::nil))::
                   (algebra.Alg.Var 9)::nil))::
                   (algebra.Alg.Var 10)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_u'6'1 ((algebra.Alg.Term 
      algebra.F.id_reduce'ii'out nil)::(algebra.Alg.Var 12)::
      (algebra.Alg.Var 8)::(algebra.Alg.Var 9)::(algebra.Alg.Var 10)::nil))
    (* u'6'2(reduce'ii'out) -> reduce'ii'out *)
   | R_xml_0_rule_13 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) 
     (algebra.Alg.Term algebra.F.id_u'6'2 ((algebra.Alg.Term 
      algebra.F.id_reduce'ii'out nil)::nil))
   
    (* reduce'ii'in(sequent(cons(x'2d(F1_),Fs_),Gs_),NF_) -> u'7'1(reduce'ii'in(sequent(Fs_,cons(F1_,Gs_)),NF_)) *)
   | R_xml_0_rule_14 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_u'7'1 ((algebra.Alg.Term 
                   algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
                   algebra.F.id_sequent ((algebra.Alg.Var 8)::
                   (algebra.Alg.Term algebra.F.id_cons 
                   ((algebra.Alg.Var 11)::(algebra.Alg.Var 9)::nil))::nil))::
                   (algebra.Alg.Var 10)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
      algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons 
      ((algebra.Alg.Term algebra.F.id_x'2d ((algebra.Alg.Var 11)::nil))::
      (algebra.Alg.Var 8)::nil))::(algebra.Alg.Var 9)::nil))::
      (algebra.Alg.Var 10)::nil))
    (* u'7'1(reduce'ii'out) -> reduce'ii'out *)
   | R_xml_0_rule_15 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) 
     (algebra.Alg.Term algebra.F.id_u'7'1 ((algebra.Alg.Term 
      algebra.F.id_reduce'ii'out nil)::nil))
   
    (* reduce'ii'in(sequent(Fs_,cons(if(A_,B_),Gs_)),NF_) -> u'8'1(reduce'ii'in(sequent(Fs_,cons(x'2b(x'2d(B_),A_),Gs_)),NF_)) *)
   | R_xml_0_rule_16 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_u'8'1 ((algebra.Alg.Term 
                   algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
                   algebra.F.id_sequent ((algebra.Alg.Var 8)::
                   (algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term 
                   algebra.F.id_x'2b ((algebra.Alg.Term algebra.F.id_x'2d 
                   ((algebra.Alg.Var 7)::nil))::(algebra.Alg.Var 6)::nil))::
                   (algebra.Alg.Var 9)::nil))::nil))::
                   (algebra.Alg.Var 10)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
      algebra.F.id_sequent ((algebra.Alg.Var 8)::(algebra.Alg.Term 
      algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_if 
      ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::nil))::
      (algebra.Alg.Var 9)::nil))::nil))::(algebra.Alg.Var 10)::nil))
    (* u'8'1(reduce'ii'out) -> reduce'ii'out *)
   | R_xml_0_rule_17 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) 
     (algebra.Alg.Term algebra.F.id_u'8'1 ((algebra.Alg.Term 
      algebra.F.id_reduce'ii'out nil)::nil))
   
    (* reduce'ii'in(sequent(Fs_,cons(iff(A_,B_),Gs_)),NF_) -> u'9'1(reduce'ii'in(sequent(Fs_,cons(x'2a(if(A_,B_),if(B_,A_)),Gs_)),NF_)) *)
   | R_xml_0_rule_18 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_u'9'1 ((algebra.Alg.Term 
                   algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
                   algebra.F.id_sequent ((algebra.Alg.Var 8)::
                   (algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term 
                   algebra.F.id_x'2a ((algebra.Alg.Term algebra.F.id_if 
                   ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::nil))::
                   (algebra.Alg.Term algebra.F.id_if ((algebra.Alg.Var 7)::
                   (algebra.Alg.Var 6)::nil))::nil))::
                   (algebra.Alg.Var 9)::nil))::nil))::
                   (algebra.Alg.Var 10)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
      algebra.F.id_sequent ((algebra.Alg.Var 8)::(algebra.Alg.Term 
      algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_iff 
      ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::nil))::
      (algebra.Alg.Var 9)::nil))::nil))::(algebra.Alg.Var 10)::nil))
    (* u'9'1(reduce'ii'out) -> reduce'ii'out *)
   | R_xml_0_rule_19 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) 
     (algebra.Alg.Term algebra.F.id_u'9'1 ((algebra.Alg.Term 
      algebra.F.id_reduce'ii'out nil)::nil))
   
    (* reduce'ii'in(sequent(cons(p(V_),Fs_),Gs_),sequent(Left_,Right_)) -> u'10'1(reduce'ii'in(sequent(Fs_,Gs_),sequent(cons(p(V_),Left_),Right_))) *)
   | R_xml_0_rule_20 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_u'10'1 ((algebra.Alg.Term 
                   algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
                   algebra.F.id_sequent ((algebra.Alg.Var 8)::
                   (algebra.Alg.Var 9)::nil))::(algebra.Alg.Term 
                   algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons 
                   ((algebra.Alg.Term algebra.F.id_p 
                   ((algebra.Alg.Var 13)::nil))::
                   (algebra.Alg.Var 14)::nil))::
                   (algebra.Alg.Var 15)::nil))::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
      algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons 
      ((algebra.Alg.Term algebra.F.id_p ((algebra.Alg.Var 13)::nil))::
      (algebra.Alg.Var 8)::nil))::(algebra.Alg.Var 9)::nil))::
      (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Var 14)::
      (algebra.Alg.Var 15)::nil))::nil))
    (* u'10'1(reduce'ii'out) -> reduce'ii'out *)
   | R_xml_0_rule_21 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) 
     (algebra.Alg.Term algebra.F.id_u'10'1 ((algebra.Alg.Term 
      algebra.F.id_reduce'ii'out nil)::nil))
   
    (* reduce'ii'in(sequent(Fs_,cons(x'2b(G1_,G2_),Gs_)),NF_) -> u'11'1(reduce'ii'in(sequent(Fs_,cons(G1_,cons(G2_,Gs_))),NF_)) *)
   | R_xml_0_rule_22 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_u'11'1 ((algebra.Alg.Term 
                   algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
                   algebra.F.id_sequent ((algebra.Alg.Var 8)::
                   (algebra.Alg.Term algebra.F.id_cons 
                   ((algebra.Alg.Var 16)::(algebra.Alg.Term 
                   algebra.F.id_cons ((algebra.Alg.Var 17)::
                   (algebra.Alg.Var 9)::nil))::nil))::nil))::
                   (algebra.Alg.Var 10)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
      algebra.F.id_sequent ((algebra.Alg.Var 8)::(algebra.Alg.Term 
      algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b 
      ((algebra.Alg.Var 16)::(algebra.Alg.Var 17)::nil))::
      (algebra.Alg.Var 9)::nil))::nil))::(algebra.Alg.Var 10)::nil))
    (* u'11'1(reduce'ii'out) -> reduce'ii'out *)
   | R_xml_0_rule_23 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) 
     (algebra.Alg.Term algebra.F.id_u'11'1 ((algebra.Alg.Term 
      algebra.F.id_reduce'ii'out nil)::nil))
   
    (* reduce'ii'in(sequent(Fs_,cons(x'2a(G1_,G2_),Gs_)),NF_) -> u'12'1(reduce'ii'in(sequent(Fs_,cons(G1_,Gs_)),NF_),Fs_,G2_,Gs_,NF_) *)
   | R_xml_0_rule_24 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_u'12'1 ((algebra.Alg.Term 
                   algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
                   algebra.F.id_sequent ((algebra.Alg.Var 8)::
                   (algebra.Alg.Term algebra.F.id_cons 
                   ((algebra.Alg.Var 16)::(algebra.Alg.Var 9)::nil))::nil))::
                   (algebra.Alg.Var 10)::nil))::(algebra.Alg.Var 8)::
                   (algebra.Alg.Var 17)::(algebra.Alg.Var 9)::
                   (algebra.Alg.Var 10)::nil)) 
     (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
      algebra.F.id_sequent ((algebra.Alg.Var 8)::(algebra.Alg.Term 
      algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2a 
      ((algebra.Alg.Var 16)::(algebra.Alg.Var 17)::nil))::
      (algebra.Alg.Var 9)::nil))::nil))::(algebra.Alg.Var 10)::nil))
   
    (* u'12'1(reduce'ii'out,Fs_,G2_,Gs_,NF_) -> u'12'2(reduce'ii'in(sequent(Fs_,cons(G2_,Gs_)),NF_)) *)
   | R_xml_0_rule_25 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_u'12'2 ((algebra.Alg.Term 
                   algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
                   algebra.F.id_sequent ((algebra.Alg.Var 8)::
                   (algebra.Alg.Term algebra.F.id_cons 
                   ((algebra.Alg.Var 17)::(algebra.Alg.Var 9)::nil))::nil))::
                   (algebra.Alg.Var 10)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_u'12'1 ((algebra.Alg.Term 
      algebra.F.id_reduce'ii'out nil)::(algebra.Alg.Var 8)::
      (algebra.Alg.Var 17)::(algebra.Alg.Var 9)::(algebra.Alg.Var 10)::nil))
    (* u'12'2(reduce'ii'out) -> reduce'ii'out *)
   | R_xml_0_rule_26 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) 
     (algebra.Alg.Term algebra.F.id_u'12'2 ((algebra.Alg.Term 
      algebra.F.id_reduce'ii'out nil)::nil))
   
    (* reduce'ii'in(sequent(Fs_,cons(x'2d(G1_),Gs_)),NF_) -> u'13'1(reduce'ii'in(sequent(cons(G1_,Fs_),Gs_),NF_)) *)
   | R_xml_0_rule_27 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_u'13'1 ((algebra.Alg.Term 
                   algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
                   algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons 
                   ((algebra.Alg.Var 16)::(algebra.Alg.Var 8)::nil))::
                   (algebra.Alg.Var 9)::nil))::
                   (algebra.Alg.Var 10)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
      algebra.F.id_sequent ((algebra.Alg.Var 8)::(algebra.Alg.Term 
      algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2d 
      ((algebra.Alg.Var 16)::nil))::(algebra.Alg.Var 9)::nil))::nil))::
      (algebra.Alg.Var 10)::nil))
    (* u'13'1(reduce'ii'out) -> reduce'ii'out *)
   | R_xml_0_rule_28 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) 
     (algebra.Alg.Term algebra.F.id_u'13'1 ((algebra.Alg.Term 
      algebra.F.id_reduce'ii'out nil)::nil))
   
    (* reduce'ii'in(sequent(nil,cons(p(V_),Gs_)),sequent(Left_,Right_)) -> u'14'1(reduce'ii'in(sequent(nil,Gs_),sequent(Left_,cons(p(V_),Right_)))) *)
   | R_xml_0_rule_29 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_u'14'1 ((algebra.Alg.Term 
                   algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
                   algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_nil 
                   nil)::(algebra.Alg.Var 9)::nil))::(algebra.Alg.Term 
                   algebra.F.id_sequent ((algebra.Alg.Var 14)::
                   (algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term 
                   algebra.F.id_p ((algebra.Alg.Var 13)::nil))::
                   (algebra.Alg.Var 15)::nil))::nil))::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
      algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_nil nil)::
      (algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_p 
      ((algebra.Alg.Var 13)::nil))::(algebra.Alg.Var 9)::nil))::nil))::
      (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Var 14)::
      (algebra.Alg.Var 15)::nil))::nil))
    (* u'14'1(reduce'ii'out) -> reduce'ii'out *)
   | R_xml_0_rule_30 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) 
     (algebra.Alg.Term algebra.F.id_u'14'1 ((algebra.Alg.Term 
      algebra.F.id_reduce'ii'out nil)::nil))
   
    (* reduce'ii'in(sequent(nil,nil),sequent(F1_,F2_)) -> u'15'1(intersect'ii'in(F1_,F2_)) *)
   | R_xml_0_rule_31 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_u'15'1 ((algebra.Alg.Term 
                   algebra.F.id_intersect'ii'in ((algebra.Alg.Var 11)::
                   (algebra.Alg.Var 12)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
      algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_nil nil)::
      (algebra.Alg.Term algebra.F.id_nil nil)::nil))::(algebra.Alg.Term 
      algebra.F.id_sequent ((algebra.Alg.Var 11)::
      (algebra.Alg.Var 12)::nil))::nil))
    (* u'15'1(intersect'ii'out) -> reduce'ii'out *)
   | R_xml_0_rule_32 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) 
     (algebra.Alg.Term algebra.F.id_u'15'1 ((algebra.Alg.Term 
      algebra.F.id_intersect'ii'out nil)::nil))
   
    (* tautology'i'in(F_) -> u'16'1(reduce'ii'in(sequent(nil,cons(F_,nil)),sequent(nil,nil))) *)
   | R_xml_0_rule_33 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_u'16'1 ((algebra.Alg.Term 
                   algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
                   algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_nil 
                   nil)::(algebra.Alg.Term algebra.F.id_cons 
                   ((algebra.Alg.Var 18)::(algebra.Alg.Term algebra.F.id_nil 
                   nil)::nil))::nil))::(algebra.Alg.Term 
                   algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_nil 
                   nil)::(algebra.Alg.Term algebra.F.id_nil 
                   nil)::nil))::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_tautology'i'in 
      ((algebra.Alg.Var 18)::nil))
    (* u'16'1(reduce'ii'out) -> tautology'i'out *)
   | R_xml_0_rule_34 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_tautology'i'out nil) 
     (algebra.Alg.Term algebra.F.id_u'16'1 ((algebra.Alg.Term 
      algebra.F.id_reduce'ii'out nil)::nil))
 .
 
 
 Definition R_xml_0_rule_as_list_0  := 
   ((algebra.Alg.Term algebra.F.id_intersect'ii'in ((algebra.Alg.Term 
     algebra.F.id_cons ((algebra.Alg.Var 1)::(algebra.Alg.Var 2)::nil))::
     (algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Var 1)::
     (algebra.Alg.Var 3)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_intersect'ii'out nil))::nil.
 
 
 Definition R_xml_0_rule_as_list_1  := 
   ((algebra.Alg.Term algebra.F.id_intersect'ii'in ((algebra.Alg.Var 4)::
     (algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Var 2)::
     (algebra.Alg.Var 5)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_u'1'1 ((algebra.Alg.Term 
     algebra.F.id_intersect'ii'in ((algebra.Alg.Var 4)::
     (algebra.Alg.Var 5)::nil))::nil)))::R_xml_0_rule_as_list_0.
 
 
 Definition R_xml_0_rule_as_list_2  := 
   ((algebra.Alg.Term algebra.F.id_u'1'1 ((algebra.Alg.Term 
     algebra.F.id_intersect'ii'out nil)::nil)),
    (algebra.Alg.Term algebra.F.id_intersect'ii'out nil))::
    R_xml_0_rule_as_list_1.
 
 
 Definition R_xml_0_rule_as_list_3  := 
   ((algebra.Alg.Term algebra.F.id_intersect'ii'in ((algebra.Alg.Term 
     algebra.F.id_cons ((algebra.Alg.Var 2)::(algebra.Alg.Var 4)::nil))::
     (algebra.Alg.Var 5)::nil)),
    (algebra.Alg.Term algebra.F.id_u'2'1 ((algebra.Alg.Term 
     algebra.F.id_intersect'ii'in ((algebra.Alg.Var 4)::
     (algebra.Alg.Var 5)::nil))::nil)))::R_xml_0_rule_as_list_2.
 
 
 Definition R_xml_0_rule_as_list_4  := 
   ((algebra.Alg.Term algebra.F.id_u'2'1 ((algebra.Alg.Term 
     algebra.F.id_intersect'ii'out nil)::nil)),
    (algebra.Alg.Term algebra.F.id_intersect'ii'out nil))::
    R_xml_0_rule_as_list_3.
 
 
 Definition R_xml_0_rule_as_list_5  := 
   ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
     algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons 
     ((algebra.Alg.Term algebra.F.id_if ((algebra.Alg.Var 6)::
     (algebra.Alg.Var 7)::nil))::(algebra.Alg.Var 8)::nil))::
     (algebra.Alg.Var 9)::nil))::(algebra.Alg.Var 10)::nil)),
    (algebra.Alg.Term algebra.F.id_u'3'1 ((algebra.Alg.Term 
     algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent 
     ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term 
     algebra.F.id_x'2b ((algebra.Alg.Term algebra.F.id_x'2d 
     ((algebra.Alg.Var 7)::nil))::(algebra.Alg.Var 6)::nil))::
     (algebra.Alg.Var 8)::nil))::(algebra.Alg.Var 9)::nil))::
     (algebra.Alg.Var 10)::nil))::nil)))::R_xml_0_rule_as_list_4.
 
 
 Definition R_xml_0_rule_as_list_6  := 
   ((algebra.Alg.Term algebra.F.id_u'3'1 ((algebra.Alg.Term 
     algebra.F.id_reduce'ii'out nil)::nil)),
    (algebra.Alg.Term algebra.F.id_reduce'ii'out nil))::
    R_xml_0_rule_as_list_5.
 
 
 Definition R_xml_0_rule_as_list_7  := 
   ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
     algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons 
     ((algebra.Alg.Term algebra.F.id_iff ((algebra.Alg.Var 6)::
     (algebra.Alg.Var 7)::nil))::(algebra.Alg.Var 8)::nil))::
     (algebra.Alg.Var 9)::nil))::(algebra.Alg.Var 10)::nil)),
    (algebra.Alg.Term algebra.F.id_u'4'1 ((algebra.Alg.Term 
     algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent 
     ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term 
     algebra.F.id_x'2a ((algebra.Alg.Term algebra.F.id_if 
     ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::nil))::(algebra.Alg.Term 
     algebra.F.id_if ((algebra.Alg.Var 7)::
     (algebra.Alg.Var 6)::nil))::nil))::(algebra.Alg.Var 8)::nil))::
     (algebra.Alg.Var 9)::nil))::(algebra.Alg.Var 10)::nil))::nil)))::
    R_xml_0_rule_as_list_6.
 
 
 Definition R_xml_0_rule_as_list_8  := 
   ((algebra.Alg.Term algebra.F.id_u'4'1 ((algebra.Alg.Term 
     algebra.F.id_reduce'ii'out nil)::nil)),
    (algebra.Alg.Term algebra.F.id_reduce'ii'out nil))::
    R_xml_0_rule_as_list_7.
 
 
 Definition R_xml_0_rule_as_list_9  := 
   ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
     algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons 
     ((algebra.Alg.Term algebra.F.id_x'2a ((algebra.Alg.Var 11)::
     (algebra.Alg.Var 12)::nil))::(algebra.Alg.Var 8)::nil))::
     (algebra.Alg.Var 9)::nil))::(algebra.Alg.Var 10)::nil)),
    (algebra.Alg.Term algebra.F.id_u'5'1 ((algebra.Alg.Term 
     algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent 
     ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Var 11)::
     (algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Var 12)::
     (algebra.Alg.Var 8)::nil))::nil))::(algebra.Alg.Var 9)::nil))::
     (algebra.Alg.Var 10)::nil))::nil)))::R_xml_0_rule_as_list_8.
 
 
 Definition R_xml_0_rule_as_list_10  := 
   ((algebra.Alg.Term algebra.F.id_u'5'1 ((algebra.Alg.Term 
     algebra.F.id_reduce'ii'out nil)::nil)),
    (algebra.Alg.Term algebra.F.id_reduce'ii'out nil))::
    R_xml_0_rule_as_list_9.
 
 
 Definition R_xml_0_rule_as_list_11  := 
   ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
     algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons 
     ((algebra.Alg.Term algebra.F.id_x'2b ((algebra.Alg.Var 11)::
     (algebra.Alg.Var 12)::nil))::(algebra.Alg.Var 8)::nil))::
     (algebra.Alg.Var 9)::nil))::(algebra.Alg.Var 10)::nil)),
    (algebra.Alg.Term algebra.F.id_u'6'1 ((algebra.Alg.Term 
     algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent 
     ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Var 11)::
     (algebra.Alg.Var 8)::nil))::(algebra.Alg.Var 9)::nil))::
     (algebra.Alg.Var 10)::nil))::(algebra.Alg.Var 12)::(algebra.Alg.Var 8)::
     (algebra.Alg.Var 9)::(algebra.Alg.Var 10)::nil)))::
    R_xml_0_rule_as_list_10.
 
 
 Definition R_xml_0_rule_as_list_12  := 
   ((algebra.Alg.Term algebra.F.id_u'6'1 ((algebra.Alg.Term 
     algebra.F.id_reduce'ii'out nil)::(algebra.Alg.Var 12)::
     (algebra.Alg.Var 8)::(algebra.Alg.Var 9)::(algebra.Alg.Var 10)::nil)),
    (algebra.Alg.Term algebra.F.id_u'6'2 ((algebra.Alg.Term 
     algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent 
     ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Var 12)::
     (algebra.Alg.Var 8)::nil))::(algebra.Alg.Var 9)::nil))::
     (algebra.Alg.Var 10)::nil))::nil)))::R_xml_0_rule_as_list_11.
 
 
 Definition R_xml_0_rule_as_list_13  := 
   ((algebra.Alg.Term algebra.F.id_u'6'2 ((algebra.Alg.Term 
     algebra.F.id_reduce'ii'out nil)::nil)),
    (algebra.Alg.Term algebra.F.id_reduce'ii'out nil))::
    R_xml_0_rule_as_list_12.
 
 
 Definition R_xml_0_rule_as_list_14  := 
   ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
     algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons 
     ((algebra.Alg.Term algebra.F.id_x'2d ((algebra.Alg.Var 11)::nil))::
     (algebra.Alg.Var 8)::nil))::(algebra.Alg.Var 9)::nil))::
     (algebra.Alg.Var 10)::nil)),
    (algebra.Alg.Term algebra.F.id_u'7'1 ((algebra.Alg.Term 
     algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent 
     ((algebra.Alg.Var 8)::(algebra.Alg.Term algebra.F.id_cons 
     ((algebra.Alg.Var 11)::(algebra.Alg.Var 9)::nil))::nil))::
     (algebra.Alg.Var 10)::nil))::nil)))::R_xml_0_rule_as_list_13.
 
 
 Definition R_xml_0_rule_as_list_15  := 
   ((algebra.Alg.Term algebra.F.id_u'7'1 ((algebra.Alg.Term 
     algebra.F.id_reduce'ii'out nil)::nil)),
    (algebra.Alg.Term algebra.F.id_reduce'ii'out nil))::
    R_xml_0_rule_as_list_14.
 
 
 Definition R_xml_0_rule_as_list_16  := 
   ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
     algebra.F.id_sequent ((algebra.Alg.Var 8)::(algebra.Alg.Term 
     algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_if 
     ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::nil))::
     (algebra.Alg.Var 9)::nil))::nil))::(algebra.Alg.Var 10)::nil)),
    (algebra.Alg.Term algebra.F.id_u'8'1 ((algebra.Alg.Term 
     algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent 
     ((algebra.Alg.Var 8)::(algebra.Alg.Term algebra.F.id_cons 
     ((algebra.Alg.Term algebra.F.id_x'2b ((algebra.Alg.Term 
     algebra.F.id_x'2d ((algebra.Alg.Var 7)::nil))::
     (algebra.Alg.Var 6)::nil))::(algebra.Alg.Var 9)::nil))::nil))::
     (algebra.Alg.Var 10)::nil))::nil)))::R_xml_0_rule_as_list_15.
 
 
 Definition R_xml_0_rule_as_list_17  := 
   ((algebra.Alg.Term algebra.F.id_u'8'1 ((algebra.Alg.Term 
     algebra.F.id_reduce'ii'out nil)::nil)),
    (algebra.Alg.Term algebra.F.id_reduce'ii'out nil))::
    R_xml_0_rule_as_list_16.
 
 
 Definition R_xml_0_rule_as_list_18  := 
   ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
     algebra.F.id_sequent ((algebra.Alg.Var 8)::(algebra.Alg.Term 
     algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_iff 
     ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::nil))::
     (algebra.Alg.Var 9)::nil))::nil))::(algebra.Alg.Var 10)::nil)),
    (algebra.Alg.Term algebra.F.id_u'9'1 ((algebra.Alg.Term 
     algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent 
     ((algebra.Alg.Var 8)::(algebra.Alg.Term algebra.F.id_cons 
     ((algebra.Alg.Term algebra.F.id_x'2a ((algebra.Alg.Term algebra.F.id_if 
     ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::nil))::(algebra.Alg.Term 
     algebra.F.id_if ((algebra.Alg.Var 7)::
     (algebra.Alg.Var 6)::nil))::nil))::(algebra.Alg.Var 9)::nil))::nil))::
     (algebra.Alg.Var 10)::nil))::nil)))::R_xml_0_rule_as_list_17.
 
 
 Definition R_xml_0_rule_as_list_19  := 
   ((algebra.Alg.Term algebra.F.id_u'9'1 ((algebra.Alg.Term 
     algebra.F.id_reduce'ii'out nil)::nil)),
    (algebra.Alg.Term algebra.F.id_reduce'ii'out nil))::
    R_xml_0_rule_as_list_18.
 
 
 Definition R_xml_0_rule_as_list_20  := 
   ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
     algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons 
     ((algebra.Alg.Term algebra.F.id_p ((algebra.Alg.Var 13)::nil))::
     (algebra.Alg.Var 8)::nil))::(algebra.Alg.Var 9)::nil))::
     (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Var 14)::
     (algebra.Alg.Var 15)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_u'10'1 ((algebra.Alg.Term 
     algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent 
     ((algebra.Alg.Var 8)::(algebra.Alg.Var 9)::nil))::(algebra.Alg.Term 
     algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons 
     ((algebra.Alg.Term algebra.F.id_p ((algebra.Alg.Var 13)::nil))::
     (algebra.Alg.Var 14)::nil))::(algebra.Alg.Var 15)::nil))::nil))::nil)))::
    R_xml_0_rule_as_list_19.
 
 
 Definition R_xml_0_rule_as_list_21  := 
   ((algebra.Alg.Term algebra.F.id_u'10'1 ((algebra.Alg.Term 
     algebra.F.id_reduce'ii'out nil)::nil)),
    (algebra.Alg.Term algebra.F.id_reduce'ii'out nil))::
    R_xml_0_rule_as_list_20.
 
 
 Definition R_xml_0_rule_as_list_22  := 
   ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
     algebra.F.id_sequent ((algebra.Alg.Var 8)::(algebra.Alg.Term 
     algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b 
     ((algebra.Alg.Var 16)::(algebra.Alg.Var 17)::nil))::
     (algebra.Alg.Var 9)::nil))::nil))::(algebra.Alg.Var 10)::nil)),
    (algebra.Alg.Term algebra.F.id_u'11'1 ((algebra.Alg.Term 
     algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent 
     ((algebra.Alg.Var 8)::(algebra.Alg.Term algebra.F.id_cons 
     ((algebra.Alg.Var 16)::(algebra.Alg.Term algebra.F.id_cons 
     ((algebra.Alg.Var 17)::(algebra.Alg.Var 9)::nil))::nil))::nil))::
     (algebra.Alg.Var 10)::nil))::nil)))::R_xml_0_rule_as_list_21.
 
 
 Definition R_xml_0_rule_as_list_23  := 
   ((algebra.Alg.Term algebra.F.id_u'11'1 ((algebra.Alg.Term 
     algebra.F.id_reduce'ii'out nil)::nil)),
    (algebra.Alg.Term algebra.F.id_reduce'ii'out nil))::
    R_xml_0_rule_as_list_22.
 
 
 Definition R_xml_0_rule_as_list_24  := 
   ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
     algebra.F.id_sequent ((algebra.Alg.Var 8)::(algebra.Alg.Term 
     algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2a 
     ((algebra.Alg.Var 16)::(algebra.Alg.Var 17)::nil))::
     (algebra.Alg.Var 9)::nil))::nil))::(algebra.Alg.Var 10)::nil)),
    (algebra.Alg.Term algebra.F.id_u'12'1 ((algebra.Alg.Term 
     algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent 
     ((algebra.Alg.Var 8)::(algebra.Alg.Term algebra.F.id_cons 
     ((algebra.Alg.Var 16)::(algebra.Alg.Var 9)::nil))::nil))::
     (algebra.Alg.Var 10)::nil))::(algebra.Alg.Var 8)::(algebra.Alg.Var 17)::
     (algebra.Alg.Var 9)::(algebra.Alg.Var 10)::nil)))::
    R_xml_0_rule_as_list_23.
 
 
 Definition R_xml_0_rule_as_list_25  := 
   ((algebra.Alg.Term algebra.F.id_u'12'1 ((algebra.Alg.Term 
     algebra.F.id_reduce'ii'out nil)::(algebra.Alg.Var 8)::
     (algebra.Alg.Var 17)::(algebra.Alg.Var 9)::(algebra.Alg.Var 10)::nil)),
    (algebra.Alg.Term algebra.F.id_u'12'2 ((algebra.Alg.Term 
     algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent 
     ((algebra.Alg.Var 8)::(algebra.Alg.Term algebra.F.id_cons 
     ((algebra.Alg.Var 17)::(algebra.Alg.Var 9)::nil))::nil))::
     (algebra.Alg.Var 10)::nil))::nil)))::R_xml_0_rule_as_list_24.
 
 
 Definition R_xml_0_rule_as_list_26  := 
   ((algebra.Alg.Term algebra.F.id_u'12'2 ((algebra.Alg.Term 
     algebra.F.id_reduce'ii'out nil)::nil)),
    (algebra.Alg.Term algebra.F.id_reduce'ii'out nil))::
    R_xml_0_rule_as_list_25.
 
 
 Definition R_xml_0_rule_as_list_27  := 
   ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
     algebra.F.id_sequent ((algebra.Alg.Var 8)::(algebra.Alg.Term 
     algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2d 
     ((algebra.Alg.Var 16)::nil))::(algebra.Alg.Var 9)::nil))::nil))::
     (algebra.Alg.Var 10)::nil)),
    (algebra.Alg.Term algebra.F.id_u'13'1 ((algebra.Alg.Term 
     algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent 
     ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Var 16)::
     (algebra.Alg.Var 8)::nil))::(algebra.Alg.Var 9)::nil))::
     (algebra.Alg.Var 10)::nil))::nil)))::R_xml_0_rule_as_list_26.
 
 
 Definition R_xml_0_rule_as_list_28  := 
   ((algebra.Alg.Term algebra.F.id_u'13'1 ((algebra.Alg.Term 
     algebra.F.id_reduce'ii'out nil)::nil)),
    (algebra.Alg.Term algebra.F.id_reduce'ii'out nil))::
    R_xml_0_rule_as_list_27.
 
 
 Definition R_xml_0_rule_as_list_29  := 
   ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
     algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_nil nil)::
     (algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_p 
     ((algebra.Alg.Var 13)::nil))::(algebra.Alg.Var 9)::nil))::nil))::
     (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Var 14)::
     (algebra.Alg.Var 15)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_u'14'1 ((algebra.Alg.Term 
     algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent 
     ((algebra.Alg.Term algebra.F.id_nil nil)::(algebra.Alg.Var 9)::nil))::
     (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Var 14)::
     (algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_p 
     ((algebra.Alg.Var 13)::nil))::
     (algebra.Alg.Var 15)::nil))::nil))::nil))::nil)))::
    R_xml_0_rule_as_list_28.
 
 
 Definition R_xml_0_rule_as_list_30  := 
   ((algebra.Alg.Term algebra.F.id_u'14'1 ((algebra.Alg.Term 
     algebra.F.id_reduce'ii'out nil)::nil)),
    (algebra.Alg.Term algebra.F.id_reduce'ii'out nil))::
    R_xml_0_rule_as_list_29.
 
 
 Definition R_xml_0_rule_as_list_31  := 
   ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
     algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_nil nil)::
     (algebra.Alg.Term algebra.F.id_nil nil)::nil))::(algebra.Alg.Term 
     algebra.F.id_sequent ((algebra.Alg.Var 11)::
     (algebra.Alg.Var 12)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_u'15'1 ((algebra.Alg.Term 
     algebra.F.id_intersect'ii'in ((algebra.Alg.Var 11)::
     (algebra.Alg.Var 12)::nil))::nil)))::R_xml_0_rule_as_list_30.
 
 
 Definition R_xml_0_rule_as_list_32  := 
   ((algebra.Alg.Term algebra.F.id_u'15'1 ((algebra.Alg.Term 
     algebra.F.id_intersect'ii'out nil)::nil)),
    (algebra.Alg.Term algebra.F.id_reduce'ii'out nil))::
    R_xml_0_rule_as_list_31.
 
 
 Definition R_xml_0_rule_as_list_33  := 
   ((algebra.Alg.Term algebra.F.id_tautology'i'in 
     ((algebra.Alg.Var 18)::nil)),
    (algebra.Alg.Term algebra.F.id_u'16'1 ((algebra.Alg.Term 
     algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent 
     ((algebra.Alg.Term algebra.F.id_nil nil)::(algebra.Alg.Term 
     algebra.F.id_cons ((algebra.Alg.Var 18)::(algebra.Alg.Term 
     algebra.F.id_nil nil)::nil))::nil))::(algebra.Alg.Term 
     algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_nil nil)::
     (algebra.Alg.Term algebra.F.id_nil nil)::nil))::nil))::nil)))::
    R_xml_0_rule_as_list_32.
 
 
 Definition R_xml_0_rule_as_list_34  := 
   ((algebra.Alg.Term algebra.F.id_u'16'1 ((algebra.Alg.Term 
     algebra.F.id_reduce'ii'out nil)::nil)),
    (algebra.Alg.Term algebra.F.id_tautology'i'out nil))::
    R_xml_0_rule_as_list_33.
 
 Definition R_xml_0_rule_as_list  := R_xml_0_rule_as_list_34.
 
 
 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 35.
   injection H;intros ;subst;constructor 34.
   injection H;intros ;subst;constructor 33.
   injection H;intros ;subst;constructor 32.
   injection H;intros ;subst;constructor 31.
   injection H;intros ;subst;constructor 30.
   injection H;intros ;subst;constructor 29.
   injection H;intros ;subst;constructor 28.
   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.
   rewrite  <- or_ext_generated.or12_equiv in H|-.
   case H;clear H;intros H.
   injection H;intros ;subst;constructor 11.
   injection H;intros ;subst;constructor 10.
   injection H;intros ;subst;constructor 9.
   injection H;intros ;subst;constructor 8.
   injection H;intros ;subst;constructor 7.
   injection H;intros ;subst;constructor 6.
   injection H;intros ;subst;constructor 5.
   injection H;intros ;subst;constructor 4.
   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_12 (x20 x21 x22 x23 x24 x25 x26 x27 x28 x29 x30 x31:Prop) :
  Prop := 
   | conj_12 :
    x20->x21->x22->x23->x24->x25->x26->x27->x28->x29->x30->x31->
     and_12 x20 x21 x22 x23 x24 x25 x26 x27 x28 x29 x30 x31
 .
 
 
 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_12 (t = (algebra.Alg.Term algebra.F.id_tautology'i'out nil) ->
            t' = (algebra.Alg.Term algebra.F.id_tautology'i'out nil)) 
     (forall x21 x23, 
      t = (algebra.Alg.Term algebra.F.id_iff (x21::x23::nil)) ->
       exists x20,
         exists x22,
           t' = (algebra.Alg.Term algebra.F.id_iff (x20::x22::nil))/\ 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
             x20 x21)/\ 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
             x22 x23)) 
     (t = (algebra.Alg.Term algebra.F.id_intersect'ii'out nil) ->
      t' = (algebra.Alg.Term algebra.F.id_intersect'ii'out nil)) 
     (forall x21, 
      t = (algebra.Alg.Term algebra.F.id_x'2d (x21::nil)) ->
       exists x20,
         t' = (algebra.Alg.Term algebra.F.id_x'2d (x20::nil))/\ 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
           x20 x21)) 
     (forall x21 x23, 
      t = (algebra.Alg.Term algebra.F.id_sequent (x21::x23::nil)) ->
       exists x20,
         exists x22,
           t' = (algebra.Alg.Term algebra.F.id_sequent (x20::x22::nil))/\ 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
             x20 x21)/\ 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
             x22 x23)) 
     (forall x21 x23, 
      t = (algebra.Alg.Term algebra.F.id_x'2a (x21::x23::nil)) ->
       exists x20,
         exists x22,
           t' = (algebra.Alg.Term algebra.F.id_x'2a (x20::x22::nil))/\ 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
             x20 x21)/\ 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
             x22 x23)) 
     (forall x21 x23, 
      t = (algebra.Alg.Term algebra.F.id_cons (x21::x23::nil)) ->
       exists x20,
         exists x22,
           t' = (algebra.Alg.Term algebra.F.id_cons (x20::x22::nil))/\ 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
             x20 x21)/\ 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
             x22 x23)) 
     (forall x21 x23, 
      t = (algebra.Alg.Term algebra.F.id_x'2b (x21::x23::nil)) ->
       exists x20,
         exists x22,
           t' = (algebra.Alg.Term algebra.F.id_x'2b (x20::x22::nil))/\ 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
             x20 x21)/\ 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
             x22 x23)) 
     (forall x21, 
      t = (algebra.Alg.Term algebra.F.id_p (x21::nil)) ->
       exists x20,
         t' = (algebra.Alg.Term algebra.F.id_p (x20::nil))/\ 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
           x20 x21)) 
     (t = (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) ->
      t' = (algebra.Alg.Term algebra.F.id_reduce'ii'out nil)) 
     (t = (algebra.Alg.Term algebra.F.id_nil nil) ->
      t' = (algebra.Alg.Term algebra.F.id_nil nil)) 
     (forall x21 x23, 
      t = (algebra.Alg.Term algebra.F.id_if (x21::x23::nil)) ->
       exists x20,
         exists x22,
           t' = (algebra.Alg.Term algebra.F.id_if (x20::x22::nil))/\ 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
             x20 x21)/\ 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
             x22 x23)).
 Proof.
   intros t t' H.
   
   induction H as [|y IH z z_to_y] using 
   closure_extension.refl_trans_clos_ind2.
   constructor 1.
   intros H;intuition;constructor 1.
   intros x21 x23 H;exists x21;exists x23;intuition;constructor 1.
   intros H;intuition;constructor 1.
   intros x21 H;exists x21;intuition;constructor 1.
   intros x21 x23 H;exists x21;exists x23;intuition;constructor 1.
   intros x21 x23 H;exists x21;exists x23;intuition;constructor 1.
   intros x21 x23 H;exists x21;exists x23;intuition;constructor 1.
   intros x21 x23 H;exists x21;exists x23;intuition;constructor 1.
   intros x21 H;exists x21;intuition;constructor 1.
   intros H;intuition;constructor 1.
   intros H;intuition;constructor 1.
   intros x21 x23 H;exists x21;exists x23;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_tautology'i'out H_id_iff H_id_intersect'ii'out H_id_x'2d 
    H_id_sequent H_id_x'2a H_id_cons H_id_x'2b H_id_p H_id_reduce'ii'out 
    H_id_nil H_id_if].
   constructor.
   
   clear H_id_iff H_id_intersect'ii'out H_id_x'2d H_id_sequent H_id_x'2a 
   H_id_cons H_id_x'2b H_id_p H_id_reduce'ii'out H_id_nil H_id_if;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_tautology'i'out H_id_intersect'ii'out H_id_x'2d H_id_sequent 
   H_id_x'2a H_id_cons H_id_x'2b H_id_p H_id_reduce'ii'out H_id_nil H_id_if;
    intros x21 x23 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 x21 |- _ =>
      destruct (H_id_iff y x23 (refl_equal _)) as [x20 [x22]];intros ;
       intuition;exists x20;exists x22;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x23 |- _ =>
      destruct (H_id_iff x21 y (refl_equal _)) as [x20 [x22]];intros ;
       intuition;exists x20;exists x22;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   clear H_id_tautology'i'out H_id_iff H_id_x'2d H_id_sequent H_id_x'2a 
   H_id_cons H_id_x'2b H_id_p H_id_reduce'ii'out H_id_nil H_id_if;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_tautology'i'out H_id_iff H_id_intersect'ii'out H_id_sequent 
   H_id_x'2a H_id_cons H_id_x'2b H_id_p H_id_reduce'ii'out H_id_nil H_id_if;
    intros x21 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 x21 |- _ =>
      destruct (H_id_x'2d y (refl_equal _)) as [x20];intros ;intuition;
       exists x20;intuition;eapply closure_extension.refl_trans_clos_R;
       eassumption
     end
   .
   
   clear H_id_tautology'i'out H_id_iff H_id_intersect'ii'out H_id_x'2d 
   H_id_x'2a H_id_cons H_id_x'2b H_id_p H_id_reduce'ii'out H_id_nil H_id_if;
    intros x21 x23 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 x21 |- _ =>
      destruct (H_id_sequent y x23 (refl_equal _)) as [x20 [x22]];intros ;
       intuition;exists x20;exists x22;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x23 |- _ =>
      destruct (H_id_sequent x21 y (refl_equal _)) as [x20 [x22]];intros ;
       intuition;exists x20;exists x22;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   clear H_id_tautology'i'out H_id_iff H_id_intersect'ii'out H_id_x'2d 
   H_id_sequent H_id_cons H_id_x'2b H_id_p H_id_reduce'ii'out H_id_nil 
   H_id_if;intros x21 x23 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 x21 |- _ =>
      destruct (H_id_x'2a y x23 (refl_equal _)) as [x20 [x22]];intros ;
       intuition;exists x20;exists x22;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x23 |- _ =>
      destruct (H_id_x'2a x21 y (refl_equal _)) as [x20 [x22]];intros ;
       intuition;exists x20;exists x22;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   clear H_id_tautology'i'out H_id_iff H_id_intersect'ii'out H_id_x'2d 
   H_id_sequent H_id_x'2a H_id_x'2b H_id_p H_id_reduce'ii'out H_id_nil 
   H_id_if;intros x21 x23 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 x21 |- _ =>
      destruct (H_id_cons y x23 (refl_equal _)) as [x20 [x22]];intros ;
       intuition;exists x20;exists x22;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x23 |- _ =>
      destruct (H_id_cons x21 y (refl_equal _)) as [x20 [x22]];intros ;
       intuition;exists x20;exists x22;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   clear H_id_tautology'i'out H_id_iff H_id_intersect'ii'out H_id_x'2d 
   H_id_sequent H_id_x'2a H_id_cons H_id_p H_id_reduce'ii'out H_id_nil 
   H_id_if;intros x21 x23 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 x21 |- _ =>
      destruct (H_id_x'2b y x23 (refl_equal _)) as [x20 [x22]];intros ;
       intuition;exists x20;exists x22;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x23 |- _ =>
      destruct (H_id_x'2b x21 y (refl_equal _)) as [x20 [x22]];intros ;
       intuition;exists x20;exists x22;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   clear H_id_tautology'i'out H_id_iff H_id_intersect'ii'out H_id_x'2d 
   H_id_sequent H_id_x'2a H_id_cons H_id_x'2b H_id_reduce'ii'out H_id_nil 
   H_id_if;intros x21 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 x21 |- _ =>
      destruct (H_id_p y (refl_equal _)) as [x20];intros ;intuition;
       exists x20;intuition;eapply closure_extension.refl_trans_clos_R;
       eassumption
     end
   .
   
   clear H_id_tautology'i'out H_id_iff H_id_intersect'ii'out H_id_x'2d 
   H_id_sequent H_id_x'2a H_id_cons H_id_x'2b H_id_p H_id_nil H_id_if;
    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_tautology'i'out H_id_iff H_id_intersect'ii'out H_id_x'2d 
   H_id_sequent H_id_x'2a H_id_cons H_id_x'2b H_id_p H_id_reduce'ii'out 
   H_id_if;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_tautology'i'out H_id_iff H_id_intersect'ii'out H_id_x'2d 
   H_id_sequent H_id_x'2a H_id_cons H_id_x'2b H_id_p H_id_reduce'ii'out 
   H_id_nil;intros x21 x23 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 x21 |- _ =>
      destruct (H_id_if y x23 (refl_equal _)) as [x20 [x22]];intros ;
       intuition;exists x20;exists x22;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x23 |- _ =>
      destruct (H_id_if x21 y (refl_equal _)) as [x20 [x22]];intros ;
       intuition;exists x20;exists x22;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
 Qed.
 
 
 Lemma id_tautology'i'out_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_tautology'i'out nil) ->
     t' = (algebra.Alg.Term algebra.F.id_tautology'i'out nil).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_iff_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x21 x23, 
     t = (algebra.Alg.Term algebra.F.id_iff (x21::x23::nil)) ->
      exists x20,
        exists x22,
          t' = (algebra.Alg.Term algebra.F.id_iff (x20::x22::nil))/\ 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
            x20 x21)/\ 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
            x22 x23).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_intersect'ii'out_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_intersect'ii'out nil) ->
     t' = (algebra.Alg.Term algebra.F.id_intersect'ii'out nil).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_x'2d_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x21, 
     t = (algebra.Alg.Term algebra.F.id_x'2d (x21::nil)) ->
      exists x20,
        t' = (algebra.Alg.Term algebra.F.id_x'2d (x20::nil))/\ 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x20 x21).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_sequent_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x21 x23, 
     t = (algebra.Alg.Term algebra.F.id_sequent (x21::x23::nil)) ->
      exists x20,
        exists x22,
          t' = (algebra.Alg.Term algebra.F.id_sequent (x20::x22::nil))/\ 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
            x20 x21)/\ 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
            x22 x23).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_x'2a_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x21 x23, 
     t = (algebra.Alg.Term algebra.F.id_x'2a (x21::x23::nil)) ->
      exists x20,
        exists x22,
          t' = (algebra.Alg.Term algebra.F.id_x'2a (x20::x22::nil))/\ 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
            x20 x21)/\ 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
            x22 x23).
 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 x21 x23, 
     t = (algebra.Alg.Term algebra.F.id_cons (x21::x23::nil)) ->
      exists x20,
        exists x22,
          t' = (algebra.Alg.Term algebra.F.id_cons (x20::x22::nil))/\ 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
            x20 x21)/\ 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
            x22 x23).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_x'2b_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x21 x23, 
     t = (algebra.Alg.Term algebra.F.id_x'2b (x21::x23::nil)) ->
      exists x20,
        exists x22,
          t' = (algebra.Alg.Term algebra.F.id_x'2b (x20::x22::nil))/\ 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
            x20 x21)/\ 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
            x22 x23).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_p_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x21, 
     t = (algebra.Alg.Term algebra.F.id_p (x21::nil)) ->
      exists x20,
        t' = (algebra.Alg.Term algebra.F.id_p (x20::nil))/\ 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x20 x21).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_reduce'ii'out_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_reduce'ii'out nil) ->
     t' = (algebra.Alg.Term algebra.F.id_reduce'ii'out nil).
 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_if_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x21 x23, 
     t = (algebra.Alg.Term algebra.F.id_if (x21::x23::nil)) ->
      exists x20,
        exists x22,
          t' = (algebra.Alg.Term algebra.F.id_if (x20::x22::nil))/\ 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
            x20 x21)/\ 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
            x22 x23).
 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_tautology'i'out nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_tautology'i'out_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_iff (?x21::?x20::nil)) |- _ =>
     let x21 := fresh "x" in 
      (let x20 := fresh "x" in 
        (let Heq := fresh "Heq" in 
          (let Hred1 := fresh "Hred" in 
            (let Hred2 := fresh "Hred" in 
              (destruct (id_iff_is_R_xml_0_constructor H (refl_equal _)) as 
               [x21 [x20 [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_intersect'ii'out nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_intersect'ii'out_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_x'2d (?x20::nil)) |- _ =>
     let x20 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_x'2d_is_R_xml_0_constructor H (refl_equal _)) as 
           [x20 [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_sequent (?x21::?x20::nil)) |- 
    _ =>
     let x21 := fresh "x" in 
      (let x20 := fresh "x" in 
        (let Heq := fresh "Heq" in 
          (let Hred1 := fresh "Hred" in 
            (let Hred2 := fresh "Hred" in 
              (destruct (id_sequent_is_R_xml_0_constructor H (refl_equal _))
                as [x21 [x20 [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_x'2a (?x21::?x20::nil)) |- 
    _ =>
     let x21 := fresh "x" in 
      (let x20 := fresh "x" in 
        (let Heq := fresh "Heq" in 
          (let Hred1 := fresh "Hred" in 
            (let Hred2 := fresh "Hred" in 
              (destruct (id_x'2a_is_R_xml_0_constructor H (refl_equal _)) as 
               [x21 [x20 [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 (?x21::?x20::nil)) |- 
    _ =>
     let x21 := fresh "x" in 
      (let x20 := 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 
               [x21 [x20 [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_x'2b (?x21::?x20::nil)) |- 
    _ =>
     let x21 := fresh "x" in 
      (let x20 := fresh "x" in 
        (let Heq := fresh "Heq" in 
          (let Hred1 := fresh "Hred" in 
            (let Hred2 := fresh "Hred" in 
              (destruct (id_x'2b_is_R_xml_0_constructor H (refl_equal _)) as 
               [x21 [x20 [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_p (?x20::nil)) |- _ =>
     let x20 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_p_is_R_xml_0_constructor H (refl_equal _)) as 
           [x20 [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_reduce'ii'out nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_reduce'ii'out_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_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_if (?x21::?x20::nil)) |- _ =>
     let x21 := fresh "x" in 
      (let x20 := fresh "x" in 
        (let Heq := fresh "Heq" in 
          (let Hred1 := fresh "Hred" in 
            (let Hred2 := fresh "Hred" in 
              (destruct (id_if_is_R_xml_0_constructor H (refl_equal _)) as 
               [x21 [x20 [Heq [Hred2 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_tautology'i'out nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_tautology'i'out_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_iff (?x21::?x20::nil)) |- _ =>
     let x21 := fresh "x" in 
      (let x20 := fresh "x" in 
        (let Heq := fresh "Heq" in 
          (let Hred1 := fresh "Hred" in 
            (let Hred2 := fresh "Hred" in 
              (destruct (id_iff_is_R_xml_0_constructor H (refl_equal _)) as 
               [x21 [x20 [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_intersect'ii'out nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_intersect'ii'out_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_x'2d (?x20::nil)) |- _ =>
     let x20 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_x'2d_is_R_xml_0_constructor H (refl_equal _)) as 
           [x20 [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_sequent (?x21::?x20::nil)) |- 
    _ =>
     let x21 := fresh "x" in 
      (let x20 := fresh "x" in 
        (let Heq := fresh "Heq" in 
          (let Hred1 := fresh "Hred" in 
            (let Hred2 := fresh "Hred" in 
              (destruct (id_sequent_is_R_xml_0_constructor H (refl_equal _))
                as [x21 [x20 [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_x'2a (?x21::?x20::nil)) |- 
    _ =>
     let x21 := fresh "x" in 
      (let x20 := fresh "x" in 
        (let Heq := fresh "Heq" in 
          (let Hred1 := fresh "Hred" in 
            (let Hred2 := fresh "Hred" in 
              (destruct (id_x'2a_is_R_xml_0_constructor H (refl_equal _)) as 
               [x21 [x20 [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 (?x21::?x20::nil)) |- 
    _ =>
     let x21 := fresh "x" in 
      (let x20 := 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 
               [x21 [x20 [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_x'2b (?x21::?x20::nil)) |- 
    _ =>
     let x21 := fresh "x" in 
      (let x20 := fresh "x" in 
        (let Heq := fresh "Heq" in 
          (let Hred1 := fresh "Hred" in 
            (let Hred2 := fresh "Hred" in 
              (destruct (id_x'2b_is_R_xml_0_constructor H (refl_equal _)) as 
               [x21 [x20 [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_p (?x20::nil)) |- _ =>
     let x20 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_p_is_R_xml_0_constructor H (refl_equal _)) as 
           [x20 [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_reduce'ii'out nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_reduce'ii'out_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_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_if (?x21::?x20::nil)) |- _ =>
     let x21 := fresh "x" in 
      (let x20 := fresh "x" in 
        (let Heq := fresh "Heq" in 
          (let Hred1 := fresh "Hred" in 
            (let Hred2 := fresh "Hred" in 
              (destruct (id_if_is_R_xml_0_constructor H (refl_equal _)) as 
               [x21 [x20 [Heq [Hred2 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_intersect'ii'in : A ->A ->A.
   
   Hypothesis P_id_tautology'i'out : A.
   
   Hypothesis P_id_u'6'1 : A ->A ->A ->A ->A ->A.
   
   Hypothesis P_id_u'3'1 : A ->A.
   
   Hypothesis P_id_u'12'1 : A ->A ->A ->A ->A ->A.
   
   Hypothesis P_id_u'2'1 : A ->A.
   
   Hypothesis P_id_u'9'1 : A ->A.
   
   Hypothesis P_id_iff : A ->A ->A.
   
   Hypothesis P_id_u'14'1 : A ->A.
   
   Hypothesis P_id_intersect'ii'out : A.
   
   Hypothesis P_id_u'7'1 : A ->A.
   
   Hypothesis P_id_x'2d : A ->A.
   
   Hypothesis P_id_u'13'1 : A ->A.
   
   Hypothesis P_id_sequent : A ->A ->A.
   
   Hypothesis P_id_u'10'1 : A ->A.
   
   Hypothesis P_id_x'2a : A ->A ->A.
   
   Hypothesis P_id_tautology'i'in : A ->A.
   
   Hypothesis P_id_cons : A ->A ->A.
   
   Hypothesis P_id_u'6'2 : A ->A.
   
   Hypothesis P_id_x'2b : A ->A ->A.
   
   Hypothesis P_id_u'12'2 : A ->A.
   
   Hypothesis P_id_reduce'ii'in : A ->A ->A.
   
   Hypothesis P_id_p : A ->A.
   
   Hypothesis P_id_u'4'1 : A ->A.
   
   Hypothesis P_id_u'15'1 : A ->A.
   
   Hypothesis P_id_u'1'1 : A ->A.
   
   Hypothesis P_id_u'8'1 : A ->A.
   
   Hypothesis P_id_reduce'ii'out : A.
   
   Hypothesis P_id_nil : A.
   
   Hypothesis P_id_if : A ->A ->A.
   
   Hypothesis P_id_u'11'1 : A ->A.
   
   Hypothesis P_id_u'5'1 : A ->A.
   
   Hypothesis P_id_u'16'1 : A ->A.
   
   Hypothesis P_id_intersect'ii'in_monotonic :
    forall x20 x22 x21 x23, 
     (A0 <= x23)/\ (x23 <= x22) ->
      (A0 <= x21)/\ (x21 <= x20) ->
       P_id_intersect'ii'in x21 x23 <= P_id_intersect'ii'in x20 x22.
   
   Hypothesis P_id_u'6'1_monotonic :
    forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, 
     (A0 <= x29)/\ (x29 <= x28) ->
      (A0 <= x27)/\ (x27 <= x26) ->
       (A0 <= x25)/\ (x25 <= x24) ->
        (A0 <= x23)/\ (x23 <= x22) ->
         (A0 <= x21)/\ (x21 <= x20) ->
          P_id_u'6'1 x21 x23 x25 x27 x29 <= P_id_u'6'1 x20 x22 x24 x26 x28.
   
   Hypothesis P_id_u'3'1_monotonic :
    forall x20 x21, 
     (A0 <= x21)/\ (x21 <= x20) ->P_id_u'3'1 x21 <= P_id_u'3'1 x20.
   
   Hypothesis P_id_u'12'1_monotonic :
    forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, 
     (A0 <= x29)/\ (x29 <= x28) ->
      (A0 <= x27)/\ (x27 <= x26) ->
       (A0 <= x25)/\ (x25 <= x24) ->
        (A0 <= x23)/\ (x23 <= x22) ->
         (A0 <= x21)/\ (x21 <= x20) ->
          P_id_u'12'1 x21 x23 x25 x27 x29 <= P_id_u'12'1 x20 x22 x24 x26 x28.
   
   Hypothesis P_id_u'2'1_monotonic :
    forall x20 x21, 
     (A0 <= x21)/\ (x21 <= x20) ->P_id_u'2'1 x21 <= P_id_u'2'1 x20.
   
   Hypothesis P_id_u'9'1_monotonic :
    forall x20 x21, 
     (A0 <= x21)/\ (x21 <= x20) ->P_id_u'9'1 x21 <= P_id_u'9'1 x20.
   
   Hypothesis P_id_iff_monotonic :
    forall x20 x22 x21 x23, 
     (A0 <= x23)/\ (x23 <= x22) ->
      (A0 <= x21)/\ (x21 <= x20) ->P_id_iff x21 x23 <= P_id_iff x20 x22.
   
   Hypothesis P_id_u'14'1_monotonic :
    forall x20 x21, 
     (A0 <= x21)/\ (x21 <= x20) ->P_id_u'14'1 x21 <= P_id_u'14'1 x20.
   
   Hypothesis P_id_u'7'1_monotonic :
    forall x20 x21, 
     (A0 <= x21)/\ (x21 <= x20) ->P_id_u'7'1 x21 <= P_id_u'7'1 x20.
   
   Hypothesis P_id_x'2d_monotonic :
    forall x20 x21, 
     (A0 <= x21)/\ (x21 <= x20) ->P_id_x'2d x21 <= P_id_x'2d x20.
   
   Hypothesis P_id_u'13'1_monotonic :
    forall x20 x21, 
     (A0 <= x21)/\ (x21 <= x20) ->P_id_u'13'1 x21 <= P_id_u'13'1 x20.
   
   Hypothesis P_id_sequent_monotonic :
    forall x20 x22 x21 x23, 
     (A0 <= x23)/\ (x23 <= x22) ->
      (A0 <= x21)/\ (x21 <= x20) ->
       P_id_sequent x21 x23 <= P_id_sequent x20 x22.
   
   Hypothesis P_id_u'10'1_monotonic :
    forall x20 x21, 
     (A0 <= x21)/\ (x21 <= x20) ->P_id_u'10'1 x21 <= P_id_u'10'1 x20.
   
   Hypothesis P_id_x'2a_monotonic :
    forall x20 x22 x21 x23, 
     (A0 <= x23)/\ (x23 <= x22) ->
      (A0 <= x21)/\ (x21 <= x20) ->P_id_x'2a x21 x23 <= P_id_x'2a x20 x22.
   
   Hypothesis P_id_tautology'i'in_monotonic :
    forall x20 x21, 
     (A0 <= x21)/\ (x21 <= x20) ->
      P_id_tautology'i'in x21 <= P_id_tautology'i'in x20.
   
   Hypothesis P_id_cons_monotonic :
    forall x20 x22 x21 x23, 
     (A0 <= x23)/\ (x23 <= x22) ->
      (A0 <= x21)/\ (x21 <= x20) ->P_id_cons x21 x23 <= P_id_cons x20 x22.
   
   Hypothesis P_id_u'6'2_monotonic :
    forall x20 x21, 
     (A0 <= x21)/\ (x21 <= x20) ->P_id_u'6'2 x21 <= P_id_u'6'2 x20.
   
   Hypothesis P_id_x'2b_monotonic :
    forall x20 x22 x21 x23, 
     (A0 <= x23)/\ (x23 <= x22) ->
      (A0 <= x21)/\ (x21 <= x20) ->P_id_x'2b x21 x23 <= P_id_x'2b x20 x22.
   
   Hypothesis P_id_u'12'2_monotonic :
    forall x20 x21, 
     (A0 <= x21)/\ (x21 <= x20) ->P_id_u'12'2 x21 <= P_id_u'12'2 x20.
   
   Hypothesis P_id_reduce'ii'in_monotonic :
    forall x20 x22 x21 x23, 
     (A0 <= x23)/\ (x23 <= x22) ->
      (A0 <= x21)/\ (x21 <= x20) ->
       P_id_reduce'ii'in x21 x23 <= P_id_reduce'ii'in x20 x22.
   
   Hypothesis P_id_p_monotonic :
    forall x20 x21, (A0 <= x21)/\ (x21 <= x20) ->P_id_p x21 <= P_id_p x20.
   
   Hypothesis P_id_u'4'1_monotonic :
    forall x20 x21, 
     (A0 <= x21)/\ (x21 <= x20) ->P_id_u'4'1 x21 <= P_id_u'4'1 x20.
   
   Hypothesis P_id_u'15'1_monotonic :
    forall x20 x21, 
     (A0 <= x21)/\ (x21 <= x20) ->P_id_u'15'1 x21 <= P_id_u'15'1 x20.
   
   Hypothesis P_id_u'1'1_monotonic :
    forall x20 x21, 
     (A0 <= x21)/\ (x21 <= x20) ->P_id_u'1'1 x21 <= P_id_u'1'1 x20.
   
   Hypothesis P_id_u'8'1_monotonic :
    forall x20 x21, 
     (A0 <= x21)/\ (x21 <= x20) ->P_id_u'8'1 x21 <= P_id_u'8'1 x20.
   
   Hypothesis P_id_if_monotonic :
    forall x20 x22 x21 x23, 
     (A0 <= x23)/\ (x23 <= x22) ->
      (A0 <= x21)/\ (x21 <= x20) ->P_id_if x21 x23 <= P_id_if x20 x22.
   
   Hypothesis P_id_u'11'1_monotonic :
    forall x20 x21, 
     (A0 <= x21)/\ (x21 <= x20) ->P_id_u'11'1 x21 <= P_id_u'11'1 x20.
   
   Hypothesis P_id_u'5'1_monotonic :
    forall x20 x21, 
     (A0 <= x21)/\ (x21 <= x20) ->P_id_u'5'1 x21 <= P_id_u'5'1 x20.
   
   Hypothesis P_id_u'16'1_monotonic :
    forall x20 x21, 
     (A0 <= x21)/\ (x21 <= x20) ->P_id_u'16'1 x21 <= P_id_u'16'1 x20.
   
   Hypothesis P_id_intersect'ii'in_bounded :
    forall x20 x21, 
     (A0 <= x20) ->(A0 <= x21) ->A0 <= P_id_intersect'ii'in x21 x20.
   
   Hypothesis P_id_tautology'i'out_bounded : A0 <= P_id_tautology'i'out .
   
   Hypothesis P_id_u'6'1_bounded :
    forall x24 x20 x22 x21 x23, 
     (A0 <= x20) ->
      (A0 <= x21) ->
       (A0 <= x22) ->
        (A0 <= x23) ->(A0 <= x24) ->A0 <= P_id_u'6'1 x24 x23 x22 x21 x20.
   
   Hypothesis P_id_u'3'1_bounded :
    forall x20, (A0 <= x20) ->A0 <= P_id_u'3'1 x20.
   
   Hypothesis P_id_u'12'1_bounded :
    forall x24 x20 x22 x21 x23, 
     (A0 <= x20) ->
      (A0 <= x21) ->
       (A0 <= x22) ->
        (A0 <= x23) ->(A0 <= x24) ->A0 <= P_id_u'12'1 x24 x23 x22 x21 x20.
   
   Hypothesis P_id_u'2'1_bounded :
    forall x20, (A0 <= x20) ->A0 <= P_id_u'2'1 x20.
   
   Hypothesis P_id_u'9'1_bounded :
    forall x20, (A0 <= x20) ->A0 <= P_id_u'9'1 x20.
   
   Hypothesis P_id_iff_bounded :
    forall x20 x21, (A0 <= x20) ->(A0 <= x21) ->A0 <= P_id_iff x21 x20.
   
   Hypothesis P_id_u'14'1_bounded :
    forall x20, (A0 <= x20) ->A0 <= P_id_u'14'1 x20.
   
   Hypothesis P_id_intersect'ii'out_bounded : A0 <= P_id_intersect'ii'out .
   
   Hypothesis P_id_u'7'1_bounded :
    forall x20, (A0 <= x20) ->A0 <= P_id_u'7'1 x20.
   
   Hypothesis P_id_x'2d_bounded :
    forall x20, (A0 <= x20) ->A0 <= P_id_x'2d x20.
   
   Hypothesis P_id_u'13'1_bounded :
    forall x20, (A0 <= x20) ->A0 <= P_id_u'13'1 x20.
   
   Hypothesis P_id_sequent_bounded :
    forall x20 x21, (A0 <= x20) ->(A0 <= x21) ->A0 <= P_id_sequent x21 x20.
   
   Hypothesis P_id_u'10'1_bounded :
    forall x20, (A0 <= x20) ->A0 <= P_id_u'10'1 x20.
   
   Hypothesis P_id_x'2a_bounded :
    forall x20 x21, (A0 <= x20) ->(A0 <= x21) ->A0 <= P_id_x'2a x21 x20.
   
   Hypothesis P_id_tautology'i'in_bounded :
    forall x20, (A0 <= x20) ->A0 <= P_id_tautology'i'in x20.
   
   Hypothesis P_id_cons_bounded :
    forall x20 x21, (A0 <= x20) ->(A0 <= x21) ->A0 <= P_id_cons x21 x20.
   
   Hypothesis P_id_u'6'2_bounded :
    forall x20, (A0 <= x20) ->A0 <= P_id_u'6'2 x20.
   
   Hypothesis P_id_x'2b_bounded :
    forall x20 x21, (A0 <= x20) ->(A0 <= x21) ->A0 <= P_id_x'2b x21 x20.
   
   Hypothesis P_id_u'12'2_bounded :
    forall x20, (A0 <= x20) ->A0 <= P_id_u'12'2 x20.
   
   Hypothesis P_id_reduce'ii'in_bounded :
    forall x20 x21, 
     (A0 <= x20) ->(A0 <= x21) ->A0 <= P_id_reduce'ii'in x21 x20.
   
   Hypothesis P_id_p_bounded : forall x20, (A0 <= x20) ->A0 <= P_id_p x20.
   
   Hypothesis P_id_u'4'1_bounded :
    forall x20, (A0 <= x20) ->A0 <= P_id_u'4'1 x20.
   
   Hypothesis P_id_u'15'1_bounded :
    forall x20, (A0 <= x20) ->A0 <= P_id_u'15'1 x20.
   
   Hypothesis P_id_u'1'1_bounded :
    forall x20, (A0 <= x20) ->A0 <= P_id_u'1'1 x20.
   
   Hypothesis P_id_u'8'1_bounded :
    forall x20, (A0 <= x20) ->A0 <= P_id_u'8'1 x20.
   
   Hypothesis P_id_reduce'ii'out_bounded : A0 <= P_id_reduce'ii'out .
   
   Hypothesis P_id_nil_bounded : A0 <= P_id_nil .
   
   Hypothesis P_id_if_bounded :
    forall x20 x21, (A0 <= x20) ->(A0 <= x21) ->A0 <= P_id_if x21 x20.
   
   Hypothesis P_id_u'11'1_bounded :
    forall x20, (A0 <= x20) ->A0 <= P_id_u'11'1 x20.
   
   Hypothesis P_id_u'5'1_bounded :
    forall x20, (A0 <= x20) ->A0 <= P_id_u'5'1 x20.
   
   Hypothesis P_id_u'16'1_bounded :
    forall x20, (A0 <= x20) ->A0 <= P_id_u'16'1 x20.
   
   Fixpoint measure t { struct t }  := 
     match t with
       | (algebra.Alg.Term algebra.F.id_intersect'ii'in (x21::x20::nil)) =>
        P_id_intersect'ii'in (measure x21) (measure x20)
       | (algebra.Alg.Term algebra.F.id_tautology'i'out nil) =>
        P_id_tautology'i'out 
       | (algebra.Alg.Term algebra.F.id_u'6'1 (x24::x23::x22::x21::x20::nil)) =>
        P_id_u'6'1 (measure x24) (measure x23) (measure x22) (measure x21) 
         (measure x20)
       | (algebra.Alg.Term algebra.F.id_u'3'1 (x20::nil)) =>
        P_id_u'3'1 (measure x20)
       | (algebra.Alg.Term algebra.F.id_u'12'1 (x24::x23::x22::x21::
          x20::nil)) =>
        P_id_u'12'1 (measure x24) (measure x23) (measure x22) (measure x21) 
         (measure x20)
       | (algebra.Alg.Term algebra.F.id_u'2'1 (x20::nil)) =>
        P_id_u'2'1 (measure x20)
       | (algebra.Alg.Term algebra.F.id_u'9'1 (x20::nil)) =>
        P_id_u'9'1 (measure x20)
       | (algebra.Alg.Term algebra.F.id_iff (x21::x20::nil)) =>
        P_id_iff (measure x21) (measure x20)
       | (algebra.Alg.Term algebra.F.id_u'14'1 (x20::nil)) =>
        P_id_u'14'1 (measure x20)
       | (algebra.Alg.Term algebra.F.id_intersect'ii'out nil) =>
        P_id_intersect'ii'out 
       | (algebra.Alg.Term algebra.F.id_u'7'1 (x20::nil)) =>
        P_id_u'7'1 (measure x20)
       | (algebra.Alg.Term algebra.F.id_x'2d (x20::nil)) =>
        P_id_x'2d (measure x20)
       | (algebra.Alg.Term algebra.F.id_u'13'1 (x20::nil)) =>
        P_id_u'13'1 (measure x20)
       | (algebra.Alg.Term algebra.F.id_sequent (x21::x20::nil)) =>
        P_id_sequent (measure x21) (measure x20)
       | (algebra.Alg.Term algebra.F.id_u'10'1 (x20::nil)) =>
        P_id_u'10'1 (measure x20)
       | (algebra.Alg.Term algebra.F.id_x'2a (x21::x20::nil)) =>
        P_id_x'2a (measure x21) (measure x20)
       | (algebra.Alg.Term algebra.F.id_tautology'i'in (x20::nil)) =>
        P_id_tautology'i'in (measure x20)
       | (algebra.Alg.Term algebra.F.id_cons (x21::x20::nil)) =>
        P_id_cons (measure x21) (measure x20)
       | (algebra.Alg.Term algebra.F.id_u'6'2 (x20::nil)) =>
        P_id_u'6'2 (measure x20)
       | (algebra.Alg.Term algebra.F.id_x'2b (x21::x20::nil)) =>
        P_id_x'2b (measure x21) (measure x20)
       | (algebra.Alg.Term algebra.F.id_u'12'2 (x20::nil)) =>
        P_id_u'12'2 (measure x20)
       | (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) =>
        P_id_reduce'ii'in (measure x21) (measure x20)
       | (algebra.Alg.Term algebra.F.id_p (x20::nil)) => P_id_p (measure x20)
       | (algebra.Alg.Term algebra.F.id_u'4'1 (x20::nil)) =>
        P_id_u'4'1 (measure x20)
       | (algebra.Alg.Term algebra.F.id_u'15'1 (x20::nil)) =>
        P_id_u'15'1 (measure x20)
       | (algebra.Alg.Term algebra.F.id_u'1'1 (x20::nil)) =>
        P_id_u'1'1 (measure x20)
       | (algebra.Alg.Term algebra.F.id_u'8'1 (x20::nil)) =>
        P_id_u'8'1 (measure x20)
       | (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) =>
        P_id_reduce'ii'out 
       | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil 
       | (algebra.Alg.Term algebra.F.id_if (x21::x20::nil)) =>
        P_id_if (measure x21) (measure x20)
       | (algebra.Alg.Term algebra.F.id_u'11'1 (x20::nil)) =>
        P_id_u'11'1 (measure x20)
       | (algebra.Alg.Term algebra.F.id_u'5'1 (x20::nil)) =>
        P_id_u'5'1 (measure x20)
       | (algebra.Alg.Term algebra.F.id_u'16'1 (x20::nil)) =>
        P_id_u'16'1 (measure x20)
       | _ => A0
       end.
   
   Lemma measure_equation :
    forall t, 
     measure t = match t with
                   | (algebra.Alg.Term algebra.F.id_intersect'ii'in (x21::
                      x20::nil)) =>
                    P_id_intersect'ii'in (measure x21) (measure x20)
                   | (algebra.Alg.Term algebra.F.id_tautology'i'out nil) =>
                    P_id_tautology'i'out 
                   | (algebra.Alg.Term algebra.F.id_u'6'1 (x24::x23::x22::
                      x21::x20::nil)) =>
                    P_id_u'6'1 (measure x24) (measure x23) (measure x22) 
                     (measure x21) (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'3'1 (x20::nil)) =>
                    P_id_u'3'1 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'12'1 (x24::x23::x22::
                      x21::x20::nil)) =>
                    P_id_u'12'1 (measure x24) (measure x23) (measure x22) 
                     (measure x21) (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'2'1 (x20::nil)) =>
                    P_id_u'2'1 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'9'1 (x20::nil)) =>
                    P_id_u'9'1 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_iff (x21::x20::nil)) =>
                    P_id_iff (measure x21) (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'14'1 (x20::nil)) =>
                    P_id_u'14'1 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_intersect'ii'out nil) =>
                    P_id_intersect'ii'out 
                   | (algebra.Alg.Term algebra.F.id_u'7'1 (x20::nil)) =>
                    P_id_u'7'1 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_x'2d (x20::nil)) =>
                    P_id_x'2d (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'13'1 (x20::nil)) =>
                    P_id_u'13'1 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_sequent (x21::x20::nil)) =>
                    P_id_sequent (measure x21) (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'10'1 (x20::nil)) =>
                    P_id_u'10'1 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_x'2a (x21::x20::nil)) =>
                    P_id_x'2a (measure x21) (measure x20)
                   | (algebra.Alg.Term algebra.F.id_tautology'i'in 
                      (x20::nil)) =>
                    P_id_tautology'i'in (measure x20)
                   | (algebra.Alg.Term algebra.F.id_cons (x21::x20::nil)) =>
                    P_id_cons (measure x21) (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'6'2 (x20::nil)) =>
                    P_id_u'6'2 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_x'2b (x21::x20::nil)) =>
                    P_id_x'2b (measure x21) (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'12'2 (x20::nil)) =>
                    P_id_u'12'2 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::
                      x20::nil)) =>
                    P_id_reduce'ii'in (measure x21) (measure x20)
                   | (algebra.Alg.Term algebra.F.id_p (x20::nil)) =>
                    P_id_p (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'4'1 (x20::nil)) =>
                    P_id_u'4'1 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'15'1 (x20::nil)) =>
                    P_id_u'15'1 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'1'1 (x20::nil)) =>
                    P_id_u'1'1 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'8'1 (x20::nil)) =>
                    P_id_u'8'1 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) =>
                    P_id_reduce'ii'out 
                   | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil 
                   | (algebra.Alg.Term algebra.F.id_if (x21::x20::nil)) =>
                    P_id_if (measure x21) (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'11'1 (x20::nil)) =>
                    P_id_u'11'1 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'5'1 (x20::nil)) =>
                    P_id_u'5'1 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'16'1 (x20::nil)) =>
                    P_id_u'16'1 (measure x20)
                   | _ => 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_intersect'ii'in => P_id_intersect'ii'in
       | algebra.F.id_tautology'i'out => P_id_tautology'i'out
       | algebra.F.id_u'6'1 => P_id_u'6'1
       | algebra.F.id_u'3'1 => P_id_u'3'1
       | algebra.F.id_u'12'1 => P_id_u'12'1
       | algebra.F.id_u'2'1 => P_id_u'2'1
       | algebra.F.id_u'9'1 => P_id_u'9'1
       | algebra.F.id_iff => P_id_iff
       | algebra.F.id_u'14'1 => P_id_u'14'1
       | algebra.F.id_intersect'ii'out => P_id_intersect'ii'out
       | algebra.F.id_u'7'1 => P_id_u'7'1
       | algebra.F.id_x'2d => P_id_x'2d
       | algebra.F.id_u'13'1 => P_id_u'13'1
       | algebra.F.id_sequent => P_id_sequent
       | algebra.F.id_u'10'1 => P_id_u'10'1
       | algebra.F.id_x'2a => P_id_x'2a
       | algebra.F.id_tautology'i'in => P_id_tautology'i'in
       | algebra.F.id_cons => P_id_cons
       | algebra.F.id_u'6'2 => P_id_u'6'2
       | algebra.F.id_x'2b => P_id_x'2b
       | algebra.F.id_u'12'2 => P_id_u'12'2
       | algebra.F.id_reduce'ii'in => P_id_reduce'ii'in
       | algebra.F.id_p => P_id_p
       | algebra.F.id_u'4'1 => P_id_u'4'1
       | algebra.F.id_u'15'1 => P_id_u'15'1
       | algebra.F.id_u'1'1 => P_id_u'1'1
       | algebra.F.id_u'8'1 => P_id_u'8'1
       | algebra.F.id_reduce'ii'out => P_id_reduce'ii'out
       | algebra.F.id_nil => P_id_nil
       | algebra.F.id_if => P_id_if
       | algebra.F.id_u'11'1 => P_id_u'11'1
       | algebra.F.id_u'5'1 => P_id_u'5'1
       | algebra.F.id_u'16'1 => P_id_u'16'1
       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_intersect'ii'in =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_tautology'i'out =>
               match l with
                 | nil => _
                 | _::_ => _
                 end
              | algebra.F.id_u'6'1 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::nil => _
                 | _::_::_::_::_::nil => _
                 | _::_::_::_::_::_::_ => _
                 end
              | algebra.F.id_u'3'1 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_u'12'1 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::nil => _
                 | _::_::_::_::_::nil => _
                 | _::_::_::_::_::_::_ => _
                 end
              | algebra.F.id_u'2'1 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_u'9'1 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_iff =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_u'14'1 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_intersect'ii'out =>
               match l with
                 | nil => _
                 | _::_ => _
                 end
              | algebra.F.id_u'7'1 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_x'2d =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_u'13'1 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_sequent =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_u'10'1 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_x'2a =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_tautology'i'in =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_cons =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_u'6'2 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_x'2b =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_u'12'2 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_reduce'ii'in =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_p =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_u'4'1 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_u'15'1 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_u'1'1 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_u'8'1 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_reduce'ii'out =>
               match l with
                 | nil => _
                 | _::_ => _
                 end
              | algebra.F.id_nil => match l with
                                      | nil => _
                                      | _::_ => _
                                      end
              | algebra.F.id_if =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_u'11'1 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_u'5'1 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_u'16'1 =>
               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_intersect'ii'in_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_tautology'i'out_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'6'1_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'3'1_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'12'1_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'2'1_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'9'1_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_iff_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'14'1_bounded;assumption.
     
     vm_compute in |-*;intros ;apply P_id_intersect'ii'out_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'7'1_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_x'2d_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'13'1_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_sequent_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'10'1_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_x'2a_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_tautology'i'in_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_cons_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'6'2_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_x'2b_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'12'2_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_reduce'ii'in_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_p_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'4'1_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'15'1_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'1'1_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'8'1_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_reduce'ii'out_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_nil_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_if_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'11'1_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'5'1_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'16'1_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 |-*;intros ;apply P_id_intersect'ii'in_monotonic;
      assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id_u'6'1_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_u'3'1_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_u'12'1_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_u'2'1_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_u'9'1_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_iff_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_u'14'1_monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id_u'7'1_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_x'2d_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_u'13'1_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_sequent_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_u'10'1_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_x'2a_monotonic;assumption.
     
     vm_compute in |-*;intros ;apply P_id_tautology'i'in_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_cons_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_u'6'2_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_x'2b_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_u'12'2_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_reduce'ii'in_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_p_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_u'4'1_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_u'15'1_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_u'1'1_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_u'8'1_monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id_if_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_u'11'1_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_u'5'1_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_u'16'1_monotonic;assumption.
     intros f.
     case f.
     vm_compute in |-*;intros ;apply P_id_intersect'ii'in_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_tautology'i'out_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'6'1_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'3'1_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'12'1_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'2'1_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'9'1_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_iff_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'14'1_bounded;assumption.
     
     vm_compute in |-*;intros ;apply P_id_intersect'ii'out_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'7'1_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_x'2d_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'13'1_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_sequent_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'10'1_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_x'2a_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_tautology'i'in_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_cons_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'6'2_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_x'2b_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'12'2_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_reduce'ii'in_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_p_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'4'1_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'15'1_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'1'1_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'8'1_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_reduce'ii'out_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_nil_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_if_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'11'1_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'5'1_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'16'1_bounded;assumption.
     intros .
     do 2 (rewrite  <- same_measure in |-*).
     apply rules_monotonic;assumption.
     assumption.
   Qed.
   
   Hypothesis P_id_U'12'2 : A ->A.
   
   Hypothesis P_id_U'6'1 : A ->A ->A ->A ->A ->A.
   
   Hypothesis P_id_REDUCE'II'IN : A ->A ->A.
   
   Hypothesis P_id_TAUTOLOGY'I'IN : A ->A.
   
   Hypothesis P_id_U'9'1 : A ->A.
   
   Hypothesis P_id_U'1'1 : A ->A.
   
   Hypothesis P_id_U'14'1 : A ->A.
   
   Hypothesis P_id_U'7'1 : A ->A.
   
   Hypothesis P_id_U'4'1 : A ->A.
   
   Hypothesis P_id_U'11'1 : A ->A.
   
   Hypothesis P_id_INTERSECT'II'IN : A ->A ->A.
   
   Hypothesis P_id_U'13'1 : A ->A.
   
   Hypothesis P_id_U'6'2 : A ->A.
   
   Hypothesis P_id_U'3'1 : A ->A.
   
   Hypothesis P_id_U'16'1 : A ->A.
   
   Hypothesis P_id_U'10'1 : A ->A.
   
   Hypothesis P_id_U'2'1 : A ->A.
   
   Hypothesis P_id_U'15'1 : A ->A.
   
   Hypothesis P_id_U'8'1 : A ->A.
   
   Hypothesis P_id_U'5'1 : A ->A.
   
   Hypothesis P_id_U'12'1 : A ->A ->A ->A ->A ->A.
   
   Hypothesis P_id_U'12'2_monotonic :
    forall x20 x21, 
     (A0 <= x21)/\ (x21 <= x20) ->P_id_U'12'2 x21 <= P_id_U'12'2 x20.
   
   Hypothesis P_id_U'6'1_monotonic :
    forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, 
     (A0 <= x29)/\ (x29 <= x28) ->
      (A0 <= x27)/\ (x27 <= x26) ->
       (A0 <= x25)/\ (x25 <= x24) ->
        (A0 <= x23)/\ (x23 <= x22) ->
         (A0 <= x21)/\ (x21 <= x20) ->
          P_id_U'6'1 x21 x23 x25 x27 x29 <= P_id_U'6'1 x20 x22 x24 x26 x28.
   
   Hypothesis P_id_REDUCE'II'IN_monotonic :
    forall x20 x22 x21 x23, 
     (A0 <= x23)/\ (x23 <= x22) ->
      (A0 <= x21)/\ (x21 <= x20) ->
       P_id_REDUCE'II'IN x21 x23 <= P_id_REDUCE'II'IN x20 x22.
   
   Hypothesis P_id_TAUTOLOGY'I'IN_monotonic :
    forall x20 x21, 
     (A0 <= x21)/\ (x21 <= x20) ->
      P_id_TAUTOLOGY'I'IN x21 <= P_id_TAUTOLOGY'I'IN x20.
   
   Hypothesis P_id_U'9'1_monotonic :
    forall x20 x21, 
     (A0 <= x21)/\ (x21 <= x20) ->P_id_U'9'1 x21 <= P_id_U'9'1 x20.
   
   Hypothesis P_id_U'1'1_monotonic :
    forall x20 x21, 
     (A0 <= x21)/\ (x21 <= x20) ->P_id_U'1'1 x21 <= P_id_U'1'1 x20.
   
   Hypothesis P_id_U'14'1_monotonic :
    forall x20 x21, 
     (A0 <= x21)/\ (x21 <= x20) ->P_id_U'14'1 x21 <= P_id_U'14'1 x20.
   
   Hypothesis P_id_U'7'1_monotonic :
    forall x20 x21, 
     (A0 <= x21)/\ (x21 <= x20) ->P_id_U'7'1 x21 <= P_id_U'7'1 x20.
   
   Hypothesis P_id_U'4'1_monotonic :
    forall x20 x21, 
     (A0 <= x21)/\ (x21 <= x20) ->P_id_U'4'1 x21 <= P_id_U'4'1 x20.
   
   Hypothesis P_id_U'11'1_monotonic :
    forall x20 x21, 
     (A0 <= x21)/\ (x21 <= x20) ->P_id_U'11'1 x21 <= P_id_U'11'1 x20.
   
   Hypothesis P_id_INTERSECT'II'IN_monotonic :
    forall x20 x22 x21 x23, 
     (A0 <= x23)/\ (x23 <= x22) ->
      (A0 <= x21)/\ (x21 <= x20) ->
       P_id_INTERSECT'II'IN x21 x23 <= P_id_INTERSECT'II'IN x20 x22.
   
   Hypothesis P_id_U'13'1_monotonic :
    forall x20 x21, 
     (A0 <= x21)/\ (x21 <= x20) ->P_id_U'13'1 x21 <= P_id_U'13'1 x20.
   
   Hypothesis P_id_U'6'2_monotonic :
    forall x20 x21, 
     (A0 <= x21)/\ (x21 <= x20) ->P_id_U'6'2 x21 <= P_id_U'6'2 x20.
   
   Hypothesis P_id_U'3'1_monotonic :
    forall x20 x21, 
     (A0 <= x21)/\ (x21 <= x20) ->P_id_U'3'1 x21 <= P_id_U'3'1 x20.
   
   Hypothesis P_id_U'16'1_monotonic :
    forall x20 x21, 
     (A0 <= x21)/\ (x21 <= x20) ->P_id_U'16'1 x21 <= P_id_U'16'1 x20.
   
   Hypothesis P_id_U'10'1_monotonic :
    forall x20 x21, 
     (A0 <= x21)/\ (x21 <= x20) ->P_id_U'10'1 x21 <= P_id_U'10'1 x20.
   
   Hypothesis P_id_U'2'1_monotonic :
    forall x20 x21, 
     (A0 <= x21)/\ (x21 <= x20) ->P_id_U'2'1 x21 <= P_id_U'2'1 x20.
   
   Hypothesis P_id_U'15'1_monotonic :
    forall x20 x21, 
     (A0 <= x21)/\ (x21 <= x20) ->P_id_U'15'1 x21 <= P_id_U'15'1 x20.
   
   Hypothesis P_id_U'8'1_monotonic :
    forall x20 x21, 
     (A0 <= x21)/\ (x21 <= x20) ->P_id_U'8'1 x21 <= P_id_U'8'1 x20.
   
   Hypothesis P_id_U'5'1_monotonic :
    forall x20 x21, 
     (A0 <= x21)/\ (x21 <= x20) ->P_id_U'5'1 x21 <= P_id_U'5'1 x20.
   
   Hypothesis P_id_U'12'1_monotonic :
    forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, 
     (A0 <= x29)/\ (x29 <= x28) ->
      (A0 <= x27)/\ (x27 <= x26) ->
       (A0 <= x25)/\ (x25 <= x24) ->
        (A0 <= x23)/\ (x23 <= x22) ->
         (A0 <= x21)/\ (x21 <= x20) ->
          P_id_U'12'1 x21 x23 x25 x27 x29 <= P_id_U'12'1 x20 x22 x24 x26 x28.
   
   Definition marked_measure t := 
     match t with
       | (algebra.Alg.Term algebra.F.id_u'12'2 (x20::nil)) =>
        P_id_U'12'2 (measure x20)
       | (algebra.Alg.Term algebra.F.id_u'6'1 (x24::x23::x22::x21::x20::nil)) =>
        P_id_U'6'1 (measure x24) (measure x23) (measure x22) (measure x21) 
         (measure x20)
       | (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) =>
        P_id_REDUCE'II'IN (measure x21) (measure x20)
       | (algebra.Alg.Term algebra.F.id_tautology'i'in (x20::nil)) =>
        P_id_TAUTOLOGY'I'IN (measure x20)
       | (algebra.Alg.Term algebra.F.id_u'9'1 (x20::nil)) =>
        P_id_U'9'1 (measure x20)
       | (algebra.Alg.Term algebra.F.id_u'1'1 (x20::nil)) =>
        P_id_U'1'1 (measure x20)
       | (algebra.Alg.Term algebra.F.id_u'14'1 (x20::nil)) =>
        P_id_U'14'1 (measure x20)
       | (algebra.Alg.Term algebra.F.id_u'7'1 (x20::nil)) =>
        P_id_U'7'1 (measure x20)
       | (algebra.Alg.Term algebra.F.id_u'4'1 (x20::nil)) =>
        P_id_U'4'1 (measure x20)
       | (algebra.Alg.Term algebra.F.id_u'11'1 (x20::nil)) =>
        P_id_U'11'1 (measure x20)
       | (algebra.Alg.Term algebra.F.id_intersect'ii'in (x21::x20::nil)) =>
        P_id_INTERSECT'II'IN (measure x21) (measure x20)
       | (algebra.Alg.Term algebra.F.id_u'13'1 (x20::nil)) =>
        P_id_U'13'1 (measure x20)
       | (algebra.Alg.Term algebra.F.id_u'6'2 (x20::nil)) =>
        P_id_U'6'2 (measure x20)
       | (algebra.Alg.Term algebra.F.id_u'3'1 (x20::nil)) =>
        P_id_U'3'1 (measure x20)
       | (algebra.Alg.Term algebra.F.id_u'16'1 (x20::nil)) =>
        P_id_U'16'1 (measure x20)
       | (algebra.Alg.Term algebra.F.id_u'10'1 (x20::nil)) =>
        P_id_U'10'1 (measure x20)
       | (algebra.Alg.Term algebra.F.id_u'2'1 (x20::nil)) =>
        P_id_U'2'1 (measure x20)
       | (algebra.Alg.Term algebra.F.id_u'15'1 (x20::nil)) =>
        P_id_U'15'1 (measure x20)
       | (algebra.Alg.Term algebra.F.id_u'8'1 (x20::nil)) =>
        P_id_U'8'1 (measure x20)
       | (algebra.Alg.Term algebra.F.id_u'5'1 (x20::nil)) =>
        P_id_U'5'1 (measure x20)
       | (algebra.Alg.Term algebra.F.id_u'12'1 (x24::x23::x22::x21::
          x20::nil)) =>
        P_id_U'12'1 (measure x24) (measure x23) (measure x22) (measure x21) 
         (measure x20)
       | _ => 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 x20;apply (P_id_U'16'1 x20).
     
     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 x20;apply (P_id_U'5'1 x20).
     
     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 x20;apply (P_id_U'11'1 x20).
     
     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 x20;apply (P_id_U'8'1 x20).
     
     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 x20;apply (P_id_U'1'1 x20).
     
     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 x20;apply (P_id_U'15'1 x20).
     
     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 x20;apply (P_id_U'4'1 x20).
     
     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 x21 x20;apply (P_id_REDUCE'II'IN x21 x20).
     
     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 x20;apply (P_id_U'12'2 x20).
     
     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 x20;apply (P_id_U'6'2 x20).
     
     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 x20;apply (P_id_TAUTOLOGY'I'IN x20).
     
     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 x20;apply (P_id_U'10'1 x20).
     
     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 x20;apply (P_id_U'13'1 x20).
     
     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 x20;apply (P_id_U'7'1 x20).
     
     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 x20;apply (P_id_U'14'1 x20).
     
     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 x20;apply (P_id_U'9'1 x20).
     
     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 x20;apply (P_id_U'2'1 x20).
     
     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 x24 x23 x22 x21 x20;
      apply (P_id_U'12'1 x24 x23 x22 x21 x20).
     
     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 x20;apply (P_id_U'3'1 x20).
     
     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 x24 x23 x22 x21 x20;
      apply (P_id_U'6'1 x24 x23 x22 x21 x20).
     
     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 x21 x20;apply (P_id_INTERSECT'II'IN x21 x20).
     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_intersect'ii'in =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_tautology'i'out =>
               match l with
                 | nil => _
                 | _::_ => _
                 end
              | algebra.F.id_u'6'1 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::nil => _
                 | _::_::_::_::_::nil => _
                 | _::_::_::_::_::_::_ => _
                 end
              | algebra.F.id_u'3'1 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_u'12'1 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::nil => _
                 | _::_::_::_::_::nil => _
                 | _::_::_::_::_::_::_ => _
                 end
              | algebra.F.id_u'2'1 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_u'9'1 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_iff =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_u'14'1 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_intersect'ii'out =>
               match l with
                 | nil => _
                 | _::_ => _
                 end
              | algebra.F.id_u'7'1 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_x'2d =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_u'13'1 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_sequent =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_u'10'1 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_x'2a =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_tautology'i'in =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_cons =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_u'6'2 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_x'2b =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_u'12'2 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_reduce'ii'in =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_p =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_u'4'1 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_u'15'1 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_u'1'1 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_u'8'1 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_reduce'ii'out =>
               match l with
                 | nil => _
                 | _::_ => _
                 end
              | algebra.F.id_nil => match l with
                                      | nil => _
                                      | _::_ => _
                                      end
              | algebra.F.id_if =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_u'11'1 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_u'5'1 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_u'16'1 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              end.
     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 .
     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 .
     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 .
     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 .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     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 .
     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 .
   Qed.
   
   Lemma marked_measure_equation :
    forall t, 
     marked_measure t = match t with
                          | (algebra.Alg.Term algebra.F.id_u'12'2 (x20::nil)) =>
                           P_id_U'12'2 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'6'1 (x24::x23::
                             x22::x21::x20::nil)) =>
                           P_id_U'6'1 (measure x24) (measure x23) 
                            (measure x22) (measure x21) (measure x20)
                          | (algebra.Alg.Term algebra.F.id_reduce'ii'in 
                             (x21::x20::nil)) =>
                           P_id_REDUCE'II'IN (measure x21) (measure x20)
                          | (algebra.Alg.Term algebra.F.id_tautology'i'in 
                             (x20::nil)) =>
                           P_id_TAUTOLOGY'I'IN (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'9'1 (x20::nil)) =>
                           P_id_U'9'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'1'1 (x20::nil)) =>
                           P_id_U'1'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'14'1 (x20::nil)) =>
                           P_id_U'14'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'7'1 (x20::nil)) =>
                           P_id_U'7'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'4'1 (x20::nil)) =>
                           P_id_U'4'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'11'1 (x20::nil)) =>
                           P_id_U'11'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_intersect'ii'in 
                             (x21::x20::nil)) =>
                           P_id_INTERSECT'II'IN (measure x21) (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'13'1 (x20::nil)) =>
                           P_id_U'13'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'6'2 (x20::nil)) =>
                           P_id_U'6'2 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'3'1 (x20::nil)) =>
                           P_id_U'3'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'16'1 (x20::nil)) =>
                           P_id_U'16'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'10'1 (x20::nil)) =>
                           P_id_U'10'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'2'1 (x20::nil)) =>
                           P_id_U'2'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'15'1 (x20::nil)) =>
                           P_id_U'15'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'8'1 (x20::nil)) =>
                           P_id_U'8'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'5'1 (x20::nil)) =>
                           P_id_U'5'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'12'1 (x24::x23::
                             x22::x21::x20::nil)) =>
                           P_id_U'12'1 (measure x24) (measure x23) 
                            (measure x22) (measure x21) (measure x20)
                          | _ => 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 |-*;intros ;apply P_id_intersect'ii'in_monotonic;
      assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id_u'6'1_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_u'3'1_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_u'12'1_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_u'2'1_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_u'9'1_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_iff_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_u'14'1_monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id_u'7'1_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_x'2d_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_u'13'1_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_sequent_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_u'10'1_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_x'2a_monotonic;assumption.
     
     vm_compute in |-*;intros ;apply P_id_tautology'i'in_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_cons_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_u'6'2_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_x'2b_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_u'12'2_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_reduce'ii'in_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_p_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_u'4'1_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_u'15'1_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_u'1'1_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_u'8'1_monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id_if_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_u'11'1_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_u'5'1_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_u'16'1_monotonic;assumption.
     clear f.
     intros f.
     case f.
     vm_compute in |-*;intros ;apply P_id_intersect'ii'in_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_tautology'i'out_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'6'1_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'3'1_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'12'1_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'2'1_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'9'1_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_iff_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'14'1_bounded;assumption.
     
     vm_compute in |-*;intros ;apply P_id_intersect'ii'out_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'7'1_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_x'2d_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'13'1_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_sequent_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'10'1_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_x'2a_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_tautology'i'in_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_cons_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'6'2_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_x'2b_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'12'2_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_reduce'ii'in_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_p_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'4'1_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'15'1_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'1'1_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'8'1_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_reduce'ii'out_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_nil_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_if_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'11'1_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'5'1_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u'16'1_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_U'16'1_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_U'5'1_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_U'11'1_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_U'8'1_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_U'1'1_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_U'15'1_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_U'4'1_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_REDUCE'II'IN_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_U'12'2_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_U'6'2_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_TAUTOLOGY'I'IN_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_U'10'1_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_U'13'1_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_U'7'1_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_U'14'1_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_U'9'1_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_U'2'1_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_U'12'1_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_U'3'1_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_U'6'1_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_INTERSECT'II'IN_monotonic;
      assumption.
     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_intersect'ii'in : Z ->Z ->Z.
   
   Hypothesis P_id_tautology'i'out : Z.
   
   Hypothesis P_id_u'6'1 : Z ->Z ->Z ->Z ->Z ->Z.
   
   Hypothesis P_id_u'3'1 : Z ->Z.
   
   Hypothesis P_id_u'12'1 : Z ->Z ->Z ->Z ->Z ->Z.
   
   Hypothesis P_id_u'2'1 : Z ->Z.
   
   Hypothesis P_id_u'9'1 : Z ->Z.
   
   Hypothesis P_id_iff : Z ->Z ->Z.
   
   Hypothesis P_id_u'14'1 : Z ->Z.
   
   Hypothesis P_id_intersect'ii'out : Z.
   
   Hypothesis P_id_u'7'1 : Z ->Z.
   
   Hypothesis P_id_x'2d : Z ->Z.
   
   Hypothesis P_id_u'13'1 : Z ->Z.
   
   Hypothesis P_id_sequent : Z ->Z ->Z.
   
   Hypothesis P_id_u'10'1 : Z ->Z.
   
   Hypothesis P_id_x'2a : Z ->Z ->Z.
   
   Hypothesis P_id_tautology'i'in : Z ->Z.
   
   Hypothesis P_id_cons : Z ->Z ->Z.
   
   Hypothesis P_id_u'6'2 : Z ->Z.
   
   Hypothesis P_id_x'2b : Z ->Z ->Z.
   
   Hypothesis P_id_u'12'2 : Z ->Z.
   
   Hypothesis P_id_reduce'ii'in : Z ->Z ->Z.
   
   Hypothesis P_id_p : Z ->Z.
   
   Hypothesis P_id_u'4'1 : Z ->Z.
   
   Hypothesis P_id_u'15'1 : Z ->Z.
   
   Hypothesis P_id_u'1'1 : Z ->Z.
   
   Hypothesis P_id_u'8'1 : Z ->Z.
   
   Hypothesis P_id_reduce'ii'out : Z.
   
   Hypothesis P_id_nil : Z.
   
   Hypothesis P_id_if : Z ->Z ->Z.
   
   Hypothesis P_id_u'11'1 : Z ->Z.
   
   Hypothesis P_id_u'5'1 : Z ->Z.
   
   Hypothesis P_id_u'16'1 : Z ->Z.
   
   Hypothesis P_id_intersect'ii'in_monotonic :
    forall x20 x22 x21 x23, 
     (min_value <= x23)/\ (x23 <= x22) ->
      (min_value <= x21)/\ (x21 <= x20) ->
       P_id_intersect'ii'in x21 x23 <= P_id_intersect'ii'in x20 x22.
   
   Hypothesis P_id_u'6'1_monotonic :
    forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, 
     (min_value <= x29)/\ (x29 <= x28) ->
      (min_value <= x27)/\ (x27 <= x26) ->
       (min_value <= x25)/\ (x25 <= x24) ->
        (min_value <= x23)/\ (x23 <= x22) ->
         (min_value <= x21)/\ (x21 <= x20) ->
          P_id_u'6'1 x21 x23 x25 x27 x29 <= P_id_u'6'1 x20 x22 x24 x26 x28.
   
   Hypothesis P_id_u'3'1_monotonic :
    forall x20 x21, 
     (min_value <= x21)/\ (x21 <= x20) ->P_id_u'3'1 x21 <= P_id_u'3'1 x20.
   
   Hypothesis P_id_u'12'1_monotonic :
    forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, 
     (min_value <= x29)/\ (x29 <= x28) ->
      (min_value <= x27)/\ (x27 <= x26) ->
       (min_value <= x25)/\ (x25 <= x24) ->
        (min_value <= x23)/\ (x23 <= x22) ->
         (min_value <= x21)/\ (x21 <= x20) ->
          P_id_u'12'1 x21 x23 x25 x27 x29 <= P_id_u'12'1 x20 x22 x24 x26 x28.
   
   Hypothesis P_id_u'2'1_monotonic :
    forall x20 x21, 
     (min_value <= x21)/\ (x21 <= x20) ->P_id_u'2'1 x21 <= P_id_u'2'1 x20.
   
   Hypothesis P_id_u'9'1_monotonic :
    forall x20 x21, 
     (min_value <= x21)/\ (x21 <= x20) ->P_id_u'9'1 x21 <= P_id_u'9'1 x20.
   
   Hypothesis P_id_iff_monotonic :
    forall x20 x22 x21 x23, 
     (min_value <= x23)/\ (x23 <= x22) ->
      (min_value <= x21)/\ (x21 <= x20) ->
       P_id_iff x21 x23 <= P_id_iff x20 x22.
   
   Hypothesis P_id_u'14'1_monotonic :
    forall x20 x21, 
     (min_value <= x21)/\ (x21 <= x20) ->P_id_u'14'1 x21 <= P_id_u'14'1 x20.
   
   Hypothesis P_id_u'7'1_monotonic :
    forall x20 x21, 
     (min_value <= x21)/\ (x21 <= x20) ->P_id_u'7'1 x21 <= P_id_u'7'1 x20.
   
   Hypothesis P_id_x'2d_monotonic :
    forall x20 x21, 
     (min_value <= x21)/\ (x21 <= x20) ->P_id_x'2d x21 <= P_id_x'2d x20.
   
   Hypothesis P_id_u'13'1_monotonic :
    forall x20 x21, 
     (min_value <= x21)/\ (x21 <= x20) ->P_id_u'13'1 x21 <= P_id_u'13'1 x20.
   
   Hypothesis P_id_sequent_monotonic :
    forall x20 x22 x21 x23, 
     (min_value <= x23)/\ (x23 <= x22) ->
      (min_value <= x21)/\ (x21 <= x20) ->
       P_id_sequent x21 x23 <= P_id_sequent x20 x22.
   
   Hypothesis P_id_u'10'1_monotonic :
    forall x20 x21, 
     (min_value <= x21)/\ (x21 <= x20) ->P_id_u'10'1 x21 <= P_id_u'10'1 x20.
   
   Hypothesis P_id_x'2a_monotonic :
    forall x20 x22 x21 x23, 
     (min_value <= x23)/\ (x23 <= x22) ->
      (min_value <= x21)/\ (x21 <= x20) ->
       P_id_x'2a x21 x23 <= P_id_x'2a x20 x22.
   
   Hypothesis P_id_tautology'i'in_monotonic :
    forall x20 x21, 
     (min_value <= x21)/\ (x21 <= x20) ->
      P_id_tautology'i'in x21 <= P_id_tautology'i'in x20.
   
   Hypothesis P_id_cons_monotonic :
    forall x20 x22 x21 x23, 
     (min_value <= x23)/\ (x23 <= x22) ->
      (min_value <= x21)/\ (x21 <= x20) ->
       P_id_cons x21 x23 <= P_id_cons x20 x22.
   
   Hypothesis P_id_u'6'2_monotonic :
    forall x20 x21, 
     (min_value <= x21)/\ (x21 <= x20) ->P_id_u'6'2 x21 <= P_id_u'6'2 x20.
   
   Hypothesis P_id_x'2b_monotonic :
    forall x20 x22 x21 x23, 
     (min_value <= x23)/\ (x23 <= x22) ->
      (min_value <= x21)/\ (x21 <= x20) ->
       P_id_x'2b x21 x23 <= P_id_x'2b x20 x22.
   
   Hypothesis P_id_u'12'2_monotonic :
    forall x20 x21, 
     (min_value <= x21)/\ (x21 <= x20) ->P_id_u'12'2 x21 <= P_id_u'12'2 x20.
   
   Hypothesis P_id_reduce'ii'in_monotonic :
    forall x20 x22 x21 x23, 
     (min_value <= x23)/\ (x23 <= x22) ->
      (min_value <= x21)/\ (x21 <= x20) ->
       P_id_reduce'ii'in x21 x23 <= P_id_reduce'ii'in x20 x22.
   
   Hypothesis P_id_p_monotonic :
    forall x20 x21, 
     (min_value <= x21)/\ (x21 <= x20) ->P_id_p x21 <= P_id_p x20.
   
   Hypothesis P_id_u'4'1_monotonic :
    forall x20 x21, 
     (min_value <= x21)/\ (x21 <= x20) ->P_id_u'4'1 x21 <= P_id_u'4'1 x20.
   
   Hypothesis P_id_u'15'1_monotonic :
    forall x20 x21, 
     (min_value <= x21)/\ (x21 <= x20) ->P_id_u'15'1 x21 <= P_id_u'15'1 x20.
   
   Hypothesis P_id_u'1'1_monotonic :
    forall x20 x21, 
     (min_value <= x21)/\ (x21 <= x20) ->P_id_u'1'1 x21 <= P_id_u'1'1 x20.
   
   Hypothesis P_id_u'8'1_monotonic :
    forall x20 x21, 
     (min_value <= x21)/\ (x21 <= x20) ->P_id_u'8'1 x21 <= P_id_u'8'1 x20.
   
   Hypothesis P_id_if_monotonic :
    forall x20 x22 x21 x23, 
     (min_value <= x23)/\ (x23 <= x22) ->
      (min_value <= x21)/\ (x21 <= x20) ->P_id_if x21 x23 <= P_id_if x20 x22.
   
   Hypothesis P_id_u'11'1_monotonic :
    forall x20 x21, 
     (min_value <= x21)/\ (x21 <= x20) ->P_id_u'11'1 x21 <= P_id_u'11'1 x20.
   
   Hypothesis P_id_u'5'1_monotonic :
    forall x20 x21, 
     (min_value <= x21)/\ (x21 <= x20) ->P_id_u'5'1 x21 <= P_id_u'5'1 x20.
   
   Hypothesis P_id_u'16'1_monotonic :
    forall x20 x21, 
     (min_value <= x21)/\ (x21 <= x20) ->P_id_u'16'1 x21 <= P_id_u'16'1 x20.
   
   Hypothesis P_id_intersect'ii'in_bounded :
    forall x20 x21, 
     (min_value <= x20) ->
      (min_value <= x21) ->min_value <= P_id_intersect'ii'in x21 x20.
   
   Hypothesis P_id_tautology'i'out_bounded :
    min_value <= P_id_tautology'i'out .
   
   Hypothesis P_id_u'6'1_bounded :
    forall x24 x20 x22 x21 x23, 
     (min_value <= x20) ->
      (min_value <= x21) ->
       (min_value <= x22) ->
        (min_value <= x23) ->
         (min_value <= x24) ->min_value <= P_id_u'6'1 x24 x23 x22 x21 x20.
   
   Hypothesis P_id_u'3'1_bounded :
    forall x20, (min_value <= x20) ->min_value <= P_id_u'3'1 x20.
   
   Hypothesis P_id_u'12'1_bounded :
    forall x24 x20 x22 x21 x23, 
     (min_value <= x20) ->
      (min_value <= x21) ->
       (min_value <= x22) ->
        (min_value <= x23) ->
         (min_value <= x24) ->min_value <= P_id_u'12'1 x24 x23 x22 x21 x20.
   
   Hypothesis P_id_u'2'1_bounded :
    forall x20, (min_value <= x20) ->min_value <= P_id_u'2'1 x20.
   
   Hypothesis P_id_u'9'1_bounded :
    forall x20, (min_value <= x20) ->min_value <= P_id_u'9'1 x20.
   
   Hypothesis P_id_iff_bounded :
    forall x20 x21, 
     (min_value <= x20) ->(min_value <= x21) ->min_value <= P_id_iff x21 x20.
   
   Hypothesis P_id_u'14'1_bounded :
    forall x20, (min_value <= x20) ->min_value <= P_id_u'14'1 x20.
   
   Hypothesis P_id_intersect'ii'out_bounded :
    min_value <= P_id_intersect'ii'out .
   
   Hypothesis P_id_u'7'1_bounded :
    forall x20, (min_value <= x20) ->min_value <= P_id_u'7'1 x20.
   
   Hypothesis P_id_x'2d_bounded :
    forall x20, (min_value <= x20) ->min_value <= P_id_x'2d x20.
   
   Hypothesis P_id_u'13'1_bounded :
    forall x20, (min_value <= x20) ->min_value <= P_id_u'13'1 x20.
   
   Hypothesis P_id_sequent_bounded :
    forall x20 x21, 
     (min_value <= x20) ->
      (min_value <= x21) ->min_value <= P_id_sequent x21 x20.
   
   Hypothesis P_id_u'10'1_bounded :
    forall x20, (min_value <= x20) ->min_value <= P_id_u'10'1 x20.
   
   Hypothesis P_id_x'2a_bounded :
    forall x20 x21, 
     (min_value <= x20) ->(min_value <= x21) ->min_value <= P_id_x'2a x21 x20.
   
   Hypothesis P_id_tautology'i'in_bounded :
    forall x20, (min_value <= x20) ->min_value <= P_id_tautology'i'in x20.
   
   Hypothesis P_id_cons_bounded :
    forall x20 x21, 
     (min_value <= x20) ->(min_value <= x21) ->min_value <= P_id_cons x21 x20.
   
   Hypothesis P_id_u'6'2_bounded :
    forall x20, (min_value <= x20) ->min_value <= P_id_u'6'2 x20.
   
   Hypothesis P_id_x'2b_bounded :
    forall x20 x21, 
     (min_value <= x20) ->(min_value <= x21) ->min_value <= P_id_x'2b x21 x20.
   
   Hypothesis P_id_u'12'2_bounded :
    forall x20, (min_value <= x20) ->min_value <= P_id_u'12'2 x20.
   
   Hypothesis P_id_reduce'ii'in_bounded :
    forall x20 x21, 
     (min_value <= x20) ->
      (min_value <= x21) ->min_value <= P_id_reduce'ii'in x21 x20.
   
   Hypothesis P_id_p_bounded :
    forall x20, (min_value <= x20) ->min_value <= P_id_p x20.
   
   Hypothesis P_id_u'4'1_bounded :
    forall x20, (min_value <= x20) ->min_value <= P_id_u'4'1 x20.
   
   Hypothesis P_id_u'15'1_bounded :
    forall x20, (min_value <= x20) ->min_value <= P_id_u'15'1 x20.
   
   Hypothesis P_id_u'1'1_bounded :
    forall x20, (min_value <= x20) ->min_value <= P_id_u'1'1 x20.
   
   Hypothesis P_id_u'8'1_bounded :
    forall x20, (min_value <= x20) ->min_value <= P_id_u'8'1 x20.
   
   Hypothesis P_id_reduce'ii'out_bounded : min_value <= P_id_reduce'ii'out .
   
   Hypothesis P_id_nil_bounded : min_value <= P_id_nil .
   
   Hypothesis P_id_if_bounded :
    forall x20 x21, 
     (min_value <= x20) ->(min_value <= x21) ->min_value <= P_id_if x21 x20.
   
   Hypothesis P_id_u'11'1_bounded :
    forall x20, (min_value <= x20) ->min_value <= P_id_u'11'1 x20.
   
   Hypothesis P_id_u'5'1_bounded :
    forall x20, (min_value <= x20) ->min_value <= P_id_u'5'1 x20.
   
   Hypothesis P_id_u'16'1_bounded :
    forall x20, (min_value <= x20) ->min_value <= P_id_u'16'1 x20.
   
   Definition measure  := 
     Interp.measure min_value P_id_intersect'ii'in P_id_tautology'i'out 
      P_id_u'6'1 P_id_u'3'1 P_id_u'12'1 P_id_u'2'1 P_id_u'9'1 P_id_iff 
      P_id_u'14'1 P_id_intersect'ii'out P_id_u'7'1 P_id_x'2d P_id_u'13'1 
      P_id_sequent P_id_u'10'1 P_id_x'2a P_id_tautology'i'in P_id_cons 
      P_id_u'6'2 P_id_x'2b P_id_u'12'2 P_id_reduce'ii'in P_id_p P_id_u'4'1 
      P_id_u'15'1 P_id_u'1'1 P_id_u'8'1 P_id_reduce'ii'out P_id_nil P_id_if 
      P_id_u'11'1 P_id_u'5'1 P_id_u'16'1.
   
   Lemma measure_equation :
    forall t, 
     measure t = match t with
                   | (algebra.Alg.Term algebra.F.id_intersect'ii'in (x21::
                      x20::nil)) =>
                    P_id_intersect'ii'in (measure x21) (measure x20)
                   | (algebra.Alg.Term algebra.F.id_tautology'i'out nil) =>
                    P_id_tautology'i'out 
                   | (algebra.Alg.Term algebra.F.id_u'6'1 (x24::x23::x22::
                      x21::x20::nil)) =>
                    P_id_u'6'1 (measure x24) (measure x23) (measure x22) 
                     (measure x21) (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'3'1 (x20::nil)) =>
                    P_id_u'3'1 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'12'1 (x24::x23::x22::
                      x21::x20::nil)) =>
                    P_id_u'12'1 (measure x24) (measure x23) (measure x22) 
                     (measure x21) (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'2'1 (x20::nil)) =>
                    P_id_u'2'1 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'9'1 (x20::nil)) =>
                    P_id_u'9'1 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_iff (x21::x20::nil)) =>
                    P_id_iff (measure x21) (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'14'1 (x20::nil)) =>
                    P_id_u'14'1 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_intersect'ii'out nil) =>
                    P_id_intersect'ii'out 
                   | (algebra.Alg.Term algebra.F.id_u'7'1 (x20::nil)) =>
                    P_id_u'7'1 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_x'2d (x20::nil)) =>
                    P_id_x'2d (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'13'1 (x20::nil)) =>
                    P_id_u'13'1 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_sequent (x21::x20::nil)) =>
                    P_id_sequent (measure x21) (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'10'1 (x20::nil)) =>
                    P_id_u'10'1 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_x'2a (x21::x20::nil)) =>
                    P_id_x'2a (measure x21) (measure x20)
                   | (algebra.Alg.Term algebra.F.id_tautology'i'in 
                      (x20::nil)) =>
                    P_id_tautology'i'in (measure x20)
                   | (algebra.Alg.Term algebra.F.id_cons (x21::x20::nil)) =>
                    P_id_cons (measure x21) (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'6'2 (x20::nil)) =>
                    P_id_u'6'2 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_x'2b (x21::x20::nil)) =>
                    P_id_x'2b (measure x21) (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'12'2 (x20::nil)) =>
                    P_id_u'12'2 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::
                      x20::nil)) =>
                    P_id_reduce'ii'in (measure x21) (measure x20)
                   | (algebra.Alg.Term algebra.F.id_p (x20::nil)) =>
                    P_id_p (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'4'1 (x20::nil)) =>
                    P_id_u'4'1 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'15'1 (x20::nil)) =>
                    P_id_u'15'1 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'1'1 (x20::nil)) =>
                    P_id_u'1'1 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'8'1 (x20::nil)) =>
                    P_id_u'8'1 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) =>
                    P_id_reduce'ii'out 
                   | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil 
                   | (algebra.Alg.Term algebra.F.id_if (x21::x20::nil)) =>
                    P_id_if (measure x21) (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'11'1 (x20::nil)) =>
                    P_id_u'11'1 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'5'1 (x20::nil)) =>
                    P_id_u'5'1 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'16'1 (x20::nil)) =>
                    P_id_u'16'1 (measure x20)
                   | _ => 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_intersect'ii'in_monotonic;assumption.
     intros ;apply P_id_u'6'1_monotonic;assumption.
     intros ;apply P_id_u'3'1_monotonic;assumption.
     intros ;apply P_id_u'12'1_monotonic;assumption.
     intros ;apply P_id_u'2'1_monotonic;assumption.
     intros ;apply P_id_u'9'1_monotonic;assumption.
     intros ;apply P_id_iff_monotonic;assumption.
     intros ;apply P_id_u'14'1_monotonic;assumption.
     intros ;apply P_id_u'7'1_monotonic;assumption.
     intros ;apply P_id_x'2d_monotonic;assumption.
     intros ;apply P_id_u'13'1_monotonic;assumption.
     intros ;apply P_id_sequent_monotonic;assumption.
     intros ;apply P_id_u'10'1_monotonic;assumption.
     intros ;apply P_id_x'2a_monotonic;assumption.
     intros ;apply P_id_tautology'i'in_monotonic;assumption.
     intros ;apply P_id_cons_monotonic;assumption.
     intros ;apply P_id_u'6'2_monotonic;assumption.
     intros ;apply P_id_x'2b_monotonic;assumption.
     intros ;apply P_id_u'12'2_monotonic;assumption.
     intros ;apply P_id_reduce'ii'in_monotonic;assumption.
     intros ;apply P_id_p_monotonic;assumption.
     intros ;apply P_id_u'4'1_monotonic;assumption.
     intros ;apply P_id_u'15'1_monotonic;assumption.
     intros ;apply P_id_u'1'1_monotonic;assumption.
     intros ;apply P_id_u'8'1_monotonic;assumption.
     intros ;apply P_id_if_monotonic;assumption.
     intros ;apply P_id_u'11'1_monotonic;assumption.
     intros ;apply P_id_u'5'1_monotonic;assumption.
     intros ;apply P_id_u'16'1_monotonic;assumption.
     intros ;apply P_id_intersect'ii'in_bounded;assumption.
     intros ;apply P_id_tautology'i'out_bounded;assumption.
     intros ;apply P_id_u'6'1_bounded;assumption.
     intros ;apply P_id_u'3'1_bounded;assumption.
     intros ;apply P_id_u'12'1_bounded;assumption.
     intros ;apply P_id_u'2'1_bounded;assumption.
     intros ;apply P_id_u'9'1_bounded;assumption.
     intros ;apply P_id_iff_bounded;assumption.
     intros ;apply P_id_u'14'1_bounded;assumption.
     intros ;apply P_id_intersect'ii'out_bounded;assumption.
     intros ;apply P_id_u'7'1_bounded;assumption.
     intros ;apply P_id_x'2d_bounded;assumption.
     intros ;apply P_id_u'13'1_bounded;assumption.
     intros ;apply P_id_sequent_bounded;assumption.
     intros ;apply P_id_u'10'1_bounded;assumption.
     intros ;apply P_id_x'2a_bounded;assumption.
     intros ;apply P_id_tautology'i'in_bounded;assumption.
     intros ;apply P_id_cons_bounded;assumption.
     intros ;apply P_id_u'6'2_bounded;assumption.
     intros ;apply P_id_x'2b_bounded;assumption.
     intros ;apply P_id_u'12'2_bounded;assumption.
     intros ;apply P_id_reduce'ii'in_bounded;assumption.
     intros ;apply P_id_p_bounded;assumption.
     intros ;apply P_id_u'4'1_bounded;assumption.
     intros ;apply P_id_u'15'1_bounded;assumption.
     intros ;apply P_id_u'1'1_bounded;assumption.
     intros ;apply P_id_u'8'1_bounded;assumption.
     intros ;apply P_id_reduce'ii'out_bounded;assumption.
     intros ;apply P_id_nil_bounded;assumption.
     intros ;apply P_id_if_bounded;assumption.
     intros ;apply P_id_u'11'1_bounded;assumption.
     intros ;apply P_id_u'5'1_bounded;assumption.
     intros ;apply P_id_u'16'1_bounded;assumption.
     apply rules_monotonic.
   Qed.
   
   Hypothesis P_id_U'12'2 : Z ->Z.
   
   Hypothesis P_id_U'6'1 : Z ->Z ->Z ->Z ->Z ->Z.
   
   Hypothesis P_id_REDUCE'II'IN : Z ->Z ->Z.
   
   Hypothesis P_id_TAUTOLOGY'I'IN : Z ->Z.
   
   Hypothesis P_id_U'9'1 : Z ->Z.
   
   Hypothesis P_id_U'1'1 : Z ->Z.
   
   Hypothesis P_id_U'14'1 : Z ->Z.
   
   Hypothesis P_id_U'7'1 : Z ->Z.
   
   Hypothesis P_id_U'4'1 : Z ->Z.
   
   Hypothesis P_id_U'11'1 : Z ->Z.
   
   Hypothesis P_id_INTERSECT'II'IN : Z ->Z ->Z.
   
   Hypothesis P_id_U'13'1 : Z ->Z.
   
   Hypothesis P_id_U'6'2 : Z ->Z.
   
   Hypothesis P_id_U'3'1 : Z ->Z.
   
   Hypothesis P_id_U'16'1 : Z ->Z.
   
   Hypothesis P_id_U'10'1 : Z ->Z.
   
   Hypothesis P_id_U'2'1 : Z ->Z.
   
   Hypothesis P_id_U'15'1 : Z ->Z.
   
   Hypothesis P_id_U'8'1 : Z ->Z.
   
   Hypothesis P_id_U'5'1 : Z ->Z.
   
   Hypothesis P_id_U'12'1 : Z ->Z ->Z ->Z ->Z ->Z.
   
   Hypothesis P_id_U'12'2_monotonic :
    forall x20 x21, 
     (min_value <= x21)/\ (x21 <= x20) ->P_id_U'12'2 x21 <= P_id_U'12'2 x20.
   
   Hypothesis P_id_U'6'1_monotonic :
    forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, 
     (min_value <= x29)/\ (x29 <= x28) ->
      (min_value <= x27)/\ (x27 <= x26) ->
       (min_value <= x25)/\ (x25 <= x24) ->
        (min_value <= x23)/\ (x23 <= x22) ->
         (min_value <= x21)/\ (x21 <= x20) ->
          P_id_U'6'1 x21 x23 x25 x27 x29 <= P_id_U'6'1 x20 x22 x24 x26 x28.
   
   Hypothesis P_id_REDUCE'II'IN_monotonic :
    forall x20 x22 x21 x23, 
     (min_value <= x23)/\ (x23 <= x22) ->
      (min_value <= x21)/\ (x21 <= x20) ->
       P_id_REDUCE'II'IN x21 x23 <= P_id_REDUCE'II'IN x20 x22.
   
   Hypothesis P_id_TAUTOLOGY'I'IN_monotonic :
    forall x20 x21, 
     (min_value <= x21)/\ (x21 <= x20) ->
      P_id_TAUTOLOGY'I'IN x21 <= P_id_TAUTOLOGY'I'IN x20.
   
   Hypothesis P_id_U'9'1_monotonic :
    forall x20 x21, 
     (min_value <= x21)/\ (x21 <= x20) ->P_id_U'9'1 x21 <= P_id_U'9'1 x20.
   
   Hypothesis P_id_U'1'1_monotonic :
    forall x20 x21, 
     (min_value <= x21)/\ (x21 <= x20) ->P_id_U'1'1 x21 <= P_id_U'1'1 x20.
   
   Hypothesis P_id_U'14'1_monotonic :
    forall x20 x21, 
     (min_value <= x21)/\ (x21 <= x20) ->P_id_U'14'1 x21 <= P_id_U'14'1 x20.
   
   Hypothesis P_id_U'7'1_monotonic :
    forall x20 x21, 
     (min_value <= x21)/\ (x21 <= x20) ->P_id_U'7'1 x21 <= P_id_U'7'1 x20.
   
   Hypothesis P_id_U'4'1_monotonic :
    forall x20 x21, 
     (min_value <= x21)/\ (x21 <= x20) ->P_id_U'4'1 x21 <= P_id_U'4'1 x20.
   
   Hypothesis P_id_U'11'1_monotonic :
    forall x20 x21, 
     (min_value <= x21)/\ (x21 <= x20) ->P_id_U'11'1 x21 <= P_id_U'11'1 x20.
   
   Hypothesis P_id_INTERSECT'II'IN_monotonic :
    forall x20 x22 x21 x23, 
     (min_value <= x23)/\ (x23 <= x22) ->
      (min_value <= x21)/\ (x21 <= x20) ->
       P_id_INTERSECT'II'IN x21 x23 <= P_id_INTERSECT'II'IN x20 x22.
   
   Hypothesis P_id_U'13'1_monotonic :
    forall x20 x21, 
     (min_value <= x21)/\ (x21 <= x20) ->P_id_U'13'1 x21 <= P_id_U'13'1 x20.
   
   Hypothesis P_id_U'6'2_monotonic :
    forall x20 x21, 
     (min_value <= x21)/\ (x21 <= x20) ->P_id_U'6'2 x21 <= P_id_U'6'2 x20.
   
   Hypothesis P_id_U'3'1_monotonic :
    forall x20 x21, 
     (min_value <= x21)/\ (x21 <= x20) ->P_id_U'3'1 x21 <= P_id_U'3'1 x20.
   
   Hypothesis P_id_U'16'1_monotonic :
    forall x20 x21, 
     (min_value <= x21)/\ (x21 <= x20) ->P_id_U'16'1 x21 <= P_id_U'16'1 x20.
   
   Hypothesis P_id_U'10'1_monotonic :
    forall x20 x21, 
     (min_value <= x21)/\ (x21 <= x20) ->P_id_U'10'1 x21 <= P_id_U'10'1 x20.
   
   Hypothesis P_id_U'2'1_monotonic :
    forall x20 x21, 
     (min_value <= x21)/\ (x21 <= x20) ->P_id_U'2'1 x21 <= P_id_U'2'1 x20.
   
   Hypothesis P_id_U'15'1_monotonic :
    forall x20 x21, 
     (min_value <= x21)/\ (x21 <= x20) ->P_id_U'15'1 x21 <= P_id_U'15'1 x20.
   
   Hypothesis P_id_U'8'1_monotonic :
    forall x20 x21, 
     (min_value <= x21)/\ (x21 <= x20) ->P_id_U'8'1 x21 <= P_id_U'8'1 x20.
   
   Hypothesis P_id_U'5'1_monotonic :
    forall x20 x21, 
     (min_value <= x21)/\ (x21 <= x20) ->P_id_U'5'1 x21 <= P_id_U'5'1 x20.
   
   Hypothesis P_id_U'12'1_monotonic :
    forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, 
     (min_value <= x29)/\ (x29 <= x28) ->
      (min_value <= x27)/\ (x27 <= x26) ->
       (min_value <= x25)/\ (x25 <= x24) ->
        (min_value <= x23)/\ (x23 <= x22) ->
         (min_value <= x21)/\ (x21 <= x20) ->
          P_id_U'12'1 x21 x23 x25 x27 x29 <= P_id_U'12'1 x20 x22 x24 x26 x28.
   
   Definition marked_measure  := 
     Interp.marked_measure min_value P_id_intersect'ii'in 
      P_id_tautology'i'out P_id_u'6'1 P_id_u'3'1 P_id_u'12'1 P_id_u'2'1 
      P_id_u'9'1 P_id_iff P_id_u'14'1 P_id_intersect'ii'out P_id_u'7'1 
      P_id_x'2d P_id_u'13'1 P_id_sequent P_id_u'10'1 P_id_x'2a 
      P_id_tautology'i'in P_id_cons P_id_u'6'2 P_id_x'2b P_id_u'12'2 
      P_id_reduce'ii'in P_id_p P_id_u'4'1 P_id_u'15'1 P_id_u'1'1 P_id_u'8'1 
      P_id_reduce'ii'out P_id_nil P_id_if P_id_u'11'1 P_id_u'5'1 P_id_u'16'1 
      P_id_U'12'2 P_id_U'6'1 P_id_REDUCE'II'IN P_id_TAUTOLOGY'I'IN 
      P_id_U'9'1 P_id_U'1'1 P_id_U'14'1 P_id_U'7'1 P_id_U'4'1 P_id_U'11'1 
      P_id_INTERSECT'II'IN P_id_U'13'1 P_id_U'6'2 P_id_U'3'1 P_id_U'16'1 
      P_id_U'10'1 P_id_U'2'1 P_id_U'15'1 P_id_U'8'1 P_id_U'5'1 P_id_U'12'1.
   
   Lemma marked_measure_equation :
    forall t, 
     marked_measure t = match t with
                          | (algebra.Alg.Term algebra.F.id_u'12'2 (x20::nil)) =>
                           P_id_U'12'2 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'6'1 (x24::x23::
                             x22::x21::x20::nil)) =>
                           P_id_U'6'1 (measure x24) (measure x23) 
                            (measure x22) (measure x21) (measure x20)
                          | (algebra.Alg.Term algebra.F.id_reduce'ii'in 
                             (x21::x20::nil)) =>
                           P_id_REDUCE'II'IN (measure x21) (measure x20)
                          | (algebra.Alg.Term algebra.F.id_tautology'i'in 
                             (x20::nil)) =>
                           P_id_TAUTOLOGY'I'IN (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'9'1 (x20::nil)) =>
                           P_id_U'9'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'1'1 (x20::nil)) =>
                           P_id_U'1'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'14'1 (x20::nil)) =>
                           P_id_U'14'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'7'1 (x20::nil)) =>
                           P_id_U'7'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'4'1 (x20::nil)) =>
                           P_id_U'4'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'11'1 (x20::nil)) =>
                           P_id_U'11'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_intersect'ii'in 
                             (x21::x20::nil)) =>
                           P_id_INTERSECT'II'IN (measure x21) (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'13'1 (x20::nil)) =>
                           P_id_U'13'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'6'2 (x20::nil)) =>
                           P_id_U'6'2 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'3'1 (x20::nil)) =>
                           P_id_U'3'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'16'1 (x20::nil)) =>
                           P_id_U'16'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'10'1 (x20::nil)) =>
                           P_id_U'10'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'2'1 (x20::nil)) =>
                           P_id_U'2'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'15'1 (x20::nil)) =>
                           P_id_U'15'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'8'1 (x20::nil)) =>
                           P_id_U'8'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'5'1 (x20::nil)) =>
                           P_id_U'5'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'12'1 (x24::x23::
                             x22::x21::x20::nil)) =>
                           P_id_U'12'1 (measure x24) (measure x23) 
                            (measure x22) (measure x21) (measure x20)
                          | _ => 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_intersect'ii'in_monotonic;assumption.
     intros ;apply P_id_u'6'1_monotonic;assumption.
     intros ;apply P_id_u'3'1_monotonic;assumption.
     intros ;apply P_id_u'12'1_monotonic;assumption.
     intros ;apply P_id_u'2'1_monotonic;assumption.
     intros ;apply P_id_u'9'1_monotonic;assumption.
     intros ;apply P_id_iff_monotonic;assumption.
     intros ;apply P_id_u'14'1_monotonic;assumption.
     intros ;apply P_id_u'7'1_monotonic;assumption.
     intros ;apply P_id_x'2d_monotonic;assumption.
     intros ;apply P_id_u'13'1_monotonic;assumption.
     intros ;apply P_id_sequent_monotonic;assumption.
     intros ;apply P_id_u'10'1_monotonic;assumption.
     intros ;apply P_id_x'2a_monotonic;assumption.
     intros ;apply P_id_tautology'i'in_monotonic;assumption.
     intros ;apply P_id_cons_monotonic;assumption.
     intros ;apply P_id_u'6'2_monotonic;assumption.
     intros ;apply P_id_x'2b_monotonic;assumption.
     intros ;apply P_id_u'12'2_monotonic;assumption.
     intros ;apply P_id_reduce'ii'in_monotonic;assumption.
     intros ;apply P_id_p_monotonic;assumption.
     intros ;apply P_id_u'4'1_monotonic;assumption.
     intros ;apply P_id_u'15'1_monotonic;assumption.
     intros ;apply P_id_u'1'1_monotonic;assumption.
     intros ;apply P_id_u'8'1_monotonic;assumption.
     intros ;apply P_id_if_monotonic;assumption.
     intros ;apply P_id_u'11'1_monotonic;assumption.
     intros ;apply P_id_u'5'1_monotonic;assumption.
     intros ;apply P_id_u'16'1_monotonic;assumption.
     intros ;apply P_id_intersect'ii'in_bounded;assumption.
     intros ;apply P_id_tautology'i'out_bounded;assumption.
     intros ;apply P_id_u'6'1_bounded;assumption.
     intros ;apply P_id_u'3'1_bounded;assumption.
     intros ;apply P_id_u'12'1_bounded;assumption.
     intros ;apply P_id_u'2'1_bounded;assumption.
     intros ;apply P_id_u'9'1_bounded;assumption.
     intros ;apply P_id_iff_bounded;assumption.
     intros ;apply P_id_u'14'1_bounded;assumption.
     intros ;apply P_id_intersect'ii'out_bounded;assumption.
     intros ;apply P_id_u'7'1_bounded;assumption.
     intros ;apply P_id_x'2d_bounded;assumption.
     intros ;apply P_id_u'13'1_bounded;assumption.
     intros ;apply P_id_sequent_bounded;assumption.
     intros ;apply P_id_u'10'1_bounded;assumption.
     intros ;apply P_id_x'2a_bounded;assumption.
     intros ;apply P_id_tautology'i'in_bounded;assumption.
     intros ;apply P_id_cons_bounded;assumption.
     intros ;apply P_id_u'6'2_bounded;assumption.
     intros ;apply P_id_x'2b_bounded;assumption.
     intros ;apply P_id_u'12'2_bounded;assumption.
     intros ;apply P_id_reduce'ii'in_bounded;assumption.
     intros ;apply P_id_p_bounded;assumption.
     intros ;apply P_id_u'4'1_bounded;assumption.
     intros ;apply P_id_u'15'1_bounded;assumption.
     intros ;apply P_id_u'1'1_bounded;assumption.
     intros ;apply P_id_u'8'1_bounded;assumption.
     intros ;apply P_id_reduce'ii'out_bounded;assumption.
     intros ;apply P_id_nil_bounded;assumption.
     intros ;apply P_id_if_bounded;assumption.
     intros ;apply P_id_u'11'1_bounded;assumption.
     intros ;apply P_id_u'5'1_bounded;assumption.
     intros ;apply P_id_u'16'1_bounded;assumption.
     apply rules_monotonic.
     intros ;apply P_id_U'12'2_monotonic;assumption.
     intros ;apply P_id_U'6'1_monotonic;assumption.
     intros ;apply P_id_REDUCE'II'IN_monotonic;assumption.
     intros ;apply P_id_TAUTOLOGY'I'IN_monotonic;assumption.
     intros ;apply P_id_U'9'1_monotonic;assumption.
     intros ;apply P_id_U'1'1_monotonic;assumption.
     intros ;apply P_id_U'14'1_monotonic;assumption.
     intros ;apply P_id_U'7'1_monotonic;assumption.
     intros ;apply P_id_U'4'1_monotonic;assumption.
     intros ;apply P_id_U'11'1_monotonic;assumption.
     intros ;apply P_id_INTERSECT'II'IN_monotonic;assumption.
     intros ;apply P_id_U'13'1_monotonic;assumption.
     intros ;apply P_id_U'6'2_monotonic;assumption.
     intros ;apply P_id_U'3'1_monotonic;assumption.
     intros ;apply P_id_U'16'1_monotonic;assumption.
     intros ;apply P_id_U'10'1_monotonic;assumption.
     intros ;apply P_id_U'2'1_monotonic;assumption.
     intros ;apply P_id_U'15'1_monotonic;assumption.
     intros ;apply P_id_U'8'1_monotonic;assumption.
     intros ;apply P_id_U'5'1_monotonic;assumption.
     intros ;apply P_id_U'12'1_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 := 
    (* <intersect'ii'in(Xs_,cons(X0_,Ys_)),u'1'1(intersect'ii'in(Xs_,Ys_))> *)
   | DP_R_xml_0_0 :
    forall x4 x20 x2 x5 x21, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                x4 x21) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_cons (x2::x5::nil)) x20) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_u'1'1 ((algebra.Alg.Term 
                   algebra.F.id_intersect'ii'in (x4::x5::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_intersect'ii'in (x21::x20::nil))
   
    (* <intersect'ii'in(Xs_,cons(X0_,Ys_)),intersect'ii'in(Xs_,Ys_)> *)
   | DP_R_xml_0_1 :
    forall x4 x20 x2 x5 x21, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                x4 x21) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_cons (x2::x5::nil)) x20) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_intersect'ii'in (x4::
                   x5::nil)) 
        (algebra.Alg.Term algebra.F.id_intersect'ii'in (x21::x20::nil))
   
    (* <intersect'ii'in(cons(X0_,Xs_),Ys_),u'2'1(intersect'ii'in(Xs_,Ys_))> *)
   | DP_R_xml_0_2 :
    forall x4 x20 x2 x5 x21, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_cons (x2::x4::nil)) x21) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x5 x20) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_u'2'1 ((algebra.Alg.Term 
                   algebra.F.id_intersect'ii'in (x4::x5::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_intersect'ii'in (x21::x20::nil))
   
    (* <intersect'ii'in(cons(X0_,Xs_),Ys_),intersect'ii'in(Xs_,Ys_)> *)
   | DP_R_xml_0_3 :
    forall x4 x20 x2 x5 x21, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_cons (x2::x4::nil)) x21) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x5 x20) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_intersect'ii'in (x4::
                   x5::nil)) 
        (algebra.Alg.Term algebra.F.id_intersect'ii'in (x21::x20::nil))
   
    (* <reduce'ii'in(sequent(cons(if(A_,B_),Fs_),Gs_),NF_),u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B_),A_),Fs_),Gs_),NF_))> *)
   | DP_R_xml_0_4 :
    forall x8 x20 x10 x6 x9 x21 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term 
        algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_if (x6::x7::nil))::
        x8::nil))::x9::nil)) x21) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x10 x20) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_u'3'1 ((algebra.Alg.Term 
                   algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
                   algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons 
                   ((algebra.Alg.Term algebra.F.id_x'2b ((algebra.Alg.Term 
                   algebra.F.id_x'2d (x7::nil))::x6::nil))::x8::nil))::
                   x9::nil))::x10::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
   
    (* <reduce'ii'in(sequent(cons(if(A_,B_),Fs_),Gs_),NF_),reduce'ii'in(sequent(cons(x'2b(x'2d(B_),A_),Fs_),Gs_),NF_)> *)
   | DP_R_xml_0_5 :
    forall x8 x20 x10 x6 x9 x21 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term 
        algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_if (x6::x7::nil))::
        x8::nil))::x9::nil)) x21) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x10 x20) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_reduce'ii'in 
                   ((algebra.Alg.Term algebra.F.id_sequent 
                   ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term 
                   algebra.F.id_x'2b ((algebra.Alg.Term algebra.F.id_x'2d 
                   (x7::nil))::x6::nil))::x8::nil))::x9::nil))::x10::nil)) 
        (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
   
    (* <reduce'ii'in(sequent(cons(iff(A_,B_),Fs_),Gs_),NF_),u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A_,B_),if(B_,A_)),Fs_),Gs_),NF_))> *)
   | DP_R_xml_0_6 :
    forall x8 x20 x10 x6 x9 x21 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term 
        algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_iff (x6::
        x7::nil))::x8::nil))::x9::nil)) x21) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x10 x20) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_u'4'1 ((algebra.Alg.Term 
                   algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
                   algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons 
                   ((algebra.Alg.Term algebra.F.id_x'2a ((algebra.Alg.Term 
                   algebra.F.id_if (x6::x7::nil))::(algebra.Alg.Term 
                   algebra.F.id_if (x7::x6::nil))::nil))::x8::nil))::
                   x9::nil))::x10::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
   
    (* <reduce'ii'in(sequent(cons(iff(A_,B_),Fs_),Gs_),NF_),reduce'ii'in(sequent(cons(x'2a(if(A_,B_),if(B_,A_)),Fs_),Gs_),NF_)> *)
   | DP_R_xml_0_7 :
    forall x8 x20 x10 x6 x9 x21 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term 
        algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_iff (x6::
        x7::nil))::x8::nil))::x9::nil)) x21) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x10 x20) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_reduce'ii'in 
                   ((algebra.Alg.Term algebra.F.id_sequent 
                   ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term 
                   algebra.F.id_x'2a ((algebra.Alg.Term algebra.F.id_if (x6::
                   x7::nil))::(algebra.Alg.Term algebra.F.id_if (x7::
                   x6::nil))::nil))::x8::nil))::x9::nil))::x10::nil)) 
        (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
   
    (* <reduce'ii'in(sequent(cons(x'2a(F1_,F2_),Fs_),Gs_),NF_),u'5'1(reduce'ii'in(sequent(cons(F1_,cons(F2_,Fs_)),Gs_),NF_))> *)
   | DP_R_xml_0_8 :
    forall x8 x20 x12 x10 x9 x21 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term 
        algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2a (x11::
        x12::nil))::x8::nil))::x9::nil)) x21) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x10 x20) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_u'5'1 ((algebra.Alg.Term 
                   algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
                   algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons 
                   (x11::(algebra.Alg.Term algebra.F.id_cons (x12::
                   x8::nil))::nil))::x9::nil))::x10::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
   
    (* <reduce'ii'in(sequent(cons(x'2a(F1_,F2_),Fs_),Gs_),NF_),reduce'ii'in(sequent(cons(F1_,cons(F2_,Fs_)),Gs_),NF_)> *)
   | DP_R_xml_0_9 :
    forall x8 x20 x12 x10 x9 x21 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term 
        algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2a (x11::
        x12::nil))::x8::nil))::x9::nil)) x21) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x10 x20) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_reduce'ii'in 
                   ((algebra.Alg.Term algebra.F.id_sequent 
                   ((algebra.Alg.Term algebra.F.id_cons (x11::
                   (algebra.Alg.Term algebra.F.id_cons (x12::
                   x8::nil))::nil))::x9::nil))::x10::nil)) 
        (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
   
    (* <reduce'ii'in(sequent(cons(x'2b(F1_,F2_),Fs_),Gs_),NF_),u'6'1(reduce'ii'in(sequent(cons(F1_,Fs_),Gs_),NF_),F2_,Fs_,Gs_,NF_)> *)
   | DP_R_xml_0_10 :
    forall x8 x20 x12 x10 x9 x21 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term 
        algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b (x11::
        x12::nil))::x8::nil))::x9::nil)) x21) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x10 x20) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_u'6'1 ((algebra.Alg.Term 
                   algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
                   algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons 
                   (x11::x8::nil))::x9::nil))::x10::nil))::x12::x8::x9::
                   x10::nil)) 
        (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
   
    (* <reduce'ii'in(sequent(cons(x'2b(F1_,F2_),Fs_),Gs_),NF_),reduce'ii'in(sequent(cons(F1_,Fs_),Gs_),NF_)> *)
   | DP_R_xml_0_11 :
    forall x8 x20 x12 x10 x9 x21 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term 
        algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b (x11::
        x12::nil))::x8::nil))::x9::nil)) x21) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x10 x20) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_reduce'ii'in 
                   ((algebra.Alg.Term algebra.F.id_sequent 
                   ((algebra.Alg.Term algebra.F.id_cons (x11::x8::nil))::
                   x9::nil))::x10::nil)) 
        (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
   
    (* <u'6'1(reduce'ii'out,F2_,Fs_,Gs_,NF_),u'6'2(reduce'ii'in(sequent(cons(F2_,Fs_),Gs_),NF_))> *)
   | DP_R_xml_0_12 :
    forall x8 x24 x20 x12 x10 x22 x9 x21 x23, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) x24) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x12 x23) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x8 x22) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x9 x21) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x10 x20) ->
          DP_R_xml_0 (algebra.Alg.Term algebra.F.id_u'6'2 ((algebra.Alg.Term 
                      algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
                      algebra.F.id_sequent ((algebra.Alg.Term 
                      algebra.F.id_cons (x12::x8::nil))::x9::nil))::
                      x10::nil))::nil)) 
           (algebra.Alg.Term algebra.F.id_u'6'1 (x24::x23::x22::x21::
            x20::nil))
   
    (* <u'6'1(reduce'ii'out,F2_,Fs_,Gs_,NF_),reduce'ii'in(sequent(cons(F2_,Fs_),Gs_),NF_)> *)
   | DP_R_xml_0_13 :
    forall x8 x24 x20 x12 x10 x22 x9 x21 x23, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) x24) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x12 x23) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x8 x22) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x9 x21) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x10 x20) ->
          DP_R_xml_0 (algebra.Alg.Term algebra.F.id_reduce'ii'in 
                      ((algebra.Alg.Term algebra.F.id_sequent 
                      ((algebra.Alg.Term algebra.F.id_cons (x12::x8::nil))::
                      x9::nil))::x10::nil)) 
           (algebra.Alg.Term algebra.F.id_u'6'1 (x24::x23::x22::x21::
            x20::nil))
   
    (* <reduce'ii'in(sequent(cons(x'2d(F1_),Fs_),Gs_),NF_),u'7'1(reduce'ii'in(sequent(Fs_,cons(F1_,Gs_)),NF_))> *)
   | DP_R_xml_0_14 :
    forall x8 x20 x10 x9 x21 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term 
        algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2d (x11::nil))::
        x8::nil))::x9::nil)) x21) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x10 x20) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_u'7'1 ((algebra.Alg.Term 
                   algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
                   algebra.F.id_sequent (x8::(algebra.Alg.Term 
                   algebra.F.id_cons (x11::x9::nil))::nil))::
                   x10::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
   
    (* <reduce'ii'in(sequent(cons(x'2d(F1_),Fs_),Gs_),NF_),reduce'ii'in(sequent(Fs_,cons(F1_,Gs_)),NF_)> *)
   | DP_R_xml_0_15 :
    forall x8 x20 x10 x9 x21 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term 
        algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2d (x11::nil))::
        x8::nil))::x9::nil)) x21) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x10 x20) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_reduce'ii'in 
                   ((algebra.Alg.Term algebra.F.id_sequent (x8::
                   (algebra.Alg.Term algebra.F.id_cons (x11::
                   x9::nil))::nil))::x10::nil)) 
        (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
   
    (* <reduce'ii'in(sequent(Fs_,cons(if(A_,B_),Gs_)),NF_),u'8'1(reduce'ii'in(sequent(Fs_,cons(x'2b(x'2d(B_),A_),Gs_)),NF_))> *)
   | DP_R_xml_0_16 :
    forall x8 x20 x10 x6 x9 x21 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term 
        algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_if (x6::x7::nil))::
        x9::nil))::nil)) x21) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x10 x20) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_u'8'1 ((algebra.Alg.Term 
                   algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
                   algebra.F.id_sequent (x8::(algebra.Alg.Term 
                   algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b 
                   ((algebra.Alg.Term algebra.F.id_x'2d (x7::nil))::
                   x6::nil))::x9::nil))::nil))::x10::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
   
    (* <reduce'ii'in(sequent(Fs_,cons(if(A_,B_),Gs_)),NF_),reduce'ii'in(sequent(Fs_,cons(x'2b(x'2d(B_),A_),Gs_)),NF_)> *)
   | DP_R_xml_0_17 :
    forall x8 x20 x10 x6 x9 x21 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term 
        algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_if (x6::x7::nil))::
        x9::nil))::nil)) x21) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x10 x20) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_reduce'ii'in 
                   ((algebra.Alg.Term algebra.F.id_sequent (x8::
                   (algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term 
                   algebra.F.id_x'2b ((algebra.Alg.Term algebra.F.id_x'2d 
                   (x7::nil))::x6::nil))::x9::nil))::nil))::x10::nil)) 
        (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
   
    (* <reduce'ii'in(sequent(Fs_,cons(iff(A_,B_),Gs_)),NF_),u'9'1(reduce'ii'in(sequent(Fs_,cons(x'2a(if(A_,B_),if(B_,A_)),Gs_)),NF_))> *)
   | DP_R_xml_0_18 :
    forall x8 x20 x10 x6 x9 x21 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term 
        algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_iff (x6::
        x7::nil))::x9::nil))::nil)) x21) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x10 x20) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_u'9'1 ((algebra.Alg.Term 
                   algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
                   algebra.F.id_sequent (x8::(algebra.Alg.Term 
                   algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2a 
                   ((algebra.Alg.Term algebra.F.id_if (x6::x7::nil))::
                   (algebra.Alg.Term algebra.F.id_if (x7::x6::nil))::nil))::
                   x9::nil))::nil))::x10::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
   
    (* <reduce'ii'in(sequent(Fs_,cons(iff(A_,B_),Gs_)),NF_),reduce'ii'in(sequent(Fs_,cons(x'2a(if(A_,B_),if(B_,A_)),Gs_)),NF_)> *)
   | DP_R_xml_0_19 :
    forall x8 x20 x10 x6 x9 x21 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term 
        algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_iff (x6::
        x7::nil))::x9::nil))::nil)) x21) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x10 x20) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_reduce'ii'in 
                   ((algebra.Alg.Term algebra.F.id_sequent (x8::
                   (algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term 
                   algebra.F.id_x'2a ((algebra.Alg.Term algebra.F.id_if (x6::
                   x7::nil))::(algebra.Alg.Term algebra.F.id_if (x7::
                   x6::nil))::nil))::x9::nil))::nil))::x10::nil)) 
        (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
   
    (* <reduce'ii'in(sequent(cons(p(V_),Fs_),Gs_),sequent(Left_,Right_)),u'10'1(reduce'ii'in(sequent(Fs_,Gs_),sequent(cons(p(V_),Left_),Right_)))> *)
   | DP_R_xml_0_20 :
    forall x8 x20 x14 x9 x21 x13 x15, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term 
        algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_p (x13::nil))::
        x8::nil))::x9::nil)) x21) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_sequent (x14::x15::nil)) x20) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_u'10'1 ((algebra.Alg.Term 
                   algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
                   algebra.F.id_sequent (x8::x9::nil))::(algebra.Alg.Term 
                   algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons 
                   ((algebra.Alg.Term algebra.F.id_p (x13::nil))::
                   x14::nil))::x15::nil))::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
   
    (* <reduce'ii'in(sequent(cons(p(V_),Fs_),Gs_),sequent(Left_,Right_)),reduce'ii'in(sequent(Fs_,Gs_),sequent(cons(p(V_),Left_),Right_))> *)
   | DP_R_xml_0_21 :
    forall x8 x20 x14 x9 x21 x13 x15, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term 
        algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_p (x13::nil))::
        x8::nil))::x9::nil)) x21) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_sequent (x14::x15::nil)) x20) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_reduce'ii'in 
                   ((algebra.Alg.Term algebra.F.id_sequent (x8::x9::nil))::
                   (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term 
                   algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_p 
                   (x13::nil))::x14::nil))::x15::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
   
    (* <reduce'ii'in(sequent(Fs_,cons(x'2b(G1_,G2_),Gs_)),NF_),u'11'1(reduce'ii'in(sequent(Fs_,cons(G1_,cons(G2_,Gs_))),NF_))> *)
   | DP_R_xml_0_22 :
    forall x16 x8 x20 x10 x17 x9 x21, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term 
        algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b (x16::
        x17::nil))::x9::nil))::nil)) x21) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x10 x20) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_u'11'1 ((algebra.Alg.Term 
                   algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
                   algebra.F.id_sequent (x8::(algebra.Alg.Term 
                   algebra.F.id_cons (x16::(algebra.Alg.Term 
                   algebra.F.id_cons (x17::x9::nil))::nil))::nil))::
                   x10::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
   
    (* <reduce'ii'in(sequent(Fs_,cons(x'2b(G1_,G2_),Gs_)),NF_),reduce'ii'in(sequent(Fs_,cons(G1_,cons(G2_,Gs_))),NF_)> *)
   | DP_R_xml_0_23 :
    forall x16 x8 x20 x10 x17 x9 x21, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term 
        algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b (x16::
        x17::nil))::x9::nil))::nil)) x21) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x10 x20) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_reduce'ii'in 
                   ((algebra.Alg.Term algebra.F.id_sequent (x8::
                   (algebra.Alg.Term algebra.F.id_cons (x16::
                   (algebra.Alg.Term algebra.F.id_cons (x17::
                   x9::nil))::nil))::nil))::x10::nil)) 
        (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
   
    (* <reduce'ii'in(sequent(Fs_,cons(x'2a(G1_,G2_),Gs_)),NF_),u'12'1(reduce'ii'in(sequent(Fs_,cons(G1_,Gs_)),NF_),Fs_,G2_,Gs_,NF_)> *)
   | DP_R_xml_0_24 :
    forall x16 x8 x20 x10 x17 x9 x21, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term 
        algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2a (x16::
        x17::nil))::x9::nil))::nil)) x21) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x10 x20) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_u'12'1 ((algebra.Alg.Term 
                   algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
                   algebra.F.id_sequent (x8::(algebra.Alg.Term 
                   algebra.F.id_cons (x16::x9::nil))::nil))::x10::nil))::x8::
                   x17::x9::x10::nil)) 
        (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
   
    (* <reduce'ii'in(sequent(Fs_,cons(x'2a(G1_,G2_),Gs_)),NF_),reduce'ii'in(sequent(Fs_,cons(G1_,Gs_)),NF_)> *)
   | DP_R_xml_0_25 :
    forall x16 x8 x20 x10 x17 x9 x21, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term 
        algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2a (x16::
        x17::nil))::x9::nil))::nil)) x21) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x10 x20) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_reduce'ii'in 
                   ((algebra.Alg.Term algebra.F.id_sequent (x8::
                   (algebra.Alg.Term algebra.F.id_cons (x16::
                   x9::nil))::nil))::x10::nil)) 
        (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
   
    (* <u'12'1(reduce'ii'out,Fs_,G2_,Gs_,NF_),u'12'2(reduce'ii'in(sequent(Fs_,cons(G2_,Gs_)),NF_))> *)
   | DP_R_xml_0_26 :
    forall x8 x24 x20 x10 x22 x17 x9 x21 x23, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) x24) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x8 x23) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x17 x22) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x9 x21) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x10 x20) ->
          DP_R_xml_0 (algebra.Alg.Term algebra.F.id_u'12'2 
                      ((algebra.Alg.Term algebra.F.id_reduce'ii'in 
                      ((algebra.Alg.Term algebra.F.id_sequent (x8::
                      (algebra.Alg.Term algebra.F.id_cons (x17::
                      x9::nil))::nil))::x10::nil))::nil)) 
           (algebra.Alg.Term algebra.F.id_u'12'1 (x24::x23::x22::x21::
            x20::nil))
   
    (* <u'12'1(reduce'ii'out,Fs_,G2_,Gs_,NF_),reduce'ii'in(sequent(Fs_,cons(G2_,Gs_)),NF_)> *)
   | DP_R_xml_0_27 :
    forall x8 x24 x20 x10 x22 x17 x9 x21 x23, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) x24) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x8 x23) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x17 x22) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x9 x21) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x10 x20) ->
          DP_R_xml_0 (algebra.Alg.Term algebra.F.id_reduce'ii'in 
                      ((algebra.Alg.Term algebra.F.id_sequent (x8::
                      (algebra.Alg.Term algebra.F.id_cons (x17::
                      x9::nil))::nil))::x10::nil)) 
           (algebra.Alg.Term algebra.F.id_u'12'1 (x24::x23::x22::x21::
            x20::nil))
   
    (* <reduce'ii'in(sequent(Fs_,cons(x'2d(G1_),Gs_)),NF_),u'13'1(reduce'ii'in(sequent(cons(G1_,Fs_),Gs_),NF_))> *)
   | DP_R_xml_0_28 :
    forall x16 x8 x20 x10 x9 x21, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term 
        algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2d (x16::nil))::
        x9::nil))::nil)) x21) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x10 x20) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_u'13'1 ((algebra.Alg.Term 
                   algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
                   algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons 
                   (x16::x8::nil))::x9::nil))::x10::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
   
    (* <reduce'ii'in(sequent(Fs_,cons(x'2d(G1_),Gs_)),NF_),reduce'ii'in(sequent(cons(G1_,Fs_),Gs_),NF_)> *)
   | DP_R_xml_0_29 :
    forall x16 x8 x20 x10 x9 x21, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term 
        algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2d (x16::nil))::
        x9::nil))::nil)) x21) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x10 x20) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_reduce'ii'in 
                   ((algebra.Alg.Term algebra.F.id_sequent 
                   ((algebra.Alg.Term algebra.F.id_cons (x16::x8::nil))::
                   x9::nil))::x10::nil)) 
        (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
   
    (* <reduce'ii'in(sequent(nil,cons(p(V_),Gs_)),sequent(Left_,Right_)),u'14'1(reduce'ii'in(sequent(nil,Gs_),sequent(Left_,cons(p(V_),Right_))))> *)
   | DP_R_xml_0_30 :
    forall x20 x14 x9 x21 x13 x15, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term 
        algebra.F.id_nil nil)::(algebra.Alg.Term algebra.F.id_cons 
        ((algebra.Alg.Term algebra.F.id_p (x13::nil))::x9::nil))::nil)) 
       x21) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_sequent (x14::x15::nil)) x20) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_u'14'1 ((algebra.Alg.Term 
                   algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
                   algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_nil 
                   nil)::x9::nil))::(algebra.Alg.Term algebra.F.id_sequent 
                   (x14::(algebra.Alg.Term algebra.F.id_cons 
                   ((algebra.Alg.Term algebra.F.id_p (x13::nil))::
                   x15::nil))::nil))::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
   
    (* <reduce'ii'in(sequent(nil,cons(p(V_),Gs_)),sequent(Left_,Right_)),reduce'ii'in(sequent(nil,Gs_),sequent(Left_,cons(p(V_),Right_)))> *)
   | DP_R_xml_0_31 :
    forall x20 x14 x9 x21 x13 x15, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term 
        algebra.F.id_nil nil)::(algebra.Alg.Term algebra.F.id_cons 
        ((algebra.Alg.Term algebra.F.id_p (x13::nil))::x9::nil))::nil)) 
       x21) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_sequent (x14::x15::nil)) x20) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_reduce'ii'in 
                   ((algebra.Alg.Term algebra.F.id_sequent 
                   ((algebra.Alg.Term algebra.F.id_nil nil)::x9::nil))::
                   (algebra.Alg.Term algebra.F.id_sequent (x14::
                   (algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term 
                   algebra.F.id_p (x13::nil))::x15::nil))::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
   
    (* <reduce'ii'in(sequent(nil,nil),sequent(F1_,F2_)),u'15'1(intersect'ii'in(F1_,F2_))> *)
   | DP_R_xml_0_32 :
    forall x20 x12 x21 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term 
        algebra.F.id_nil nil)::(algebra.Alg.Term algebra.F.id_nil nil)::nil)) 
       x21) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_sequent (x11::x12::nil)) x20) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_u'15'1 ((algebra.Alg.Term 
                   algebra.F.id_intersect'ii'in (x11::x12::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
   
    (* <reduce'ii'in(sequent(nil,nil),sequent(F1_,F2_)),intersect'ii'in(F1_,F2_)> *)
   | DP_R_xml_0_33 :
    forall x20 x12 x21 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term 
        algebra.F.id_nil nil)::(algebra.Alg.Term algebra.F.id_nil nil)::nil)) 
       x21) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_sequent (x11::x12::nil)) x20) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_intersect'ii'in (x11::
                   x12::nil)) 
        (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
   
    (* <tautology'i'in(F_),u'16'1(reduce'ii'in(sequent(nil,cons(F_,nil)),sequent(nil,nil)))> *)
   | DP_R_xml_0_34 :
    forall x20 x18, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                x18 x20) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_u'16'1 ((algebra.Alg.Term 
                  algebra.F.id_reduce'ii'in ((algebra.Alg.Term 
                  algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_nil 
                  nil)::(algebra.Alg.Term algebra.F.id_cons (x18::
                  (algebra.Alg.Term algebra.F.id_nil nil)::nil))::nil))::
                  (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term 
                  algebra.F.id_nil nil)::(algebra.Alg.Term algebra.F.id_nil 
                  nil)::nil))::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_tautology'i'in (x20::nil))
   
    (* <tautology'i'in(F_),reduce'ii'in(sequent(nil,cons(F_,nil)),sequent(nil,nil))> *)
   | DP_R_xml_0_35 :
    forall x20 x18, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                x18 x20) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_reduce'ii'in 
                  ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term 
                  algebra.F.id_nil nil)::(algebra.Alg.Term algebra.F.id_cons 
                  (x18::(algebra.Alg.Term algebra.F.id_nil 
                  nil)::nil))::nil))::(algebra.Alg.Term algebra.F.id_sequent 
                  ((algebra.Alg.Term algebra.F.id_nil nil)::
                  (algebra.Alg.Term algebra.F.id_nil nil)::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_tautology'i'in (x20::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 := 
     (* <tautology'i'in(F_),u'16'1(reduce'ii'in(sequent(nil,cons(F_,nil)),sequent(nil,nil)))> *)
    | DP_R_xml_0_non_scc_1_0 :
     forall x20 x18, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x18 x20) ->
       DP_R_xml_0_non_scc_1 (algebra.Alg.Term algebra.F.id_u'16'1 
                             ((algebra.Alg.Term algebra.F.id_reduce'ii'in 
                             ((algebra.Alg.Term algebra.F.id_sequent 
                             ((algebra.Alg.Term algebra.F.id_nil nil)::
                             (algebra.Alg.Term algebra.F.id_cons (x18::
                             (algebra.Alg.Term algebra.F.id_nil 
                             nil)::nil))::nil))::(algebra.Alg.Term 
                             algebra.F.id_sequent ((algebra.Alg.Term 
                             algebra.F.id_nil nil)::(algebra.Alg.Term 
                             algebra.F.id_nil nil)::nil))::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_tautology'i'in (x20::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 := 
     (* <reduce'ii'in(sequent(nil,nil),sequent(F1_,F2_)),u'15'1(intersect'ii'in(F1_,F2_))> *)
    | DP_R_xml_0_non_scc_2_0 :
     forall x20 x12 x21 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term 
         algebra.F.id_nil nil)::(algebra.Alg.Term algebra.F.id_nil 
         nil)::nil)) x21) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_sequent (x11::x12::nil)) x20) ->
        DP_R_xml_0_non_scc_2 (algebra.Alg.Term algebra.F.id_u'15'1 
                              ((algebra.Alg.Term 
                              algebra.F.id_intersect'ii'in (x11::
                              x12::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::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 := 
     (* <reduce'ii'in(sequent(nil,cons(p(V_),Gs_)),sequent(Left_,Right_)),u'14'1(reduce'ii'in(sequent(nil,Gs_),sequent(Left_,cons(p(V_),Right_))))> *)
    | DP_R_xml_0_non_scc_3_0 :
     forall x20 x14 x9 x21 x13 x15, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term 
         algebra.F.id_nil nil)::(algebra.Alg.Term algebra.F.id_cons 
         ((algebra.Alg.Term algebra.F.id_p (x13::nil))::x9::nil))::nil)) 
        x21) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_sequent (x14::x15::nil)) x20) ->
        DP_R_xml_0_non_scc_3 (algebra.Alg.Term algebra.F.id_u'14'1 
                              ((algebra.Alg.Term algebra.F.id_reduce'ii'in 
                              ((algebra.Alg.Term algebra.F.id_sequent 
                              ((algebra.Alg.Term algebra.F.id_nil nil)::
                              x9::nil))::(algebra.Alg.Term 
                              algebra.F.id_sequent (x14::(algebra.Alg.Term 
                              algebra.F.id_cons ((algebra.Alg.Term 
                              algebra.F.id_p (x13::nil))::
                              x15::nil))::nil))::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::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 := 
     (* <reduce'ii'in(sequent(Fs_,cons(x'2d(G1_),Gs_)),NF_),u'13'1(reduce'ii'in(sequent(cons(G1_,Fs_),Gs_),NF_))> *)
    | DP_R_xml_0_non_scc_4_0 :
     forall x16 x8 x20 x10 x9 x21, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term 
         algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2d (x16::nil))::
         x9::nil))::nil)) x21) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x10 x20) ->
        DP_R_xml_0_non_scc_4 (algebra.Alg.Term algebra.F.id_u'13'1 
                              ((algebra.Alg.Term algebra.F.id_reduce'ii'in 
                              ((algebra.Alg.Term algebra.F.id_sequent 
                              ((algebra.Alg.Term algebra.F.id_cons (x16::
                              x8::nil))::x9::nil))::x10::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::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 := 
     (* <u'12'1(reduce'ii'out,Fs_,G2_,Gs_,NF_),u'12'2(reduce'ii'in(sequent(Fs_,cons(G2_,Gs_)),NF_))> *)
    | DP_R_xml_0_non_scc_5_0 :
     forall x8 x24 x20 x10 x22 x17 x9 x21 x23, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) x24) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x8 x23) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x17 x22) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x9 x21) ->
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     x10 x20) ->
           DP_R_xml_0_non_scc_5 (algebra.Alg.Term algebra.F.id_u'12'2 
                                 ((algebra.Alg.Term 
                                 algebra.F.id_reduce'ii'in 
                                 ((algebra.Alg.Term algebra.F.id_sequent 
                                 (x8::(algebra.Alg.Term algebra.F.id_cons 
                                 (x17::x9::nil))::nil))::x10::nil))::nil)) 
            (algebra.Alg.Term algebra.F.id_u'12'1 (x24::x23::x22::x21::
             x20::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 := 
     (* <reduce'ii'in(sequent(Fs_,cons(x'2b(G1_,G2_),Gs_)),NF_),u'11'1(reduce'ii'in(sequent(Fs_,cons(G1_,cons(G2_,Gs_))),NF_))> *)
    | DP_R_xml_0_non_scc_6_0 :
     forall x16 x8 x20 x10 x17 x9 x21, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term 
         algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b (x16::
         x17::nil))::x9::nil))::nil)) x21) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x10 x20) ->
        DP_R_xml_0_non_scc_6 (algebra.Alg.Term algebra.F.id_u'11'1 
                              ((algebra.Alg.Term algebra.F.id_reduce'ii'in 
                              ((algebra.Alg.Term algebra.F.id_sequent (x8::
                              (algebra.Alg.Term algebra.F.id_cons (x16::
                              (algebra.Alg.Term algebra.F.id_cons (x17::
                              x9::nil))::nil))::nil))::x10::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::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 := 
     (* <reduce'ii'in(sequent(cons(p(V_),Fs_),Gs_),sequent(Left_,Right_)),u'10'1(reduce'ii'in(sequent(Fs_,Gs_),sequent(cons(p(V_),Left_),Right_)))> *)
    | DP_R_xml_0_non_scc_7_0 :
     forall x8 x20 x14 x9 x21 x13 x15, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term 
         algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_p (x13::nil))::
         x8::nil))::x9::nil)) x21) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_sequent (x14::x15::nil)) x20) ->
        DP_R_xml_0_non_scc_7 (algebra.Alg.Term algebra.F.id_u'10'1 
                              ((algebra.Alg.Term algebra.F.id_reduce'ii'in 
                              ((algebra.Alg.Term algebra.F.id_sequent (x8::
                              x9::nil))::(algebra.Alg.Term 
                              algebra.F.id_sequent ((algebra.Alg.Term 
                              algebra.F.id_cons ((algebra.Alg.Term 
                              algebra.F.id_p (x13::nil))::x14::nil))::
                              x15::nil))::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::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 := 
     (* <reduce'ii'in(sequent(Fs_,cons(iff(A_,B_),Gs_)),NF_),u'9'1(reduce'ii'in(sequent(Fs_,cons(x'2a(if(A_,B_),if(B_,A_)),Gs_)),NF_))> *)
    | DP_R_xml_0_non_scc_8_0 :
     forall x8 x20 x10 x6 x9 x21 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term 
         algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_iff (x6::
         x7::nil))::x9::nil))::nil)) x21) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x10 x20) ->
        DP_R_xml_0_non_scc_8 (algebra.Alg.Term algebra.F.id_u'9'1 
                              ((algebra.Alg.Term algebra.F.id_reduce'ii'in 
                              ((algebra.Alg.Term algebra.F.id_sequent (x8::
                              (algebra.Alg.Term algebra.F.id_cons 
                              ((algebra.Alg.Term algebra.F.id_x'2a 
                              ((algebra.Alg.Term algebra.F.id_if (x6::
                              x7::nil))::(algebra.Alg.Term algebra.F.id_if 
                              (x7::x6::nil))::nil))::x9::nil))::nil))::
                              x10::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::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 := 
     (* <reduce'ii'in(sequent(Fs_,cons(if(A_,B_),Gs_)),NF_),u'8'1(reduce'ii'in(sequent(Fs_,cons(x'2b(x'2d(B_),A_),Gs_)),NF_))> *)
    | DP_R_xml_0_non_scc_9_0 :
     forall x8 x20 x10 x6 x9 x21 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term 
         algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_if (x6::
         x7::nil))::x9::nil))::nil)) x21) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x10 x20) ->
        DP_R_xml_0_non_scc_9 (algebra.Alg.Term algebra.F.id_u'8'1 
                              ((algebra.Alg.Term algebra.F.id_reduce'ii'in 
                              ((algebra.Alg.Term algebra.F.id_sequent (x8::
                              (algebra.Alg.Term algebra.F.id_cons 
                              ((algebra.Alg.Term algebra.F.id_x'2b 
                              ((algebra.Alg.Term algebra.F.id_x'2d 
                              (x7::nil))::x6::nil))::x9::nil))::nil))::
                              x10::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::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 := 
     (* <reduce'ii'in(sequent(cons(x'2d(F1_),Fs_),Gs_),NF_),u'7'1(reduce'ii'in(sequent(Fs_,cons(F1_,Gs_)),NF_))> *)
    | DP_R_xml_0_non_scc_10_0 :
     forall x8 x20 x10 x9 x21 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term 
         algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2d (x11::nil))::
         x8::nil))::x9::nil)) x21) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x10 x20) ->
        DP_R_xml_0_non_scc_10 (algebra.Alg.Term algebra.F.id_u'7'1 
                               ((algebra.Alg.Term algebra.F.id_reduce'ii'in 
                               ((algebra.Alg.Term algebra.F.id_sequent (x8::
                               (algebra.Alg.Term algebra.F.id_cons (x11::
                               x9::nil))::nil))::x10::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::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 := 
     (* <u'6'1(reduce'ii'out,F2_,Fs_,Gs_,NF_),u'6'2(reduce'ii'in(sequent(cons(F2_,Fs_),Gs_),NF_))> *)
    | DP_R_xml_0_non_scc_11_0 :
     forall x8 x24 x20 x12 x10 x22 x9 x21 x23, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) x24) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x12 x23) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x8 x22) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x9 x21) ->
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     x10 x20) ->
           DP_R_xml_0_non_scc_11 (algebra.Alg.Term algebra.F.id_u'6'2 
                                  ((algebra.Alg.Term 
                                  algebra.F.id_reduce'ii'in 
                                  ((algebra.Alg.Term algebra.F.id_sequent 
                                  ((algebra.Alg.Term algebra.F.id_cons (x12::
                                  x8::nil))::x9::nil))::x10::nil))::nil)) 
            (algebra.Alg.Term algebra.F.id_u'6'1 (x24::x23::x22::x21::
             x20::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_11 :
   forall x y, 
    (DP_R_xml_0_non_scc_11 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
      (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_12  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <reduce'ii'in(sequent(cons(x'2a(F1_,F2_),Fs_),Gs_),NF_),u'5'1(reduce'ii'in(sequent(cons(F1_,cons(F2_,Fs_)),Gs_),NF_))> *)
    | DP_R_xml_0_non_scc_12_0 :
     forall x8 x20 x12 x10 x9 x21 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term 
         algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2a (x11::
         x12::nil))::x8::nil))::x9::nil)) x21) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x10 x20) ->
        DP_R_xml_0_non_scc_12 (algebra.Alg.Term algebra.F.id_u'5'1 
                               ((algebra.Alg.Term algebra.F.id_reduce'ii'in 
                               ((algebra.Alg.Term algebra.F.id_sequent 
                               ((algebra.Alg.Term algebra.F.id_cons (x11::
                               (algebra.Alg.Term algebra.F.id_cons (x12::
                               x8::nil))::nil))::x9::nil))::x10::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_12 :
   forall x y, 
    (DP_R_xml_0_non_scc_12 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
      (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_13  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <reduce'ii'in(sequent(cons(iff(A_,B_),Fs_),Gs_),NF_),u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A_,B_),if(B_,A_)),Fs_),Gs_),NF_))> *)
    | DP_R_xml_0_non_scc_13_0 :
     forall x8 x20 x10 x6 x9 x21 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term 
         algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_iff (x6::
         x7::nil))::x8::nil))::x9::nil)) x21) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x10 x20) ->
        DP_R_xml_0_non_scc_13 (algebra.Alg.Term algebra.F.id_u'4'1 
                               ((algebra.Alg.Term algebra.F.id_reduce'ii'in 
                               ((algebra.Alg.Term algebra.F.id_sequent 
                               ((algebra.Alg.Term algebra.F.id_cons 
                               ((algebra.Alg.Term algebra.F.id_x'2a 
                               ((algebra.Alg.Term algebra.F.id_if (x6::
                               x7::nil))::(algebra.Alg.Term algebra.F.id_if 
                               (x7::x6::nil))::nil))::x8::nil))::x9::nil))::
                               x10::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_13 :
   forall x y, 
    (DP_R_xml_0_non_scc_13 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
      (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_14  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <reduce'ii'in(sequent(cons(if(A_,B_),Fs_),Gs_),NF_),u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B_),A_),Fs_),Gs_),NF_))> *)
    | DP_R_xml_0_non_scc_14_0 :
     forall x8 x20 x10 x6 x9 x21 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term 
         algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_if (x6::
         x7::nil))::x8::nil))::x9::nil)) x21) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x10 x20) ->
        DP_R_xml_0_non_scc_14 (algebra.Alg.Term algebra.F.id_u'3'1 
                               ((algebra.Alg.Term algebra.F.id_reduce'ii'in 
                               ((algebra.Alg.Term algebra.F.id_sequent 
                               ((algebra.Alg.Term algebra.F.id_cons 
                               ((algebra.Alg.Term algebra.F.id_x'2b 
                               ((algebra.Alg.Term algebra.F.id_x'2d 
                               (x7::nil))::x6::nil))::x8::nil))::x9::nil))::
                               x10::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_14 :
   forall x y, 
    (DP_R_xml_0_non_scc_14 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_15  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <intersect'ii'in(cons(X0_,Xs_),Ys_),u'2'1(intersect'ii'in(Xs_,Ys_))> *)
    | DP_R_xml_0_non_scc_15_0 :
     forall x4 x20 x2 x5 x21, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_cons (x2::x4::nil)) x21) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x5 x20) ->
        DP_R_xml_0_non_scc_15 (algebra.Alg.Term algebra.F.id_u'2'1 
                               ((algebra.Alg.Term 
                               algebra.F.id_intersect'ii'in (x4::
                               x5::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_intersect'ii'in (x21::x20::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_15 :
   forall x y, 
    (DP_R_xml_0_non_scc_15 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
      (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_16  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <intersect'ii'in(Xs_,cons(X0_,Ys_)),u'1'1(intersect'ii'in(Xs_,Ys_))> *)
    | DP_R_xml_0_non_scc_16_0 :
     forall x4 x20 x2 x5 x21, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x4 x21) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_cons (x2::x5::nil)) x20) ->
        DP_R_xml_0_non_scc_16 (algebra.Alg.Term algebra.F.id_u'1'1 
                               ((algebra.Alg.Term 
                               algebra.F.id_intersect'ii'in (x4::
                               x5::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_intersect'ii'in (x21::x20::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_16 :
   forall x y, 
    (DP_R_xml_0_non_scc_16 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (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_17  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <intersect'ii'in(cons(X0_,Xs_),Ys_),intersect'ii'in(Xs_,Ys_)> *)
    | DP_R_xml_0_scc_17_0 :
     forall x4 x20 x2 x5 x21, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_cons (x2::x4::nil)) x21) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x5 x20) ->
        DP_R_xml_0_scc_17 (algebra.Alg.Term algebra.F.id_intersect'ii'in 
                           (x4::x5::nil)) 
         (algebra.Alg.Term algebra.F.id_intersect'ii'in (x21::x20::nil))
    
     (* <intersect'ii'in(Xs_,cons(X0_,Ys_)),intersect'ii'in(Xs_,Ys_)> *)
    | DP_R_xml_0_scc_17_1 :
     forall x4 x20 x2 x5 x21, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x4 x21) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_cons (x2::x5::nil)) x20) ->
        DP_R_xml_0_scc_17 (algebra.Alg.Term algebra.F.id_intersect'ii'in 
                           (x4::x5::nil)) 
         (algebra.Alg.Term algebra.F.id_intersect'ii'in (x21::x20::nil))
  .
  
  
  Module WF_DP_R_xml_0_scc_17.
   Open Scope Z_scope.
   
   Import ring_extention.
   
   Notation Local "a <= b" := (Zle a b).
   
   Notation Local "a < b" := (Zlt a b).
   
   Definition P_id_intersect'ii'in (x20:Z) (x21:Z) := 1* x21.
   
   Definition P_id_tautology'i'out  := 0.
   
   Definition P_id_u'6'1 (x20:Z) (x21:Z) (x22:Z) (x23:Z) (x24:Z) := 
     1 + 2* x22 + 1* x23.
   
   Definition P_id_u'3'1 (x20:Z) := 0.
   
   Definition P_id_u'12'1 (x20:Z) (x21:Z) (x22:Z) (x23:Z) (x24:Z) := 
     1* x22 + 1* x23.
   
   Definition P_id_u'2'1 (x20:Z) := 1* x20.
   
   Definition P_id_u'9'1 (x20:Z) := 0.
   
   Definition P_id_iff (x20:Z) (x21:Z) := 0.
   
   Definition P_id_u'14'1 (x20:Z) := 3.
   
   Definition P_id_intersect'ii'out  := 0.
   
   Definition P_id_u'7'1 (x20:Z) := 1.
   
   Definition P_id_x'2d (x20:Z) := 0.
   
   Definition P_id_u'13'1 (x20:Z) := 1.
   
   Definition P_id_sequent (x20:Z) (x21:Z) := 1* x20 + 2* x21.
   
   Definition P_id_u'10'1 (x20:Z) := 0.
   
   Definition P_id_x'2a (x20:Z) (x21:Z) := 1* x21.
   
   Definition P_id_tautology'i'in (x20:Z) := 2.
   
   Definition P_id_cons (x20:Z) (x21:Z) := 1 + 1* x20 + 2* x21.
   
   Definition P_id_u'6'2 (x20:Z) := 1.
   
   Definition P_id_x'2b (x20:Z) (x21:Z) := 0.
   
   Definition P_id_u'12'2 (x20:Z) := 0.
   
   Definition P_id_reduce'ii'in (x20:Z) (x21:Z) := 1* x20.
   
   Definition P_id_p (x20:Z) := 3 + 1* x20.
   
   Definition P_id_u'4'1 (x20:Z) := 1.
   
   Definition P_id_u'15'1 (x20:Z) := 0.
   
   Definition P_id_u'1'1 (x20:Z) := 1.
   
   Definition P_id_u'8'1 (x20:Z) := 2.
   
   Definition P_id_reduce'ii'out  := 0.
   
   Definition P_id_nil  := 0.
   
   Definition P_id_if (x20:Z) (x21:Z) := 0.
   
   Definition P_id_u'11'1 (x20:Z) := 2.
   
   Definition P_id_u'5'1 (x20:Z) := 0.
   
   Definition P_id_u'16'1 (x20:Z) := 1.
   
   Lemma P_id_intersect'ii'in_monotonic :
    forall x20 x22 x21 x23, 
     (0 <= x23)/\ (x23 <= x22) ->
      (0 <= x21)/\ (x21 <= x20) ->
       P_id_intersect'ii'in x21 x23 <= P_id_intersect'ii'in x20 x22.
   Proof.
     intros x23 x22 x21 x20.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'6'1_monotonic :
    forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, 
     (0 <= x29)/\ (x29 <= x28) ->
      (0 <= x27)/\ (x27 <= x26) ->
       (0 <= x25)/\ (x25 <= x24) ->
        (0 <= x23)/\ (x23 <= x22) ->
         (0 <= x21)/\ (x21 <= x20) ->
          P_id_u'6'1 x21 x23 x25 x27 x29 <= P_id_u'6'1 x20 x22 x24 x26 x28.
   Proof.
     intros x29 x28 x27 x26 x25 x24 x23 x22 x21 x20.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     intros [H_7 H_6].
     intros [H_9 H_8].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'3'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_u'3'1 x21 <= P_id_u'3'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'12'1_monotonic :
    forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, 
     (0 <= x29)/\ (x29 <= x28) ->
      (0 <= x27)/\ (x27 <= x26) ->
       (0 <= x25)/\ (x25 <= x24) ->
        (0 <= x23)/\ (x23 <= x22) ->
         (0 <= x21)/\ (x21 <= x20) ->
          P_id_u'12'1 x21 x23 x25 x27 x29 <= P_id_u'12'1 x20 x22 x24 x26 x28.
   Proof.
     intros x29 x28 x27 x26 x25 x24 x23 x22 x21 x20.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     intros [H_7 H_6].
     intros [H_9 H_8].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'2'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_u'2'1 x21 <= P_id_u'2'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'9'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_u'9'1 x21 <= P_id_u'9'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_iff_monotonic :
    forall x20 x22 x21 x23, 
     (0 <= x23)/\ (x23 <= x22) ->
      (0 <= x21)/\ (x21 <= x20) ->P_id_iff x21 x23 <= P_id_iff x20 x22.
   Proof.
     intros x23 x22 x21 x20.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'14'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_u'14'1 x21 <= P_id_u'14'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'7'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_u'7'1 x21 <= P_id_u'7'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_x'2d_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_x'2d x21 <= P_id_x'2d x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'13'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_u'13'1 x21 <= P_id_u'13'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_sequent_monotonic :
    forall x20 x22 x21 x23, 
     (0 <= x23)/\ (x23 <= x22) ->
      (0 <= x21)/\ (x21 <= x20) ->
       P_id_sequent x21 x23 <= P_id_sequent x20 x22.
   Proof.
     intros x23 x22 x21 x20.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'10'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_u'10'1 x21 <= P_id_u'10'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_x'2a_monotonic :
    forall x20 x22 x21 x23, 
     (0 <= x23)/\ (x23 <= x22) ->
      (0 <= x21)/\ (x21 <= x20) ->P_id_x'2a x21 x23 <= P_id_x'2a x20 x22.
   Proof.
     intros x23 x22 x21 x20.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_tautology'i'in_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->
      P_id_tautology'i'in x21 <= P_id_tautology'i'in x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_cons_monotonic :
    forall x20 x22 x21 x23, 
     (0 <= x23)/\ (x23 <= x22) ->
      (0 <= x21)/\ (x21 <= x20) ->P_id_cons x21 x23 <= P_id_cons x20 x22.
   Proof.
     intros x23 x22 x21 x20.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'6'2_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_u'6'2 x21 <= P_id_u'6'2 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_x'2b_monotonic :
    forall x20 x22 x21 x23, 
     (0 <= x23)/\ (x23 <= x22) ->
      (0 <= x21)/\ (x21 <= x20) ->P_id_x'2b x21 x23 <= P_id_x'2b x20 x22.
   Proof.
     intros x23 x22 x21 x20.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'12'2_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_u'12'2 x21 <= P_id_u'12'2 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_reduce'ii'in_monotonic :
    forall x20 x22 x21 x23, 
     (0 <= x23)/\ (x23 <= x22) ->
      (0 <= x21)/\ (x21 <= x20) ->
       P_id_reduce'ii'in x21 x23 <= P_id_reduce'ii'in x20 x22.
   Proof.
     intros x23 x22 x21 x20.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_p_monotonic :
    forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_p x21 <= P_id_p x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'4'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_u'4'1 x21 <= P_id_u'4'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'15'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_u'15'1 x21 <= P_id_u'15'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'1'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_u'1'1 x21 <= P_id_u'1'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'8'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_u'8'1 x21 <= P_id_u'8'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_if_monotonic :
    forall x20 x22 x21 x23, 
     (0 <= x23)/\ (x23 <= x22) ->
      (0 <= x21)/\ (x21 <= x20) ->P_id_if x21 x23 <= P_id_if x20 x22.
   Proof.
     intros x23 x22 x21 x20.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'11'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_u'11'1 x21 <= P_id_u'11'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'5'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_u'5'1 x21 <= P_id_u'5'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'16'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_u'16'1 x21 <= P_id_u'16'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_intersect'ii'in_bounded :
    forall x20 x21, 
     (0 <= x20) ->(0 <= x21) ->0 <= P_id_intersect'ii'in x21 x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_tautology'i'out_bounded : 0 <= P_id_tautology'i'out .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'6'1_bounded :
    forall x24 x20 x22 x21 x23, 
     (0 <= x20) ->
      (0 <= x21) ->
       (0 <= x22) ->
        (0 <= x23) ->(0 <= x24) ->0 <= P_id_u'6'1 x24 x23 x22 x21 x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'3'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'3'1 x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'12'1_bounded :
    forall x24 x20 x22 x21 x23, 
     (0 <= x20) ->
      (0 <= x21) ->
       (0 <= x22) ->
        (0 <= x23) ->(0 <= x24) ->0 <= P_id_u'12'1 x24 x23 x22 x21 x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'2'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'2'1 x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'9'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'9'1 x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_iff_bounded :
    forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_iff x21 x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'14'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'14'1 x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_intersect'ii'out_bounded : 0 <= P_id_intersect'ii'out .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'7'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'7'1 x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_x'2d_bounded : forall x20, (0 <= x20) ->0 <= P_id_x'2d x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'13'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'13'1 x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_sequent_bounded :
    forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_sequent x21 x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'10'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'10'1 x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_x'2a_bounded :
    forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_x'2a x21 x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_tautology'i'in_bounded :
    forall x20, (0 <= x20) ->0 <= P_id_tautology'i'in x20.
   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 x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_cons x21 x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'6'2_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'6'2 x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_x'2b_bounded :
    forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_x'2b x21 x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'12'2_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'12'2 x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_reduce'ii'in_bounded :
    forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_reduce'ii'in x21 x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_p_bounded : forall x20, (0 <= x20) ->0 <= P_id_p x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'4'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'4'1 x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'15'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'15'1 x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'1'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'1'1 x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'8'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'8'1 x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_reduce'ii'out_bounded : 0 <= P_id_reduce'ii'out .
   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_if_bounded :
    forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_if x21 x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'11'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'11'1 x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'5'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'5'1 x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'16'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'16'1 x20.
   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_intersect'ii'in P_id_tautology'i'out P_id_u'6'1 
      P_id_u'3'1 P_id_u'12'1 P_id_u'2'1 P_id_u'9'1 P_id_iff P_id_u'14'1 
      P_id_intersect'ii'out P_id_u'7'1 P_id_x'2d P_id_u'13'1 P_id_sequent 
      P_id_u'10'1 P_id_x'2a P_id_tautology'i'in P_id_cons P_id_u'6'2 
      P_id_x'2b P_id_u'12'2 P_id_reduce'ii'in P_id_p P_id_u'4'1 P_id_u'15'1 
      P_id_u'1'1 P_id_u'8'1 P_id_reduce'ii'out P_id_nil P_id_if P_id_u'11'1 
      P_id_u'5'1 P_id_u'16'1.
   
   Lemma measure_equation :
    forall t, 
     measure t = match t with
                   | (algebra.Alg.Term algebra.F.id_intersect'ii'in (x21::
                      x20::nil)) =>
                    P_id_intersect'ii'in (measure x21) (measure x20)
                   | (algebra.Alg.Term algebra.F.id_tautology'i'out nil) =>
                    P_id_tautology'i'out 
                   | (algebra.Alg.Term algebra.F.id_u'6'1 (x24::x23::x22::
                      x21::x20::nil)) =>
                    P_id_u'6'1 (measure x24) (measure x23) (measure x22) 
                     (measure x21) (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'3'1 (x20::nil)) =>
                    P_id_u'3'1 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'12'1 (x24::x23::x22::
                      x21::x20::nil)) =>
                    P_id_u'12'1 (measure x24) (measure x23) (measure x22) 
                     (measure x21) (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'2'1 (x20::nil)) =>
                    P_id_u'2'1 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'9'1 (x20::nil)) =>
                    P_id_u'9'1 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_iff (x21::x20::nil)) =>
                    P_id_iff (measure x21) (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'14'1 (x20::nil)) =>
                    P_id_u'14'1 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_intersect'ii'out nil) =>
                    P_id_intersect'ii'out 
                   | (algebra.Alg.Term algebra.F.id_u'7'1 (x20::nil)) =>
                    P_id_u'7'1 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_x'2d (x20::nil)) =>
                    P_id_x'2d (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'13'1 (x20::nil)) =>
                    P_id_u'13'1 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_sequent (x21::x20::nil)) =>
                    P_id_sequent (measure x21) (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'10'1 (x20::nil)) =>
                    P_id_u'10'1 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_x'2a (x21::x20::nil)) =>
                    P_id_x'2a (measure x21) (measure x20)
                   | (algebra.Alg.Term algebra.F.id_tautology'i'in 
                      (x20::nil)) =>
                    P_id_tautology'i'in (measure x20)
                   | (algebra.Alg.Term algebra.F.id_cons (x21::x20::nil)) =>
                    P_id_cons (measure x21) (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'6'2 (x20::nil)) =>
                    P_id_u'6'2 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_x'2b (x21::x20::nil)) =>
                    P_id_x'2b (measure x21) (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'12'2 (x20::nil)) =>
                    P_id_u'12'2 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::
                      x20::nil)) =>
                    P_id_reduce'ii'in (measure x21) (measure x20)
                   | (algebra.Alg.Term algebra.F.id_p (x20::nil)) =>
                    P_id_p (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'4'1 (x20::nil)) =>
                    P_id_u'4'1 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'15'1 (x20::nil)) =>
                    P_id_u'15'1 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'1'1 (x20::nil)) =>
                    P_id_u'1'1 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'8'1 (x20::nil)) =>
                    P_id_u'8'1 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) =>
                    P_id_reduce'ii'out 
                   | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil 
                   | (algebra.Alg.Term algebra.F.id_if (x21::x20::nil)) =>
                    P_id_if (measure x21) (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'11'1 (x20::nil)) =>
                    P_id_u'11'1 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'5'1 (x20::nil)) =>
                    P_id_u'5'1 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'16'1 (x20::nil)) =>
                    P_id_u'16'1 (measure x20)
                   | _ => 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_intersect'ii'in_monotonic;assumption.
     intros ;apply P_id_u'6'1_monotonic;assumption.
     intros ;apply P_id_u'3'1_monotonic;assumption.
     intros ;apply P_id_u'12'1_monotonic;assumption.
     intros ;apply P_id_u'2'1_monotonic;assumption.
     intros ;apply P_id_u'9'1_monotonic;assumption.
     intros ;apply P_id_iff_monotonic;assumption.
     intros ;apply P_id_u'14'1_monotonic;assumption.
     intros ;apply P_id_u'7'1_monotonic;assumption.
     intros ;apply P_id_x'2d_monotonic;assumption.
     intros ;apply P_id_u'13'1_monotonic;assumption.
     intros ;apply P_id_sequent_monotonic;assumption.
     intros ;apply P_id_u'10'1_monotonic;assumption.
     intros ;apply P_id_x'2a_monotonic;assumption.
     intros ;apply P_id_tautology'i'in_monotonic;assumption.
     intros ;apply P_id_cons_monotonic;assumption.
     intros ;apply P_id_u'6'2_monotonic;assumption.
     intros ;apply P_id_x'2b_monotonic;assumption.
     intros ;apply P_id_u'12'2_monotonic;assumption.
     intros ;apply P_id_reduce'ii'in_monotonic;assumption.
     intros ;apply P_id_p_monotonic;assumption.
     intros ;apply P_id_u'4'1_monotonic;assumption.
     intros ;apply P_id_u'15'1_monotonic;assumption.
     intros ;apply P_id_u'1'1_monotonic;assumption.
     intros ;apply P_id_u'8'1_monotonic;assumption.
     intros ;apply P_id_if_monotonic;assumption.
     intros ;apply P_id_u'11'1_monotonic;assumption.
     intros ;apply P_id_u'5'1_monotonic;assumption.
     intros ;apply P_id_u'16'1_monotonic;assumption.
     intros ;apply P_id_intersect'ii'in_bounded;assumption.
     intros ;apply P_id_tautology'i'out_bounded;assumption.
     intros ;apply P_id_u'6'1_bounded;assumption.
     intros ;apply P_id_u'3'1_bounded;assumption.
     intros ;apply P_id_u'12'1_bounded;assumption.
     intros ;apply P_id_u'2'1_bounded;assumption.
     intros ;apply P_id_u'9'1_bounded;assumption.
     intros ;apply P_id_iff_bounded;assumption.
     intros ;apply P_id_u'14'1_bounded;assumption.
     intros ;apply P_id_intersect'ii'out_bounded;assumption.
     intros ;apply P_id_u'7'1_bounded;assumption.
     intros ;apply P_id_x'2d_bounded;assumption.
     intros ;apply P_id_u'13'1_bounded;assumption.
     intros ;apply P_id_sequent_bounded;assumption.
     intros ;apply P_id_u'10'1_bounded;assumption.
     intros ;apply P_id_x'2a_bounded;assumption.
     intros ;apply P_id_tautology'i'in_bounded;assumption.
     intros ;apply P_id_cons_bounded;assumption.
     intros ;apply P_id_u'6'2_bounded;assumption.
     intros ;apply P_id_x'2b_bounded;assumption.
     intros ;apply P_id_u'12'2_bounded;assumption.
     intros ;apply P_id_reduce'ii'in_bounded;assumption.
     intros ;apply P_id_p_bounded;assumption.
     intros ;apply P_id_u'4'1_bounded;assumption.
     intros ;apply P_id_u'15'1_bounded;assumption.
     intros ;apply P_id_u'1'1_bounded;assumption.
     intros ;apply P_id_u'8'1_bounded;assumption.
     intros ;apply P_id_reduce'ii'out_bounded;assumption.
     intros ;apply P_id_nil_bounded;assumption.
     intros ;apply P_id_if_bounded;assumption.
     intros ;apply P_id_u'11'1_bounded;assumption.
     intros ;apply P_id_u'5'1_bounded;assumption.
     intros ;apply P_id_u'16'1_bounded;assumption.
     apply rules_monotonic.
   Qed.
   
   Definition P_id_U'12'2 (x20:Z) := 0.
   
   Definition P_id_U'6'1 (x20:Z) (x21:Z) (x22:Z) (x23:Z) (x24:Z) := 0.
   
   Definition P_id_REDUCE'II'IN (x20:Z) (x21:Z) := 0.
   
   Definition P_id_TAUTOLOGY'I'IN (x20:Z) := 0.
   
   Definition P_id_U'9'1 (x20:Z) := 0.
   
   Definition P_id_U'1'1 (x20:Z) := 0.
   
   Definition P_id_U'14'1 (x20:Z) := 0.
   
   Definition P_id_U'7'1 (x20:Z) := 0.
   
   Definition P_id_U'4'1 (x20:Z) := 0.
   
   Definition P_id_U'11'1 (x20:Z) := 0.
   
   Definition P_id_INTERSECT'II'IN (x20:Z) (x21:Z) := 1* x20 + 1* x21.
   
   Definition P_id_U'13'1 (x20:Z) := 0.
   
   Definition P_id_U'6'2 (x20:Z) := 0.
   
   Definition P_id_U'3'1 (x20:Z) := 0.
   
   Definition P_id_U'16'1 (x20:Z) := 0.
   
   Definition P_id_U'10'1 (x20:Z) := 0.
   
   Definition P_id_U'2'1 (x20:Z) := 0.
   
   Definition P_id_U'15'1 (x20:Z) := 0.
   
   Definition P_id_U'8'1 (x20:Z) := 0.
   
   Definition P_id_U'5'1 (x20:Z) := 0.
   
   Definition P_id_U'12'1 (x20:Z) (x21:Z) (x22:Z) (x23:Z) (x24:Z) := 0.
   
   Lemma P_id_U'12'2_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_U'12'2 x21 <= P_id_U'12'2 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U'6'1_monotonic :
    forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, 
     (0 <= x29)/\ (x29 <= x28) ->
      (0 <= x27)/\ (x27 <= x26) ->
       (0 <= x25)/\ (x25 <= x24) ->
        (0 <= x23)/\ (x23 <= x22) ->
         (0 <= x21)/\ (x21 <= x20) ->
          P_id_U'6'1 x21 x23 x25 x27 x29 <= P_id_U'6'1 x20 x22 x24 x26 x28.
   Proof.
     intros x29 x28 x27 x26 x25 x24 x23 x22 x21 x20.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     intros [H_7 H_6].
     intros [H_9 H_8].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_REDUCE'II'IN_monotonic :
    forall x20 x22 x21 x23, 
     (0 <= x23)/\ (x23 <= x22) ->
      (0 <= x21)/\ (x21 <= x20) ->
       P_id_REDUCE'II'IN x21 x23 <= P_id_REDUCE'II'IN x20 x22.
   Proof.
     intros x23 x22 x21 x20.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_TAUTOLOGY'I'IN_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->
      P_id_TAUTOLOGY'I'IN x21 <= P_id_TAUTOLOGY'I'IN x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U'9'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_U'9'1 x21 <= P_id_U'9'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U'1'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_U'1'1 x21 <= P_id_U'1'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U'14'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_U'14'1 x21 <= P_id_U'14'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U'7'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_U'7'1 x21 <= P_id_U'7'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U'4'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_U'4'1 x21 <= P_id_U'4'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U'11'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_U'11'1 x21 <= P_id_U'11'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_INTERSECT'II'IN_monotonic :
    forall x20 x22 x21 x23, 
     (0 <= x23)/\ (x23 <= x22) ->
      (0 <= x21)/\ (x21 <= x20) ->
       P_id_INTERSECT'II'IN x21 x23 <= P_id_INTERSECT'II'IN x20 x22.
   Proof.
     intros x23 x22 x21 x20.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U'13'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_U'13'1 x21 <= P_id_U'13'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U'6'2_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_U'6'2 x21 <= P_id_U'6'2 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U'3'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_U'3'1 x21 <= P_id_U'3'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U'16'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_U'16'1 x21 <= P_id_U'16'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U'10'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_U'10'1 x21 <= P_id_U'10'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U'2'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_U'2'1 x21 <= P_id_U'2'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U'15'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_U'15'1 x21 <= P_id_U'15'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U'8'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_U'8'1 x21 <= P_id_U'8'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U'5'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_U'5'1 x21 <= P_id_U'5'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U'12'1_monotonic :
    forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, 
     (0 <= x29)/\ (x29 <= x28) ->
      (0 <= x27)/\ (x27 <= x26) ->
       (0 <= x25)/\ (x25 <= x24) ->
        (0 <= x23)/\ (x23 <= x22) ->
         (0 <= x21)/\ (x21 <= x20) ->
          P_id_U'12'1 x21 x23 x25 x27 x29 <= P_id_U'12'1 x20 x22 x24 x26 x28.
   Proof.
     intros x29 x28 x27 x26 x25 x24 x23 x22 x21 x20.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     intros [H_7 H_6].
     intros [H_9 H_8].
     
     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_intersect'ii'in P_id_tautology'i'out 
      P_id_u'6'1 P_id_u'3'1 P_id_u'12'1 P_id_u'2'1 P_id_u'9'1 P_id_iff 
      P_id_u'14'1 P_id_intersect'ii'out P_id_u'7'1 P_id_x'2d P_id_u'13'1 
      P_id_sequent P_id_u'10'1 P_id_x'2a P_id_tautology'i'in P_id_cons 
      P_id_u'6'2 P_id_x'2b P_id_u'12'2 P_id_reduce'ii'in P_id_p P_id_u'4'1 
      P_id_u'15'1 P_id_u'1'1 P_id_u'8'1 P_id_reduce'ii'out P_id_nil P_id_if 
      P_id_u'11'1 P_id_u'5'1 P_id_u'16'1 P_id_U'12'2 P_id_U'6'1 
      P_id_REDUCE'II'IN P_id_TAUTOLOGY'I'IN P_id_U'9'1 P_id_U'1'1 
      P_id_U'14'1 P_id_U'7'1 P_id_U'4'1 P_id_U'11'1 P_id_INTERSECT'II'IN 
      P_id_U'13'1 P_id_U'6'2 P_id_U'3'1 P_id_U'16'1 P_id_U'10'1 P_id_U'2'1 
      P_id_U'15'1 P_id_U'8'1 P_id_U'5'1 P_id_U'12'1.
   
   Lemma marked_measure_equation :
    forall t, 
     marked_measure t = match t with
                          | (algebra.Alg.Term algebra.F.id_u'12'2 (x20::nil)) =>
                           P_id_U'12'2 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'6'1 (x24::x23::
                             x22::x21::x20::nil)) =>
                           P_id_U'6'1 (measure x24) (measure x23) 
                            (measure x22) (measure x21) (measure x20)
                          | (algebra.Alg.Term algebra.F.id_reduce'ii'in 
                             (x21::x20::nil)) =>
                           P_id_REDUCE'II'IN (measure x21) (measure x20)
                          | (algebra.Alg.Term algebra.F.id_tautology'i'in 
                             (x20::nil)) =>
                           P_id_TAUTOLOGY'I'IN (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'9'1 (x20::nil)) =>
                           P_id_U'9'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'1'1 (x20::nil)) =>
                           P_id_U'1'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'14'1 (x20::nil)) =>
                           P_id_U'14'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'7'1 (x20::nil)) =>
                           P_id_U'7'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'4'1 (x20::nil)) =>
                           P_id_U'4'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'11'1 (x20::nil)) =>
                           P_id_U'11'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_intersect'ii'in 
                             (x21::x20::nil)) =>
                           P_id_INTERSECT'II'IN (measure x21) (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'13'1 (x20::nil)) =>
                           P_id_U'13'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'6'2 (x20::nil)) =>
                           P_id_U'6'2 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'3'1 (x20::nil)) =>
                           P_id_U'3'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'16'1 (x20::nil)) =>
                           P_id_U'16'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'10'1 (x20::nil)) =>
                           P_id_U'10'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'2'1 (x20::nil)) =>
                           P_id_U'2'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'15'1 (x20::nil)) =>
                           P_id_U'15'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'8'1 (x20::nil)) =>
                           P_id_U'8'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'5'1 (x20::nil)) =>
                           P_id_U'5'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'12'1 (x24::x23::
                             x22::x21::x20::nil)) =>
                           P_id_U'12'1 (measure x24) (measure x23) 
                            (measure x22) (measure x21) (measure x20)
                          | _ => 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_intersect'ii'in_monotonic;assumption.
     intros ;apply P_id_u'6'1_monotonic;assumption.
     intros ;apply P_id_u'3'1_monotonic;assumption.
     intros ;apply P_id_u'12'1_monotonic;assumption.
     intros ;apply P_id_u'2'1_monotonic;assumption.
     intros ;apply P_id_u'9'1_monotonic;assumption.
     intros ;apply P_id_iff_monotonic;assumption.
     intros ;apply P_id_u'14'1_monotonic;assumption.
     intros ;apply P_id_u'7'1_monotonic;assumption.
     intros ;apply P_id_x'2d_monotonic;assumption.
     intros ;apply P_id_u'13'1_monotonic;assumption.
     intros ;apply P_id_sequent_monotonic;assumption.
     intros ;apply P_id_u'10'1_monotonic;assumption.
     intros ;apply P_id_x'2a_monotonic;assumption.
     intros ;apply P_id_tautology'i'in_monotonic;assumption.
     intros ;apply P_id_cons_monotonic;assumption.
     intros ;apply P_id_u'6'2_monotonic;assumption.
     intros ;apply P_id_x'2b_monotonic;assumption.
     intros ;apply P_id_u'12'2_monotonic;assumption.
     intros ;apply P_id_reduce'ii'in_monotonic;assumption.
     intros ;apply P_id_p_monotonic;assumption.
     intros ;apply P_id_u'4'1_monotonic;assumption.
     intros ;apply P_id_u'15'1_monotonic;assumption.
     intros ;apply P_id_u'1'1_monotonic;assumption.
     intros ;apply P_id_u'8'1_monotonic;assumption.
     intros ;apply P_id_if_monotonic;assumption.
     intros ;apply P_id_u'11'1_monotonic;assumption.
     intros ;apply P_id_u'5'1_monotonic;assumption.
     intros ;apply P_id_u'16'1_monotonic;assumption.
     intros ;apply P_id_intersect'ii'in_bounded;assumption.
     intros ;apply P_id_tautology'i'out_bounded;assumption.
     intros ;apply P_id_u'6'1_bounded;assumption.
     intros ;apply P_id_u'3'1_bounded;assumption.
     intros ;apply P_id_u'12'1_bounded;assumption.
     intros ;apply P_id_u'2'1_bounded;assumption.
     intros ;apply P_id_u'9'1_bounded;assumption.
     intros ;apply P_id_iff_bounded;assumption.
     intros ;apply P_id_u'14'1_bounded;assumption.
     intros ;apply P_id_intersect'ii'out_bounded;assumption.
     intros ;apply P_id_u'7'1_bounded;assumption.
     intros ;apply P_id_x'2d_bounded;assumption.
     intros ;apply P_id_u'13'1_bounded;assumption.
     intros ;apply P_id_sequent_bounded;assumption.
     intros ;apply P_id_u'10'1_bounded;assumption.
     intros ;apply P_id_x'2a_bounded;assumption.
     intros ;apply P_id_tautology'i'in_bounded;assumption.
     intros ;apply P_id_cons_bounded;assumption.
     intros ;apply P_id_u'6'2_bounded;assumption.
     intros ;apply P_id_x'2b_bounded;assumption.
     intros ;apply P_id_u'12'2_bounded;assumption.
     intros ;apply P_id_reduce'ii'in_bounded;assumption.
     intros ;apply P_id_p_bounded;assumption.
     intros ;apply P_id_u'4'1_bounded;assumption.
     intros ;apply P_id_u'15'1_bounded;assumption.
     intros ;apply P_id_u'1'1_bounded;assumption.
     intros ;apply P_id_u'8'1_bounded;assumption.
     intros ;apply P_id_reduce'ii'out_bounded;assumption.
     intros ;apply P_id_nil_bounded;assumption.
     intros ;apply P_id_if_bounded;assumption.
     intros ;apply P_id_u'11'1_bounded;assumption.
     intros ;apply P_id_u'5'1_bounded;assumption.
     intros ;apply P_id_u'16'1_bounded;assumption.
     apply rules_monotonic.
     intros ;apply P_id_U'12'2_monotonic;assumption.
     intros ;apply P_id_U'6'1_monotonic;assumption.
     intros ;apply P_id_REDUCE'II'IN_monotonic;assumption.
     intros ;apply P_id_TAUTOLOGY'I'IN_monotonic;assumption.
     intros ;apply P_id_U'9'1_monotonic;assumption.
     intros ;apply P_id_U'1'1_monotonic;assumption.
     intros ;apply P_id_U'14'1_monotonic;assumption.
     intros ;apply P_id_U'7'1_monotonic;assumption.
     intros ;apply P_id_U'4'1_monotonic;assumption.
     intros ;apply P_id_U'11'1_monotonic;assumption.
     intros ;apply P_id_INTERSECT'II'IN_monotonic;assumption.
     intros ;apply P_id_U'13'1_monotonic;assumption.
     intros ;apply P_id_U'6'2_monotonic;assumption.
     intros ;apply P_id_U'3'1_monotonic;assumption.
     intros ;apply P_id_U'16'1_monotonic;assumption.
     intros ;apply P_id_U'10'1_monotonic;assumption.
     intros ;apply P_id_U'2'1_monotonic;assumption.
     intros ;apply P_id_U'15'1_monotonic;assumption.
     intros ;apply P_id_U'8'1_monotonic;assumption.
     intros ;apply P_id_U'5'1_monotonic;assumption.
     intros ;apply P_id_U'12'1_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.DP_R_xml_0_scc_17.
   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_17.
  
  Definition wf_DP_R_xml_0_scc_17  := WF_DP_R_xml_0_scc_17.wf.
  
  
  Lemma acc_DP_R_xml_0_scc_17 :
   forall x y, (DP_R_xml_0_scc_17 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_17).
    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_16;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_15;
          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_17.
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_18  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <reduce'ii'in(sequent(nil,nil),sequent(F1_,F2_)),intersect'ii'in(F1_,F2_)> *)
    | DP_R_xml_0_non_scc_18_0 :
     forall x20 x12 x21 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term 
         algebra.F.id_nil nil)::(algebra.Alg.Term algebra.F.id_nil 
         nil)::nil)) x21) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_sequent (x11::x12::nil)) x20) ->
        DP_R_xml_0_non_scc_18 (algebra.Alg.Term algebra.F.id_intersect'ii'in 
                               (x11::x12::nil)) 
         (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_18 :
   forall x y, 
    (DP_R_xml_0_non_scc_18 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_17;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_16;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_15;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((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_19  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <reduce'ii'in(sequent(cons(x'2b(F1_,F2_),Fs_),Gs_),NF_),u'6'1(reduce'ii'in(sequent(cons(F1_,Fs_),Gs_),NF_),F2_,Fs_,Gs_,NF_)> *)
    | DP_R_xml_0_scc_19_0 :
     forall x8 x20 x12 x10 x9 x21 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term 
         algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b (x11::
         x12::nil))::x8::nil))::x9::nil)) x21) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x10 x20) ->
        DP_R_xml_0_scc_19 (algebra.Alg.Term algebra.F.id_u'6'1 
                           ((algebra.Alg.Term algebra.F.id_reduce'ii'in 
                           ((algebra.Alg.Term algebra.F.id_sequent 
                           ((algebra.Alg.Term algebra.F.id_cons (x11::
                           x8::nil))::x9::nil))::x10::nil))::x12::x8::x9::
                           x10::nil)) 
         (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
    
     (* <u'6'1(reduce'ii'out,F2_,Fs_,Gs_,NF_),reduce'ii'in(sequent(cons(F2_,Fs_),Gs_),NF_)> *)
    | DP_R_xml_0_scc_19_1 :
     forall x8 x24 x20 x12 x10 x22 x9 x21 x23, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) x24) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x12 x23) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x8 x22) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x9 x21) ->
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     x10 x20) ->
           DP_R_xml_0_scc_19 (algebra.Alg.Term algebra.F.id_reduce'ii'in 
                              ((algebra.Alg.Term algebra.F.id_sequent 
                              ((algebra.Alg.Term algebra.F.id_cons (x12::
                              x8::nil))::x9::nil))::x10::nil)) 
            (algebra.Alg.Term algebra.F.id_u'6'1 (x24::x23::x22::x21::
             x20::nil))
    
     (* <reduce'ii'in(sequent(cons(if(A_,B_),Fs_),Gs_),NF_),reduce'ii'in(sequent(cons(x'2b(x'2d(B_),A_),Fs_),Gs_),NF_)> *)
    | DP_R_xml_0_scc_19_2 :
     forall x8 x20 x10 x6 x9 x21 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term 
         algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_if (x6::
         x7::nil))::x8::nil))::x9::nil)) x21) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x10 x20) ->
        DP_R_xml_0_scc_19 (algebra.Alg.Term algebra.F.id_reduce'ii'in 
                           ((algebra.Alg.Term algebra.F.id_sequent 
                           ((algebra.Alg.Term algebra.F.id_cons 
                           ((algebra.Alg.Term algebra.F.id_x'2b 
                           ((algebra.Alg.Term algebra.F.id_x'2d (x7::nil))::
                           x6::nil))::x8::nil))::x9::nil))::x10::nil)) 
         (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
    
     (* <reduce'ii'in(sequent(cons(x'2b(F1_,F2_),Fs_),Gs_),NF_),reduce'ii'in(sequent(cons(F1_,Fs_),Gs_),NF_)> *)
    | DP_R_xml_0_scc_19_3 :
     forall x8 x20 x12 x10 x9 x21 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term 
         algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b (x11::
         x12::nil))::x8::nil))::x9::nil)) x21) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x10 x20) ->
        DP_R_xml_0_scc_19 (algebra.Alg.Term algebra.F.id_reduce'ii'in 
                           ((algebra.Alg.Term algebra.F.id_sequent 
                           ((algebra.Alg.Term algebra.F.id_cons (x11::
                           x8::nil))::x9::nil))::x10::nil)) 
         (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
    
     (* <reduce'ii'in(sequent(cons(iff(A_,B_),Fs_),Gs_),NF_),reduce'ii'in(sequent(cons(x'2a(if(A_,B_),if(B_,A_)),Fs_),Gs_),NF_)> *)
    | DP_R_xml_0_scc_19_4 :
     forall x8 x20 x10 x6 x9 x21 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term 
         algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_iff (x6::
         x7::nil))::x8::nil))::x9::nil)) x21) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x10 x20) ->
        DP_R_xml_0_scc_19 (algebra.Alg.Term algebra.F.id_reduce'ii'in 
                           ((algebra.Alg.Term algebra.F.id_sequent 
                           ((algebra.Alg.Term algebra.F.id_cons 
                           ((algebra.Alg.Term algebra.F.id_x'2a 
                           ((algebra.Alg.Term algebra.F.id_if (x6::
                           x7::nil))::(algebra.Alg.Term algebra.F.id_if (x7::
                           x6::nil))::nil))::x8::nil))::x9::nil))::x10::nil)) 
         (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
    
     (* <reduce'ii'in(sequent(cons(x'2a(F1_,F2_),Fs_),Gs_),NF_),reduce'ii'in(sequent(cons(F1_,cons(F2_,Fs_)),Gs_),NF_)> *)
    | DP_R_xml_0_scc_19_5 :
     forall x8 x20 x12 x10 x9 x21 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term 
         algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2a (x11::
         x12::nil))::x8::nil))::x9::nil)) x21) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x10 x20) ->
        DP_R_xml_0_scc_19 (algebra.Alg.Term algebra.F.id_reduce'ii'in 
                           ((algebra.Alg.Term algebra.F.id_sequent 
                           ((algebra.Alg.Term algebra.F.id_cons (x11::
                           (algebra.Alg.Term algebra.F.id_cons (x12::
                           x8::nil))::nil))::x9::nil))::x10::nil)) 
         (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
    
     (* <reduce'ii'in(sequent(cons(x'2d(F1_),Fs_),Gs_),NF_),reduce'ii'in(sequent(Fs_,cons(F1_,Gs_)),NF_)> *)
    | DP_R_xml_0_scc_19_6 :
     forall x8 x20 x10 x9 x21 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term 
         algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2d (x11::nil))::
         x8::nil))::x9::nil)) x21) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x10 x20) ->
        DP_R_xml_0_scc_19 (algebra.Alg.Term algebra.F.id_reduce'ii'in 
                           ((algebra.Alg.Term algebra.F.id_sequent (x8::
                           (algebra.Alg.Term algebra.F.id_cons (x11::
                           x9::nil))::nil))::x10::nil)) 
         (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
    
     (* <reduce'ii'in(sequent(Fs_,cons(if(A_,B_),Gs_)),NF_),reduce'ii'in(sequent(Fs_,cons(x'2b(x'2d(B_),A_),Gs_)),NF_)> *)
    | DP_R_xml_0_scc_19_7 :
     forall x8 x20 x10 x6 x9 x21 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term 
         algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_if (x6::
         x7::nil))::x9::nil))::nil)) x21) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x10 x20) ->
        DP_R_xml_0_scc_19 (algebra.Alg.Term algebra.F.id_reduce'ii'in 
                           ((algebra.Alg.Term algebra.F.id_sequent (x8::
                           (algebra.Alg.Term algebra.F.id_cons 
                           ((algebra.Alg.Term algebra.F.id_x'2b 
                           ((algebra.Alg.Term algebra.F.id_x'2d (x7::nil))::
                           x6::nil))::x9::nil))::nil))::x10::nil)) 
         (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
    
     (* <reduce'ii'in(sequent(cons(p(V_),Fs_),Gs_),sequent(Left_,Right_)),reduce'ii'in(sequent(Fs_,Gs_),sequent(cons(p(V_),Left_),Right_))> *)
    | DP_R_xml_0_scc_19_8 :
     forall x8 x20 x14 x9 x21 x13 x15, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term 
         algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_p (x13::nil))::
         x8::nil))::x9::nil)) x21) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_sequent (x14::x15::nil)) x20) ->
        DP_R_xml_0_scc_19 (algebra.Alg.Term algebra.F.id_reduce'ii'in 
                           ((algebra.Alg.Term algebra.F.id_sequent (x8::
                           x9::nil))::(algebra.Alg.Term algebra.F.id_sequent 
                           ((algebra.Alg.Term algebra.F.id_cons 
                           ((algebra.Alg.Term algebra.F.id_p (x13::nil))::
                           x14::nil))::x15::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
    
     (* <reduce'ii'in(sequent(Fs_,cons(iff(A_,B_),Gs_)),NF_),reduce'ii'in(sequent(Fs_,cons(x'2a(if(A_,B_),if(B_,A_)),Gs_)),NF_)> *)
    | DP_R_xml_0_scc_19_9 :
     forall x8 x20 x10 x6 x9 x21 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term 
         algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_iff (x6::
         x7::nil))::x9::nil))::nil)) x21) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x10 x20) ->
        DP_R_xml_0_scc_19 (algebra.Alg.Term algebra.F.id_reduce'ii'in 
                           ((algebra.Alg.Term algebra.F.id_sequent (x8::
                           (algebra.Alg.Term algebra.F.id_cons 
                           ((algebra.Alg.Term algebra.F.id_x'2a 
                           ((algebra.Alg.Term algebra.F.id_if (x6::
                           x7::nil))::(algebra.Alg.Term algebra.F.id_if (x7::
                           x6::nil))::nil))::x9::nil))::nil))::x10::nil)) 
         (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
    
     (* <reduce'ii'in(sequent(Fs_,cons(x'2a(G1_,G2_),Gs_)),NF_),u'12'1(reduce'ii'in(sequent(Fs_,cons(G1_,Gs_)),NF_),Fs_,G2_,Gs_,NF_)> *)
    | DP_R_xml_0_scc_19_10 :
     forall x16 x8 x20 x10 x17 x9 x21, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term 
         algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2a (x16::
         x17::nil))::x9::nil))::nil)) x21) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x10 x20) ->
        DP_R_xml_0_scc_19 (algebra.Alg.Term algebra.F.id_u'12'1 
                           ((algebra.Alg.Term algebra.F.id_reduce'ii'in 
                           ((algebra.Alg.Term algebra.F.id_sequent (x8::
                           (algebra.Alg.Term algebra.F.id_cons (x16::
                           x9::nil))::nil))::x10::nil))::x8::x17::x9::
                           x10::nil)) 
         (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
    
     (* <u'12'1(reduce'ii'out,Fs_,G2_,Gs_,NF_),reduce'ii'in(sequent(Fs_,cons(G2_,Gs_)),NF_)> *)
    | DP_R_xml_0_scc_19_11 :
     forall x8 x24 x20 x10 x22 x17 x9 x21 x23, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) x24) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x8 x23) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x17 x22) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x9 x21) ->
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     x10 x20) ->
           DP_R_xml_0_scc_19 (algebra.Alg.Term algebra.F.id_reduce'ii'in 
                              ((algebra.Alg.Term algebra.F.id_sequent (x8::
                              (algebra.Alg.Term algebra.F.id_cons (x17::
                              x9::nil))::nil))::x10::nil)) 
            (algebra.Alg.Term algebra.F.id_u'12'1 (x24::x23::x22::x21::
             x20::nil))
    
     (* <reduce'ii'in(sequent(Fs_,cons(x'2b(G1_,G2_),Gs_)),NF_),reduce'ii'in(sequent(Fs_,cons(G1_,cons(G2_,Gs_))),NF_)> *)
    | DP_R_xml_0_scc_19_12 :
     forall x16 x8 x20 x10 x17 x9 x21, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term 
         algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b (x16::
         x17::nil))::x9::nil))::nil)) x21) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x10 x20) ->
        DP_R_xml_0_scc_19 (algebra.Alg.Term algebra.F.id_reduce'ii'in 
                           ((algebra.Alg.Term algebra.F.id_sequent (x8::
                           (algebra.Alg.Term algebra.F.id_cons (x16::
                           (algebra.Alg.Term algebra.F.id_cons (x17::
                           x9::nil))::nil))::nil))::x10::nil)) 
         (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
    
     (* <reduce'ii'in(sequent(Fs_,cons(x'2a(G1_,G2_),Gs_)),NF_),reduce'ii'in(sequent(Fs_,cons(G1_,Gs_)),NF_)> *)
    | DP_R_xml_0_scc_19_13 :
     forall x16 x8 x20 x10 x17 x9 x21, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term 
         algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2a (x16::
         x17::nil))::x9::nil))::nil)) x21) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x10 x20) ->
        DP_R_xml_0_scc_19 (algebra.Alg.Term algebra.F.id_reduce'ii'in 
                           ((algebra.Alg.Term algebra.F.id_sequent (x8::
                           (algebra.Alg.Term algebra.F.id_cons (x16::
                           x9::nil))::nil))::x10::nil)) 
         (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
    
     (* <reduce'ii'in(sequent(Fs_,cons(x'2d(G1_),Gs_)),NF_),reduce'ii'in(sequent(cons(G1_,Fs_),Gs_),NF_)> *)
    | DP_R_xml_0_scc_19_14 :
     forall x16 x8 x20 x10 x9 x21, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term 
         algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2d (x16::nil))::
         x9::nil))::nil)) x21) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x10 x20) ->
        DP_R_xml_0_scc_19 (algebra.Alg.Term algebra.F.id_reduce'ii'in 
                           ((algebra.Alg.Term algebra.F.id_sequent 
                           ((algebra.Alg.Term algebra.F.id_cons (x16::
                           x8::nil))::x9::nil))::x10::nil)) 
         (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
    
     (* <reduce'ii'in(sequent(nil,cons(p(V_),Gs_)),sequent(Left_,Right_)),reduce'ii'in(sequent(nil,Gs_),sequent(Left_,cons(p(V_),Right_)))> *)
    | DP_R_xml_0_scc_19_15 :
     forall x20 x14 x9 x21 x13 x15, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term 
         algebra.F.id_nil nil)::(algebra.Alg.Term algebra.F.id_cons 
         ((algebra.Alg.Term algebra.F.id_p (x13::nil))::x9::nil))::nil)) 
        x21) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_sequent (x14::x15::nil)) x20) ->
        DP_R_xml_0_scc_19 (algebra.Alg.Term algebra.F.id_reduce'ii'in 
                           ((algebra.Alg.Term algebra.F.id_sequent 
                           ((algebra.Alg.Term algebra.F.id_nil nil)::
                           x9::nil))::(algebra.Alg.Term algebra.F.id_sequent 
                           (x14::(algebra.Alg.Term algebra.F.id_cons 
                           ((algebra.Alg.Term algebra.F.id_p (x13::nil))::
                           x15::nil))::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
  .
  
  
  Module WF_DP_R_xml_0_scc_19.
   Inductive DP_R_xml_0_scc_19_large  :
    algebra.Alg.term ->algebra.Alg.term ->Prop := 
      (* <reduce'ii'in(sequent(cons(x'2b(F1_,F2_),Fs_),Gs_),NF_),u'6'1(reduce'ii'in(sequent(cons(F1_,Fs_),Gs_),NF_),F2_,Fs_,Gs_,NF_)> *)
     | DP_R_xml_0_scc_19_large_0 :
      forall x8 x20 x12 x10 x9 x21 x11, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term 
          algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b (x11::
          x12::nil))::x8::nil))::x9::nil)) x21) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x10 x20) ->
         DP_R_xml_0_scc_19_large (algebra.Alg.Term algebra.F.id_u'6'1 
                                  ((algebra.Alg.Term 
                                  algebra.F.id_reduce'ii'in 
                                  ((algebra.Alg.Term algebra.F.id_sequent 
                                  ((algebra.Alg.Term algebra.F.id_cons (x11::
                                  x8::nil))::x9::nil))::x10::nil))::x12::x8::
                                  x9::x10::nil)) 
          (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
     
      (* <u'6'1(reduce'ii'out,F2_,Fs_,Gs_,NF_),reduce'ii'in(sequent(cons(F2_,Fs_),Gs_),NF_)> *)
     | DP_R_xml_0_scc_19_large_1 :
      forall x8 x24 x20 x12 x10 x22 x9 x21 x23, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) x24) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x12 x23) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x8 x22) ->
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     x9 x21) ->
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                      x10 x20) ->
            DP_R_xml_0_scc_19_large (algebra.Alg.Term 
                                     algebra.F.id_reduce'ii'in 
                                     ((algebra.Alg.Term algebra.F.id_sequent 
                                     ((algebra.Alg.Term algebra.F.id_cons 
                                     (x12::x8::nil))::x9::nil))::x10::nil)) 
             (algebra.Alg.Term algebra.F.id_u'6'1 (x24::x23::x22::x21::
              x20::nil))
     
      (* <reduce'ii'in(sequent(cons(if(A_,B_),Fs_),Gs_),NF_),reduce'ii'in(sequent(cons(x'2b(x'2d(B_),A_),Fs_),Gs_),NF_)> *)
     | DP_R_xml_0_scc_19_large_2 :
      forall x8 x20 x10 x6 x9 x21 x7, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term 
          algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_if (x6::
          x7::nil))::x8::nil))::x9::nil)) x21) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x10 x20) ->
         DP_R_xml_0_scc_19_large (algebra.Alg.Term algebra.F.id_reduce'ii'in 
                                  ((algebra.Alg.Term algebra.F.id_sequent 
                                  ((algebra.Alg.Term algebra.F.id_cons 
                                  ((algebra.Alg.Term algebra.F.id_x'2b 
                                  ((algebra.Alg.Term algebra.F.id_x'2d 
                                  (x7::nil))::x6::nil))::x8::nil))::
                                  x9::nil))::x10::nil)) 
          (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
     
      (* <reduce'ii'in(sequent(cons(x'2b(F1_,F2_),Fs_),Gs_),NF_),reduce'ii'in(sequent(cons(F1_,Fs_),Gs_),NF_)> *)
     | DP_R_xml_0_scc_19_large_3 :
      forall x8 x20 x12 x10 x9 x21 x11, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term 
          algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b (x11::
          x12::nil))::x8::nil))::x9::nil)) x21) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x10 x20) ->
         DP_R_xml_0_scc_19_large (algebra.Alg.Term algebra.F.id_reduce'ii'in 
                                  ((algebra.Alg.Term algebra.F.id_sequent 
                                  ((algebra.Alg.Term algebra.F.id_cons (x11::
                                  x8::nil))::x9::nil))::x10::nil)) 
          (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
     
      (* <reduce'ii'in(sequent(cons(x'2a(F1_,F2_),Fs_),Gs_),NF_),reduce'ii'in(sequent(cons(F1_,cons(F2_,Fs_)),Gs_),NF_)> *)
     | DP_R_xml_0_scc_19_large_4 :
      forall x8 x20 x12 x10 x9 x21 x11, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term 
          algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2a (x11::
          x12::nil))::x8::nil))::x9::nil)) x21) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x10 x20) ->
         DP_R_xml_0_scc_19_large (algebra.Alg.Term algebra.F.id_reduce'ii'in 
                                  ((algebra.Alg.Term algebra.F.id_sequent 
                                  ((algebra.Alg.Term algebra.F.id_cons (x11::
                                  (algebra.Alg.Term algebra.F.id_cons (x12::
                                  x8::nil))::nil))::x9::nil))::x10::nil)) 
          (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
     
      (* <reduce'ii'in(sequent(cons(x'2d(F1_),Fs_),Gs_),NF_),reduce'ii'in(sequent(Fs_,cons(F1_,Gs_)),NF_)> *)
     | DP_R_xml_0_scc_19_large_5 :
      forall x8 x20 x10 x9 x21 x11, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term 
          algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2d 
          (x11::nil))::x8::nil))::x9::nil)) x21) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x10 x20) ->
         DP_R_xml_0_scc_19_large (algebra.Alg.Term algebra.F.id_reduce'ii'in 
                                  ((algebra.Alg.Term algebra.F.id_sequent 
                                  (x8::(algebra.Alg.Term algebra.F.id_cons 
                                  (x11::x9::nil))::nil))::x10::nil)) 
          (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
     
      (* <reduce'ii'in(sequent(Fs_,cons(if(A_,B_),Gs_)),NF_),reduce'ii'in(sequent(Fs_,cons(x'2b(x'2d(B_),A_),Gs_)),NF_)> *)
     | DP_R_xml_0_scc_19_large_6 :
      forall x8 x20 x10 x6 x9 x21 x7, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term 
          algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_if (x6::
          x7::nil))::x9::nil))::nil)) x21) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x10 x20) ->
         DP_R_xml_0_scc_19_large (algebra.Alg.Term algebra.F.id_reduce'ii'in 
                                  ((algebra.Alg.Term algebra.F.id_sequent 
                                  (x8::(algebra.Alg.Term algebra.F.id_cons 
                                  ((algebra.Alg.Term algebra.F.id_x'2b 
                                  ((algebra.Alg.Term algebra.F.id_x'2d 
                                  (x7::nil))::x6::nil))::x9::nil))::nil))::
                                  x10::nil)) 
          (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
     
      (* <reduce'ii'in(sequent(Fs_,cons(x'2a(G1_,G2_),Gs_)),NF_),u'12'1(reduce'ii'in(sequent(Fs_,cons(G1_,Gs_)),NF_),Fs_,G2_,Gs_,NF_)> *)
     | DP_R_xml_0_scc_19_large_7 :
      forall x16 x8 x20 x10 x17 x9 x21, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term 
          algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2a (x16::
          x17::nil))::x9::nil))::nil)) x21) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x10 x20) ->
         DP_R_xml_0_scc_19_large (algebra.Alg.Term algebra.F.id_u'12'1 
                                  ((algebra.Alg.Term 
                                  algebra.F.id_reduce'ii'in 
                                  ((algebra.Alg.Term algebra.F.id_sequent 
                                  (x8::(algebra.Alg.Term algebra.F.id_cons 
                                  (x16::x9::nil))::nil))::x10::nil))::x8::
                                  x17::x9::x10::nil)) 
          (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
     
      (* <u'12'1(reduce'ii'out,Fs_,G2_,Gs_,NF_),reduce'ii'in(sequent(Fs_,cons(G2_,Gs_)),NF_)> *)
     | DP_R_xml_0_scc_19_large_8 :
      forall x8 x24 x20 x10 x22 x17 x9 x21 x23, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) x24) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x8 x23) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x17 x22) ->
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     x9 x21) ->
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                      x10 x20) ->
            DP_R_xml_0_scc_19_large (algebra.Alg.Term 
                                     algebra.F.id_reduce'ii'in 
                                     ((algebra.Alg.Term algebra.F.id_sequent 
                                     (x8::(algebra.Alg.Term 
                                     algebra.F.id_cons (x17::
                                     x9::nil))::nil))::x10::nil)) 
             (algebra.Alg.Term algebra.F.id_u'12'1 (x24::x23::x22::x21::
              x20::nil))
     
      (* <reduce'ii'in(sequent(Fs_,cons(x'2b(G1_,G2_),Gs_)),NF_),reduce'ii'in(sequent(Fs_,cons(G1_,cons(G2_,Gs_))),NF_)> *)
     | DP_R_xml_0_scc_19_large_9 :
      forall x16 x8 x20 x10 x17 x9 x21, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term 
          algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b (x16::
          x17::nil))::x9::nil))::nil)) x21) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x10 x20) ->
         DP_R_xml_0_scc_19_large (algebra.Alg.Term algebra.F.id_reduce'ii'in 
                                  ((algebra.Alg.Term algebra.F.id_sequent 
                                  (x8::(algebra.Alg.Term algebra.F.id_cons 
                                  (x16::(algebra.Alg.Term algebra.F.id_cons 
                                  (x17::x9::nil))::nil))::nil))::x10::nil)) 
          (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
     
      (* <reduce'ii'in(sequent(Fs_,cons(x'2a(G1_,G2_),Gs_)),NF_),reduce'ii'in(sequent(Fs_,cons(G1_,Gs_)),NF_)> *)
     | DP_R_xml_0_scc_19_large_10 :
      forall x16 x8 x20 x10 x17 x9 x21, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term 
          algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2a (x16::
          x17::nil))::x9::nil))::nil)) x21) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x10 x20) ->
         DP_R_xml_0_scc_19_large (algebra.Alg.Term algebra.F.id_reduce'ii'in 
                                  ((algebra.Alg.Term algebra.F.id_sequent 
                                  (x8::(algebra.Alg.Term algebra.F.id_cons 
                                  (x16::x9::nil))::nil))::x10::nil)) 
          (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
     
      (* <reduce'ii'in(sequent(Fs_,cons(x'2d(G1_),Gs_)),NF_),reduce'ii'in(sequent(cons(G1_,Fs_),Gs_),NF_)> *)
     | DP_R_xml_0_scc_19_large_11 :
      forall x16 x8 x20 x10 x9 x21, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term 
          algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2d 
          (x16::nil))::x9::nil))::nil)) x21) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x10 x20) ->
         DP_R_xml_0_scc_19_large (algebra.Alg.Term algebra.F.id_reduce'ii'in 
                                  ((algebra.Alg.Term algebra.F.id_sequent 
                                  ((algebra.Alg.Term algebra.F.id_cons (x16::
                                  x8::nil))::x9::nil))::x10::nil)) 
          (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
   .
   
   
   Inductive DP_R_xml_0_scc_19_strict  :
    algebra.Alg.term ->algebra.Alg.term ->Prop := 
      (* <reduce'ii'in(sequent(cons(iff(A_,B_),Fs_),Gs_),NF_),reduce'ii'in(sequent(cons(x'2a(if(A_,B_),if(B_,A_)),Fs_),Gs_),NF_)> *)
     | DP_R_xml_0_scc_19_strict_0 :
      forall x8 x20 x10 x6 x9 x21 x7, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term 
          algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_iff (x6::
          x7::nil))::x8::nil))::x9::nil)) x21) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x10 x20) ->
         DP_R_xml_0_scc_19_strict (algebra.Alg.Term 
                                   algebra.F.id_reduce'ii'in 
                                   ((algebra.Alg.Term algebra.F.id_sequent 
                                   ((algebra.Alg.Term algebra.F.id_cons 
                                   ((algebra.Alg.Term algebra.F.id_x'2a 
                                   ((algebra.Alg.Term algebra.F.id_if (x6::
                                   x7::nil))::(algebra.Alg.Term 
                                   algebra.F.id_if (x7::x6::nil))::nil))::
                                   x8::nil))::x9::nil))::x10::nil)) 
          (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
     
      (* <reduce'ii'in(sequent(cons(p(V_),Fs_),Gs_),sequent(Left_,Right_)),reduce'ii'in(sequent(Fs_,Gs_),sequent(cons(p(V_),Left_),Right_))> *)
     | DP_R_xml_0_scc_19_strict_1 :
      forall x8 x20 x14 x9 x21 x13 x15, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term 
          algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_p (x13::nil))::
          x8::nil))::x9::nil)) x21) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_sequent (x14::x15::nil)) x20) ->
         DP_R_xml_0_scc_19_strict (algebra.Alg.Term 
                                   algebra.F.id_reduce'ii'in 
                                   ((algebra.Alg.Term algebra.F.id_sequent 
                                   (x8::x9::nil))::(algebra.Alg.Term 
                                   algebra.F.id_sequent ((algebra.Alg.Term 
                                   algebra.F.id_cons ((algebra.Alg.Term 
                                   algebra.F.id_p (x13::nil))::x14::nil))::
                                   x15::nil))::nil)) 
          (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
     
      (* <reduce'ii'in(sequent(Fs_,cons(iff(A_,B_),Gs_)),NF_),reduce'ii'in(sequent(Fs_,cons(x'2a(if(A_,B_),if(B_,A_)),Gs_)),NF_)> *)
     | DP_R_xml_0_scc_19_strict_2 :
      forall x8 x20 x10 x6 x9 x21 x7, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term 
          algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_iff (x6::
          x7::nil))::x9::nil))::nil)) x21) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x10 x20) ->
         DP_R_xml_0_scc_19_strict (algebra.Alg.Term 
                                   algebra.F.id_reduce'ii'in 
                                   ((algebra.Alg.Term algebra.F.id_sequent 
                                   (x8::(algebra.Alg.Term algebra.F.id_cons 
                                   ((algebra.Alg.Term algebra.F.id_x'2a 
                                   ((algebra.Alg.Term algebra.F.id_if (x6::
                                   x7::nil))::(algebra.Alg.Term 
                                   algebra.F.id_if (x7::x6::nil))::nil))::
                                   x9::nil))::nil))::x10::nil)) 
          (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
     
      (* <reduce'ii'in(sequent(nil,cons(p(V_),Gs_)),sequent(Left_,Right_)),reduce'ii'in(sequent(nil,Gs_),sequent(Left_,cons(p(V_),Right_)))> *)
     | DP_R_xml_0_scc_19_strict_3 :
      forall x20 x14 x9 x21 x13 x15, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term 
          algebra.F.id_nil nil)::(algebra.Alg.Term algebra.F.id_cons 
          ((algebra.Alg.Term algebra.F.id_p (x13::nil))::x9::nil))::nil)) 
         x21) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_sequent (x14::x15::nil)) x20) ->
         DP_R_xml_0_scc_19_strict (algebra.Alg.Term 
                                   algebra.F.id_reduce'ii'in 
                                   ((algebra.Alg.Term algebra.F.id_sequent 
                                   ((algebra.Alg.Term algebra.F.id_nil nil)::
                                   x9::nil))::(algebra.Alg.Term 
                                   algebra.F.id_sequent (x14::
                                   (algebra.Alg.Term algebra.F.id_cons 
                                   ((algebra.Alg.Term algebra.F.id_p 
                                   (x13::nil))::x15::nil))::nil))::nil)) 
          (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
   .
   
   
   Module WF_DP_R_xml_0_scc_19_large.
    Inductive DP_R_xml_0_scc_19_large_large  :
     algebra.Alg.term ->algebra.Alg.term ->Prop := 
       (* <reduce'ii'in(sequent(cons(x'2b(F1_,F2_),Fs_),Gs_),NF_),u'6'1(reduce'ii'in(sequent(cons(F1_,Fs_),Gs_),NF_),F2_,Fs_,Gs_,NF_)> *)
      | DP_R_xml_0_scc_19_large_large_0 :
       forall x8 x20 x12 x10 x9 x21 x11, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term 
           algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b (x11::
           x12::nil))::x8::nil))::x9::nil)) x21) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x10 x20) ->
          DP_R_xml_0_scc_19_large_large (algebra.Alg.Term algebra.F.id_u'6'1 
                                         ((algebra.Alg.Term 
                                         algebra.F.id_reduce'ii'in 
                                         ((algebra.Alg.Term 
                                         algebra.F.id_sequent 
                                         ((algebra.Alg.Term 
                                         algebra.F.id_cons (x11::x8::nil))::
                                         x9::nil))::x10::nil))::x12::x8::x9::
                                         x10::nil)) 
           (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
      
       (* <u'6'1(reduce'ii'out,F2_,Fs_,Gs_,NF_),reduce'ii'in(sequent(cons(F2_,Fs_),Gs_),NF_)> *)
      | DP_R_xml_0_scc_19_large_large_1 :
       forall x8 x24 x20 x12 x10 x22 x9 x21 x23, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) x24) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x12 x23) ->
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     x8 x22) ->
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                      x9 x21) ->
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                       x10 x20) ->
             DP_R_xml_0_scc_19_large_large (algebra.Alg.Term 
                                            algebra.F.id_reduce'ii'in 
                                            ((algebra.Alg.Term 
                                            algebra.F.id_sequent 
                                            ((algebra.Alg.Term 
                                            algebra.F.id_cons (x12::
                                            x8::nil))::x9::nil))::x10::nil)) 
              (algebra.Alg.Term algebra.F.id_u'6'1 (x24::x23::x22::x21::
               x20::nil))
      
       (* <reduce'ii'in(sequent(cons(x'2b(F1_,F2_),Fs_),Gs_),NF_),reduce'ii'in(sequent(cons(F1_,Fs_),Gs_),NF_)> *)
      | DP_R_xml_0_scc_19_large_large_2 :
       forall x8 x20 x12 x10 x9 x21 x11, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term 
           algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b (x11::
           x12::nil))::x8::nil))::x9::nil)) x21) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x10 x20) ->
          DP_R_xml_0_scc_19_large_large (algebra.Alg.Term 
                                         algebra.F.id_reduce'ii'in 
                                         ((algebra.Alg.Term 
                                         algebra.F.id_sequent 
                                         ((algebra.Alg.Term 
                                         algebra.F.id_cons (x11::x8::nil))::
                                         x9::nil))::x10::nil)) 
           (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
      
       (* <u'12'1(reduce'ii'out,Fs_,G2_,Gs_,NF_),reduce'ii'in(sequent(Fs_,cons(G2_,Gs_)),NF_)> *)
      | DP_R_xml_0_scc_19_large_large_3 :
       forall x8 x24 x20 x10 x22 x17 x9 x21 x23, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) x24) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x8 x23) ->
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     x17 x22) ->
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                      x9 x21) ->
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                       x10 x20) ->
             DP_R_xml_0_scc_19_large_large (algebra.Alg.Term 
                                            algebra.F.id_reduce'ii'in 
                                            ((algebra.Alg.Term 
                                            algebra.F.id_sequent (x8::
                                            (algebra.Alg.Term 
                                            algebra.F.id_cons (x17::
                                            x9::nil))::nil))::x10::nil)) 
              (algebra.Alg.Term algebra.F.id_u'12'1 (x24::x23::x22::x21::
               x20::nil))
      
       (* <reduce'ii'in(sequent(Fs_,cons(x'2b(G1_,G2_),Gs_)),NF_),reduce'ii'in(sequent(Fs_,cons(G1_,cons(G2_,Gs_))),NF_)> *)
      | DP_R_xml_0_scc_19_large_large_4 :
       forall x16 x8 x20 x10 x17 x9 x21, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term 
           algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b (x16::
           x17::nil))::x9::nil))::nil)) x21) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x10 x20) ->
          DP_R_xml_0_scc_19_large_large (algebra.Alg.Term 
                                         algebra.F.id_reduce'ii'in 
                                         ((algebra.Alg.Term 
                                         algebra.F.id_sequent (x8::
                                         (algebra.Alg.Term algebra.F.id_cons 
                                         (x16::(algebra.Alg.Term 
                                         algebra.F.id_cons (x17::
                                         x9::nil))::nil))::nil))::x10::nil)) 
           (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
    .
    
    
    Inductive DP_R_xml_0_scc_19_large_strict  :
     algebra.Alg.term ->algebra.Alg.term ->Prop := 
       (* <reduce'ii'in(sequent(cons(if(A_,B_),Fs_),Gs_),NF_),reduce'ii'in(sequent(cons(x'2b(x'2d(B_),A_),Fs_),Gs_),NF_)> *)
      | DP_R_xml_0_scc_19_large_strict_0 :
       forall x8 x20 x10 x6 x9 x21 x7, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term 
           algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_if (x6::
           x7::nil))::x8::nil))::x9::nil)) x21) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x10 x20) ->
          DP_R_xml_0_scc_19_large_strict (algebra.Alg.Term 
                                          algebra.F.id_reduce'ii'in 
                                          ((algebra.Alg.Term 
                                          algebra.F.id_sequent 
                                          ((algebra.Alg.Term 
                                          algebra.F.id_cons 
                                          ((algebra.Alg.Term 
                                          algebra.F.id_x'2b 
                                          ((algebra.Alg.Term 
                                          algebra.F.id_x'2d (x7::nil))::
                                          x6::nil))::x8::nil))::x9::nil))::
                                          x10::nil)) 
           (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
      
       (* <reduce'ii'in(sequent(cons(x'2a(F1_,F2_),Fs_),Gs_),NF_),reduce'ii'in(sequent(cons(F1_,cons(F2_,Fs_)),Gs_),NF_)> *)
      | DP_R_xml_0_scc_19_large_strict_1 :
       forall x8 x20 x12 x10 x9 x21 x11, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term 
           algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2a (x11::
           x12::nil))::x8::nil))::x9::nil)) x21) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x10 x20) ->
          DP_R_xml_0_scc_19_large_strict (algebra.Alg.Term 
                                          algebra.F.id_reduce'ii'in 
                                          ((algebra.Alg.Term 
                                          algebra.F.id_sequent 
                                          ((algebra.Alg.Term 
                                          algebra.F.id_cons (x11::
                                          (algebra.Alg.Term 
                                          algebra.F.id_cons (x12::
                                          x8::nil))::nil))::x9::nil))::
                                          x10::nil)) 
           (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
      
       (* <reduce'ii'in(sequent(cons(x'2d(F1_),Fs_),Gs_),NF_),reduce'ii'in(sequent(Fs_,cons(F1_,Gs_)),NF_)> *)
      | DP_R_xml_0_scc_19_large_strict_2 :
       forall x8 x20 x10 x9 x21 x11, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term 
           algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2d 
           (x11::nil))::x8::nil))::x9::nil)) x21) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x10 x20) ->
          DP_R_xml_0_scc_19_large_strict (algebra.Alg.Term 
                                          algebra.F.id_reduce'ii'in 
                                          ((algebra.Alg.Term 
                                          algebra.F.id_sequent (x8::
                                          (algebra.Alg.Term 
                                          algebra.F.id_cons (x11::
                                          x9::nil))::nil))::x10::nil)) 
           (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
      
       (* <reduce'ii'in(sequent(Fs_,cons(if(A_,B_),Gs_)),NF_),reduce'ii'in(sequent(Fs_,cons(x'2b(x'2d(B_),A_),Gs_)),NF_)> *)
      | DP_R_xml_0_scc_19_large_strict_3 :
       forall x8 x20 x10 x6 x9 x21 x7, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term 
           algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_if (x6::
           x7::nil))::x9::nil))::nil)) x21) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x10 x20) ->
          DP_R_xml_0_scc_19_large_strict (algebra.Alg.Term 
                                          algebra.F.id_reduce'ii'in 
                                          ((algebra.Alg.Term 
                                          algebra.F.id_sequent (x8::
                                          (algebra.Alg.Term 
                                          algebra.F.id_cons 
                                          ((algebra.Alg.Term 
                                          algebra.F.id_x'2b 
                                          ((algebra.Alg.Term 
                                          algebra.F.id_x'2d (x7::nil))::
                                          x6::nil))::x9::nil))::nil))::
                                          x10::nil)) 
           (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
      
       (* <reduce'ii'in(sequent(Fs_,cons(x'2a(G1_,G2_),Gs_)),NF_),u'12'1(reduce'ii'in(sequent(Fs_,cons(G1_,Gs_)),NF_),Fs_,G2_,Gs_,NF_)> *)
      | DP_R_xml_0_scc_19_large_strict_4 :
       forall x16 x8 x20 x10 x17 x9 x21, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term 
           algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2a (x16::
           x17::nil))::x9::nil))::nil)) x21) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x10 x20) ->
          DP_R_xml_0_scc_19_large_strict (algebra.Alg.Term 
                                          algebra.F.id_u'12'1 
                                          ((algebra.Alg.Term 
                                          algebra.F.id_reduce'ii'in 
                                          ((algebra.Alg.Term 
                                          algebra.F.id_sequent (x8::
                                          (algebra.Alg.Term 
                                          algebra.F.id_cons (x16::
                                          x9::nil))::nil))::x10::nil))::x8::
                                          x17::x9::x10::nil)) 
           (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
      
       (* <reduce'ii'in(sequent(Fs_,cons(x'2a(G1_,G2_),Gs_)),NF_),reduce'ii'in(sequent(Fs_,cons(G1_,Gs_)),NF_)> *)
      | DP_R_xml_0_scc_19_large_strict_5 :
       forall x16 x8 x20 x10 x17 x9 x21, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term 
           algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2a (x16::
           x17::nil))::x9::nil))::nil)) x21) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x10 x20) ->
          DP_R_xml_0_scc_19_large_strict (algebra.Alg.Term 
                                          algebra.F.id_reduce'ii'in 
                                          ((algebra.Alg.Term 
                                          algebra.F.id_sequent (x8::
                                          (algebra.Alg.Term 
                                          algebra.F.id_cons (x16::
                                          x9::nil))::nil))::x10::nil)) 
           (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
      
       (* <reduce'ii'in(sequent(Fs_,cons(x'2d(G1_),Gs_)),NF_),reduce'ii'in(sequent(cons(G1_,Fs_),Gs_),NF_)> *)
      | DP_R_xml_0_scc_19_large_strict_6 :
       forall x16 x8 x20 x10 x9 x21, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term 
           algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2d 
           (x16::nil))::x9::nil))::nil)) x21) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x10 x20) ->
          DP_R_xml_0_scc_19_large_strict (algebra.Alg.Term 
                                          algebra.F.id_reduce'ii'in 
                                          ((algebra.Alg.Term 
                                          algebra.F.id_sequent 
                                          ((algebra.Alg.Term 
                                          algebra.F.id_cons (x16::x8::nil))::
                                          x9::nil))::x10::nil)) 
           (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
    .
    
    
    Module WF_DP_R_xml_0_scc_19_large_large.
     Inductive DP_R_xml_0_scc_19_large_large_scc_1  :
      algebra.Alg.term ->algebra.Alg.term ->Prop := 
        (* <u'6'1(reduce'ii'out,F2_,Fs_,Gs_,NF_),reduce'ii'in(sequent(cons(F2_,Fs_),Gs_),NF_)> *)
       | DP_R_xml_0_scc_19_large_large_scc_1_0 :
        forall x8 x24 x20 x12 x10 x22 x9 x21 x23, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    
           (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) x24) ->
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     x12 x23) ->
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                      x8 x22) ->
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                       x9 x21) ->
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                        x10 x20) ->
              DP_R_xml_0_scc_19_large_large_scc_1 (algebra.Alg.Term 
                                                   algebra.F.id_reduce'ii'in 
                                                   ((algebra.Alg.Term 
                                                   algebra.F.id_sequent 
                                                   ((algebra.Alg.Term 
                                                   algebra.F.id_cons (x12::
                                                   x8::nil))::x9::nil))::
                                                   x10::nil)) 
               (algebra.Alg.Term algebra.F.id_u'6'1 (x24::x23::x22::x21::
                x20::nil))
       
        (* <reduce'ii'in(sequent(cons(x'2b(F1_,F2_),Fs_),Gs_),NF_),u'6'1(reduce'ii'in(sequent(cons(F1_,Fs_),Gs_),NF_),F2_,Fs_,Gs_,NF_)> *)
       | DP_R_xml_0_scc_19_large_large_scc_1_1 :
        forall x8 x20 x12 x10 x9 x21 x11, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    
           (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term 
            algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b (x11::
            x12::nil))::x8::nil))::x9::nil)) x21) ->
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     x10 x20) ->
           DP_R_xml_0_scc_19_large_large_scc_1 (algebra.Alg.Term 
                                                algebra.F.id_u'6'1 
                                                ((algebra.Alg.Term 
                                                algebra.F.id_reduce'ii'in 
                                                ((algebra.Alg.Term 
                                                algebra.F.id_sequent 
                                                ((algebra.Alg.Term 
                                                algebra.F.id_cons (x11::
                                                x8::nil))::x9::nil))::
                                                x10::nil))::x12::x8::x9::
                                                x10::nil)) 
            (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
       
        (* <reduce'ii'in(sequent(cons(x'2b(F1_,F2_),Fs_),Gs_),NF_),reduce'ii'in(sequent(cons(F1_,Fs_),Gs_),NF_)> *)
       | DP_R_xml_0_scc_19_large_large_scc_1_2 :
        forall x8 x20 x12 x10 x9 x21 x11, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    
           (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term 
            algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b (x11::
            x12::nil))::x8::nil))::x9::nil)) x21) ->
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     x10 x20) ->
           DP_R_xml_0_scc_19_large_large_scc_1 (algebra.Alg.Term 
                                                algebra.F.id_reduce'ii'in 
                                                ((algebra.Alg.Term 
                                                algebra.F.id_sequent 
                                                ((algebra.Alg.Term 
                                                algebra.F.id_cons (x11::
                                                x8::nil))::x9::nil))::
                                                x10::nil)) 
            (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
       
        (* <reduce'ii'in(sequent(Fs_,cons(x'2b(G1_,G2_),Gs_)),NF_),reduce'ii'in(sequent(Fs_,cons(G1_,cons(G2_,Gs_))),NF_)> *)
       | DP_R_xml_0_scc_19_large_large_scc_1_3 :
        forall x16 x8 x20 x10 x17 x9 x21, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    
           (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term 
            algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b (x16::
            x17::nil))::x9::nil))::nil)) x21) ->
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     x10 x20) ->
           DP_R_xml_0_scc_19_large_large_scc_1 (algebra.Alg.Term 
                                                algebra.F.id_reduce'ii'in 
                                                ((algebra.Alg.Term 
                                                algebra.F.id_sequent (x8::
                                                (algebra.Alg.Term 
                                                algebra.F.id_cons (x16::
                                                (algebra.Alg.Term 
                                                algebra.F.id_cons (x17::
                                                x9::nil))::nil))::nil))::
                                                x10::nil)) 
            (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
     .
     
     
     Module WF_DP_R_xml_0_scc_19_large_large_scc_1.
      Inductive DP_R_xml_0_scc_19_large_large_scc_1_large  :
       algebra.Alg.term ->algebra.Alg.term ->Prop := 
         (* <reduce'ii'in(sequent(Fs_,cons(x'2b(G1_,G2_),Gs_)),NF_),reduce'ii'in(sequent(Fs_,cons(G1_,cons(G2_,Gs_))),NF_)> *)
        | DP_R_xml_0_scc_19_large_large_scc_1_large_0 :
         forall x16 x8 x20 x10 x17 x9 x21, 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     
            (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term 
             algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b (x16::
             x17::nil))::x9::nil))::nil)) x21) ->
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                      x10 x20) ->
            DP_R_xml_0_scc_19_large_large_scc_1_large (algebra.Alg.Term 
                                                       algebra.F.id_reduce'ii'in 
                                                       ((algebra.Alg.Term 
                                                       algebra.F.id_sequent 
                                                       (x8::
                                                       (algebra.Alg.Term 
                                                       algebra.F.id_cons 
                                                       (x16::
                                                       (algebra.Alg.Term 
                                                       algebra.F.id_cons 
                                                       (x17::
                                                       x9::nil))::nil))::nil))::
                                                       x10::nil)) 
             (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
      .
      
      
      Inductive DP_R_xml_0_scc_19_large_large_scc_1_strict  :
       algebra.Alg.term ->algebra.Alg.term ->Prop := 
         (* <u'6'1(reduce'ii'out,F2_,Fs_,Gs_,NF_),reduce'ii'in(sequent(cons(F2_,Fs_),Gs_),NF_)> *)
        | DP_R_xml_0_scc_19_large_large_scc_1_strict_0 :
         forall x8 x24 x20 x12 x10 x22 x9 x21 x23, 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     
            (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) x24) ->
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                      x12 x23) ->
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                       x8 x22) ->
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                        x9 x21) ->
              (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                         x10 x20) ->
               DP_R_xml_0_scc_19_large_large_scc_1_strict (algebra.Alg.Term 
                                                           algebra.F.id_reduce'ii'in 
                                                           ((algebra.Alg.Term 
                                                           algebra.F.id_sequent 
                                                           ((algebra.Alg.Term 
                                                           algebra.F.id_cons 
                                                           (x12::x8::nil))::
                                                           x9::nil))::
                                                           x10::nil)) 
                (algebra.Alg.Term algebra.F.id_u'6'1 (x24::x23::x22::x21::
                 x20::nil))
        
         (* <reduce'ii'in(sequent(cons(x'2b(F1_,F2_),Fs_),Gs_),NF_),u'6'1(reduce'ii'in(sequent(cons(F1_,Fs_),Gs_),NF_),F2_,Fs_,Gs_,NF_)> *)
        | DP_R_xml_0_scc_19_large_large_scc_1_strict_1 :
         forall x8 x20 x12 x10 x9 x21 x11, 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     
            (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term 
             algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b (x11::
             x12::nil))::x8::nil))::x9::nil)) x21) ->
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                      x10 x20) ->
            DP_R_xml_0_scc_19_large_large_scc_1_strict (algebra.Alg.Term 
                                                        algebra.F.id_u'6'1 
                                                        ((algebra.Alg.Term 
                                                        algebra.F.id_reduce'ii'in 
                                                        ((algebra.Alg.Term 
                                                        algebra.F.id_sequent 
                                                        ((algebra.Alg.Term 
                                                        algebra.F.id_cons 
                                                        (x11::x8::nil))::
                                                        x9::nil))::
                                                        x10::nil))::x12::x8::
                                                        x9::x10::nil)) 
             (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
        
         (* <reduce'ii'in(sequent(cons(x'2b(F1_,F2_),Fs_),Gs_),NF_),reduce'ii'in(sequent(cons(F1_,Fs_),Gs_),NF_)> *)
        | DP_R_xml_0_scc_19_large_large_scc_1_strict_2 :
         forall x8 x20 x12 x10 x9 x21 x11, 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     
            (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term 
             algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b (x11::
             x12::nil))::x8::nil))::x9::nil)) x21) ->
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                      x10 x20) ->
            DP_R_xml_0_scc_19_large_large_scc_1_strict (algebra.Alg.Term 
                                                        algebra.F.id_reduce'ii'in 
                                                        ((algebra.Alg.Term 
                                                        algebra.F.id_sequent 
                                                        ((algebra.Alg.Term 
                                                        algebra.F.id_cons 
                                                        (x11::x8::nil))::
                                                        x9::nil))::x10::nil)) 
             (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil))
      .
      
      
      Module WF_DP_R_xml_0_scc_19_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_intersect'ii'in (x20:Z) (x21:Z) := 1.
       
       Definition P_id_tautology'i'out  := 1.
       
       Definition P_id_u'6'1 (x20:Z) (x21:Z) (x22:Z) (x23:Z) (x24:Z) := 
         1* x24.
       
       Definition P_id_u'3'1 (x20:Z) := 0.
       
       Definition P_id_u'12'1 (x20:Z) (x21:Z) (x22:Z) (x23:Z) (x24:Z) := 
         1* x24.
       
       Definition P_id_u'2'1 (x20:Z) := 0.
       
       Definition P_id_u'9'1 (x20:Z) := 1* x20.
       
       Definition P_id_iff (x20:Z) (x21:Z) := 0.
       
       Definition P_id_u'14'1 (x20:Z) := 0.
       
       Definition P_id_intersect'ii'out  := 0.
       
       Definition P_id_u'7'1 (x20:Z) := 1* x20.
       
       Definition P_id_x'2d (x20:Z) := 0.
       
       Definition P_id_u'13'1 (x20:Z) := 0.
       
       Definition P_id_sequent (x20:Z) (x21:Z) := 2* x21.
       
       Definition P_id_u'10'1 (x20:Z) := 1* x20.
       
       Definition P_id_x'2a (x20:Z) (x21:Z) := 0.
       
       Definition P_id_tautology'i'in (x20:Z) := 2 + 3* x20.
       
       Definition P_id_cons (x20:Z) (x21:Z) := 2* x20.
       
       Definition P_id_u'6'2 (x20:Z) := 0.
       
       Definition P_id_x'2b (x20:Z) (x21:Z) := 2 + 2* x20.
       
       Definition P_id_u'12'2 (x20:Z) := 0.
       
       Definition P_id_reduce'ii'in (x20:Z) (x21:Z) := 1* x21.
       
       Definition P_id_p (x20:Z) := 0.
       
       Definition P_id_u'4'1 (x20:Z) := 1* x20.
       
       Definition P_id_u'15'1 (x20:Z) := 0.
       
       Definition P_id_u'1'1 (x20:Z) := 1* x20.
       
       Definition P_id_u'8'1 (x20:Z) := 1* x20.
       
       Definition P_id_reduce'ii'out  := 0.
       
       Definition P_id_nil  := 0.
       
       Definition P_id_if (x20:Z) (x21:Z) := 0.
       
       Definition P_id_u'11'1 (x20:Z) := 1* x20.
       
       Definition P_id_u'5'1 (x20:Z) := 0.
       
       Definition P_id_u'16'1 (x20:Z) := 1.
       
       Lemma P_id_intersect'ii'in_monotonic :
        forall x20 x22 x21 x23, 
         (0 <= x23)/\ (x23 <= x22) ->
          (0 <= x21)/\ (x21 <= x20) ->
           P_id_intersect'ii'in x21 x23 <= P_id_intersect'ii'in x20 x22.
       Proof.
         intros x23 x22 x21 x20.
         intros [H_1 H_0].
         intros [H_3 H_2].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_u'6'1_monotonic :
        forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, 
         (0 <= x29)/\ (x29 <= x28) ->
          (0 <= x27)/\ (x27 <= x26) ->
           (0 <= x25)/\ (x25 <= x24) ->
            (0 <= x23)/\ (x23 <= x22) ->
             (0 <= x21)/\ (x21 <= x20) ->
              P_id_u'6'1 x21 x23 x25 x27 x29 <= P_id_u'6'1 x20 x22 x24 
                                                 x26 x28.
       Proof.
         intros x29 x28 x27 x26 x25 x24 x23 x22 x21 x20.
         intros [H_1 H_0].
         intros [H_3 H_2].
         intros [H_5 H_4].
         intros [H_7 H_6].
         intros [H_9 H_8].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_u'3'1_monotonic :
        forall x20 x21, 
         (0 <= x21)/\ (x21 <= x20) ->P_id_u'3'1 x21 <= P_id_u'3'1 x20.
       Proof.
         intros x21 x20.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_u'12'1_monotonic :
        forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, 
         (0 <= x29)/\ (x29 <= x28) ->
          (0 <= x27)/\ (x27 <= x26) ->
           (0 <= x25)/\ (x25 <= x24) ->
            (0 <= x23)/\ (x23 <= x22) ->
             (0 <= x21)/\ (x21 <= x20) ->
              P_id_u'12'1 x21 x23 x25 x27 x29 <= P_id_u'12'1 x20 x22 
                                                  x24 x26 x28.
       Proof.
         intros x29 x28 x27 x26 x25 x24 x23 x22 x21 x20.
         intros [H_1 H_0].
         intros [H_3 H_2].
         intros [H_5 H_4].
         intros [H_7 H_6].
         intros [H_9 H_8].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_u'2'1_monotonic :
        forall x20 x21, 
         (0 <= x21)/\ (x21 <= x20) ->P_id_u'2'1 x21 <= P_id_u'2'1 x20.
       Proof.
         intros x21 x20.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_u'9'1_monotonic :
        forall x20 x21, 
         (0 <= x21)/\ (x21 <= x20) ->P_id_u'9'1 x21 <= P_id_u'9'1 x20.
       Proof.
         intros x21 x20.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_iff_monotonic :
        forall x20 x22 x21 x23, 
         (0 <= x23)/\ (x23 <= x22) ->
          (0 <= x21)/\ (x21 <= x20) ->P_id_iff x21 x23 <= P_id_iff x20 x22.
       Proof.
         intros x23 x22 x21 x20.
         intros [H_1 H_0].
         intros [H_3 H_2].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_u'14'1_monotonic :
        forall x20 x21, 
         (0 <= x21)/\ (x21 <= x20) ->P_id_u'14'1 x21 <= P_id_u'14'1 x20.
       Proof.
         intros x21 x20.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_u'7'1_monotonic :
        forall x20 x21, 
         (0 <= x21)/\ (x21 <= x20) ->P_id_u'7'1 x21 <= P_id_u'7'1 x20.
       Proof.
         intros x21 x20.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_x'2d_monotonic :
        forall x20 x21, 
         (0 <= x21)/\ (x21 <= x20) ->P_id_x'2d x21 <= P_id_x'2d x20.
       Proof.
         intros x21 x20.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_u'13'1_monotonic :
        forall x20 x21, 
         (0 <= x21)/\ (x21 <= x20) ->P_id_u'13'1 x21 <= P_id_u'13'1 x20.
       Proof.
         intros x21 x20.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_sequent_monotonic :
        forall x20 x22 x21 x23, 
         (0 <= x23)/\ (x23 <= x22) ->
          (0 <= x21)/\ (x21 <= x20) ->
           P_id_sequent x21 x23 <= P_id_sequent x20 x22.
       Proof.
         intros x23 x22 x21 x20.
         intros [H_1 H_0].
         intros [H_3 H_2].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_u'10'1_monotonic :
        forall x20 x21, 
         (0 <= x21)/\ (x21 <= x20) ->P_id_u'10'1 x21 <= P_id_u'10'1 x20.
       Proof.
         intros x21 x20.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_x'2a_monotonic :
        forall x20 x22 x21 x23, 
         (0 <= x23)/\ (x23 <= x22) ->
          (0 <= x21)/\ (x21 <= x20) ->P_id_x'2a x21 x23 <= P_id_x'2a x20 x22.
       Proof.
         intros x23 x22 x21 x20.
         intros [H_1 H_0].
         intros [H_3 H_2].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_tautology'i'in_monotonic :
        forall x20 x21, 
         (0 <= x21)/\ (x21 <= x20) ->
          P_id_tautology'i'in x21 <= P_id_tautology'i'in x20.
       Proof.
         intros x21 x20.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_cons_monotonic :
        forall x20 x22 x21 x23, 
         (0 <= x23)/\ (x23 <= x22) ->
          (0 <= x21)/\ (x21 <= x20) ->P_id_cons x21 x23 <= P_id_cons x20 x22.
       Proof.
         intros x23 x22 x21 x20.
         intros [H_1 H_0].
         intros [H_3 H_2].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_u'6'2_monotonic :
        forall x20 x21, 
         (0 <= x21)/\ (x21 <= x20) ->P_id_u'6'2 x21 <= P_id_u'6'2 x20.
       Proof.
         intros x21 x20.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_x'2b_monotonic :
        forall x20 x22 x21 x23, 
         (0 <= x23)/\ (x23 <= x22) ->
          (0 <= x21)/\ (x21 <= x20) ->P_id_x'2b x21 x23 <= P_id_x'2b x20 x22.
       Proof.
         intros x23 x22 x21 x20.
         intros [H_1 H_0].
         intros [H_3 H_2].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_u'12'2_monotonic :
        forall x20 x21, 
         (0 <= x21)/\ (x21 <= x20) ->P_id_u'12'2 x21 <= P_id_u'12'2 x20.
       Proof.
         intros x21 x20.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_reduce'ii'in_monotonic :
        forall x20 x22 x21 x23, 
         (0 <= x23)/\ (x23 <= x22) ->
          (0 <= x21)/\ (x21 <= x20) ->
           P_id_reduce'ii'in x21 x23 <= P_id_reduce'ii'in x20 x22.
       Proof.
         intros x23 x22 x21 x20.
         intros [H_1 H_0].
         intros [H_3 H_2].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_p_monotonic :
        forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_p x21 <= P_id_p x20.
       Proof.
         intros x21 x20.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_u'4'1_monotonic :
        forall x20 x21, 
         (0 <= x21)/\ (x21 <= x20) ->P_id_u'4'1 x21 <= P_id_u'4'1 x20.
       Proof.
         intros x21 x20.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_u'15'1_monotonic :
        forall x20 x21, 
         (0 <= x21)/\ (x21 <= x20) ->P_id_u'15'1 x21 <= P_id_u'15'1 x20.
       Proof.
         intros x21 x20.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_u'1'1_monotonic :
        forall x20 x21, 
         (0 <= x21)/\ (x21 <= x20) ->P_id_u'1'1 x21 <= P_id_u'1'1 x20.
       Proof.
         intros x21 x20.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_u'8'1_monotonic :
        forall x20 x21, 
         (0 <= x21)/\ (x21 <= x20) ->P_id_u'8'1 x21 <= P_id_u'8'1 x20.
       Proof.
         intros x21 x20.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_if_monotonic :
        forall x20 x22 x21 x23, 
         (0 <= x23)/\ (x23 <= x22) ->
          (0 <= x21)/\ (x21 <= x20) ->P_id_if x21 x23 <= P_id_if x20 x22.
       Proof.
         intros x23 x22 x21 x20.
         intros [H_1 H_0].
         intros [H_3 H_2].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_u'11'1_monotonic :
        forall x20 x21, 
         (0 <= x21)/\ (x21 <= x20) ->P_id_u'11'1 x21 <= P_id_u'11'1 x20.
       Proof.
         intros x21 x20.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_u'5'1_monotonic :
        forall x20 x21, 
         (0 <= x21)/\ (x21 <= x20) ->P_id_u'5'1 x21 <= P_id_u'5'1 x20.
       Proof.
         intros x21 x20.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_u'16'1_monotonic :
        forall x20 x21, 
         (0 <= x21)/\ (x21 <= x20) ->P_id_u'16'1 x21 <= P_id_u'16'1 x20.
       Proof.
         intros x21 x20.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_intersect'ii'in_bounded :
        forall x20 x21, 
         (0 <= x20) ->(0 <= x21) ->0 <= P_id_intersect'ii'in x21 x20.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_tautology'i'out_bounded : 0 <= P_id_tautology'i'out .
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_u'6'1_bounded :
        forall x24 x20 x22 x21 x23, 
         (0 <= x20) ->
          (0 <= x21) ->
           (0 <= x22) ->
            (0 <= x23) ->(0 <= x24) ->0 <= P_id_u'6'1 x24 x23 x22 x21 x20.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_u'3'1_bounded :
        forall x20, (0 <= x20) ->0 <= P_id_u'3'1 x20.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_u'12'1_bounded :
        forall x24 x20 x22 x21 x23, 
         (0 <= x20) ->
          (0 <= x21) ->
           (0 <= x22) ->
            (0 <= x23) ->(0 <= x24) ->0 <= P_id_u'12'1 x24 x23 x22 x21 x20.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_u'2'1_bounded :
        forall x20, (0 <= x20) ->0 <= P_id_u'2'1 x20.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_u'9'1_bounded :
        forall x20, (0 <= x20) ->0 <= P_id_u'9'1 x20.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_iff_bounded :
        forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_iff x21 x20.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_u'14'1_bounded :
        forall x20, (0 <= x20) ->0 <= P_id_u'14'1 x20.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_intersect'ii'out_bounded : 0 <= P_id_intersect'ii'out .
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_u'7'1_bounded :
        forall x20, (0 <= x20) ->0 <= P_id_u'7'1 x20.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_x'2d_bounded : forall x20, (0 <= x20) ->0 <= P_id_x'2d x20.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_u'13'1_bounded :
        forall x20, (0 <= x20) ->0 <= P_id_u'13'1 x20.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_sequent_bounded :
        forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_sequent x21 x20.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_u'10'1_bounded :
        forall x20, (0 <= x20) ->0 <= P_id_u'10'1 x20.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_x'2a_bounded :
        forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_x'2a x21 x20.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_tautology'i'in_bounded :
        forall x20, (0 <= x20) ->0 <= P_id_tautology'i'in x20.
       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 x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_cons x21 x20.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_u'6'2_bounded :
        forall x20, (0 <= x20) ->0 <= P_id_u'6'2 x20.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_x'2b_bounded :
        forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_x'2b x21 x20.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_u'12'2_bounded :
        forall x20, (0 <= x20) ->0 <= P_id_u'12'2 x20.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_reduce'ii'in_bounded :
        forall x20 x21, 
         (0 <= x20) ->(0 <= x21) ->0 <= P_id_reduce'ii'in x21 x20.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_p_bounded : forall x20, (0 <= x20) ->0 <= P_id_p x20.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_u'4'1_bounded :
        forall x20, (0 <= x20) ->0 <= P_id_u'4'1 x20.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_u'15'1_bounded :
        forall x20, (0 <= x20) ->0 <= P_id_u'15'1 x20.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_u'1'1_bounded :
        forall x20, (0 <= x20) ->0 <= P_id_u'1'1 x20.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_u'8'1_bounded :
        forall x20, (0 <= x20) ->0 <= P_id_u'8'1 x20.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_reduce'ii'out_bounded : 0 <= P_id_reduce'ii'out .
       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_if_bounded :
        forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_if x21 x20.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_u'11'1_bounded :
        forall x20, (0 <= x20) ->0 <= P_id_u'11'1 x20.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_u'5'1_bounded :
        forall x20, (0 <= x20) ->0 <= P_id_u'5'1 x20.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_u'16'1_bounded :
        forall x20, (0 <= x20) ->0 <= P_id_u'16'1 x20.
       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_intersect'ii'in P_id_tautology'i'out 
          P_id_u'6'1 P_id_u'3'1 P_id_u'12'1 P_id_u'2'1 P_id_u'9'1 P_id_iff 
          P_id_u'14'1 P_id_intersect'ii'out P_id_u'7'1 P_id_x'2d P_id_u'13'1 
          P_id_sequent P_id_u'10'1 P_id_x'2a P_id_tautology'i'in P_id_cons 
          P_id_u'6'2 P_id_x'2b P_id_u'12'2 P_id_reduce'ii'in P_id_p 
          P_id_u'4'1 P_id_u'15'1 P_id_u'1'1 P_id_u'8'1 P_id_reduce'ii'out 
          P_id_nil P_id_if P_id_u'11'1 P_id_u'5'1 P_id_u'16'1.
       
       Lemma measure_equation :
        forall t, 
         measure t = match t with
                       | (algebra.Alg.Term algebra.F.id_intersect'ii'in 
                          (x21::x20::nil)) =>
                        P_id_intersect'ii'in (measure x21) (measure x20)
                       | (algebra.Alg.Term algebra.F.id_tautology'i'out nil) =>
                        P_id_tautology'i'out 
                       | (algebra.Alg.Term algebra.F.id_u'6'1 (x24::x23::
                          x22::x21::x20::nil)) =>
                        P_id_u'6'1 (measure x24) (measure x23) (measure x22) 
                         (measure x21) (measure x20)
                       | (algebra.Alg.Term algebra.F.id_u'3'1 (x20::nil)) =>
                        P_id_u'3'1 (measure x20)
                       | (algebra.Alg.Term algebra.F.id_u'12'1 (x24::x23::
                          x22::x21::x20::nil)) =>
                        P_id_u'12'1 (measure x24) (measure x23) 
                         (measure x22) (measure x21) (measure x20)
                       | (algebra.Alg.Term algebra.F.id_u'2'1 (x20::nil)) =>
                        P_id_u'2'1 (measure x20)
                       | (algebra.Alg.Term algebra.F.id_u'9'1 (x20::nil)) =>
                        P_id_u'9'1 (measure x20)
                       | (algebra.Alg.Term algebra.F.id_iff (x21::x20::nil)) =>
                        P_id_iff (measure x21) (measure x20)
                       | (algebra.Alg.Term algebra.F.id_u'14'1 (x20::nil)) =>
                        P_id_u'14'1 (measure x20)
                       | (algebra.Alg.Term algebra.F.id_intersect'ii'out nil) =>
                        P_id_intersect'ii'out 
                       | (algebra.Alg.Term algebra.F.id_u'7'1 (x20::nil)) =>
                        P_id_u'7'1 (measure x20)
                       | (algebra.Alg.Term algebra.F.id_x'2d (x20::nil)) =>
                        P_id_x'2d (measure x20)
                       | (algebra.Alg.Term algebra.F.id_u'13'1 (x20::nil)) =>
                        P_id_u'13'1 (measure x20)
                       | (algebra.Alg.Term algebra.F.id_sequent (x21::
                          x20::nil)) =>
                        P_id_sequent (measure x21) (measure x20)
                       | (algebra.Alg.Term algebra.F.id_u'10'1 (x20::nil)) =>
                        P_id_u'10'1 (measure x20)
                       | (algebra.Alg.Term algebra.F.id_x'2a (x21::x20::nil)) =>
                        P_id_x'2a (measure x21) (measure x20)
                       | (algebra.Alg.Term algebra.F.id_tautology'i'in 
                          (x20::nil)) =>
                        P_id_tautology'i'in (measure x20)
                       | (algebra.Alg.Term algebra.F.id_cons (x21::x20::nil)) =>
                        P_id_cons (measure x21) (measure x20)
                       | (algebra.Alg.Term algebra.F.id_u'6'2 (x20::nil)) =>
                        P_id_u'6'2 (measure x20)
                       | (algebra.Alg.Term algebra.F.id_x'2b (x21::x20::nil)) =>
                        P_id_x'2b (measure x21) (measure x20)
                       | (algebra.Alg.Term algebra.F.id_u'12'2 (x20::nil)) =>
                        P_id_u'12'2 (measure x20)
                       | (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::
                          x20::nil)) =>
                        P_id_reduce'ii'in (measure x21) (measure x20)
                       | (algebra.Alg.Term algebra.F.id_p (x20::nil)) =>
                        P_id_p (measure x20)
                       | (algebra.Alg.Term algebra.F.id_u'4'1 (x20::nil)) =>
                        P_id_u'4'1 (measure x20)
                       | (algebra.Alg.Term algebra.F.id_u'15'1 (x20::nil)) =>
                        P_id_u'15'1 (measure x20)
                       | (algebra.Alg.Term algebra.F.id_u'1'1 (x20::nil)) =>
                        P_id_u'1'1 (measure x20)
                       | (algebra.Alg.Term algebra.F.id_u'8'1 (x20::nil)) =>
                        P_id_u'8'1 (measure x20)
                       | (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) =>
                        P_id_reduce'ii'out 
                       | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil 
                       | (algebra.Alg.Term algebra.F.id_if (x21::x20::nil)) =>
                        P_id_if (measure x21) (measure x20)
                       | (algebra.Alg.Term algebra.F.id_u'11'1 (x20::nil)) =>
                        P_id_u'11'1 (measure x20)
                       | (algebra.Alg.Term algebra.F.id_u'5'1 (x20::nil)) =>
                        P_id_u'5'1 (measure x20)
                       | (algebra.Alg.Term algebra.F.id_u'16'1 (x20::nil)) =>
                        P_id_u'16'1 (measure x20)
                       | _ => 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_intersect'ii'in_monotonic;assumption.
         intros ;apply P_id_u'6'1_monotonic;assumption.
         intros ;apply P_id_u'3'1_monotonic;assumption.
         intros ;apply P_id_u'12'1_monotonic;assumption.
         intros ;apply P_id_u'2'1_monotonic;assumption.
         intros ;apply P_id_u'9'1_monotonic;assumption.
         intros ;apply P_id_iff_monotonic;assumption.
         intros ;apply P_id_u'14'1_monotonic;assumption.
         intros ;apply P_id_u'7'1_monotonic;assumption.
         intros ;apply P_id_x'2d_monotonic;assumption.
         intros ;apply P_id_u'13'1_monotonic;assumption.
         intros ;apply P_id_sequent_monotonic;assumption.
         intros ;apply P_id_u'10'1_monotonic;assumption.
         intros ;apply P_id_x'2a_monotonic;assumption.
         intros ;apply P_id_tautology'i'in_monotonic;assumption.
         intros ;apply P_id_cons_monotonic;assumption.
         intros ;apply P_id_u'6'2_monotonic;assumption.
         intros ;apply P_id_x'2b_monotonic;assumption.
         intros ;apply P_id_u'12'2_monotonic;assumption.
         intros ;apply P_id_reduce'ii'in_monotonic;assumption.
         intros ;apply P_id_p_monotonic;assumption.
         intros ;apply P_id_u'4'1_monotonic;assumption.
         intros ;apply P_id_u'15'1_monotonic;assumption.
         intros ;apply P_id_u'1'1_monotonic;assumption.
         intros ;apply P_id_u'8'1_monotonic;assumption.
         intros ;apply P_id_if_monotonic;assumption.
         intros ;apply P_id_u'11'1_monotonic;assumption.
         intros ;apply P_id_u'5'1_monotonic;assumption.
         intros ;apply P_id_u'16'1_monotonic;assumption.
         intros ;apply P_id_intersect'ii'in_bounded;assumption.
         intros ;apply P_id_tautology'i'out_bounded;assumption.
         intros ;apply P_id_u'6'1_bounded;assumption.
         intros ;apply P_id_u'3'1_bounded;assumption.
         intros ;apply P_id_u'12'1_bounded;assumption.
         intros ;apply P_id_u'2'1_bounded;assumption.
         intros ;apply P_id_u'9'1_bounded;assumption.
         intros ;apply P_id_iff_bounded;assumption.
         intros ;apply P_id_u'14'1_bounded;assumption.
         intros ;apply P_id_intersect'ii'out_bounded;assumption.
         intros ;apply P_id_u'7'1_bounded;assumption.
         intros ;apply P_id_x'2d_bounded;assumption.
         intros ;apply P_id_u'13'1_bounded;assumption.
         intros ;apply P_id_sequent_bounded;assumption.
         intros ;apply P_id_u'10'1_bounded;assumption.
         intros ;apply P_id_x'2a_bounded;assumption.
         intros ;apply P_id_tautology'i'in_bounded;assumption.
         intros ;apply P_id_cons_bounded;assumption.
         intros ;apply P_id_u'6'2_bounded;assumption.
         intros ;apply P_id_x'2b_bounded;assumption.
         intros ;apply P_id_u'12'2_bounded;assumption.
         intros ;apply P_id_reduce'ii'in_bounded;assumption.
         intros ;apply P_id_p_bounded;assumption.
         intros ;apply P_id_u'4'1_bounded;assumption.
         intros ;apply P_id_u'15'1_bounded;assumption.
         intros ;apply P_id_u'1'1_bounded;assumption.
         intros ;apply P_id_u'8'1_bounded;assumption.
         intros ;apply P_id_reduce'ii'out_bounded;assumption.
         intros ;apply P_id_nil_bounded;assumption.
         intros ;apply P_id_if_bounded;assumption.
         intros ;apply P_id_u'11'1_bounded;assumption.
         intros ;apply P_id_u'5'1_bounded;assumption.
         intros ;apply P_id_u'16'1_bounded;assumption.
         apply rules_monotonic.
       Qed.
       
       Definition P_id_U'12'2 (x20:Z) := 0.
       
       Definition P_id_U'6'1 (x20:Z) (x21:Z) (x22:Z) (x23:Z) (x24:Z) := 0.
       
       Definition P_id_REDUCE'II'IN (x20:Z) (x21:Z) := 2* x20.
       
       Definition P_id_TAUTOLOGY'I'IN (x20:Z) := 0.
       
       Definition P_id_U'9'1 (x20:Z) := 0.
       
       Definition P_id_U'1'1 (x20:Z) := 0.
       
       Definition P_id_U'14'1 (x20:Z) := 0.
       
       Definition P_id_U'7'1 (x20:Z) := 0.
       
       Definition P_id_U'4'1 (x20:Z) := 0.
       
       Definition P_id_U'11'1 (x20:Z) := 0.
       
       Definition P_id_INTERSECT'II'IN (x20:Z) (x21:Z) := 0.
       
       Definition P_id_U'13'1 (x20:Z) := 0.
       
       Definition P_id_U'6'2 (x20:Z) := 0.
       
       Definition P_id_U'3'1 (x20:Z) := 0.
       
       Definition P_id_U'16'1 (x20:Z) := 0.
       
       Definition P_id_U'10'1 (x20:Z) := 0.
       
       Definition P_id_U'2'1 (x20:Z) := 0.
       
       Definition P_id_U'15'1 (x20:Z) := 0.
       
       Definition P_id_U'8'1 (x20:Z) := 0.
       
       Definition P_id_U'5'1 (x20:Z) := 0.
       
       Definition P_id_U'12'1 (x20:Z) (x21:Z) (x22:Z) (x23:Z) (x24:Z) := 0.
       
       Lemma P_id_U'12'2_monotonic :
        forall x20 x21, 
         (0 <= x21)/\ (x21 <= x20) ->P_id_U'12'2 x21 <= P_id_U'12'2 x20.
       Proof.
         intros x21 x20.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U'6'1_monotonic :
        forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, 
         (0 <= x29)/\ (x29 <= x28) ->
          (0 <= x27)/\ (x27 <= x26) ->
           (0 <= x25)/\ (x25 <= x24) ->
            (0 <= x23)/\ (x23 <= x22) ->
             (0 <= x21)/\ (x21 <= x20) ->
              P_id_U'6'1 x21 x23 x25 x27 x29 <= P_id_U'6'1 x20 x22 x24 
                                                 x26 x28.
       Proof.
         intros x29 x28 x27 x26 x25 x24 x23 x22 x21 x20.
         intros [H_1 H_0].
         intros [H_3 H_2].
         intros [H_5 H_4].
         intros [H_7 H_6].
         intros [H_9 H_8].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_REDUCE'II'IN_monotonic :
        forall x20 x22 x21 x23, 
         (0 <= x23)/\ (x23 <= x22) ->
          (0 <= x21)/\ (x21 <= x20) ->
           P_id_REDUCE'II'IN x21 x23 <= P_id_REDUCE'II'IN x20 x22.
       Proof.
         intros x23 x22 x21 x20.
         intros [H_1 H_0].
         intros [H_3 H_2].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_TAUTOLOGY'I'IN_monotonic :
        forall x20 x21, 
         (0 <= x21)/\ (x21 <= x20) ->
          P_id_TAUTOLOGY'I'IN x21 <= P_id_TAUTOLOGY'I'IN x20.
       Proof.
         intros x21 x20.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U'9'1_monotonic :
        forall x20 x21, 
         (0 <= x21)/\ (x21 <= x20) ->P_id_U'9'1 x21 <= P_id_U'9'1 x20.
       Proof.
         intros x21 x20.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U'1'1_monotonic :
        forall x20 x21, 
         (0 <= x21)/\ (x21 <= x20) ->P_id_U'1'1 x21 <= P_id_U'1'1 x20.
       Proof.
         intros x21 x20.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U'14'1_monotonic :
        forall x20 x21, 
         (0 <= x21)/\ (x21 <= x20) ->P_id_U'14'1 x21 <= P_id_U'14'1 x20.
       Proof.
         intros x21 x20.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U'7'1_monotonic :
        forall x20 x21, 
         (0 <= x21)/\ (x21 <= x20) ->P_id_U'7'1 x21 <= P_id_U'7'1 x20.
       Proof.
         intros x21 x20.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U'4'1_monotonic :
        forall x20 x21, 
         (0 <= x21)/\ (x21 <= x20) ->P_id_U'4'1 x21 <= P_id_U'4'1 x20.
       Proof.
         intros x21 x20.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U'11'1_monotonic :
        forall x20 x21, 
         (0 <= x21)/\ (x21 <= x20) ->P_id_U'11'1 x21 <= P_id_U'11'1 x20.
       Proof.
         intros x21 x20.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_INTERSECT'II'IN_monotonic :
        forall x20 x22 x21 x23, 
         (0 <= x23)/\ (x23 <= x22) ->
          (0 <= x21)/\ (x21 <= x20) ->
           P_id_INTERSECT'II'IN x21 x23 <= P_id_INTERSECT'II'IN x20 x22.
       Proof.
         intros x23 x22 x21 x20.
         intros [H_1 H_0].
         intros [H_3 H_2].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U'13'1_monotonic :
        forall x20 x21, 
         (0 <= x21)/\ (x21 <= x20) ->P_id_U'13'1 x21 <= P_id_U'13'1 x20.
       Proof.
         intros x21 x20.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U'6'2_monotonic :
        forall x20 x21, 
         (0 <= x21)/\ (x21 <= x20) ->P_id_U'6'2 x21 <= P_id_U'6'2 x20.
       Proof.
         intros x21 x20.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U'3'1_monotonic :
        forall x20 x21, 
         (0 <= x21)/\ (x21 <= x20) ->P_id_U'3'1 x21 <= P_id_U'3'1 x20.
       Proof.
         intros x21 x20.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U'16'1_monotonic :
        forall x20 x21, 
         (0 <= x21)/\ (x21 <= x20) ->P_id_U'16'1 x21 <= P_id_U'16'1 x20.
       Proof.
         intros x21 x20.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U'10'1_monotonic :
        forall x20 x21, 
         (0 <= x21)/\ (x21 <= x20) ->P_id_U'10'1 x21 <= P_id_U'10'1 x20.
       Proof.
         intros x21 x20.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U'2'1_monotonic :
        forall x20 x21, 
         (0 <= x21)/\ (x21 <= x20) ->P_id_U'2'1 x21 <= P_id_U'2'1 x20.
       Proof.
         intros x21 x20.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U'15'1_monotonic :
        forall x20 x21, 
         (0 <= x21)/\ (x21 <= x20) ->P_id_U'15'1 x21 <= P_id_U'15'1 x20.
       Proof.
         intros x21 x20.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U'8'1_monotonic :
        forall x20 x21, 
         (0 <= x21)/\ (x21 <= x20) ->P_id_U'8'1 x21 <= P_id_U'8'1 x20.
       Proof.
         intros x21 x20.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U'5'1_monotonic :
        forall x20 x21, 
         (0 <= x21)/\ (x21 <= x20) ->P_id_U'5'1 x21 <= P_id_U'5'1 x20.
       Proof.
         intros x21 x20.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_U'12'1_monotonic :
        forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, 
         (0 <= x29)/\ (x29 <= x28) ->
          (0 <= x27)/\ (x27 <= x26) ->
           (0 <= x25)/\ (x25 <= x24) ->
            (0 <= x23)/\ (x23 <= x22) ->
             (0 <= x21)/\ (x21 <= x20) ->
              P_id_U'12'1 x21 x23 x25 x27 x29 <= P_id_U'12'1 x20 x22 
                                                  x24 x26 x28.
       Proof.
         intros x29 x28 x27 x26 x25 x24 x23 x22 x21 x20.
         intros [H_1 H_0].
         intros [H_3 H_2].
         intros [H_5 H_4].
         intros [H_7 H_6].
         intros [H_9 H_8].
         
         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_intersect'ii'in P_id_tautology'i'out 
          P_id_u'6'1 P_id_u'3'1 P_id_u'12'1 P_id_u'2'1 P_id_u'9'1 P_id_iff 
          P_id_u'14'1 P_id_intersect'ii'out P_id_u'7'1 P_id_x'2d P_id_u'13'1 
          P_id_sequent P_id_u'10'1 P_id_x'2a P_id_tautology'i'in P_id_cons 
          P_id_u'6'2 P_id_x'2b P_id_u'12'2 P_id_reduce'ii'in P_id_p 
          P_id_u'4'1 P_id_u'15'1 P_id_u'1'1 P_id_u'8'1 P_id_reduce'ii'out 
          P_id_nil P_id_if P_id_u'11'1 P_id_u'5'1 P_id_u'16'1 P_id_U'12'2 
          P_id_U'6'1 P_id_REDUCE'II'IN P_id_TAUTOLOGY'I'IN P_id_U'9'1 
          P_id_U'1'1 P_id_U'14'1 P_id_U'7'1 P_id_U'4'1 P_id_U'11'1 
          P_id_INTERSECT'II'IN P_id_U'13'1 P_id_U'6'2 P_id_U'3'1 P_id_U'16'1 
          P_id_U'10'1 P_id_U'2'1 P_id_U'15'1 P_id_U'8'1 P_id_U'5'1 
          P_id_U'12'1.
       
       Lemma marked_measure_equation :
        forall t, 
         marked_measure t = match t with
                              | (algebra.Alg.Term algebra.F.id_u'12'2 
                                 (x20::nil)) =>
                               P_id_U'12'2 (measure x20)
                              | (algebra.Alg.Term algebra.F.id_u'6'1 (x24::
                                 x23::x22::x21::x20::nil)) =>
                               P_id_U'6'1 (measure x24) (measure x23) 
                                (measure x22) (measure x21) (measure x20)
                              | (algebra.Alg.Term algebra.F.id_reduce'ii'in 
                                 (x21::x20::nil)) =>
                               P_id_REDUCE'II'IN (measure x21) (measure x20)
                              | (algebra.Alg.Term 
                                 algebra.F.id_tautology'i'in (x20::nil)) =>
                               P_id_TAUTOLOGY'I'IN (measure x20)
                              | (algebra.Alg.Term algebra.F.id_u'9'1 
                                 (x20::nil)) =>
                               P_id_U'9'1 (measure x20)
                              | (algebra.Alg.Term algebra.F.id_u'1'1 
                                 (x20::nil)) =>
                               P_id_U'1'1 (measure x20)
                              | (algebra.Alg.Term algebra.F.id_u'14'1 
                                 (x20::nil)) =>
                               P_id_U'14'1 (measure x20)
                              | (algebra.Alg.Term algebra.F.id_u'7'1 
                                 (x20::nil)) =>
                               P_id_U'7'1 (measure x20)
                              | (algebra.Alg.Term algebra.F.id_u'4'1 
                                 (x20::nil)) =>
                               P_id_U'4'1 (measure x20)
                              | (algebra.Alg.Term algebra.F.id_u'11'1 
                                 (x20::nil)) =>
                               P_id_U'11'1 (measure x20)
                              | (algebra.Alg.Term 
                                 algebra.F.id_intersect'ii'in (x21::
                                 x20::nil)) =>
                               P_id_INTERSECT'II'IN (measure x21) 
                                (measure x20)
                              | (algebra.Alg.Term algebra.F.id_u'13'1 
                                 (x20::nil)) =>
                               P_id_U'13'1 (measure x20)
                              | (algebra.Alg.Term algebra.F.id_u'6'2 
                                 (x20::nil)) =>
                               P_id_U'6'2 (measure x20)
                              | (algebra.Alg.Term algebra.F.id_u'3'1 
                                 (x20::nil)) =>
                               P_id_U'3'1 (measure x20)
                              | (algebra.Alg.Term algebra.F.id_u'16'1 
                                 (x20::nil)) =>
                               P_id_U'16'1 (measure x20)
                              | (algebra.Alg.Term algebra.F.id_u'10'1 
                                 (x20::nil)) =>
                               P_id_U'10'1 (measure x20)
                              | (algebra.Alg.Term algebra.F.id_u'2'1 
                                 (x20::nil)) =>
                               P_id_U'2'1 (measure x20)
                              | (algebra.Alg.Term algebra.F.id_u'15'1 
                                 (x20::nil)) =>
                               P_id_U'15'1 (measure x20)
                              | (algebra.Alg.Term algebra.F.id_u'8'1 
                                 (x20::nil)) =>
                               P_id_U'8'1 (measure x20)
                              | (algebra.Alg.Term algebra.F.id_u'5'1 
                                 (x20::nil)) =>
                               P_id_U'5'1 (measure x20)
                              | (algebra.Alg.Term algebra.F.id_u'12'1 (x24::
                                 x23::x22::x21::x20::nil)) =>
                               P_id_U'12'1 (measure x24) (measure x23) 
                                (measure x22) (measure x21) (measure x20)
                              | _ => 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_intersect'ii'in_monotonic;assumption.
         intros ;apply P_id_u'6'1_monotonic;assumption.
         intros ;apply P_id_u'3'1_monotonic;assumption.
         intros ;apply P_id_u'12'1_monotonic;assumption.
         intros ;apply P_id_u'2'1_monotonic;assumption.
         intros ;apply P_id_u'9'1_monotonic;assumption.
         intros ;apply P_id_iff_monotonic;assumption.
         intros ;apply P_id_u'14'1_monotonic;assumption.
         intros ;apply P_id_u'7'1_monotonic;assumption.
         intros ;apply P_id_x'2d_monotonic;assumption.
         intros ;apply P_id_u'13'1_monotonic;assumption.
         intros ;apply P_id_sequent_monotonic;assumption.
         intros ;apply P_id_u'10'1_monotonic;assumption.
         intros ;apply P_id_x'2a_monotonic;assumption.
         intros ;apply P_id_tautology'i'in_monotonic;assumption.
         intros ;apply P_id_cons_monotonic;assumption.
         intros ;apply P_id_u'6'2_monotonic;assumption.
         intros ;apply P_id_x'2b_monotonic;assumption.
         intros ;apply P_id_u'12'2_monotonic;assumption.
         intros ;apply P_id_reduce'ii'in_monotonic;assumption.
         intros ;apply P_id_p_monotonic;assumption.
         intros ;apply P_id_u'4'1_monotonic;assumption.
         intros ;apply P_id_u'15'1_monotonic;assumption.
         intros ;apply P_id_u'1'1_monotonic;assumption.
         intros ;apply P_id_u'8'1_monotonic;assumption.
         intros ;apply P_id_if_monotonic;assumption.
         intros ;apply P_id_u'11'1_monotonic;assumption.
         intros ;apply P_id_u'5'1_monotonic;assumption.
         intros ;apply P_id_u'16'1_monotonic;assumption.
         intros ;apply P_id_intersect'ii'in_bounded;assumption.
         intros ;apply P_id_tautology'i'out_bounded;assumption.
         intros ;apply P_id_u'6'1_bounded;assumption.
         intros ;apply P_id_u'3'1_bounded;assumption.
         intros ;apply P_id_u'12'1_bounded;assumption.
         intros ;apply P_id_u'2'1_bounded;assumption.
         intros ;apply P_id_u'9'1_bounded;assumption.
         intros ;apply P_id_iff_bounded;assumption.
         intros ;apply P_id_u'14'1_bounded;assumption.
         intros ;apply P_id_intersect'ii'out_bounded;assumption.
         intros ;apply P_id_u'7'1_bounded;assumption.
         intros ;apply P_id_x'2d_bounded;assumption.
         intros ;apply P_id_u'13'1_bounded;assumption.
         intros ;apply P_id_sequent_bounded;assumption.
         intros ;apply P_id_u'10'1_bounded;assumption.
         intros ;apply P_id_x'2a_bounded;assumption.
         intros ;apply P_id_tautology'i'in_bounded;assumption.
         intros ;apply P_id_cons_bounded;assumption.
         intros ;apply P_id_u'6'2_bounded;assumption.
         intros ;apply P_id_x'2b_bounded;assumption.
         intros ;apply P_id_u'12'2_bounded;assumption.
         intros ;apply P_id_reduce'ii'in_bounded;assumption.
         intros ;apply P_id_p_bounded;assumption.
         intros ;apply P_id_u'4'1_bounded;assumption.
         intros ;apply P_id_u'15'1_bounded;assumption.
         intros ;apply P_id_u'1'1_bounded;assumption.
         intros ;apply P_id_u'8'1_bounded;assumption.
         intros ;apply P_id_reduce'ii'out_bounded;assumption.
         intros ;apply P_id_nil_bounded;assumption.
         intros ;apply P_id_if_bounded;assumption.
         intros ;apply P_id_u'11'1_bounded;assumption.
         intros ;apply P_id_u'5'1_bounded;assumption.
         intros ;apply P_id_u'16'1_bounded;assumption.
         apply rules_monotonic.
         intros ;apply P_id_U'12'2_monotonic;assumption.
         intros ;apply P_id_U'6'1_monotonic;assumption.
         intros ;apply P_id_REDUCE'II'IN_monotonic;assumption.
         intros ;apply P_id_TAUTOLOGY'I'IN_monotonic;assumption.
         intros ;apply P_id_U'9'1_monotonic;assumption.
         intros ;apply P_id_U'1'1_monotonic;assumption.
         intros ;apply P_id_U'14'1_monotonic;assumption.
         intros ;apply P_id_U'7'1_monotonic;assumption.
         intros ;apply P_id_U'4'1_monotonic;assumption.
         intros ;apply P_id_U'11'1_monotonic;assumption.
         intros ;apply P_id_INTERSECT'II'IN_monotonic;assumption.
         intros ;apply P_id_U'13'1_monotonic;assumption.
         intros ;apply P_id_U'6'2_monotonic;assumption.
         intros ;apply P_id_U'3'1_monotonic;assumption.
         intros ;apply P_id_U'16'1_monotonic;assumption.
         intros ;apply P_id_U'10'1_monotonic;assumption.
         intros ;apply P_id_U'2'1_monotonic;assumption.
         intros ;apply P_id_U'15'1_monotonic;assumption.
         intros ;apply P_id_U'8'1_monotonic;assumption.
         intros ;apply P_id_U'5'1_monotonic;assumption.
         intros ;apply P_id_U'12'1_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_19_large_large_scc_1.DP_R_xml_0_scc_19_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_19_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_intersect'ii'in (x20:Z) (x21:Z) := 0.
      
      Definition P_id_tautology'i'out  := 0.
      
      Definition P_id_u'6'1 (x20:Z) (x21:Z) (x22:Z) (x23:Z) (x24:Z) := 0.
      
      Definition P_id_u'3'1 (x20:Z) := 0.
      
      Definition P_id_u'12'1 (x20:Z) (x21:Z) (x22:Z) (x23:Z) (x24:Z) := 0.
      
      Definition P_id_u'2'1 (x20:Z) := 0.
      
      Definition P_id_u'9'1 (x20:Z) := 0.
      
      Definition P_id_iff (x20:Z) (x21:Z) := 0.
      
      Definition P_id_u'14'1 (x20:Z) := 0.
      
      Definition P_id_intersect'ii'out  := 0.
      
      Definition P_id_u'7'1 (x20:Z) := 0.
      
      Definition P_id_x'2d (x20:Z) := 0.
      
      Definition P_id_u'13'1 (x20:Z) := 0.
      
      Definition P_id_sequent (x20:Z) (x21:Z) := 1* x20.
      
      Definition P_id_u'10'1 (x20:Z) := 0.
      
      Definition P_id_x'2a (x20:Z) (x21:Z) := 0.
      
      Definition P_id_tautology'i'in (x20:Z) := 3 + 3* x20.
      
      Definition P_id_cons (x20:Z) (x21:Z) := 1* x20.
      
      Definition P_id_u'6'2 (x20:Z) := 0.
      
      Definition P_id_x'2b (x20:Z) (x21:Z) := 2 + 1* x20 + 2* x21.
      
      Definition P_id_u'12'2 (x20:Z) := 0.
      
      Definition P_id_reduce'ii'in (x20:Z) (x21:Z) := 0.
      
      Definition P_id_p (x20:Z) := 0.
      
      Definition P_id_u'4'1 (x20:Z) := 0.
      
      Definition P_id_u'15'1 (x20:Z) := 0.
      
      Definition P_id_u'1'1 (x20:Z) := 0.
      
      Definition P_id_u'8'1 (x20:Z) := 0.
      
      Definition P_id_reduce'ii'out  := 0.
      
      Definition P_id_nil  := 0.
      
      Definition P_id_if (x20:Z) (x21:Z) := 0.
      
      Definition P_id_u'11'1 (x20:Z) := 0.
      
      Definition P_id_u'5'1 (x20:Z) := 0.
      
      Definition P_id_u'16'1 (x20:Z) := 0.
      
      Lemma P_id_intersect'ii'in_monotonic :
       forall x20 x22 x21 x23, 
        (0 <= x23)/\ (x23 <= x22) ->
         (0 <= x21)/\ (x21 <= x20) ->
          P_id_intersect'ii'in x21 x23 <= P_id_intersect'ii'in x20 x22.
      Proof.
        intros x23 x22 x21 x20.
        intros [H_1 H_0].
        intros [H_3 H_2].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_u'6'1_monotonic :
       forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, 
        (0 <= x29)/\ (x29 <= x28) ->
         (0 <= x27)/\ (x27 <= x26) ->
          (0 <= x25)/\ (x25 <= x24) ->
           (0 <= x23)/\ (x23 <= x22) ->
            (0 <= x21)/\ (x21 <= x20) ->
             P_id_u'6'1 x21 x23 x25 x27 x29 <= P_id_u'6'1 x20 x22 x24 x26 x28.
      Proof.
        intros x29 x28 x27 x26 x25 x24 x23 x22 x21 x20.
        intros [H_1 H_0].
        intros [H_3 H_2].
        intros [H_5 H_4].
        intros [H_7 H_6].
        intros [H_9 H_8].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_u'3'1_monotonic :
       forall x20 x21, 
        (0 <= x21)/\ (x21 <= x20) ->P_id_u'3'1 x21 <= P_id_u'3'1 x20.
      Proof.
        intros x21 x20.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_u'12'1_monotonic :
       forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, 
        (0 <= x29)/\ (x29 <= x28) ->
         (0 <= x27)/\ (x27 <= x26) ->
          (0 <= x25)/\ (x25 <= x24) ->
           (0 <= x23)/\ (x23 <= x22) ->
            (0 <= x21)/\ (x21 <= x20) ->
             P_id_u'12'1 x21 x23 x25 x27 x29 <= P_id_u'12'1 x20 x22 x24 
                                                 x26 x28.
      Proof.
        intros x29 x28 x27 x26 x25 x24 x23 x22 x21 x20.
        intros [H_1 H_0].
        intros [H_3 H_2].
        intros [H_5 H_4].
        intros [H_7 H_6].
        intros [H_9 H_8].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_u'2'1_monotonic :
       forall x20 x21, 
        (0 <= x21)/\ (x21 <= x20) ->P_id_u'2'1 x21 <= P_id_u'2'1 x20.
      Proof.
        intros x21 x20.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_u'9'1_monotonic :
       forall x20 x21, 
        (0 <= x21)/\ (x21 <= x20) ->P_id_u'9'1 x21 <= P_id_u'9'1 x20.
      Proof.
        intros x21 x20.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_iff_monotonic :
       forall x20 x22 x21 x23, 
        (0 <= x23)/\ (x23 <= x22) ->
         (0 <= x21)/\ (x21 <= x20) ->P_id_iff x21 x23 <= P_id_iff x20 x22.
      Proof.
        intros x23 x22 x21 x20.
        intros [H_1 H_0].
        intros [H_3 H_2].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_u'14'1_monotonic :
       forall x20 x21, 
        (0 <= x21)/\ (x21 <= x20) ->P_id_u'14'1 x21 <= P_id_u'14'1 x20.
      Proof.
        intros x21 x20.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_u'7'1_monotonic :
       forall x20 x21, 
        (0 <= x21)/\ (x21 <= x20) ->P_id_u'7'1 x21 <= P_id_u'7'1 x20.
      Proof.
        intros x21 x20.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_x'2d_monotonic :
       forall x20 x21, 
        (0 <= x21)/\ (x21 <= x20) ->P_id_x'2d x21 <= P_id_x'2d x20.
      Proof.
        intros x21 x20.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_u'13'1_monotonic :
       forall x20 x21, 
        (0 <= x21)/\ (x21 <= x20) ->P_id_u'13'1 x21 <= P_id_u'13'1 x20.
      Proof.
        intros x21 x20.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_sequent_monotonic :
       forall x20 x22 x21 x23, 
        (0 <= x23)/\ (x23 <= x22) ->
         (0 <= x21)/\ (x21 <= x20) ->
          P_id_sequent x21 x23 <= P_id_sequent x20 x22.
      Proof.
        intros x23 x22 x21 x20.
        intros [H_1 H_0].
        intros [H_3 H_2].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_u'10'1_monotonic :
       forall x20 x21, 
        (0 <= x21)/\ (x21 <= x20) ->P_id_u'10'1 x21 <= P_id_u'10'1 x20.
      Proof.
        intros x21 x20.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_x'2a_monotonic :
       forall x20 x22 x21 x23, 
        (0 <= x23)/\ (x23 <= x22) ->
         (0 <= x21)/\ (x21 <= x20) ->P_id_x'2a x21 x23 <= P_id_x'2a x20 x22.
      Proof.
        intros x23 x22 x21 x20.
        intros [H_1 H_0].
        intros [H_3 H_2].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_tautology'i'in_monotonic :
       forall x20 x21, 
        (0 <= x21)/\ (x21 <= x20) ->
         P_id_tautology'i'in x21 <= P_id_tautology'i'in x20.
      Proof.
        intros x21 x20.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_cons_monotonic :
       forall x20 x22 x21 x23, 
        (0 <= x23)/\ (x23 <= x22) ->
         (0 <= x21)/\ (x21 <= x20) ->P_id_cons x21 x23 <= P_id_cons x20 x22.
      Proof.
        intros x23 x22 x21 x20.
        intros [H_1 H_0].
        intros [H_3 H_2].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_u'6'2_monotonic :
       forall x20 x21, 
        (0 <= x21)/\ (x21 <= x20) ->P_id_u'6'2 x21 <= P_id_u'6'2 x20.
      Proof.
        intros x21 x20.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_x'2b_monotonic :
       forall x20 x22 x21 x23, 
        (0 <= x23)/\ (x23 <= x22) ->
         (0 <= x21)/\ (x21 <= x20) ->P_id_x'2b x21 x23 <= P_id_x'2b x20 x22.
      Proof.
        intros x23 x22 x21 x20.
        intros [H_1 H_0].
        intros [H_3 H_2].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_u'12'2_monotonic :
       forall x20 x21, 
        (0 <= x21)/\ (x21 <= x20) ->P_id_u'12'2 x21 <= P_id_u'12'2 x20.
      Proof.
        intros x21 x20.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_reduce'ii'in_monotonic :
       forall x20 x22 x21 x23, 
        (0 <= x23)/\ (x23 <= x22) ->
         (0 <= x21)/\ (x21 <= x20) ->
          P_id_reduce'ii'in x21 x23 <= P_id_reduce'ii'in x20 x22.
      Proof.
        intros x23 x22 x21 x20.
        intros [H_1 H_0].
        intros [H_3 H_2].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_p_monotonic :
       forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_p x21 <= P_id_p x20.
      Proof.
        intros x21 x20.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_u'4'1_monotonic :
       forall x20 x21, 
        (0 <= x21)/\ (x21 <= x20) ->P_id_u'4'1 x21 <= P_id_u'4'1 x20.
      Proof.
        intros x21 x20.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_u'15'1_monotonic :
       forall x20 x21, 
        (0 <= x21)/\ (x21 <= x20) ->P_id_u'15'1 x21 <= P_id_u'15'1 x20.
      Proof.
        intros x21 x20.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_u'1'1_monotonic :
       forall x20 x21, 
        (0 <= x21)/\ (x21 <= x20) ->P_id_u'1'1 x21 <= P_id_u'1'1 x20.
      Proof.
        intros x21 x20.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_u'8'1_monotonic :
       forall x20 x21, 
        (0 <= x21)/\ (x21 <= x20) ->P_id_u'8'1 x21 <= P_id_u'8'1 x20.
      Proof.
        intros x21 x20.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_if_monotonic :
       forall x20 x22 x21 x23, 
        (0 <= x23)/\ (x23 <= x22) ->
         (0 <= x21)/\ (x21 <= x20) ->P_id_if x21 x23 <= P_id_if x20 x22.
      Proof.
        intros x23 x22 x21 x20.
        intros [H_1 H_0].
        intros [H_3 H_2].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_u'11'1_monotonic :
       forall x20 x21, 
        (0 <= x21)/\ (x21 <= x20) ->P_id_u'11'1 x21 <= P_id_u'11'1 x20.
      Proof.
        intros x21 x20.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_u'5'1_monotonic :
       forall x20 x21, 
        (0 <= x21)/\ (x21 <= x20) ->P_id_u'5'1 x21 <= P_id_u'5'1 x20.
      Proof.
        intros x21 x20.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_u'16'1_monotonic :
       forall x20 x21, 
        (0 <= x21)/\ (x21 <= x20) ->P_id_u'16'1 x21 <= P_id_u'16'1 x20.
      Proof.
        intros x21 x20.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_intersect'ii'in_bounded :
       forall x20 x21, 
        (0 <= x20) ->(0 <= x21) ->0 <= P_id_intersect'ii'in x21 x20.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_tautology'i'out_bounded : 0 <= P_id_tautology'i'out .
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_u'6'1_bounded :
       forall x24 x20 x22 x21 x23, 
        (0 <= x20) ->
         (0 <= x21) ->
          (0 <= x22) ->
           (0 <= x23) ->(0 <= x24) ->0 <= P_id_u'6'1 x24 x23 x22 x21 x20.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_u'3'1_bounded :
       forall x20, (0 <= x20) ->0 <= P_id_u'3'1 x20.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_u'12'1_bounded :
       forall x24 x20 x22 x21 x23, 
        (0 <= x20) ->
         (0 <= x21) ->
          (0 <= x22) ->
           (0 <= x23) ->(0 <= x24) ->0 <= P_id_u'12'1 x24 x23 x22 x21 x20.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_u'2'1_bounded :
       forall x20, (0 <= x20) ->0 <= P_id_u'2'1 x20.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_u'9'1_bounded :
       forall x20, (0 <= x20) ->0 <= P_id_u'9'1 x20.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_iff_bounded :
       forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_iff x21 x20.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_u'14'1_bounded :
       forall x20, (0 <= x20) ->0 <= P_id_u'14'1 x20.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_intersect'ii'out_bounded : 0 <= P_id_intersect'ii'out .
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_u'7'1_bounded :
       forall x20, (0 <= x20) ->0 <= P_id_u'7'1 x20.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_x'2d_bounded : forall x20, (0 <= x20) ->0 <= P_id_x'2d x20.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_u'13'1_bounded :
       forall x20, (0 <= x20) ->0 <= P_id_u'13'1 x20.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_sequent_bounded :
       forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_sequent x21 x20.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_u'10'1_bounded :
       forall x20, (0 <= x20) ->0 <= P_id_u'10'1 x20.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_x'2a_bounded :
       forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_x'2a x21 x20.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_tautology'i'in_bounded :
       forall x20, (0 <= x20) ->0 <= P_id_tautology'i'in x20.
      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 x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_cons x21 x20.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_u'6'2_bounded :
       forall x20, (0 <= x20) ->0 <= P_id_u'6'2 x20.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_x'2b_bounded :
       forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_x'2b x21 x20.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_u'12'2_bounded :
       forall x20, (0 <= x20) ->0 <= P_id_u'12'2 x20.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_reduce'ii'in_bounded :
       forall x20 x21, 
        (0 <= x20) ->(0 <= x21) ->0 <= P_id_reduce'ii'in x21 x20.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_p_bounded : forall x20, (0 <= x20) ->0 <= P_id_p x20.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_u'4'1_bounded :
       forall x20, (0 <= x20) ->0 <= P_id_u'4'1 x20.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_u'15'1_bounded :
       forall x20, (0 <= x20) ->0 <= P_id_u'15'1 x20.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_u'1'1_bounded :
       forall x20, (0 <= x20) ->0 <= P_id_u'1'1 x20.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_u'8'1_bounded :
       forall x20, (0 <= x20) ->0 <= P_id_u'8'1 x20.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_reduce'ii'out_bounded : 0 <= P_id_reduce'ii'out .
      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_if_bounded :
       forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_if x21 x20.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_u'11'1_bounded :
       forall x20, (0 <= x20) ->0 <= P_id_u'11'1 x20.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_u'5'1_bounded :
       forall x20, (0 <= x20) ->0 <= P_id_u'5'1 x20.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_u'16'1_bounded :
       forall x20, (0 <= x20) ->0 <= P_id_u'16'1 x20.
      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_intersect'ii'in P_id_tautology'i'out 
         P_id_u'6'1 P_id_u'3'1 P_id_u'12'1 P_id_u'2'1 P_id_u'9'1 P_id_iff 
         P_id_u'14'1 P_id_intersect'ii'out P_id_u'7'1 P_id_x'2d P_id_u'13'1 
         P_id_sequent P_id_u'10'1 P_id_x'2a P_id_tautology'i'in P_id_cons 
         P_id_u'6'2 P_id_x'2b P_id_u'12'2 P_id_reduce'ii'in P_id_p 
         P_id_u'4'1 P_id_u'15'1 P_id_u'1'1 P_id_u'8'1 P_id_reduce'ii'out 
         P_id_nil P_id_if P_id_u'11'1 P_id_u'5'1 P_id_u'16'1.
      
      Lemma measure_equation :
       forall t, 
        measure t = match t with
                      | (algebra.Alg.Term algebra.F.id_intersect'ii'in (x21::
                         x20::nil)) =>
                       P_id_intersect'ii'in (measure x21) (measure x20)
                      | (algebra.Alg.Term algebra.F.id_tautology'i'out nil) =>
                       P_id_tautology'i'out 
                      | (algebra.Alg.Term algebra.F.id_u'6'1 (x24::x23::x22::
                         x21::x20::nil)) =>
                       P_id_u'6'1 (measure x24) (measure x23) (measure x22) 
                        (measure x21) (measure x20)
                      | (algebra.Alg.Term algebra.F.id_u'3'1 (x20::nil)) =>
                       P_id_u'3'1 (measure x20)
                      | (algebra.Alg.Term algebra.F.id_u'12'1 (x24::x23::
                         x22::x21::x20::nil)) =>
                       P_id_u'12'1 (measure x24) (measure x23) (measure x22) 
                        (measure x21) (measure x20)
                      | (algebra.Alg.Term algebra.F.id_u'2'1 (x20::nil)) =>
                       P_id_u'2'1 (measure x20)
                      | (algebra.Alg.Term algebra.F.id_u'9'1 (x20::nil)) =>
                       P_id_u'9'1 (measure x20)
                      | (algebra.Alg.Term algebra.F.id_iff (x21::x20::nil)) =>
                       P_id_iff (measure x21) (measure x20)
                      | (algebra.Alg.Term algebra.F.id_u'14'1 (x20::nil)) =>
                       P_id_u'14'1 (measure x20)
                      | (algebra.Alg.Term algebra.F.id_intersect'ii'out nil) =>
                       P_id_intersect'ii'out 
                      | (algebra.Alg.Term algebra.F.id_u'7'1 (x20::nil)) =>
                       P_id_u'7'1 (measure x20)
                      | (algebra.Alg.Term algebra.F.id_x'2d (x20::nil)) =>
                       P_id_x'2d (measure x20)
                      | (algebra.Alg.Term algebra.F.id_u'13'1 (x20::nil)) =>
                       P_id_u'13'1 (measure x20)
                      | (algebra.Alg.Term algebra.F.id_sequent (x21::
                         x20::nil)) =>
                       P_id_sequent (measure x21) (measure x20)
                      | (algebra.Alg.Term algebra.F.id_u'10'1 (x20::nil)) =>
                       P_id_u'10'1 (measure x20)
                      | (algebra.Alg.Term algebra.F.id_x'2a (x21::x20::nil)) =>
                       P_id_x'2a (measure x21) (measure x20)
                      | (algebra.Alg.Term algebra.F.id_tautology'i'in 
                         (x20::nil)) =>
                       P_id_tautology'i'in (measure x20)
                      | (algebra.Alg.Term algebra.F.id_cons (x21::x20::nil)) =>
                       P_id_cons (measure x21) (measure x20)
                      | (algebra.Alg.Term algebra.F.id_u'6'2 (x20::nil)) =>
                       P_id_u'6'2 (measure x20)
                      | (algebra.Alg.Term algebra.F.id_x'2b (x21::x20::nil)) =>
                       P_id_x'2b (measure x21) (measure x20)
                      | (algebra.Alg.Term algebra.F.id_u'12'2 (x20::nil)) =>
                       P_id_u'12'2 (measure x20)
                      | (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::
                         x20::nil)) =>
                       P_id_reduce'ii'in (measure x21) (measure x20)
                      | (algebra.Alg.Term algebra.F.id_p (x20::nil)) =>
                       P_id_p (measure x20)
                      | (algebra.Alg.Term algebra.F.id_u'4'1 (x20::nil)) =>
                       P_id_u'4'1 (measure x20)
                      | (algebra.Alg.Term algebra.F.id_u'15'1 (x20::nil)) =>
                       P_id_u'15'1 (measure x20)
                      | (algebra.Alg.Term algebra.F.id_u'1'1 (x20::nil)) =>
                       P_id_u'1'1 (measure x20)
                      | (algebra.Alg.Term algebra.F.id_u'8'1 (x20::nil)) =>
                       P_id_u'8'1 (measure x20)
                      | (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) =>
                       P_id_reduce'ii'out 
                      | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil 
                      | (algebra.Alg.Term algebra.F.id_if (x21::x20::nil)) =>
                       P_id_if (measure x21) (measure x20)
                      | (algebra.Alg.Term algebra.F.id_u'11'1 (x20::nil)) =>
                       P_id_u'11'1 (measure x20)
                      | (algebra.Alg.Term algebra.F.id_u'5'1 (x20::nil)) =>
                       P_id_u'5'1 (measure x20)
                      | (algebra.Alg.Term algebra.F.id_u'16'1 (x20::nil)) =>
                       P_id_u'16'1 (measure x20)
                      | _ => 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_intersect'ii'in_monotonic;assumption.
        intros ;apply P_id_u'6'1_monotonic;assumption.
        intros ;apply P_id_u'3'1_monotonic;assumption.
        intros ;apply P_id_u'12'1_monotonic;assumption.
        intros ;apply P_id_u'2'1_monotonic;assumption.
        intros ;apply P_id_u'9'1_monotonic;assumption.
        intros ;apply P_id_iff_monotonic;assumption.
        intros ;apply P_id_u'14'1_monotonic;assumption.
        intros ;apply P_id_u'7'1_monotonic;assumption.
        intros ;apply P_id_x'2d_monotonic;assumption.
        intros ;apply P_id_u'13'1_monotonic;assumption.
        intros ;apply P_id_sequent_monotonic;assumption.
        intros ;apply P_id_u'10'1_monotonic;assumption.
        intros ;apply P_id_x'2a_monotonic;assumption.
        intros ;apply P_id_tautology'i'in_monotonic;assumption.
        intros ;apply P_id_cons_monotonic;assumption.
        intros ;apply P_id_u'6'2_monotonic;assumption.
        intros ;apply P_id_x'2b_monotonic;assumption.
        intros ;apply P_id_u'12'2_monotonic;assumption.
        intros ;apply P_id_reduce'ii'in_monotonic;assumption.
        intros ;apply P_id_p_monotonic;assumption.
        intros ;apply P_id_u'4'1_monotonic;assumption.
        intros ;apply P_id_u'15'1_monotonic;assumption.
        intros ;apply P_id_u'1'1_monotonic;assumption.
        intros ;apply P_id_u'8'1_monotonic;assumption.
        intros ;apply P_id_if_monotonic;assumption.
        intros ;apply P_id_u'11'1_monotonic;assumption.
        intros ;apply P_id_u'5'1_monotonic;assumption.
        intros ;apply P_id_u'16'1_monotonic;assumption.
        intros ;apply P_id_intersect'ii'in_bounded;assumption.
        intros ;apply P_id_tautology'i'out_bounded;assumption.
        intros ;apply P_id_u'6'1_bounded;assumption.
        intros ;apply P_id_u'3'1_bounded;assumption.
        intros ;apply P_id_u'12'1_bounded;assumption.
        intros ;apply P_id_u'2'1_bounded;assumption.
        intros ;apply P_id_u'9'1_bounded;assumption.
        intros ;apply P_id_iff_bounded;assumption.
        intros ;apply P_id_u'14'1_bounded;assumption.
        intros ;apply P_id_intersect'ii'out_bounded;assumption.
        intros ;apply P_id_u'7'1_bounded;assumption.
        intros ;apply P_id_x'2d_bounded;assumption.
        intros ;apply P_id_u'13'1_bounded;assumption.
        intros ;apply P_id_sequent_bounded;assumption.
        intros ;apply P_id_u'10'1_bounded;assumption.
        intros ;apply P_id_x'2a_bounded;assumption.
        intros ;apply P_id_tautology'i'in_bounded;assumption.
        intros ;apply P_id_cons_bounded;assumption.
        intros ;apply P_id_u'6'2_bounded;assumption.
        intros ;apply P_id_x'2b_bounded;assumption.
        intros ;apply P_id_u'12'2_bounded;assumption.
        intros ;apply P_id_reduce'ii'in_bounded;assumption.
        intros ;apply P_id_p_bounded;assumption.
        intros ;apply P_id_u'4'1_bounded;assumption.
        intros ;apply P_id_u'15'1_bounded;assumption.
        intros ;apply P_id_u'1'1_bounded;assumption.
        intros ;apply P_id_u'8'1_bounded;assumption.
        intros ;apply P_id_reduce'ii'out_bounded;assumption.
        intros ;apply P_id_nil_bounded;assumption.
        intros ;apply P_id_if_bounded;assumption.
        intros ;apply P_id_u'11'1_bounded;assumption.
        intros ;apply P_id_u'5'1_bounded;assumption.
        intros ;apply P_id_u'16'1_bounded;assumption.
        apply rules_monotonic.
      Qed.
      
      Definition P_id_U'12'2 (x20:Z) := 0.
      
      Definition P_id_U'6'1 (x20:Z) (x21:Z) (x22:Z) (x23:Z) (x24:Z) := 
        2 + 3* x21 + 1* x24.
      
      Definition P_id_REDUCE'II'IN (x20:Z) (x21:Z) := 2* x20 + 1* x21.
      
      Definition P_id_TAUTOLOGY'I'IN (x20:Z) := 0.
      
      Definition P_id_U'9'1 (x20:Z) := 0.
      
      Definition P_id_U'1'1 (x20:Z) := 0.
      
      Definition P_id_U'14'1 (x20:Z) := 0.
      
      Definition P_id_U'7'1 (x20:Z) := 0.
      
      Definition P_id_U'4'1 (x20:Z) := 0.
      
      Definition P_id_U'11'1 (x20:Z) := 0.
      
      Definition P_id_INTERSECT'II'IN (x20:Z) (x21:Z) := 0.
      
      Definition P_id_U'13'1 (x20:Z) := 0.
      
      Definition P_id_U'6'2 (x20:Z) := 0.
      
      Definition P_id_U'3'1 (x20:Z) := 0.
      
      Definition P_id_U'16'1 (x20:Z) := 0.
      
      Definition P_id_U'10'1 (x20:Z) := 0.
      
      Definition P_id_U'2'1 (x20:Z) := 0.
      
      Definition P_id_U'15'1 (x20:Z) := 0.
      
      Definition P_id_U'8'1 (x20:Z) := 0.
      
      Definition P_id_U'5'1 (x20:Z) := 0.
      
      Definition P_id_U'12'1 (x20:Z) (x21:Z) (x22:Z) (x23:Z) (x24:Z) := 0.
      
      Lemma P_id_U'12'2_monotonic :
       forall x20 x21, 
        (0 <= x21)/\ (x21 <= x20) ->P_id_U'12'2 x21 <= P_id_U'12'2 x20.
      Proof.
        intros x21 x20.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U'6'1_monotonic :
       forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, 
        (0 <= x29)/\ (x29 <= x28) ->
         (0 <= x27)/\ (x27 <= x26) ->
          (0 <= x25)/\ (x25 <= x24) ->
           (0 <= x23)/\ (x23 <= x22) ->
            (0 <= x21)/\ (x21 <= x20) ->
             P_id_U'6'1 x21 x23 x25 x27 x29 <= P_id_U'6'1 x20 x22 x24 x26 x28.
      Proof.
        intros x29 x28 x27 x26 x25 x24 x23 x22 x21 x20.
        intros [H_1 H_0].
        intros [H_3 H_2].
        intros [H_5 H_4].
        intros [H_7 H_6].
        intros [H_9 H_8].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_REDUCE'II'IN_monotonic :
       forall x20 x22 x21 x23, 
        (0 <= x23)/\ (x23 <= x22) ->
         (0 <= x21)/\ (x21 <= x20) ->
          P_id_REDUCE'II'IN x21 x23 <= P_id_REDUCE'II'IN x20 x22.
      Proof.
        intros x23 x22 x21 x20.
        intros [H_1 H_0].
        intros [H_3 H_2].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_TAUTOLOGY'I'IN_monotonic :
       forall x20 x21, 
        (0 <= x21)/\ (x21 <= x20) ->
         P_id_TAUTOLOGY'I'IN x21 <= P_id_TAUTOLOGY'I'IN x20.
      Proof.
        intros x21 x20.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U'9'1_monotonic :
       forall x20 x21, 
        (0 <= x21)/\ (x21 <= x20) ->P_id_U'9'1 x21 <= P_id_U'9'1 x20.
      Proof.
        intros x21 x20.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U'1'1_monotonic :
       forall x20 x21, 
        (0 <= x21)/\ (x21 <= x20) ->P_id_U'1'1 x21 <= P_id_U'1'1 x20.
      Proof.
        intros x21 x20.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U'14'1_monotonic :
       forall x20 x21, 
        (0 <= x21)/\ (x21 <= x20) ->P_id_U'14'1 x21 <= P_id_U'14'1 x20.
      Proof.
        intros x21 x20.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U'7'1_monotonic :
       forall x20 x21, 
        (0 <= x21)/\ (x21 <= x20) ->P_id_U'7'1 x21 <= P_id_U'7'1 x20.
      Proof.
        intros x21 x20.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U'4'1_monotonic :
       forall x20 x21, 
        (0 <= x21)/\ (x21 <= x20) ->P_id_U'4'1 x21 <= P_id_U'4'1 x20.
      Proof.
        intros x21 x20.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U'11'1_monotonic :
       forall x20 x21, 
        (0 <= x21)/\ (x21 <= x20) ->P_id_U'11'1 x21 <= P_id_U'11'1 x20.
      Proof.
        intros x21 x20.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_INTERSECT'II'IN_monotonic :
       forall x20 x22 x21 x23, 
        (0 <= x23)/\ (x23 <= x22) ->
         (0 <= x21)/\ (x21 <= x20) ->
          P_id_INTERSECT'II'IN x21 x23 <= P_id_INTERSECT'II'IN x20 x22.
      Proof.
        intros x23 x22 x21 x20.
        intros [H_1 H_0].
        intros [H_3 H_2].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U'13'1_monotonic :
       forall x20 x21, 
        (0 <= x21)/\ (x21 <= x20) ->P_id_U'13'1 x21 <= P_id_U'13'1 x20.
      Proof.
        intros x21 x20.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U'6'2_monotonic :
       forall x20 x21, 
        (0 <= x21)/\ (x21 <= x20) ->P_id_U'6'2 x21 <= P_id_U'6'2 x20.
      Proof.
        intros x21 x20.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U'3'1_monotonic :
       forall x20 x21, 
        (0 <= x21)/\ (x21 <= x20) ->P_id_U'3'1 x21 <= P_id_U'3'1 x20.
      Proof.
        intros x21 x20.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U'16'1_monotonic :
       forall x20 x21, 
        (0 <= x21)/\ (x21 <= x20) ->P_id_U'16'1 x21 <= P_id_U'16'1 x20.
      Proof.
        intros x21 x20.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U'10'1_monotonic :
       forall x20 x21, 
        (0 <= x21)/\ (x21 <= x20) ->P_id_U'10'1 x21 <= P_id_U'10'1 x20.
      Proof.
        intros x21 x20.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U'2'1_monotonic :
       forall x20 x21, 
        (0 <= x21)/\ (x21 <= x20) ->P_id_U'2'1 x21 <= P_id_U'2'1 x20.
      Proof.
        intros x21 x20.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U'15'1_monotonic :
       forall x20 x21, 
        (0 <= x21)/\ (x21 <= x20) ->P_id_U'15'1 x21 <= P_id_U'15'1 x20.
      Proof.
        intros x21 x20.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U'8'1_monotonic :
       forall x20 x21, 
        (0 <= x21)/\ (x21 <= x20) ->P_id_U'8'1 x21 <= P_id_U'8'1 x20.
      Proof.
        intros x21 x20.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U'5'1_monotonic :
       forall x20 x21, 
        (0 <= x21)/\ (x21 <= x20) ->P_id_U'5'1 x21 <= P_id_U'5'1 x20.
      Proof.
        intros x21 x20.
        intros [H_1 H_0].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_U'12'1_monotonic :
       forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, 
        (0 <= x29)/\ (x29 <= x28) ->
         (0 <= x27)/\ (x27 <= x26) ->
          (0 <= x25)/\ (x25 <= x24) ->
           (0 <= x23)/\ (x23 <= x22) ->
            (0 <= x21)/\ (x21 <= x20) ->
             P_id_U'12'1 x21 x23 x25 x27 x29 <= P_id_U'12'1 x20 x22 x24 
                                                 x26 x28.
      Proof.
        intros x29 x28 x27 x26 x25 x24 x23 x22 x21 x20.
        intros [H_1 H_0].
        intros [H_3 H_2].
        intros [H_5 H_4].
        intros [H_7 H_6].
        intros [H_9 H_8].
        
        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_intersect'ii'in P_id_tautology'i'out 
         P_id_u'6'1 P_id_u'3'1 P_id_u'12'1 P_id_u'2'1 P_id_u'9'1 P_id_iff 
         P_id_u'14'1 P_id_intersect'ii'out P_id_u'7'1 P_id_x'2d P_id_u'13'1 
         P_id_sequent P_id_u'10'1 P_id_x'2a P_id_tautology'i'in P_id_cons 
         P_id_u'6'2 P_id_x'2b P_id_u'12'2 P_id_reduce'ii'in P_id_p 
         P_id_u'4'1 P_id_u'15'1 P_id_u'1'1 P_id_u'8'1 P_id_reduce'ii'out 
         P_id_nil P_id_if P_id_u'11'1 P_id_u'5'1 P_id_u'16'1 P_id_U'12'2 
         P_id_U'6'1 P_id_REDUCE'II'IN P_id_TAUTOLOGY'I'IN P_id_U'9'1 
         P_id_U'1'1 P_id_U'14'1 P_id_U'7'1 P_id_U'4'1 P_id_U'11'1 
         P_id_INTERSECT'II'IN P_id_U'13'1 P_id_U'6'2 P_id_U'3'1 P_id_U'16'1 
         P_id_U'10'1 P_id_U'2'1 P_id_U'15'1 P_id_U'8'1 P_id_U'5'1 P_id_U'12'1
        .
      
      Lemma marked_measure_equation :
       forall t, 
        marked_measure t = match t with
                             | (algebra.Alg.Term algebra.F.id_u'12'2 
                                (x20::nil)) =>
                              P_id_U'12'2 (measure x20)
                             | (algebra.Alg.Term algebra.F.id_u'6'1 (x24::
                                x23::x22::x21::x20::nil)) =>
                              P_id_U'6'1 (measure x24) (measure x23) 
                               (measure x22) (measure x21) (measure x20)
                             | (algebra.Alg.Term algebra.F.id_reduce'ii'in 
                                (x21::x20::nil)) =>
                              P_id_REDUCE'II'IN (measure x21) (measure x20)
                             | (algebra.Alg.Term algebra.F.id_tautology'i'in 
                                (x20::nil)) =>
                              P_id_TAUTOLOGY'I'IN (measure x20)
                             | (algebra.Alg.Term algebra.F.id_u'9'1 
                                (x20::nil)) =>
                              P_id_U'9'1 (measure x20)
                             | (algebra.Alg.Term algebra.F.id_u'1'1 
                                (x20::nil)) =>
                              P_id_U'1'1 (measure x20)
                             | (algebra.Alg.Term algebra.F.id_u'14'1 
                                (x20::nil)) =>
                              P_id_U'14'1 (measure x20)
                             | (algebra.Alg.Term algebra.F.id_u'7'1 
                                (x20::nil)) =>
                              P_id_U'7'1 (measure x20)
                             | (algebra.Alg.Term algebra.F.id_u'4'1 
                                (x20::nil)) =>
                              P_id_U'4'1 (measure x20)
                             | (algebra.Alg.Term algebra.F.id_u'11'1 
                                (x20::nil)) =>
                              P_id_U'11'1 (measure x20)
                             | (algebra.Alg.Term 
                                algebra.F.id_intersect'ii'in (x21::x20::nil)) =>
                              P_id_INTERSECT'II'IN (measure x21) 
                               (measure x20)
                             | (algebra.Alg.Term algebra.F.id_u'13'1 
                                (x20::nil)) =>
                              P_id_U'13'1 (measure x20)
                             | (algebra.Alg.Term algebra.F.id_u'6'2 
                                (x20::nil)) =>
                              P_id_U'6'2 (measure x20)
                             | (algebra.Alg.Term algebra.F.id_u'3'1 
                                (x20::nil)) =>
                              P_id_U'3'1 (measure x20)
                             | (algebra.Alg.Term algebra.F.id_u'16'1 
                                (x20::nil)) =>
                              P_id_U'16'1 (measure x20)
                             | (algebra.Alg.Term algebra.F.id_u'10'1 
                                (x20::nil)) =>
                              P_id_U'10'1 (measure x20)
                             | (algebra.Alg.Term algebra.F.id_u'2'1 
                                (x20::nil)) =>
                              P_id_U'2'1 (measure x20)
                             | (algebra.Alg.Term algebra.F.id_u'15'1 
                                (x20::nil)) =>
                              P_id_U'15'1 (measure x20)
                             | (algebra.Alg.Term algebra.F.id_u'8'1 
                                (x20::nil)) =>
                              P_id_U'8'1 (measure x20)
                             | (algebra.Alg.Term algebra.F.id_u'5'1 
                                (x20::nil)) =>
                              P_id_U'5'1 (measure x20)
                             | (algebra.Alg.Term algebra.F.id_u'12'1 (x24::
                                x23::x22::x21::x20::nil)) =>
                              P_id_U'12'1 (measure x24) (measure x23) 
                               (measure x22) (measure x21) (measure x20)
                             | _ => 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_intersect'ii'in_monotonic;assumption.
        intros ;apply P_id_u'6'1_monotonic;assumption.
        intros ;apply P_id_u'3'1_monotonic;assumption.
        intros ;apply P_id_u'12'1_monotonic;assumption.
        intros ;apply P_id_u'2'1_monotonic;assumption.
        intros ;apply P_id_u'9'1_monotonic;assumption.
        intros ;apply P_id_iff_monotonic;assumption.
        intros ;apply P_id_u'14'1_monotonic;assumption.
        intros ;apply P_id_u'7'1_monotonic;assumption.
        intros ;apply P_id_x'2d_monotonic;assumption.
        intros ;apply P_id_u'13'1_monotonic;assumption.
        intros ;apply P_id_sequent_monotonic;assumption.
        intros ;apply P_id_u'10'1_monotonic;assumption.
        intros ;apply P_id_x'2a_monotonic;assumption.
        intros ;apply P_id_tautology'i'in_monotonic;assumption.
        intros ;apply P_id_cons_monotonic;assumption.
        intros ;apply P_id_u'6'2_monotonic;assumption.
        intros ;apply P_id_x'2b_monotonic;assumption.
        intros ;apply P_id_u'12'2_monotonic;assumption.
        intros ;apply P_id_reduce'ii'in_monotonic;assumption.
        intros ;apply P_id_p_monotonic;assumption.
        intros ;apply P_id_u'4'1_monotonic;assumption.
        intros ;apply P_id_u'15'1_monotonic;assumption.
        intros ;apply P_id_u'1'1_monotonic;assumption.
        intros ;apply P_id_u'8'1_monotonic;assumption.
        intros ;apply P_id_if_monotonic;assumption.
        intros ;apply P_id_u'11'1_monotonic;assumption.
        intros ;apply P_id_u'5'1_monotonic;assumption.
        intros ;apply P_id_u'16'1_monotonic;assumption.
        intros ;apply P_id_intersect'ii'in_bounded;assumption.
        intros ;apply P_id_tautology'i'out_bounded;assumption.
        intros ;apply P_id_u'6'1_bounded;assumption.
        intros ;apply P_id_u'3'1_bounded;assumption.
        intros ;apply P_id_u'12'1_bounded;assumption.
        intros ;apply P_id_u'2'1_bounded;assumption.
        intros ;apply P_id_u'9'1_bounded;assumption.
        intros ;apply P_id_iff_bounded;assumption.
        intros ;apply P_id_u'14'1_bounded;assumption.
        intros ;apply P_id_intersect'ii'out_bounded;assumption.
        intros ;apply P_id_u'7'1_bounded;assumption.
        intros ;apply P_id_x'2d_bounded;assumption.
        intros ;apply P_id_u'13'1_bounded;assumption.
        intros ;apply P_id_sequent_bounded;assumption.
        intros ;apply P_id_u'10'1_bounded;assumption.
        intros ;apply P_id_x'2a_bounded;assumption.
        intros ;apply P_id_tautology'i'in_bounded;assumption.
        intros ;apply P_id_cons_bounded;assumption.
        intros ;apply P_id_u'6'2_bounded;assumption.
        intros ;apply P_id_x'2b_bounded;assumption.
        intros ;apply P_id_u'12'2_bounded;assumption.
        intros ;apply P_id_reduce'ii'in_bounded;assumption.
        intros ;apply P_id_p_bounded;assumption.
        intros ;apply P_id_u'4'1_bounded;assumption.
        intros ;apply P_id_u'15'1_bounded;assumption.
        intros ;apply P_id_u'1'1_bounded;assumption.
        intros ;apply P_id_u'8'1_bounded;assumption.
        intros ;apply P_id_reduce'ii'out_bounded;assumption.
        intros ;apply P_id_nil_bounded;assumption.
        intros ;apply P_id_if_bounded;assumption.
        intros ;apply P_id_u'11'1_bounded;assumption.
        intros ;apply P_id_u'5'1_bounded;assumption.
        intros ;apply P_id_u'16'1_bounded;assumption.
        apply rules_monotonic.
        intros ;apply P_id_U'12'2_monotonic;assumption.
        intros ;apply P_id_U'6'1_monotonic;assumption.
        intros ;apply P_id_REDUCE'II'IN_monotonic;assumption.
        intros ;apply P_id_TAUTOLOGY'I'IN_monotonic;assumption.
        intros ;apply P_id_U'9'1_monotonic;assumption.
        intros ;apply P_id_U'1'1_monotonic;assumption.
        intros ;apply P_id_U'14'1_monotonic;assumption.
        intros ;apply P_id_U'7'1_monotonic;assumption.
        intros ;apply P_id_U'4'1_monotonic;assumption.
        intros ;apply P_id_U'11'1_monotonic;assumption.
        intros ;apply P_id_INTERSECT'II'IN_monotonic;assumption.
        intros ;apply P_id_U'13'1_monotonic;assumption.
        intros ;apply P_id_U'6'2_monotonic;assumption.
        intros ;apply P_id_U'3'1_monotonic;assumption.
        intros ;apply P_id_U'16'1_monotonic;assumption.
        intros ;apply P_id_U'10'1_monotonic;assumption.
        intros ;apply P_id_U'2'1_monotonic;assumption.
        intros ;apply P_id_U'15'1_monotonic;assumption.
        intros ;apply P_id_U'8'1_monotonic;assumption.
        intros ;apply P_id_U'5'1_monotonic;assumption.
        intros ;apply P_id_U'12'1_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_19_large_large_scc_1_strict_in_lt :
       Relation_Definitions.inclusion _ 
        DP_R_xml_0_scc_19_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_19_large_large_scc_1_large_in_le :
       Relation_Definitions.inclusion _ 
        DP_R_xml_0_scc_19_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_19_large_large_scc_1_large  := 
        WF_DP_R_xml_0_scc_19_large_large_scc_1_large.wf.
      
      
      Lemma wf :
       well_founded WF_DP_R_xml_0_scc_19_large_large.DP_R_xml_0_scc_19_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_19_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_19_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_19_large_large_scc_1_large_in_le;
            econstructor eassumption])).
        apply wf_DP_R_xml_0_scc_19_large_large_scc_1_large.
      Qed.
     End WF_DP_R_xml_0_scc_19_large_large_scc_1.
     
     Definition wf_DP_R_xml_0_scc_19_large_large_scc_1  := 
       WF_DP_R_xml_0_scc_19_large_large_scc_1.wf.
     
     
     Lemma acc_DP_R_xml_0_scc_19_large_large_scc_1 :
      forall x y, 
       (DP_R_xml_0_scc_19_large_large_scc_1 x y) ->
        Acc WF_DP_R_xml_0_scc_19_large.DP_R_xml_0_scc_19_large_large x.
     Proof.
       intros x.
       pattern x.
       apply (@Acc_ind _ DP_R_xml_0_scc_19_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_19_large_large_scc_1.
     Qed.
     
     
     Inductive DP_R_xml_0_scc_19_large_large_non_scc_2  :
      algebra.Alg.term ->algebra.Alg.term ->Prop := 
        (* <u'12'1(reduce'ii'out,Fs_,G2_,Gs_,NF_),reduce'ii'in(sequent(Fs_,cons(G2_,Gs_)),NF_)> *)
       | DP_R_xml_0_scc_19_large_large_non_scc_2_0 :
        forall x8 x24 x20 x10 x22 x17 x9 x21 x23, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    
           (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) x24) ->
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     x8 x23) ->
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                      x17 x22) ->
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                       x9 x21) ->
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                        x10 x20) ->
              DP_R_xml_0_scc_19_large_large_non_scc_2 (algebra.Alg.Term 
                                                       algebra.F.id_reduce'ii'in 
                                                       ((algebra.Alg.Term 
                                                       algebra.F.id_sequent 
                                                       (x8::
                                                       (algebra.Alg.Term 
                                                       algebra.F.id_cons 
                                                       (x17::
                                                       x9::nil))::nil))::
                                                       x10::nil)) 
               (algebra.Alg.Term algebra.F.id_u'12'1 (x24::x23::x22::x21::
                x20::nil))
     .
     
     
     Lemma acc_DP_R_xml_0_scc_19_large_large_non_scc_2 :
      forall x y, 
       (DP_R_xml_0_scc_19_large_large_non_scc_2 x y) ->
        Acc WF_DP_R_xml_0_scc_19_large.DP_R_xml_0_scc_19_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_19_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_19_large.DP_R_xml_0_scc_19_large_large.
     Proof.
       constructor;intros _y _h;inversion _h;clear _h;subst;
        (eapply acc_DP_R_xml_0_scc_19_large_large_non_scc_2;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((eapply acc_DP_R_xml_0_scc_19_large_large_non_scc_1;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
         ((eapply acc_DP_R_xml_0_scc_19_large_large_non_scc_0;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
          ((eapply acc_DP_R_xml_0_scc_19_large_large_scc_1;
             econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
           ((eapply acc_DP_R_xml_0_scc_19_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_19_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_intersect'ii'in (x20:Z) (x21:Z) := 0.
    
    Definition P_id_tautology'i'out  := 0.
    
    Definition P_id_u'6'1 (x20:Z) (x21:Z) (x22:Z) (x23:Z) (x24:Z) := 0.
    
    Definition P_id_u'3'1 (x20:Z) := 0.
    
    Definition P_id_u'12'1 (x20:Z) (x21:Z) (x22:Z) (x23:Z) (x24:Z) := 0.
    
    Definition P_id_u'2'1 (x20:Z) := 0.
    
    Definition P_id_u'9'1 (x20:Z) := 0.
    
    Definition P_id_iff (x20:Z) (x21:Z) := 0.
    
    Definition P_id_u'14'1 (x20:Z) := 0.
    
    Definition P_id_intersect'ii'out  := 0.
    
    Definition P_id_u'7'1 (x20:Z) := 0.
    
    Definition P_id_x'2d (x20:Z) := 1 + 1* x20.
    
    Definition P_id_u'13'1 (x20:Z) := 0.
    
    Definition P_id_sequent (x20:Z) (x21:Z) := 1* x20 + 1* x21.
    
    Definition P_id_u'10'1 (x20:Z) := 0.
    
    Definition P_id_x'2a (x20:Z) (x21:Z) := 3 + 1* x20 + 1* x21.
    
    Definition P_id_tautology'i'in (x20:Z) := 2* x20.
    
    Definition P_id_cons (x20:Z) (x21:Z) := 1* x20 + 1* x21.
    
    Definition P_id_u'6'2 (x20:Z) := 0.
    
    Definition P_id_x'2b (x20:Z) (x21:Z) := 1* x20 + 2* x21.
    
    Definition P_id_u'12'2 (x20:Z) := 0.
    
    Definition P_id_reduce'ii'in (x20:Z) (x21:Z) := 0.
    
    Definition P_id_p (x20:Z) := 0.
    
    Definition P_id_u'4'1 (x20:Z) := 0.
    
    Definition P_id_u'15'1 (x20:Z) := 0.
    
    Definition P_id_u'1'1 (x20:Z) := 0.
    
    Definition P_id_u'8'1 (x20:Z) := 0.
    
    Definition P_id_reduce'ii'out  := 0.
    
    Definition P_id_nil  := 0.
    
    Definition P_id_if (x20:Z) (x21:Z) := 2 + 2* x20 + 2* x21.
    
    Definition P_id_u'11'1 (x20:Z) := 0.
    
    Definition P_id_u'5'1 (x20:Z) := 0.
    
    Definition P_id_u'16'1 (x20:Z) := 0.
    
    Lemma P_id_intersect'ii'in_monotonic :
     forall x20 x22 x21 x23, 
      (0 <= x23)/\ (x23 <= x22) ->
       (0 <= x21)/\ (x21 <= x20) ->
        P_id_intersect'ii'in x21 x23 <= P_id_intersect'ii'in x20 x22.
    Proof.
      intros x23 x22 x21 x20.
      intros [H_1 H_0].
      intros [H_3 H_2].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_u'6'1_monotonic :
     forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, 
      (0 <= x29)/\ (x29 <= x28) ->
       (0 <= x27)/\ (x27 <= x26) ->
        (0 <= x25)/\ (x25 <= x24) ->
         (0 <= x23)/\ (x23 <= x22) ->
          (0 <= x21)/\ (x21 <= x20) ->
           P_id_u'6'1 x21 x23 x25 x27 x29 <= P_id_u'6'1 x20 x22 x24 x26 x28.
    Proof.
      intros x29 x28 x27 x26 x25 x24 x23 x22 x21 x20.
      intros [H_1 H_0].
      intros [H_3 H_2].
      intros [H_5 H_4].
      intros [H_7 H_6].
      intros [H_9 H_8].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_u'3'1_monotonic :
     forall x20 x21, 
      (0 <= x21)/\ (x21 <= x20) ->P_id_u'3'1 x21 <= P_id_u'3'1 x20.
    Proof.
      intros x21 x20.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_u'12'1_monotonic :
     forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, 
      (0 <= x29)/\ (x29 <= x28) ->
       (0 <= x27)/\ (x27 <= x26) ->
        (0 <= x25)/\ (x25 <= x24) ->
         (0 <= x23)/\ (x23 <= x22) ->
          (0 <= x21)/\ (x21 <= x20) ->
           P_id_u'12'1 x21 x23 x25 x27 x29 <= P_id_u'12'1 x20 x22 x24 x26 x28.
    Proof.
      intros x29 x28 x27 x26 x25 x24 x23 x22 x21 x20.
      intros [H_1 H_0].
      intros [H_3 H_2].
      intros [H_5 H_4].
      intros [H_7 H_6].
      intros [H_9 H_8].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_u'2'1_monotonic :
     forall x20 x21, 
      (0 <= x21)/\ (x21 <= x20) ->P_id_u'2'1 x21 <= P_id_u'2'1 x20.
    Proof.
      intros x21 x20.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_u'9'1_monotonic :
     forall x20 x21, 
      (0 <= x21)/\ (x21 <= x20) ->P_id_u'9'1 x21 <= P_id_u'9'1 x20.
    Proof.
      intros x21 x20.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_iff_monotonic :
     forall x20 x22 x21 x23, 
      (0 <= x23)/\ (x23 <= x22) ->
       (0 <= x21)/\ (x21 <= x20) ->P_id_iff x21 x23 <= P_id_iff x20 x22.
    Proof.
      intros x23 x22 x21 x20.
      intros [H_1 H_0].
      intros [H_3 H_2].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_u'14'1_monotonic :
     forall x20 x21, 
      (0 <= x21)/\ (x21 <= x20) ->P_id_u'14'1 x21 <= P_id_u'14'1 x20.
    Proof.
      intros x21 x20.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_u'7'1_monotonic :
     forall x20 x21, 
      (0 <= x21)/\ (x21 <= x20) ->P_id_u'7'1 x21 <= P_id_u'7'1 x20.
    Proof.
      intros x21 x20.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_x'2d_monotonic :
     forall x20 x21, 
      (0 <= x21)/\ (x21 <= x20) ->P_id_x'2d x21 <= P_id_x'2d x20.
    Proof.
      intros x21 x20.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_u'13'1_monotonic :
     forall x20 x21, 
      (0 <= x21)/\ (x21 <= x20) ->P_id_u'13'1 x21 <= P_id_u'13'1 x20.
    Proof.
      intros x21 x20.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_sequent_monotonic :
     forall x20 x22 x21 x23, 
      (0 <= x23)/\ (x23 <= x22) ->
       (0 <= x21)/\ (x21 <= x20) ->
        P_id_sequent x21 x23 <= P_id_sequent x20 x22.
    Proof.
      intros x23 x22 x21 x20.
      intros [H_1 H_0].
      intros [H_3 H_2].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_u'10'1_monotonic :
     forall x20 x21, 
      (0 <= x21)/\ (x21 <= x20) ->P_id_u'10'1 x21 <= P_id_u'10'1 x20.
    Proof.
      intros x21 x20.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_x'2a_monotonic :
     forall x20 x22 x21 x23, 
      (0 <= x23)/\ (x23 <= x22) ->
       (0 <= x21)/\ (x21 <= x20) ->P_id_x'2a x21 x23 <= P_id_x'2a x20 x22.
    Proof.
      intros x23 x22 x21 x20.
      intros [H_1 H_0].
      intros [H_3 H_2].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_tautology'i'in_monotonic :
     forall x20 x21, 
      (0 <= x21)/\ (x21 <= x20) ->
       P_id_tautology'i'in x21 <= P_id_tautology'i'in x20.
    Proof.
      intros x21 x20.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_cons_monotonic :
     forall x20 x22 x21 x23, 
      (0 <= x23)/\ (x23 <= x22) ->
       (0 <= x21)/\ (x21 <= x20) ->P_id_cons x21 x23 <= P_id_cons x20 x22.
    Proof.
      intros x23 x22 x21 x20.
      intros [H_1 H_0].
      intros [H_3 H_2].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_u'6'2_monotonic :
     forall x20 x21, 
      (0 <= x21)/\ (x21 <= x20) ->P_id_u'6'2 x21 <= P_id_u'6'2 x20.
    Proof.
      intros x21 x20.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_x'2b_monotonic :
     forall x20 x22 x21 x23, 
      (0 <= x23)/\ (x23 <= x22) ->
       (0 <= x21)/\ (x21 <= x20) ->P_id_x'2b x21 x23 <= P_id_x'2b x20 x22.
    Proof.
      intros x23 x22 x21 x20.
      intros [H_1 H_0].
      intros [H_3 H_2].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_u'12'2_monotonic :
     forall x20 x21, 
      (0 <= x21)/\ (x21 <= x20) ->P_id_u'12'2 x21 <= P_id_u'12'2 x20.
    Proof.
      intros x21 x20.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_reduce'ii'in_monotonic :
     forall x20 x22 x21 x23, 
      (0 <= x23)/\ (x23 <= x22) ->
       (0 <= x21)/\ (x21 <= x20) ->
        P_id_reduce'ii'in x21 x23 <= P_id_reduce'ii'in x20 x22.
    Proof.
      intros x23 x22 x21 x20.
      intros [H_1 H_0].
      intros [H_3 H_2].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_p_monotonic :
     forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_p x21 <= P_id_p x20.
    Proof.
      intros x21 x20.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_u'4'1_monotonic :
     forall x20 x21, 
      (0 <= x21)/\ (x21 <= x20) ->P_id_u'4'1 x21 <= P_id_u'4'1 x20.
    Proof.
      intros x21 x20.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_u'15'1_monotonic :
     forall x20 x21, 
      (0 <= x21)/\ (x21 <= x20) ->P_id_u'15'1 x21 <= P_id_u'15'1 x20.
    Proof.
      intros x21 x20.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_u'1'1_monotonic :
     forall x20 x21, 
      (0 <= x21)/\ (x21 <= x20) ->P_id_u'1'1 x21 <= P_id_u'1'1 x20.
    Proof.
      intros x21 x20.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_u'8'1_monotonic :
     forall x20 x21, 
      (0 <= x21)/\ (x21 <= x20) ->P_id_u'8'1 x21 <= P_id_u'8'1 x20.
    Proof.
      intros x21 x20.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_if_monotonic :
     forall x20 x22 x21 x23, 
      (0 <= x23)/\ (x23 <= x22) ->
       (0 <= x21)/\ (x21 <= x20) ->P_id_if x21 x23 <= P_id_if x20 x22.
    Proof.
      intros x23 x22 x21 x20.
      intros [H_1 H_0].
      intros [H_3 H_2].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_u'11'1_monotonic :
     forall x20 x21, 
      (0 <= x21)/\ (x21 <= x20) ->P_id_u'11'1 x21 <= P_id_u'11'1 x20.
    Proof.
      intros x21 x20.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_u'5'1_monotonic :
     forall x20 x21, 
      (0 <= x21)/\ (x21 <= x20) ->P_id_u'5'1 x21 <= P_id_u'5'1 x20.
    Proof.
      intros x21 x20.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_u'16'1_monotonic :
     forall x20 x21, 
      (0 <= x21)/\ (x21 <= x20) ->P_id_u'16'1 x21 <= P_id_u'16'1 x20.
    Proof.
      intros x21 x20.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_intersect'ii'in_bounded :
     forall x20 x21, 
      (0 <= x20) ->(0 <= x21) ->0 <= P_id_intersect'ii'in x21 x20.
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_tautology'i'out_bounded : 0 <= P_id_tautology'i'out .
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_u'6'1_bounded :
     forall x24 x20 x22 x21 x23, 
      (0 <= x20) ->
       (0 <= x21) ->
        (0 <= x22) ->
         (0 <= x23) ->(0 <= x24) ->0 <= P_id_u'6'1 x24 x23 x22 x21 x20.
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_u'3'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'3'1 x20.
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_u'12'1_bounded :
     forall x24 x20 x22 x21 x23, 
      (0 <= x20) ->
       (0 <= x21) ->
        (0 <= x22) ->
         (0 <= x23) ->(0 <= x24) ->0 <= P_id_u'12'1 x24 x23 x22 x21 x20.
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_u'2'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'2'1 x20.
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_u'9'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'9'1 x20.
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_iff_bounded :
     forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_iff x21 x20.
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_u'14'1_bounded :
     forall x20, (0 <= x20) ->0 <= P_id_u'14'1 x20.
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_intersect'ii'out_bounded : 0 <= P_id_intersect'ii'out .
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_u'7'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'7'1 x20.
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_x'2d_bounded : forall x20, (0 <= x20) ->0 <= P_id_x'2d x20.
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_u'13'1_bounded :
     forall x20, (0 <= x20) ->0 <= P_id_u'13'1 x20.
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_sequent_bounded :
     forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_sequent x21 x20.
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_u'10'1_bounded :
     forall x20, (0 <= x20) ->0 <= P_id_u'10'1 x20.
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_x'2a_bounded :
     forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_x'2a x21 x20.
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_tautology'i'in_bounded :
     forall x20, (0 <= x20) ->0 <= P_id_tautology'i'in x20.
    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 x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_cons x21 x20.
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_u'6'2_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'6'2 x20.
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_x'2b_bounded :
     forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_x'2b x21 x20.
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_u'12'2_bounded :
     forall x20, (0 <= x20) ->0 <= P_id_u'12'2 x20.
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_reduce'ii'in_bounded :
     forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_reduce'ii'in x21 x20.
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_p_bounded : forall x20, (0 <= x20) ->0 <= P_id_p x20.
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_u'4'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'4'1 x20.
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_u'15'1_bounded :
     forall x20, (0 <= x20) ->0 <= P_id_u'15'1 x20.
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_u'1'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'1'1 x20.
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_u'8'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'8'1 x20.
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_reduce'ii'out_bounded : 0 <= P_id_reduce'ii'out .
    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_if_bounded :
     forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_if x21 x20.
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_u'11'1_bounded :
     forall x20, (0 <= x20) ->0 <= P_id_u'11'1 x20.
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_u'5'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'5'1 x20.
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_u'16'1_bounded :
     forall x20, (0 <= x20) ->0 <= P_id_u'16'1 x20.
    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_intersect'ii'in P_id_tautology'i'out P_id_u'6'1 
       P_id_u'3'1 P_id_u'12'1 P_id_u'2'1 P_id_u'9'1 P_id_iff P_id_u'14'1 
       P_id_intersect'ii'out P_id_u'7'1 P_id_x'2d P_id_u'13'1 P_id_sequent 
       P_id_u'10'1 P_id_x'2a P_id_tautology'i'in P_id_cons P_id_u'6'2 
       P_id_x'2b P_id_u'12'2 P_id_reduce'ii'in P_id_p P_id_u'4'1 P_id_u'15'1 
       P_id_u'1'1 P_id_u'8'1 P_id_reduce'ii'out P_id_nil P_id_if P_id_u'11'1 
       P_id_u'5'1 P_id_u'16'1.
    
    Lemma measure_equation :
     forall t, 
      measure t = match t with
                    | (algebra.Alg.Term algebra.F.id_intersect'ii'in (x21::
                       x20::nil)) =>
                     P_id_intersect'ii'in (measure x21) (measure x20)
                    | (algebra.Alg.Term algebra.F.id_tautology'i'out nil) =>
                     P_id_tautology'i'out 
                    | (algebra.Alg.Term algebra.F.id_u'6'1 (x24::x23::x22::
                       x21::x20::nil)) =>
                     P_id_u'6'1 (measure x24) (measure x23) (measure x22) 
                      (measure x21) (measure x20)
                    | (algebra.Alg.Term algebra.F.id_u'3'1 (x20::nil)) =>
                     P_id_u'3'1 (measure x20)
                    | (algebra.Alg.Term algebra.F.id_u'12'1 (x24::x23::x22::
                       x21::x20::nil)) =>
                     P_id_u'12'1 (measure x24) (measure x23) (measure x22) 
                      (measure x21) (measure x20)
                    | (algebra.Alg.Term algebra.F.id_u'2'1 (x20::nil)) =>
                     P_id_u'2'1 (measure x20)
                    | (algebra.Alg.Term algebra.F.id_u'9'1 (x20::nil)) =>
                     P_id_u'9'1 (measure x20)
                    | (algebra.Alg.Term algebra.F.id_iff (x21::x20::nil)) =>
                     P_id_iff (measure x21) (measure x20)
                    | (algebra.Alg.Term algebra.F.id_u'14'1 (x20::nil)) =>
                     P_id_u'14'1 (measure x20)
                    | (algebra.Alg.Term algebra.F.id_intersect'ii'out nil) =>
                     P_id_intersect'ii'out 
                    | (algebra.Alg.Term algebra.F.id_u'7'1 (x20::nil)) =>
                     P_id_u'7'1 (measure x20)
                    | (algebra.Alg.Term algebra.F.id_x'2d (x20::nil)) =>
                     P_id_x'2d (measure x20)
                    | (algebra.Alg.Term algebra.F.id_u'13'1 (x20::nil)) =>
                     P_id_u'13'1 (measure x20)
                    | (algebra.Alg.Term algebra.F.id_sequent (x21::x20::nil)) =>
                     P_id_sequent (measure x21) (measure x20)
                    | (algebra.Alg.Term algebra.F.id_u'10'1 (x20::nil)) =>
                     P_id_u'10'1 (measure x20)
                    | (algebra.Alg.Term algebra.F.id_x'2a (x21::x20::nil)) =>
                     P_id_x'2a (measure x21) (measure x20)
                    | (algebra.Alg.Term algebra.F.id_tautology'i'in 
                       (x20::nil)) =>
                     P_id_tautology'i'in (measure x20)
                    | (algebra.Alg.Term algebra.F.id_cons (x21::x20::nil)) =>
                     P_id_cons (measure x21) (measure x20)
                    | (algebra.Alg.Term algebra.F.id_u'6'2 (x20::nil)) =>
                     P_id_u'6'2 (measure x20)
                    | (algebra.Alg.Term algebra.F.id_x'2b (x21::x20::nil)) =>
                     P_id_x'2b (measure x21) (measure x20)
                    | (algebra.Alg.Term algebra.F.id_u'12'2 (x20::nil)) =>
                     P_id_u'12'2 (measure x20)
                    | (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::
                       x20::nil)) =>
                     P_id_reduce'ii'in (measure x21) (measure x20)
                    | (algebra.Alg.Term algebra.F.id_p (x20::nil)) =>
                     P_id_p (measure x20)
                    | (algebra.Alg.Term algebra.F.id_u'4'1 (x20::nil)) =>
                     P_id_u'4'1 (measure x20)
                    | (algebra.Alg.Term algebra.F.id_u'15'1 (x20::nil)) =>
                     P_id_u'15'1 (measure x20)
                    | (algebra.Alg.Term algebra.F.id_u'1'1 (x20::nil)) =>
                     P_id_u'1'1 (measure x20)
                    | (algebra.Alg.Term algebra.F.id_u'8'1 (x20::nil)) =>
                     P_id_u'8'1 (measure x20)
                    | (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) =>
                     P_id_reduce'ii'out 
                    | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil 
                    | (algebra.Alg.Term algebra.F.id_if (x21::x20::nil)) =>
                     P_id_if (measure x21) (measure x20)
                    | (algebra.Alg.Term algebra.F.id_u'11'1 (x20::nil)) =>
                     P_id_u'11'1 (measure x20)
                    | (algebra.Alg.Term algebra.F.id_u'5'1 (x20::nil)) =>
                     P_id_u'5'1 (measure x20)
                    | (algebra.Alg.Term algebra.F.id_u'16'1 (x20::nil)) =>
                     P_id_u'16'1 (measure x20)
                    | _ => 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_intersect'ii'in_monotonic;assumption.
      intros ;apply P_id_u'6'1_monotonic;assumption.
      intros ;apply P_id_u'3'1_monotonic;assumption.
      intros ;apply P_id_u'12'1_monotonic;assumption.
      intros ;apply P_id_u'2'1_monotonic;assumption.
      intros ;apply P_id_u'9'1_monotonic;assumption.
      intros ;apply P_id_iff_monotonic;assumption.
      intros ;apply P_id_u'14'1_monotonic;assumption.
      intros ;apply P_id_u'7'1_monotonic;assumption.
      intros ;apply P_id_x'2d_monotonic;assumption.
      intros ;apply P_id_u'13'1_monotonic;assumption.
      intros ;apply P_id_sequent_monotonic;assumption.
      intros ;apply P_id_u'10'1_monotonic;assumption.
      intros ;apply P_id_x'2a_monotonic;assumption.
      intros ;apply P_id_tautology'i'in_monotonic;assumption.
      intros ;apply P_id_cons_monotonic;assumption.
      intros ;apply P_id_u'6'2_monotonic;assumption.
      intros ;apply P_id_x'2b_monotonic;assumption.
      intros ;apply P_id_u'12'2_monotonic;assumption.
      intros ;apply P_id_reduce'ii'in_monotonic;assumption.
      intros ;apply P_id_p_monotonic;assumption.
      intros ;apply P_id_u'4'1_monotonic;assumption.
      intros ;apply P_id_u'15'1_monotonic;assumption.
      intros ;apply P_id_u'1'1_monotonic;assumption.
      intros ;apply P_id_u'8'1_monotonic;assumption.
      intros ;apply P_id_if_monotonic;assumption.
      intros ;apply P_id_u'11'1_monotonic;assumption.
      intros ;apply P_id_u'5'1_monotonic;assumption.
      intros ;apply P_id_u'16'1_monotonic;assumption.
      intros ;apply P_id_intersect'ii'in_bounded;assumption.
      intros ;apply P_id_tautology'i'out_bounded;assumption.
      intros ;apply P_id_u'6'1_bounded;assumption.
      intros ;apply P_id_u'3'1_bounded;assumption.
      intros ;apply P_id_u'12'1_bounded;assumption.
      intros ;apply P_id_u'2'1_bounded;assumption.
      intros ;apply P_id_u'9'1_bounded;assumption.
      intros ;apply P_id_iff_bounded;assumption.
      intros ;apply P_id_u'14'1_bounded;assumption.
      intros ;apply P_id_intersect'ii'out_bounded;assumption.
      intros ;apply P_id_u'7'1_bounded;assumption.
      intros ;apply P_id_x'2d_bounded;assumption.
      intros ;apply P_id_u'13'1_bounded;assumption.
      intros ;apply P_id_sequent_bounded;assumption.
      intros ;apply P_id_u'10'1_bounded;assumption.
      intros ;apply P_id_x'2a_bounded;assumption.
      intros ;apply P_id_tautology'i'in_bounded;assumption.
      intros ;apply P_id_cons_bounded;assumption.
      intros ;apply P_id_u'6'2_bounded;assumption.
      intros ;apply P_id_x'2b_bounded;assumption.
      intros ;apply P_id_u'12'2_bounded;assumption.
      intros ;apply P_id_reduce'ii'in_bounded;assumption.
      intros ;apply P_id_p_bounded;assumption.
      intros ;apply P_id_u'4'1_bounded;assumption.
      intros ;apply P_id_u'15'1_bounded;assumption.
      intros ;apply P_id_u'1'1_bounded;assumption.
      intros ;apply P_id_u'8'1_bounded;assumption.
      intros ;apply P_id_reduce'ii'out_bounded;assumption.
      intros ;apply P_id_nil_bounded;assumption.
      intros ;apply P_id_if_bounded;assumption.
      intros ;apply P_id_u'11'1_bounded;assumption.
      intros ;apply P_id_u'5'1_bounded;assumption.
      intros ;apply P_id_u'16'1_bounded;assumption.
      apply rules_monotonic.
    Qed.
    
    Definition P_id_U'12'2 (x20:Z) := 0.
    
    Definition P_id_U'6'1 (x20:Z) (x21:Z) (x22:Z) (x23:Z) (x24:Z) := 
      3* x21 + 2* x22 + 2* x23.
    
    Definition P_id_REDUCE'II'IN (x20:Z) (x21:Z) := 2* x20.
    
    Definition P_id_TAUTOLOGY'I'IN (x20:Z) := 0.
    
    Definition P_id_U'9'1 (x20:Z) := 0.
    
    Definition P_id_U'1'1 (x20:Z) := 0.
    
    Definition P_id_U'14'1 (x20:Z) := 0.
    
    Definition P_id_U'7'1 (x20:Z) := 0.
    
    Definition P_id_U'4'1 (x20:Z) := 0.
    
    Definition P_id_U'11'1 (x20:Z) := 0.
    
    Definition P_id_INTERSECT'II'IN (x20:Z) (x21:Z) := 0.
    
    Definition P_id_U'13'1 (x20:Z) := 0.
    
    Definition P_id_U'6'2 (x20:Z) := 0.
    
    Definition P_id_U'3'1 (x20:Z) := 0.
    
    Definition P_id_U'16'1 (x20:Z) := 0.
    
    Definition P_id_U'10'1 (x20:Z) := 0.
    
    Definition P_id_U'2'1 (x20:Z) := 0.
    
    Definition P_id_U'15'1 (x20:Z) := 0.
    
    Definition P_id_U'8'1 (x20:Z) := 0.
    
    Definition P_id_U'5'1 (x20:Z) := 0.
    
    Definition P_id_U'12'1 (x20:Z) (x21:Z) (x22:Z) (x23:Z) (x24:Z) := 
      2* x21 + 2* x22 + 2* x23.
    
    Lemma P_id_U'12'2_monotonic :
     forall x20 x21, 
      (0 <= x21)/\ (x21 <= x20) ->P_id_U'12'2 x21 <= P_id_U'12'2 x20.
    Proof.
      intros x21 x20.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_U'6'1_monotonic :
     forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, 
      (0 <= x29)/\ (x29 <= x28) ->
       (0 <= x27)/\ (x27 <= x26) ->
        (0 <= x25)/\ (x25 <= x24) ->
         (0 <= x23)/\ (x23 <= x22) ->
          (0 <= x21)/\ (x21 <= x20) ->
           P_id_U'6'1 x21 x23 x25 x27 x29 <= P_id_U'6'1 x20 x22 x24 x26 x28.
    Proof.
      intros x29 x28 x27 x26 x25 x24 x23 x22 x21 x20.
      intros [H_1 H_0].
      intros [H_3 H_2].
      intros [H_5 H_4].
      intros [H_7 H_6].
      intros [H_9 H_8].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_REDUCE'II'IN_monotonic :
     forall x20 x22 x21 x23, 
      (0 <= x23)/\ (x23 <= x22) ->
       (0 <= x21)/\ (x21 <= x20) ->
        P_id_REDUCE'II'IN x21 x23 <= P_id_REDUCE'II'IN x20 x22.
    Proof.
      intros x23 x22 x21 x20.
      intros [H_1 H_0].
      intros [H_3 H_2].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_TAUTOLOGY'I'IN_monotonic :
     forall x20 x21, 
      (0 <= x21)/\ (x21 <= x20) ->
       P_id_TAUTOLOGY'I'IN x21 <= P_id_TAUTOLOGY'I'IN x20.
    Proof.
      intros x21 x20.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_U'9'1_monotonic :
     forall x20 x21, 
      (0 <= x21)/\ (x21 <= x20) ->P_id_U'9'1 x21 <= P_id_U'9'1 x20.
    Proof.
      intros x21 x20.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_U'1'1_monotonic :
     forall x20 x21, 
      (0 <= x21)/\ (x21 <= x20) ->P_id_U'1'1 x21 <= P_id_U'1'1 x20.
    Proof.
      intros x21 x20.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_U'14'1_monotonic :
     forall x20 x21, 
      (0 <= x21)/\ (x21 <= x20) ->P_id_U'14'1 x21 <= P_id_U'14'1 x20.
    Proof.
      intros x21 x20.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_U'7'1_monotonic :
     forall x20 x21, 
      (0 <= x21)/\ (x21 <= x20) ->P_id_U'7'1 x21 <= P_id_U'7'1 x20.
    Proof.
      intros x21 x20.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_U'4'1_monotonic :
     forall x20 x21, 
      (0 <= x21)/\ (x21 <= x20) ->P_id_U'4'1 x21 <= P_id_U'4'1 x20.
    Proof.
      intros x21 x20.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_U'11'1_monotonic :
     forall x20 x21, 
      (0 <= x21)/\ (x21 <= x20) ->P_id_U'11'1 x21 <= P_id_U'11'1 x20.
    Proof.
      intros x21 x20.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_INTERSECT'II'IN_monotonic :
     forall x20 x22 x21 x23, 
      (0 <= x23)/\ (x23 <= x22) ->
       (0 <= x21)/\ (x21 <= x20) ->
        P_id_INTERSECT'II'IN x21 x23 <= P_id_INTERSECT'II'IN x20 x22.
    Proof.
      intros x23 x22 x21 x20.
      intros [H_1 H_0].
      intros [H_3 H_2].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_U'13'1_monotonic :
     forall x20 x21, 
      (0 <= x21)/\ (x21 <= x20) ->P_id_U'13'1 x21 <= P_id_U'13'1 x20.
    Proof.
      intros x21 x20.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_U'6'2_monotonic :
     forall x20 x21, 
      (0 <= x21)/\ (x21 <= x20) ->P_id_U'6'2 x21 <= P_id_U'6'2 x20.
    Proof.
      intros x21 x20.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_U'3'1_monotonic :
     forall x20 x21, 
      (0 <= x21)/\ (x21 <= x20) ->P_id_U'3'1 x21 <= P_id_U'3'1 x20.
    Proof.
      intros x21 x20.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_U'16'1_monotonic :
     forall x20 x21, 
      (0 <= x21)/\ (x21 <= x20) ->P_id_U'16'1 x21 <= P_id_U'16'1 x20.
    Proof.
      intros x21 x20.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_U'10'1_monotonic :
     forall x20 x21, 
      (0 <= x21)/\ (x21 <= x20) ->P_id_U'10'1 x21 <= P_id_U'10'1 x20.
    Proof.
      intros x21 x20.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_U'2'1_monotonic :
     forall x20 x21, 
      (0 <= x21)/\ (x21 <= x20) ->P_id_U'2'1 x21 <= P_id_U'2'1 x20.
    Proof.
      intros x21 x20.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_U'15'1_monotonic :
     forall x20 x21, 
      (0 <= x21)/\ (x21 <= x20) ->P_id_U'15'1 x21 <= P_id_U'15'1 x20.
    Proof.
      intros x21 x20.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_U'8'1_monotonic :
     forall x20 x21, 
      (0 <= x21)/\ (x21 <= x20) ->P_id_U'8'1 x21 <= P_id_U'8'1 x20.
    Proof.
      intros x21 x20.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_U'5'1_monotonic :
     forall x20 x21, 
      (0 <= x21)/\ (x21 <= x20) ->P_id_U'5'1 x21 <= P_id_U'5'1 x20.
    Proof.
      intros x21 x20.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_U'12'1_monotonic :
     forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, 
      (0 <= x29)/\ (x29 <= x28) ->
       (0 <= x27)/\ (x27 <= x26) ->
        (0 <= x25)/\ (x25 <= x24) ->
         (0 <= x23)/\ (x23 <= x22) ->
          (0 <= x21)/\ (x21 <= x20) ->
           P_id_U'12'1 x21 x23 x25 x27 x29 <= P_id_U'12'1 x20 x22 x24 x26 x28.
    Proof.
      intros x29 x28 x27 x26 x25 x24 x23 x22 x21 x20.
      intros [H_1 H_0].
      intros [H_3 H_2].
      intros [H_5 H_4].
      intros [H_7 H_6].
      intros [H_9 H_8].
      
      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_intersect'ii'in P_id_tautology'i'out 
       P_id_u'6'1 P_id_u'3'1 P_id_u'12'1 P_id_u'2'1 P_id_u'9'1 P_id_iff 
       P_id_u'14'1 P_id_intersect'ii'out P_id_u'7'1 P_id_x'2d P_id_u'13'1 
       P_id_sequent P_id_u'10'1 P_id_x'2a P_id_tautology'i'in P_id_cons 
       P_id_u'6'2 P_id_x'2b P_id_u'12'2 P_id_reduce'ii'in P_id_p P_id_u'4'1 
       P_id_u'15'1 P_id_u'1'1 P_id_u'8'1 P_id_reduce'ii'out P_id_nil 
       P_id_if P_id_u'11'1 P_id_u'5'1 P_id_u'16'1 P_id_U'12'2 P_id_U'6'1 
       P_id_REDUCE'II'IN P_id_TAUTOLOGY'I'IN P_id_U'9'1 P_id_U'1'1 
       P_id_U'14'1 P_id_U'7'1 P_id_U'4'1 P_id_U'11'1 P_id_INTERSECT'II'IN 
       P_id_U'13'1 P_id_U'6'2 P_id_U'3'1 P_id_U'16'1 P_id_U'10'1 P_id_U'2'1 
       P_id_U'15'1 P_id_U'8'1 P_id_U'5'1 P_id_U'12'1.
    
    Lemma marked_measure_equation :
     forall t, 
      marked_measure t = match t with
                           | (algebra.Alg.Term algebra.F.id_u'12'2 
                              (x20::nil)) =>
                            P_id_U'12'2 (measure x20)
                           | (algebra.Alg.Term algebra.F.id_u'6'1 (x24::x23::
                              x22::x21::x20::nil)) =>
                            P_id_U'6'1 (measure x24) (measure x23) 
                             (measure x22) (measure x21) (measure x20)
                           | (algebra.Alg.Term algebra.F.id_reduce'ii'in 
                              (x21::x20::nil)) =>
                            P_id_REDUCE'II'IN (measure x21) (measure x20)
                           | (algebra.Alg.Term algebra.F.id_tautology'i'in 
                              (x20::nil)) =>
                            P_id_TAUTOLOGY'I'IN (measure x20)
                           | (algebra.Alg.Term algebra.F.id_u'9'1 (x20::nil)) =>
                            P_id_U'9'1 (measure x20)
                           | (algebra.Alg.Term algebra.F.id_u'1'1 (x20::nil)) =>
                            P_id_U'1'1 (measure x20)
                           | (algebra.Alg.Term algebra.F.id_u'14'1 
                              (x20::nil)) =>
                            P_id_U'14'1 (measure x20)
                           | (algebra.Alg.Term algebra.F.id_u'7'1 (x20::nil)) =>
                            P_id_U'7'1 (measure x20)
                           | (algebra.Alg.Term algebra.F.id_u'4'1 (x20::nil)) =>
                            P_id_U'4'1 (measure x20)
                           | (algebra.Alg.Term algebra.F.id_u'11'1 
                              (x20::nil)) =>
                            P_id_U'11'1 (measure x20)
                           | (algebra.Alg.Term algebra.F.id_intersect'ii'in 
                              (x21::x20::nil)) =>
                            P_id_INTERSECT'II'IN (measure x21) (measure x20)
                           | (algebra.Alg.Term algebra.F.id_u'13'1 
                              (x20::nil)) =>
                            P_id_U'13'1 (measure x20)
                           | (algebra.Alg.Term algebra.F.id_u'6'2 (x20::nil)) =>
                            P_id_U'6'2 (measure x20)
                           | (algebra.Alg.Term algebra.F.id_u'3'1 (x20::nil)) =>
                            P_id_U'3'1 (measure x20)
                           | (algebra.Alg.Term algebra.F.id_u'16'1 
                              (x20::nil)) =>
                            P_id_U'16'1 (measure x20)
                           | (algebra.Alg.Term algebra.F.id_u'10'1 
                              (x20::nil)) =>
                            P_id_U'10'1 (measure x20)
                           | (algebra.Alg.Term algebra.F.id_u'2'1 (x20::nil)) =>
                            P_id_U'2'1 (measure x20)
                           | (algebra.Alg.Term algebra.F.id_u'15'1 
                              (x20::nil)) =>
                            P_id_U'15'1 (measure x20)
                           | (algebra.Alg.Term algebra.F.id_u'8'1 (x20::nil)) =>
                            P_id_U'8'1 (measure x20)
                           | (algebra.Alg.Term algebra.F.id_u'5'1 (x20::nil)) =>
                            P_id_U'5'1 (measure x20)
                           | (algebra.Alg.Term algebra.F.id_u'12'1 (x24::
                              x23::x22::x21::x20::nil)) =>
                            P_id_U'12'1 (measure x24) (measure x23) 
                             (measure x22) (measure x21) (measure x20)
                           | _ => 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_intersect'ii'in_monotonic;assumption.
      intros ;apply P_id_u'6'1_monotonic;assumption.
      intros ;apply P_id_u'3'1_monotonic;assumption.
      intros ;apply P_id_u'12'1_monotonic;assumption.
      intros ;apply P_id_u'2'1_monotonic;assumption.
      intros ;apply P_id_u'9'1_monotonic;assumption.
      intros ;apply P_id_iff_monotonic;assumption.
      intros ;apply P_id_u'14'1_monotonic;assumption.
      intros ;apply P_id_u'7'1_monotonic;assumption.
      intros ;apply P_id_x'2d_monotonic;assumption.
      intros ;apply P_id_u'13'1_monotonic;assumption.
      intros ;apply P_id_sequent_monotonic;assumption.
      intros ;apply P_id_u'10'1_monotonic;assumption.
      intros ;apply P_id_x'2a_monotonic;assumption.
      intros ;apply P_id_tautology'i'in_monotonic;assumption.
      intros ;apply P_id_cons_monotonic;assumption.
      intros ;apply P_id_u'6'2_monotonic;assumption.
      intros ;apply P_id_x'2b_monotonic;assumption.
      intros ;apply P_id_u'12'2_monotonic;assumption.
      intros ;apply P_id_reduce'ii'in_monotonic;assumption.
      intros ;apply P_id_p_monotonic;assumption.
      intros ;apply P_id_u'4'1_monotonic;assumption.
      intros ;apply P_id_u'15'1_monotonic;assumption.
      intros ;apply P_id_u'1'1_monotonic;assumption.
      intros ;apply P_id_u'8'1_monotonic;assumption.
      intros ;apply P_id_if_monotonic;assumption.
      intros ;apply P_id_u'11'1_monotonic;assumption.
      intros ;apply P_id_u'5'1_monotonic;assumption.
      intros ;apply P_id_u'16'1_monotonic;assumption.
      intros ;apply P_id_intersect'ii'in_bounded;assumption.
      intros ;apply P_id_tautology'i'out_bounded;assumption.
      intros ;apply P_id_u'6'1_bounded;assumption.
      intros ;apply P_id_u'3'1_bounded;assumption.
      intros ;apply P_id_u'12'1_bounded;assumption.
      intros ;apply P_id_u'2'1_bounded;assumption.
      intros ;apply P_id_u'9'1_bounded;assumption.
      intros ;apply P_id_iff_bounded;assumption.
      intros ;apply P_id_u'14'1_bounded;assumption.
      intros ;apply P_id_intersect'ii'out_bounded;assumption.
      intros ;apply P_id_u'7'1_bounded;assumption.
      intros ;apply P_id_x'2d_bounded;assumption.
      intros ;apply P_id_u'13'1_bounded;assumption.
      intros ;apply P_id_sequent_bounded;assumption.
      intros ;apply P_id_u'10'1_bounded;assumption.
      intros ;apply P_id_x'2a_bounded;assumption.
      intros ;apply P_id_tautology'i'in_bounded;assumption.
      intros ;apply P_id_cons_bounded;assumption.
      intros ;apply P_id_u'6'2_bounded;assumption.
      intros ;apply P_id_x'2b_bounded;assumption.
      intros ;apply P_id_u'12'2_bounded;assumption.
      intros ;apply P_id_reduce'ii'in_bounded;assumption.
      intros ;apply P_id_p_bounded;assumption.
      intros ;apply P_id_u'4'1_bounded;assumption.
      intros ;apply P_id_u'15'1_bounded;assumption.
      intros ;apply P_id_u'1'1_bounded;assumption.
      intros ;apply P_id_u'8'1_bounded;assumption.
      intros ;apply P_id_reduce'ii'out_bounded;assumption.
      intros ;apply P_id_nil_bounded;assumption.
      intros ;apply P_id_if_bounded;assumption.
      intros ;apply P_id_u'11'1_bounded;assumption.
      intros ;apply P_id_u'5'1_bounded;assumption.
      intros ;apply P_id_u'16'1_bounded;assumption.
      apply rules_monotonic.
      intros ;apply P_id_U'12'2_monotonic;assumption.
      intros ;apply P_id_U'6'1_monotonic;assumption.
      intros ;apply P_id_REDUCE'II'IN_monotonic;assumption.
      intros ;apply P_id_TAUTOLOGY'I'IN_monotonic;assumption.
      intros ;apply P_id_U'9'1_monotonic;assumption.
      intros ;apply P_id_U'1'1_monotonic;assumption.
      intros ;apply P_id_U'14'1_monotonic;assumption.
      intros ;apply P_id_U'7'1_monotonic;assumption.
      intros ;apply P_id_U'4'1_monotonic;assumption.
      intros ;apply P_id_U'11'1_monotonic;assumption.
      intros ;apply P_id_INTERSECT'II'IN_monotonic;assumption.
      intros ;apply P_id_U'13'1_monotonic;assumption.
      intros ;apply P_id_U'6'2_monotonic;assumption.
      intros ;apply P_id_U'3'1_monotonic;assumption.
      intros ;apply P_id_U'16'1_monotonic;assumption.
      intros ;apply P_id_U'10'1_monotonic;assumption.
      intros ;apply P_id_U'2'1_monotonic;assumption.
      intros ;apply P_id_U'15'1_monotonic;assumption.
      intros ;apply P_id_U'8'1_monotonic;assumption.
      intros ;apply P_id_U'5'1_monotonic;assumption.
      intros ;apply P_id_U'12'1_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_19_large_strict_in_lt :
     Relation_Definitions.inclusion _ DP_R_xml_0_scc_19_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_19_large_large_in_le :
     Relation_Definitions.inclusion _ DP_R_xml_0_scc_19_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_19_large_large  := 
      WF_DP_R_xml_0_scc_19_large_large.wf.
    
    
    Lemma wf : well_founded WF_DP_R_xml_0_scc_19.DP_R_xml_0_scc_19_large.
    Proof.
      intros x.
      apply (well_founded_ind wf_lt).
      clear x.
      intros x.
      pattern x.
      apply (@Acc_ind _ DP_R_xml_0_scc_19_large_large).
      clear x.
      intros x _ IHx IHx'.
      constructor.
      intros y H.
      
      destruct H;
       (apply IHx';apply DP_R_xml_0_scc_19_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_19_large_large_in_le;econstructor eassumption])).
      apply wf_DP_R_xml_0_scc_19_large_large.
    Qed.
   End WF_DP_R_xml_0_scc_19_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_intersect'ii'in (x20:Z) (x21:Z) := 0.
   
   Definition P_id_tautology'i'out  := 0.
   
   Definition P_id_u'6'1 (x20:Z) (x21:Z) (x22:Z) (x23:Z) (x24:Z) := 0.
   
   Definition P_id_u'3'1 (x20:Z) := 0.
   
   Definition P_id_u'12'1 (x20:Z) (x21:Z) (x22:Z) (x23:Z) (x24:Z) := 0.
   
   Definition P_id_u'2'1 (x20:Z) := 0.
   
   Definition P_id_u'9'1 (x20:Z) := 0.
   
   Definition P_id_iff (x20:Z) (x21:Z) := 3 + 3* x20 + 3* x21.
   
   Definition P_id_u'14'1 (x20:Z) := 0.
   
   Definition P_id_intersect'ii'out  := 0.
   
   Definition P_id_u'7'1 (x20:Z) := 0.
   
   Definition P_id_x'2d (x20:Z) := 2* x20.
   
   Definition P_id_u'13'1 (x20:Z) := 0.
   
   Definition P_id_sequent (x20:Z) (x21:Z) := 1* x20 + 1* x21.
   
   Definition P_id_u'10'1 (x20:Z) := 0.
   
   Definition P_id_x'2a (x20:Z) (x21:Z) := 1* x20 + 1* x21.
   
   Definition P_id_tautology'i'in (x20:Z) := 2 + 1* x20.
   
   Definition P_id_cons (x20:Z) (x21:Z) := 1* x20 + 1* x21.
   
   Definition P_id_u'6'2 (x20:Z) := 0.
   
   Definition P_id_x'2b (x20:Z) (x21:Z) := 1* x20 + 1* x21.
   
   Definition P_id_u'12'2 (x20:Z) := 0.
   
   Definition P_id_reduce'ii'in (x20:Z) (x21:Z) := 0.
   
   Definition P_id_p (x20:Z) := 3.
   
   Definition P_id_u'4'1 (x20:Z) := 0.
   
   Definition P_id_u'15'1 (x20:Z) := 0.
   
   Definition P_id_u'1'1 (x20:Z) := 0.
   
   Definition P_id_u'8'1 (x20:Z) := 0.
   
   Definition P_id_reduce'ii'out  := 0.
   
   Definition P_id_nil  := 0.
   
   Definition P_id_if (x20:Z) (x21:Z) := 1* x20 + 2* x21.
   
   Definition P_id_u'11'1 (x20:Z) := 0.
   
   Definition P_id_u'5'1 (x20:Z) := 0.
   
   Definition P_id_u'16'1 (x20:Z) := 1.
   
   Lemma P_id_intersect'ii'in_monotonic :
    forall x20 x22 x21 x23, 
     (0 <= x23)/\ (x23 <= x22) ->
      (0 <= x21)/\ (x21 <= x20) ->
       P_id_intersect'ii'in x21 x23 <= P_id_intersect'ii'in x20 x22.
   Proof.
     intros x23 x22 x21 x20.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'6'1_monotonic :
    forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, 
     (0 <= x29)/\ (x29 <= x28) ->
      (0 <= x27)/\ (x27 <= x26) ->
       (0 <= x25)/\ (x25 <= x24) ->
        (0 <= x23)/\ (x23 <= x22) ->
         (0 <= x21)/\ (x21 <= x20) ->
          P_id_u'6'1 x21 x23 x25 x27 x29 <= P_id_u'6'1 x20 x22 x24 x26 x28.
   Proof.
     intros x29 x28 x27 x26 x25 x24 x23 x22 x21 x20.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     intros [H_7 H_6].
     intros [H_9 H_8].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'3'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_u'3'1 x21 <= P_id_u'3'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'12'1_monotonic :
    forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, 
     (0 <= x29)/\ (x29 <= x28) ->
      (0 <= x27)/\ (x27 <= x26) ->
       (0 <= x25)/\ (x25 <= x24) ->
        (0 <= x23)/\ (x23 <= x22) ->
         (0 <= x21)/\ (x21 <= x20) ->
          P_id_u'12'1 x21 x23 x25 x27 x29 <= P_id_u'12'1 x20 x22 x24 x26 x28.
   Proof.
     intros x29 x28 x27 x26 x25 x24 x23 x22 x21 x20.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     intros [H_7 H_6].
     intros [H_9 H_8].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'2'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_u'2'1 x21 <= P_id_u'2'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'9'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_u'9'1 x21 <= P_id_u'9'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_iff_monotonic :
    forall x20 x22 x21 x23, 
     (0 <= x23)/\ (x23 <= x22) ->
      (0 <= x21)/\ (x21 <= x20) ->P_id_iff x21 x23 <= P_id_iff x20 x22.
   Proof.
     intros x23 x22 x21 x20.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'14'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_u'14'1 x21 <= P_id_u'14'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'7'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_u'7'1 x21 <= P_id_u'7'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_x'2d_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_x'2d x21 <= P_id_x'2d x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'13'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_u'13'1 x21 <= P_id_u'13'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_sequent_monotonic :
    forall x20 x22 x21 x23, 
     (0 <= x23)/\ (x23 <= x22) ->
      (0 <= x21)/\ (x21 <= x20) ->
       P_id_sequent x21 x23 <= P_id_sequent x20 x22.
   Proof.
     intros x23 x22 x21 x20.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'10'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_u'10'1 x21 <= P_id_u'10'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_x'2a_monotonic :
    forall x20 x22 x21 x23, 
     (0 <= x23)/\ (x23 <= x22) ->
      (0 <= x21)/\ (x21 <= x20) ->P_id_x'2a x21 x23 <= P_id_x'2a x20 x22.
   Proof.
     intros x23 x22 x21 x20.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_tautology'i'in_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->
      P_id_tautology'i'in x21 <= P_id_tautology'i'in x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_cons_monotonic :
    forall x20 x22 x21 x23, 
     (0 <= x23)/\ (x23 <= x22) ->
      (0 <= x21)/\ (x21 <= x20) ->P_id_cons x21 x23 <= P_id_cons x20 x22.
   Proof.
     intros x23 x22 x21 x20.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'6'2_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_u'6'2 x21 <= P_id_u'6'2 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_x'2b_monotonic :
    forall x20 x22 x21 x23, 
     (0 <= x23)/\ (x23 <= x22) ->
      (0 <= x21)/\ (x21 <= x20) ->P_id_x'2b x21 x23 <= P_id_x'2b x20 x22.
   Proof.
     intros x23 x22 x21 x20.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'12'2_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_u'12'2 x21 <= P_id_u'12'2 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_reduce'ii'in_monotonic :
    forall x20 x22 x21 x23, 
     (0 <= x23)/\ (x23 <= x22) ->
      (0 <= x21)/\ (x21 <= x20) ->
       P_id_reduce'ii'in x21 x23 <= P_id_reduce'ii'in x20 x22.
   Proof.
     intros x23 x22 x21 x20.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_p_monotonic :
    forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_p x21 <= P_id_p x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'4'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_u'4'1 x21 <= P_id_u'4'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'15'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_u'15'1 x21 <= P_id_u'15'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'1'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_u'1'1 x21 <= P_id_u'1'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'8'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_u'8'1 x21 <= P_id_u'8'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_if_monotonic :
    forall x20 x22 x21 x23, 
     (0 <= x23)/\ (x23 <= x22) ->
      (0 <= x21)/\ (x21 <= x20) ->P_id_if x21 x23 <= P_id_if x20 x22.
   Proof.
     intros x23 x22 x21 x20.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'11'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_u'11'1 x21 <= P_id_u'11'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'5'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_u'5'1 x21 <= P_id_u'5'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'16'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_u'16'1 x21 <= P_id_u'16'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_intersect'ii'in_bounded :
    forall x20 x21, 
     (0 <= x20) ->(0 <= x21) ->0 <= P_id_intersect'ii'in x21 x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_tautology'i'out_bounded : 0 <= P_id_tautology'i'out .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'6'1_bounded :
    forall x24 x20 x22 x21 x23, 
     (0 <= x20) ->
      (0 <= x21) ->
       (0 <= x22) ->
        (0 <= x23) ->(0 <= x24) ->0 <= P_id_u'6'1 x24 x23 x22 x21 x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'3'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'3'1 x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'12'1_bounded :
    forall x24 x20 x22 x21 x23, 
     (0 <= x20) ->
      (0 <= x21) ->
       (0 <= x22) ->
        (0 <= x23) ->(0 <= x24) ->0 <= P_id_u'12'1 x24 x23 x22 x21 x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'2'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'2'1 x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'9'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'9'1 x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_iff_bounded :
    forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_iff x21 x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'14'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'14'1 x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_intersect'ii'out_bounded : 0 <= P_id_intersect'ii'out .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'7'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'7'1 x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_x'2d_bounded : forall x20, (0 <= x20) ->0 <= P_id_x'2d x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'13'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'13'1 x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_sequent_bounded :
    forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_sequent x21 x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'10'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'10'1 x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_x'2a_bounded :
    forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_x'2a x21 x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_tautology'i'in_bounded :
    forall x20, (0 <= x20) ->0 <= P_id_tautology'i'in x20.
   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 x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_cons x21 x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'6'2_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'6'2 x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_x'2b_bounded :
    forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_x'2b x21 x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'12'2_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'12'2 x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_reduce'ii'in_bounded :
    forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_reduce'ii'in x21 x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_p_bounded : forall x20, (0 <= x20) ->0 <= P_id_p x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'4'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'4'1 x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'15'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'15'1 x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'1'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'1'1 x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'8'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'8'1 x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_reduce'ii'out_bounded : 0 <= P_id_reduce'ii'out .
   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_if_bounded :
    forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_if x21 x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'11'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'11'1 x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'5'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'5'1 x20.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_u'16'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'16'1 x20.
   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_intersect'ii'in P_id_tautology'i'out P_id_u'6'1 
      P_id_u'3'1 P_id_u'12'1 P_id_u'2'1 P_id_u'9'1 P_id_iff P_id_u'14'1 
      P_id_intersect'ii'out P_id_u'7'1 P_id_x'2d P_id_u'13'1 P_id_sequent 
      P_id_u'10'1 P_id_x'2a P_id_tautology'i'in P_id_cons P_id_u'6'2 
      P_id_x'2b P_id_u'12'2 P_id_reduce'ii'in P_id_p P_id_u'4'1 P_id_u'15'1 
      P_id_u'1'1 P_id_u'8'1 P_id_reduce'ii'out P_id_nil P_id_if P_id_u'11'1 
      P_id_u'5'1 P_id_u'16'1.
   
   Lemma measure_equation :
    forall t, 
     measure t = match t with
                   | (algebra.Alg.Term algebra.F.id_intersect'ii'in (x21::
                      x20::nil)) =>
                    P_id_intersect'ii'in (measure x21) (measure x20)
                   | (algebra.Alg.Term algebra.F.id_tautology'i'out nil) =>
                    P_id_tautology'i'out 
                   | (algebra.Alg.Term algebra.F.id_u'6'1 (x24::x23::x22::
                      x21::x20::nil)) =>
                    P_id_u'6'1 (measure x24) (measure x23) (measure x22) 
                     (measure x21) (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'3'1 (x20::nil)) =>
                    P_id_u'3'1 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'12'1 (x24::x23::x22::
                      x21::x20::nil)) =>
                    P_id_u'12'1 (measure x24) (measure x23) (measure x22) 
                     (measure x21) (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'2'1 (x20::nil)) =>
                    P_id_u'2'1 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'9'1 (x20::nil)) =>
                    P_id_u'9'1 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_iff (x21::x20::nil)) =>
                    P_id_iff (measure x21) (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'14'1 (x20::nil)) =>
                    P_id_u'14'1 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_intersect'ii'out nil) =>
                    P_id_intersect'ii'out 
                   | (algebra.Alg.Term algebra.F.id_u'7'1 (x20::nil)) =>
                    P_id_u'7'1 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_x'2d (x20::nil)) =>
                    P_id_x'2d (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'13'1 (x20::nil)) =>
                    P_id_u'13'1 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_sequent (x21::x20::nil)) =>
                    P_id_sequent (measure x21) (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'10'1 (x20::nil)) =>
                    P_id_u'10'1 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_x'2a (x21::x20::nil)) =>
                    P_id_x'2a (measure x21) (measure x20)
                   | (algebra.Alg.Term algebra.F.id_tautology'i'in 
                      (x20::nil)) =>
                    P_id_tautology'i'in (measure x20)
                   | (algebra.Alg.Term algebra.F.id_cons (x21::x20::nil)) =>
                    P_id_cons (measure x21) (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'6'2 (x20::nil)) =>
                    P_id_u'6'2 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_x'2b (x21::x20::nil)) =>
                    P_id_x'2b (measure x21) (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'12'2 (x20::nil)) =>
                    P_id_u'12'2 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::
                      x20::nil)) =>
                    P_id_reduce'ii'in (measure x21) (measure x20)
                   | (algebra.Alg.Term algebra.F.id_p (x20::nil)) =>
                    P_id_p (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'4'1 (x20::nil)) =>
                    P_id_u'4'1 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'15'1 (x20::nil)) =>
                    P_id_u'15'1 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'1'1 (x20::nil)) =>
                    P_id_u'1'1 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'8'1 (x20::nil)) =>
                    P_id_u'8'1 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) =>
                    P_id_reduce'ii'out 
                   | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil 
                   | (algebra.Alg.Term algebra.F.id_if (x21::x20::nil)) =>
                    P_id_if (measure x21) (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'11'1 (x20::nil)) =>
                    P_id_u'11'1 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'5'1 (x20::nil)) =>
                    P_id_u'5'1 (measure x20)
                   | (algebra.Alg.Term algebra.F.id_u'16'1 (x20::nil)) =>
                    P_id_u'16'1 (measure x20)
                   | _ => 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_intersect'ii'in_monotonic;assumption.
     intros ;apply P_id_u'6'1_monotonic;assumption.
     intros ;apply P_id_u'3'1_monotonic;assumption.
     intros ;apply P_id_u'12'1_monotonic;assumption.
     intros ;apply P_id_u'2'1_monotonic;assumption.
     intros ;apply P_id_u'9'1_monotonic;assumption.
     intros ;apply P_id_iff_monotonic;assumption.
     intros ;apply P_id_u'14'1_monotonic;assumption.
     intros ;apply P_id_u'7'1_monotonic;assumption.
     intros ;apply P_id_x'2d_monotonic;assumption.
     intros ;apply P_id_u'13'1_monotonic;assumption.
     intros ;apply P_id_sequent_monotonic;assumption.
     intros ;apply P_id_u'10'1_monotonic;assumption.
     intros ;apply P_id_x'2a_monotonic;assumption.
     intros ;apply P_id_tautology'i'in_monotonic;assumption.
     intros ;apply P_id_cons_monotonic;assumption.
     intros ;apply P_id_u'6'2_monotonic;assumption.
     intros ;apply P_id_x'2b_monotonic;assumption.
     intros ;apply P_id_u'12'2_monotonic;assumption.
     intros ;apply P_id_reduce'ii'in_monotonic;assumption.
     intros ;apply P_id_p_monotonic;assumption.
     intros ;apply P_id_u'4'1_monotonic;assumption.
     intros ;apply P_id_u'15'1_monotonic;assumption.
     intros ;apply P_id_u'1'1_monotonic;assumption.
     intros ;apply P_id_u'8'1_monotonic;assumption.
     intros ;apply P_id_if_monotonic;assumption.
     intros ;apply P_id_u'11'1_monotonic;assumption.
     intros ;apply P_id_u'5'1_monotonic;assumption.
     intros ;apply P_id_u'16'1_monotonic;assumption.
     intros ;apply P_id_intersect'ii'in_bounded;assumption.
     intros ;apply P_id_tautology'i'out_bounded;assumption.
     intros ;apply P_id_u'6'1_bounded;assumption.
     intros ;apply P_id_u'3'1_bounded;assumption.
     intros ;apply P_id_u'12'1_bounded;assumption.
     intros ;apply P_id_u'2'1_bounded;assumption.
     intros ;apply P_id_u'9'1_bounded;assumption.
     intros ;apply P_id_iff_bounded;assumption.
     intros ;apply P_id_u'14'1_bounded;assumption.
     intros ;apply P_id_intersect'ii'out_bounded;assumption.
     intros ;apply P_id_u'7'1_bounded;assumption.
     intros ;apply P_id_x'2d_bounded;assumption.
     intros ;apply P_id_u'13'1_bounded;assumption.
     intros ;apply P_id_sequent_bounded;assumption.
     intros ;apply P_id_u'10'1_bounded;assumption.
     intros ;apply P_id_x'2a_bounded;assumption.
     intros ;apply P_id_tautology'i'in_bounded;assumption.
     intros ;apply P_id_cons_bounded;assumption.
     intros ;apply P_id_u'6'2_bounded;assumption.
     intros ;apply P_id_x'2b_bounded;assumption.
     intros ;apply P_id_u'12'2_bounded;assumption.
     intros ;apply P_id_reduce'ii'in_bounded;assumption.
     intros ;apply P_id_p_bounded;assumption.
     intros ;apply P_id_u'4'1_bounded;assumption.
     intros ;apply P_id_u'15'1_bounded;assumption.
     intros ;apply P_id_u'1'1_bounded;assumption.
     intros ;apply P_id_u'8'1_bounded;assumption.
     intros ;apply P_id_reduce'ii'out_bounded;assumption.
     intros ;apply P_id_nil_bounded;assumption.
     intros ;apply P_id_if_bounded;assumption.
     intros ;apply P_id_u'11'1_bounded;assumption.
     intros ;apply P_id_u'5'1_bounded;assumption.
     intros ;apply P_id_u'16'1_bounded;assumption.
     apply rules_monotonic.
   Qed.
   
   Definition P_id_U'12'2 (x20:Z) := 0.
   
   Definition P_id_U'6'1 (x20:Z) (x21:Z) (x22:Z) (x23:Z) (x24:Z) := 
     2* x21 + 2* x22 + 2* x23 + 1* x24.
   
   Definition P_id_REDUCE'II'IN (x20:Z) (x21:Z) := 2* x20 + 1* x21.
   
   Definition P_id_TAUTOLOGY'I'IN (x20:Z) := 0.
   
   Definition P_id_U'9'1 (x20:Z) := 0.
   
   Definition P_id_U'1'1 (x20:Z) := 0.
   
   Definition P_id_U'14'1 (x20:Z) := 0.
   
   Definition P_id_U'7'1 (x20:Z) := 0.
   
   Definition P_id_U'4'1 (x20:Z) := 0.
   
   Definition P_id_U'11'1 (x20:Z) := 0.
   
   Definition P_id_INTERSECT'II'IN (x20:Z) (x21:Z) := 0.
   
   Definition P_id_U'13'1 (x20:Z) := 0.
   
   Definition P_id_U'6'2 (x20:Z) := 0.
   
   Definition P_id_U'3'1 (x20:Z) := 0.
   
   Definition P_id_U'16'1 (x20:Z) := 0.
   
   Definition P_id_U'10'1 (x20:Z) := 0.
   
   Definition P_id_U'2'1 (x20:Z) := 0.
   
   Definition P_id_U'15'1 (x20:Z) := 0.
   
   Definition P_id_U'8'1 (x20:Z) := 0.
   
   Definition P_id_U'5'1 (x20:Z) := 0.
   
   Definition P_id_U'12'1 (x20:Z) (x21:Z) (x22:Z) (x23:Z) (x24:Z) := 
     2* x21 + 2* x22 + 2* x23 + 1* x24.
   
   Lemma P_id_U'12'2_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_U'12'2 x21 <= P_id_U'12'2 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U'6'1_monotonic :
    forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, 
     (0 <= x29)/\ (x29 <= x28) ->
      (0 <= x27)/\ (x27 <= x26) ->
       (0 <= x25)/\ (x25 <= x24) ->
        (0 <= x23)/\ (x23 <= x22) ->
         (0 <= x21)/\ (x21 <= x20) ->
          P_id_U'6'1 x21 x23 x25 x27 x29 <= P_id_U'6'1 x20 x22 x24 x26 x28.
   Proof.
     intros x29 x28 x27 x26 x25 x24 x23 x22 x21 x20.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     intros [H_7 H_6].
     intros [H_9 H_8].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_REDUCE'II'IN_monotonic :
    forall x20 x22 x21 x23, 
     (0 <= x23)/\ (x23 <= x22) ->
      (0 <= x21)/\ (x21 <= x20) ->
       P_id_REDUCE'II'IN x21 x23 <= P_id_REDUCE'II'IN x20 x22.
   Proof.
     intros x23 x22 x21 x20.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_TAUTOLOGY'I'IN_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->
      P_id_TAUTOLOGY'I'IN x21 <= P_id_TAUTOLOGY'I'IN x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U'9'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_U'9'1 x21 <= P_id_U'9'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U'1'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_U'1'1 x21 <= P_id_U'1'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U'14'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_U'14'1 x21 <= P_id_U'14'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U'7'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_U'7'1 x21 <= P_id_U'7'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U'4'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_U'4'1 x21 <= P_id_U'4'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U'11'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_U'11'1 x21 <= P_id_U'11'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_INTERSECT'II'IN_monotonic :
    forall x20 x22 x21 x23, 
     (0 <= x23)/\ (x23 <= x22) ->
      (0 <= x21)/\ (x21 <= x20) ->
       P_id_INTERSECT'II'IN x21 x23 <= P_id_INTERSECT'II'IN x20 x22.
   Proof.
     intros x23 x22 x21 x20.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U'13'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_U'13'1 x21 <= P_id_U'13'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U'6'2_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_U'6'2 x21 <= P_id_U'6'2 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U'3'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_U'3'1 x21 <= P_id_U'3'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U'16'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_U'16'1 x21 <= P_id_U'16'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U'10'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_U'10'1 x21 <= P_id_U'10'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U'2'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_U'2'1 x21 <= P_id_U'2'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U'15'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_U'15'1 x21 <= P_id_U'15'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U'8'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_U'8'1 x21 <= P_id_U'8'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U'5'1_monotonic :
    forall x20 x21, 
     (0 <= x21)/\ (x21 <= x20) ->P_id_U'5'1 x21 <= P_id_U'5'1 x20.
   Proof.
     intros x21 x20.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U'12'1_monotonic :
    forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, 
     (0 <= x29)/\ (x29 <= x28) ->
      (0 <= x27)/\ (x27 <= x26) ->
       (0 <= x25)/\ (x25 <= x24) ->
        (0 <= x23)/\ (x23 <= x22) ->
         (0 <= x21)/\ (x21 <= x20) ->
          P_id_U'12'1 x21 x23 x25 x27 x29 <= P_id_U'12'1 x20 x22 x24 x26 x28.
   Proof.
     intros x29 x28 x27 x26 x25 x24 x23 x22 x21 x20.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     intros [H_7 H_6].
     intros [H_9 H_8].
     
     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_intersect'ii'in P_id_tautology'i'out 
      P_id_u'6'1 P_id_u'3'1 P_id_u'12'1 P_id_u'2'1 P_id_u'9'1 P_id_iff 
      P_id_u'14'1 P_id_intersect'ii'out P_id_u'7'1 P_id_x'2d P_id_u'13'1 
      P_id_sequent P_id_u'10'1 P_id_x'2a P_id_tautology'i'in P_id_cons 
      P_id_u'6'2 P_id_x'2b P_id_u'12'2 P_id_reduce'ii'in P_id_p P_id_u'4'1 
      P_id_u'15'1 P_id_u'1'1 P_id_u'8'1 P_id_reduce'ii'out P_id_nil P_id_if 
      P_id_u'11'1 P_id_u'5'1 P_id_u'16'1 P_id_U'12'2 P_id_U'6'1 
      P_id_REDUCE'II'IN P_id_TAUTOLOGY'I'IN P_id_U'9'1 P_id_U'1'1 
      P_id_U'14'1 P_id_U'7'1 P_id_U'4'1 P_id_U'11'1 P_id_INTERSECT'II'IN 
      P_id_U'13'1 P_id_U'6'2 P_id_U'3'1 P_id_U'16'1 P_id_U'10'1 P_id_U'2'1 
      P_id_U'15'1 P_id_U'8'1 P_id_U'5'1 P_id_U'12'1.
   
   Lemma marked_measure_equation :
    forall t, 
     marked_measure t = match t with
                          | (algebra.Alg.Term algebra.F.id_u'12'2 (x20::nil)) =>
                           P_id_U'12'2 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'6'1 (x24::x23::
                             x22::x21::x20::nil)) =>
                           P_id_U'6'1 (measure x24) (measure x23) 
                            (measure x22) (measure x21) (measure x20)
                          | (algebra.Alg.Term algebra.F.id_reduce'ii'in 
                             (x21::x20::nil)) =>
                           P_id_REDUCE'II'IN (measure x21) (measure x20)
                          | (algebra.Alg.Term algebra.F.id_tautology'i'in 
                             (x20::nil)) =>
                           P_id_TAUTOLOGY'I'IN (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'9'1 (x20::nil)) =>
                           P_id_U'9'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'1'1 (x20::nil)) =>
                           P_id_U'1'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'14'1 (x20::nil)) =>
                           P_id_U'14'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'7'1 (x20::nil)) =>
                           P_id_U'7'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'4'1 (x20::nil)) =>
                           P_id_U'4'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'11'1 (x20::nil)) =>
                           P_id_U'11'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_intersect'ii'in 
                             (x21::x20::nil)) =>
                           P_id_INTERSECT'II'IN (measure x21) (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'13'1 (x20::nil)) =>
                           P_id_U'13'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'6'2 (x20::nil)) =>
                           P_id_U'6'2 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'3'1 (x20::nil)) =>
                           P_id_U'3'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'16'1 (x20::nil)) =>
                           P_id_U'16'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'10'1 (x20::nil)) =>
                           P_id_U'10'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'2'1 (x20::nil)) =>
                           P_id_U'2'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'15'1 (x20::nil)) =>
                           P_id_U'15'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'8'1 (x20::nil)) =>
                           P_id_U'8'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'5'1 (x20::nil)) =>
                           P_id_U'5'1 (measure x20)
                          | (algebra.Alg.Term algebra.F.id_u'12'1 (x24::x23::
                             x22::x21::x20::nil)) =>
                           P_id_U'12'1 (measure x24) (measure x23) 
                            (measure x22) (measure x21) (measure x20)
                          | _ => 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_intersect'ii'in_monotonic;assumption.
     intros ;apply P_id_u'6'1_monotonic;assumption.
     intros ;apply P_id_u'3'1_monotonic;assumption.
     intros ;apply P_id_u'12'1_monotonic;assumption.
     intros ;apply P_id_u'2'1_monotonic;assumption.
     intros ;apply P_id_u'9'1_monotonic;assumption.
     intros ;apply P_id_iff_monotonic;assumption.
     intros ;apply P_id_u'14'1_monotonic;assumption.
     intros ;apply P_id_u'7'1_monotonic;assumption.
     intros ;apply P_id_x'2d_monotonic;assumption.
     intros ;apply P_id_u'13'1_monotonic;assumption.
     intros ;apply P_id_sequent_monotonic;assumption.
     intros ;apply P_id_u'10'1_monotonic;assumption.
     intros ;apply P_id_x'2a_monotonic;assumption.
     intros ;apply P_id_tautology'i'in_monotonic;assumption.
     intros ;apply P_id_cons_monotonic;assumption.
     intros ;apply P_id_u'6'2_monotonic;assumption.
     intros ;apply P_id_x'2b_monotonic;assumption.
     intros ;apply P_id_u'12'2_monotonic;assumption.
     intros ;apply P_id_reduce'ii'in_monotonic;assumption.
     intros ;apply P_id_p_monotonic;assumption.
     intros ;apply P_id_u'4'1_monotonic;assumption.
     intros ;apply P_id_u'15'1_monotonic;assumption.
     intros ;apply P_id_u'1'1_monotonic;assumption.
     intros ;apply P_id_u'8'1_monotonic;assumption.
     intros ;apply P_id_if_monotonic;assumption.
     intros ;apply P_id_u'11'1_monotonic;assumption.
     intros ;apply P_id_u'5'1_monotonic;assumption.
     intros ;apply P_id_u'16'1_monotonic;assumption.
     intros ;apply P_id_intersect'ii'in_bounded;assumption.
     intros ;apply P_id_tautology'i'out_bounded;assumption.
     intros ;apply P_id_u'6'1_bounded;assumption.
     intros ;apply P_id_u'3'1_bounded;assumption.
     intros ;apply P_id_u'12'1_bounded;assumption.
     intros ;apply P_id_u'2'1_bounded;assumption.
     intros ;apply P_id_u'9'1_bounded;assumption.
     intros ;apply P_id_iff_bounded;assumption.
     intros ;apply P_id_u'14'1_bounded;assumption.
     intros ;apply P_id_intersect'ii'out_bounded;assumption.
     intros ;apply P_id_u'7'1_bounded;assumption.
     intros ;apply P_id_x'2d_bounded;assumption.
     intros ;apply P_id_u'13'1_bounded;assumption.
     intros ;apply P_id_sequent_bounded;assumption.
     intros ;apply P_id_u'10'1_bounded;assumption.
     intros ;apply P_id_x'2a_bounded;assumption.
     intros ;apply P_id_tautology'i'in_bounded;assumption.
     intros ;apply P_id_cons_bounded;assumption.
     intros ;apply P_id_u'6'2_bounded;assumption.
     intros ;apply P_id_x'2b_bounded;assumption.
     intros ;apply P_id_u'12'2_bounded;assumption.
     intros ;apply P_id_reduce'ii'in_bounded;assumption.
     intros ;apply P_id_p_bounded;assumption.
     intros ;apply P_id_u'4'1_bounded;assumption.
     intros ;apply P_id_u'15'1_bounded;assumption.
     intros ;apply P_id_u'1'1_bounded;assumption.
     intros ;apply P_id_u'8'1_bounded;assumption.
     intros ;apply P_id_reduce'ii'out_bounded;assumption.
     intros ;apply P_id_nil_bounded;assumption.
     intros ;apply P_id_if_bounded;assumption.
     intros ;apply P_id_u'11'1_bounded;assumption.
     intros ;apply P_id_u'5'1_bounded;assumption.
     intros ;apply P_id_u'16'1_bounded;assumption.
     apply rules_monotonic.
     intros ;apply P_id_U'12'2_monotonic;assumption.
     intros ;apply P_id_U'6'1_monotonic;assumption.
     intros ;apply P_id_REDUCE'II'IN_monotonic;assumption.
     intros ;apply P_id_TAUTOLOGY'I'IN_monotonic;assumption.
     intros ;apply P_id_U'9'1_monotonic;assumption.
     intros ;apply P_id_U'1'1_monotonic;assumption.
     intros ;apply P_id_U'14'1_monotonic;assumption.
     intros ;apply P_id_U'7'1_monotonic;assumption.
     intros ;apply P_id_U'4'1_monotonic;assumption.
     intros ;apply P_id_U'11'1_monotonic;assumption.
     intros ;apply P_id_INTERSECT'II'IN_monotonic;assumption.
     intros ;apply P_id_U'13'1_monotonic;assumption.
     intros ;apply P_id_U'6'2_monotonic;assumption.
     intros ;apply P_id_U'3'1_monotonic;assumption.
     intros ;apply P_id_U'16'1_monotonic;assumption.
     intros ;apply P_id_U'10'1_monotonic;assumption.
     intros ;apply P_id_U'2'1_monotonic;assumption.
     intros ;apply P_id_U'15'1_monotonic;assumption.
     intros ;apply P_id_U'8'1_monotonic;assumption.
     intros ;apply P_id_U'5'1_monotonic;assumption.
     intros ;apply P_id_U'12'1_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_19_strict_in_lt :
    Relation_Definitions.inclusion _ DP_R_xml_0_scc_19_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_19_large_in_le :
    Relation_Definitions.inclusion _ DP_R_xml_0_scc_19_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_19_large  := WF_DP_R_xml_0_scc_19_large.wf.
   
   
   Lemma wf : well_founded WF_DP_R_xml_0.DP_R_xml_0_scc_19.
   Proof.
     intros x.
     apply (well_founded_ind wf_lt).
     clear x.
     intros x.
     pattern x.
     apply (@Acc_ind _ DP_R_xml_0_scc_19_large).
     clear x.
     intros x _ IHx IHx'.
     constructor.
     intros y H.
     
     destruct H;
      (apply IHx';apply DP_R_xml_0_scc_19_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_19_large_in_le;econstructor eassumption])).
     apply wf_DP_R_xml_0_scc_19_large.
   Qed.
  End WF_DP_R_xml_0_scc_19.
  
  Definition wf_DP_R_xml_0_scc_19  := WF_DP_R_xml_0_scc_19.wf.
  
  
  Lemma acc_DP_R_xml_0_scc_19 :
   forall x y, (DP_R_xml_0_scc_19 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_19).
    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_18;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_14;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((eapply acc_DP_R_xml_0_non_scc_13;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
         ((eapply acc_DP_R_xml_0_non_scc_12;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
          ((eapply acc_DP_R_xml_0_non_scc_11;
             econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
           ((eapply acc_DP_R_xml_0_non_scc_10;
              econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
            ((eapply acc_DP_R_xml_0_non_scc_9;
               econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
             ((eapply acc_DP_R_xml_0_non_scc_8;
                econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
              ((eapply acc_DP_R_xml_0_non_scc_7;
                 econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
               ((eapply acc_DP_R_xml_0_non_scc_6;
                  econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
                ((eapply acc_DP_R_xml_0_non_scc_5;
                   econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
                 ((eapply acc_DP_R_xml_0_non_scc_4;
                    econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
                  ((eapply acc_DP_R_xml_0_non_scc_3;
                     econstructor 
                     (eassumption)||(algebra.Alg_ext.star_refl' ))||
                   ((eapply acc_DP_R_xml_0_non_scc_2;
                      econstructor 
                      (eassumption)||(algebra.Alg_ext.star_refl' ))||
                    ((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_19.
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_20  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <tautology'i'in(F_),reduce'ii'in(sequent(nil,cons(F_,nil)),sequent(nil,nil))> *)
    | DP_R_xml_0_non_scc_20_0 :
     forall x20 x18, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x18 x20) ->
       DP_R_xml_0_non_scc_20 (algebra.Alg.Term algebra.F.id_reduce'ii'in 
                              ((algebra.Alg.Term algebra.F.id_sequent 
                              ((algebra.Alg.Term algebra.F.id_nil nil)::
                              (algebra.Alg.Term algebra.F.id_cons (x18::
                              (algebra.Alg.Term algebra.F.id_nil 
                              nil)::nil))::nil))::(algebra.Alg.Term 
                              algebra.F.id_sequent ((algebra.Alg.Term 
                              algebra.F.id_nil nil)::(algebra.Alg.Term 
                              algebra.F.id_nil nil)::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_tautology'i'in (x20::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_20 :
   forall x y, 
    (DP_R_xml_0_non_scc_20 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_19;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_18;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_14;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((eapply acc_DP_R_xml_0_non_scc_13;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
         ((eapply acc_DP_R_xml_0_non_scc_12;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
          ((eapply acc_DP_R_xml_0_non_scc_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_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' ))||
                  ((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_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_20;
       econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
     ((eapply acc_DP_R_xml_0_non_scc_19;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_18;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_17;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((eapply acc_DP_R_xml_0_non_scc_16;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
         ((eapply acc_DP_R_xml_0_non_scc_15;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
          ((eapply acc_DP_R_xml_0_non_scc_14;
             econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
           ((eapply acc_DP_R_xml_0_non_scc_13;
              econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
            ((eapply acc_DP_R_xml_0_non_scc_12;
               econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
             ((eapply acc_DP_R_xml_0_non_scc_11;
                econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
              ((eapply acc_DP_R_xml_0_non_scc_10;
                 econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
               ((eapply acc_DP_R_xml_0_non_scc_9;
                  econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
                ((eapply acc_DP_R_xml_0_non_scc_8;
                   econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
                 ((eapply acc_DP_R_xml_0_non_scc_7;
                    econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
                  ((eapply acc_DP_R_xml_0_non_scc_6;
                     econstructor 
                     (eassumption)||(algebra.Alg_ext.star_refl' ))||
                   ((eapply acc_DP_R_xml_0_non_scc_5;
                      econstructor 
                      (eassumption)||(algebra.Alg_ext.star_refl' ))||
                    ((eapply acc_DP_R_xml_0_non_scc_4;
                       econstructor 
                       (eassumption)||(algebra.Alg_ext.star_refl' ))||
                     ((eapply acc_DP_R_xml_0_non_scc_3;
                        econstructor 
                        (eassumption)||(algebra.Alg_ext.star_refl' ))||
                      ((eapply acc_DP_R_xml_0_non_scc_2;
                         econstructor 
                         (eassumption)||(algebra.Alg_ext.star_refl' ))||
                       ((eapply acc_DP_R_xml_0_non_scc_1;
                          econstructor 
                          (eassumption)||(algebra.Alg_ext.star_refl' ))||
                        ((eapply acc_DP_R_xml_0_non_scc_0;
                           econstructor 
                           (eassumption)||(algebra.Alg_ext.star_refl' ))||
                         ((eapply acc_DP_R_xml_0_scc_19;
                            econstructor 
                            (eassumption)||(algebra.Alg_ext.star_refl' ))||
                          ((eapply acc_DP_R_xml_0_scc_18;
                             econstructor 
                             (eassumption)||(algebra.Alg_ext.star_refl' ))||
                           ((eapply acc_DP_R_xml_0_scc_17;
                              econstructor 
                              (eassumption)||(algebra.Alg_ext.star_refl' ))||
                            ((eapply acc_DP_R_xml_0_scc_16;
                               econstructor 
                               (eassumption)||(algebra.Alg_ext.star_refl' ))||
                             ((eapply acc_DP_R_xml_0_scc_15;
                                econstructor 
                                (eassumption)||(algebra.Alg_ext.star_refl' ))||
                              ((eapply acc_DP_R_xml_0_scc_14;
                                 econstructor 
                                 (eassumption)||(algebra.Alg_ext.star_refl' ))||
                               ((eapply acc_DP_R_xml_0_scc_13;
                                  econstructor 
                                  (eassumption)||
                                  (algebra.Alg_ext.star_refl' ))||
                                ((eapply acc_DP_R_xml_0_scc_12;
                                   econstructor 
                                   (eassumption)||
                                   (algebra.Alg_ext.star_refl' ))||
                                 ((eapply acc_DP_R_xml_0_scc_11;
                                    econstructor 
                                    (eassumption)||
                                    (algebra.Alg_ext.star_refl' ))||
                                  ((eapply acc_DP_R_xml_0_scc_10;
                                     econstructor 
                                     (eassumption)||
                                     (algebra.Alg_ext.star_refl' ))||
                                   ((eapply acc_DP_R_xml_0_scc_9;
                                      econstructor 
                                      (eassumption)||
                                      (algebra.Alg_ext.star_refl' ))||
                                    ((eapply acc_DP_R_xml_0_scc_8;
                                       econstructor 
                                       (eassumption)||
                                       (algebra.Alg_ext.star_refl' ))||
                                     ((eapply acc_DP_R_xml_0_scc_7;
                                        econstructor 
                                        (eassumption)||
                                        (algebra.Alg_ext.star_refl' ))||
                                      ((eapply acc_DP_R_xml_0_scc_6;
                                         econstructor 
                                         (eassumption)||
                                         (algebra.Alg_ext.star_refl' ))||
                                       ((eapply acc_DP_R_xml_0_scc_5;
                                          econstructor 
                                          (eassumption)||
                                          (algebra.Alg_ext.star_refl' ))||
                                        ((eapply acc_DP_R_xml_0_scc_4;
                                           econstructor 
                                           (eassumption)||
                                           (algebra.Alg_ext.star_refl' ))||
                                         ((eapply acc_DP_R_xml_0_scc_3;
                                            econstructor 
                                            (eassumption)||
                                            (algebra.Alg_ext.star_refl' ))||
                                          ((eapply acc_DP_R_xml_0_scc_2;
                                             econstructor 
                                             (eassumption)||
                                             (algebra.Alg_ext.star_refl' ))||
                                           ((eapply acc_DP_R_xml_0_scc_1;
                                              econstructor 
                                              (eassumption)||
                                              (algebra.Alg_ext.star_refl' ))||
                                            ((eapply acc_DP_R_xml_0_scc_0;
                                               econstructor 
                                               (eassumption)||
                                               (algebra.Alg_ext.star_refl' ))||
                                             ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
                                              (fail)))))))))))))))))))))))))))))))))))))))))).
  Qed.
 End WF_DP_R_xml_0.
 
 Definition wf_H  := WF_DP_R_xml_0.wf.
 
 Lemma wf :
  well_founded (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules).
 Proof.
   apply ddp.dp_criterion.
   apply R_xml_0_deep_rew.R_xml_0_non_var.
   apply R_xml_0_deep_rew.R_xml_0_reg.
   
   intros ;
    apply (ddp.constructor_defined_dec _ _ 
            R_xml_0_deep_rew.R_xml_0_rules_included).
   refine (Inclusion.wf_incl _ _ _ _ wf_H).
   intros x y H.
   destruct (R_xml_0_dp_step_spec H) as [f [l1 [l2 [H1 [H2 H3]]]]].
   
   destruct (ddp.dp_list_complete _ _ 
              R_xml_0_deep_rew.R_xml_0_rules_included _ _ H3)
    as [x' [y' [sigma [h1 [h2 h3]]]]].
   clear H3.
   subst.
   vm_compute in h3|-.
   let e := type of h3 in (dp_concl_tac h2 h3 ltac:(fun _ => idtac) e).
 Qed.
End WF_R_xml_0_deep_rew.


(* 
*** Local Variables: ***
*** coq-prog-name: "coqtop" ***
*** coq-prog-args: ("-emacs-U" "-I" "$COCCINELLE/examples" "-I" "$COCCINELLE/term_algebra" "-I" "$COCCINELLE/term_orderings" "-I" "$COCCINELLE/basis" "-I" "$COCCINELLE/list_extensions" "-I" "$COCCINELLE/examples/cime_trace/") ***
*** compile-command: "coqc -I $COCCINELLE/term_algebra -I $COCCINELLE/term_orderings -I $COCCINELLE/basis -I $COCCINELLE/list_extensions -I $COCCINELLE/examples/cime_trace/ -I $COCCINELLE/examples/  c_output/strat/tpdb-5.0___TRS___secret05___cime5.trs/a3pat.v" ***
*** End: ***
 *)